Entropy-Based Design and Analysis of Fluids Engineering Systems

Entropy-BasedDesign andAnalysis of

FluidsEngineering

Systems

Greg F. NatererJosé A. Camberos

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Library of Congress Cataloging‑in‑Publication Data

Naterer, Greg F.Entropy‑based design and analysis of fluids engineering systems / Greg F.

Naterer, José A. Camberos.p. cm.

ISBN 978‑0‑8493‑7262‑9 (hardback : alk. paper) 1. Entropy. 2. Heat‑‑Transmission. 3. Fluid dynamics. I. Camberos, José A. II.

Title.

TJ265.N38 2007620.1’06‑‑dc22 2007028612

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Dedication

To my wife Josie, our children Jordan, Julia, and Veronica, and my mother and father.

G.N.

To my parents for bringing me into this world, my wife Tina, and our children Antonio,

Isabella, and Esteban.

J.C.

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Contents

Foreword...................................................................................................................xi

Preface.................................................................................................................... xiii

Acknowledgments...................................................................................................xv

Authors...................................................................................................................xvii

Chapter 1 Introduction...........................................................................................1

1.1. Background......................................................................................................11.2. Governing.Equations.of.Fluid.Flowand.Heat.Transfer....................................4

1.2.1. Vector.and.Tensor.Notations.................................................................41.2.2. Mass.and.Momentum.Equations..........................................................51.2.3. Energy.Transport.Equations.................................................................7

1.3. Mathematical.Properties.of.Entropy.and.Exergy............................................81.3.1. Concavity.Property.of.Entropy.............................................................91.3.2. Distance.Functional.with.Respect.to.Equilibrium.Conditions........... 14

1.4. Governing.Equations.of.Entropyand.the.Second.Law................................... 171.4.1. Closed.System..................................................................................... 171.4.2. Open.System.......................................................................................20

1.5. Formulation.of.Entropy.Production.and.Exergy.Destruction........................221.5.1. Closed.System.....................................................................................221.5.2. Linear.Advection.Equation.(without.Diffusion).................................231.5.3. Linear.Advection.Equation.(with.Diffusion)......................................241.5.4. Navier–Stokes.Equations....................................................................25

References.................................................................................................................29

Chapter 2 Statistical.and.Numerical.Formulations.of.the.Second.Law............... 33

2.1. Introduction.................................................................................................... 332.2. Conservation.Laws.as.Moments.of.the.Boltzmann.Equation........................342.3. Extended.Probability.Distributions................................................................362.4. Selected.Multivariate.Probability.Distribution.Functions............................. 38

2.4.1. Maxwell–Boltzmann.Probability.Distribution.Function.................... 392.4.2. Central.Distribution.Probability.Distribution.Function..................... 392.4.3. Chapman–Enskog.Probability.Distribution.Function........................402.4.4. Skew-Normal.Probability.Distribution.Function................................ 41

2.5. Concave.Entropy.Functions........................................................................... 432.6. Statistical.Formulation.of.the.Second.Law....................................................462.7. Numerical.Formulation.of.the.Second.Law...................................................48

2.7.1. Discretization.of.the.Problem.Domain...............................................48

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2.7.2. Discretization.of.the.Conservation.Equations.................................... 512.7.3. Discretization.of.the.Second.Law....................................................... 53

References................................................................................................................. 55

Chapter 3 Predicted.Irreversibilities.of.Incompressible.Flows............................ 57

3.1. Introduction.................................................................................................... 573.2. Entropy.Transport.Equation.for.Incompressible.Flows................................. 583.3. Formulation.of.Loss.Coefficients.in.Terms.of.Entropy.Production............... 61

3.3.1. Entropy.Production.in.Bernoulli’s.Equation....................................... 613.3.2. Loss.Coefficients.in.a.Plane.Diffuser................................................. 633.3.3. Case.Study.of.Channel.and.Diffuser.Design......................................64

3.4. Upper.Entropy.Bounds.in.Closed.Systems.................................................... 703.4.1. Upper.Bounds.of.Thermal.Irreversibility........................................... 713.4.2. Optimal.Aspect.Ratio.of.Upper.Entropy.Bounds............................... 753.4.3. Case.Study.of.Mixing.Tank.Design.................................................... 76

3.5. Case.Study.of.Automotive.Fuel.Cell.Design................................................. 793.5.1. Electrochemical.Irreversibilities.in.a.Porous.Electrode..................... 793.5.2. Formulation.of.Channel.Flow.Irreversibilities................................... 823.5.3. Proton.Exchange.Membrane.Fuel.Cell.(PEMFC).and.Solid.

Oxide.Fuel.Cell.(SOFC).Design.........................................................853.6. Case.Study.of.Fluid.Machinery.Design.........................................................90References.................................................................................................................92

Chapter 4 Measured.Irreversibilities.of.Incompressible.Flows...........................95

4.1. Introduction....................................................................................................954.2. Experimental.Techniques.of.Irreversibility.Measurement............................95

4.2.1. Velocity.Field.Measurement...............................................................954.2.2. Temperature.Field.Measurement........................................................974.2.3. Postprocessing.for.Entropy.Production.Measurement.......................99

4.3. Case.Study.of.Magnetic.Stirring.Tank.Design............................................ 1004.4. Case.Study.of.Natural.Convection.in.Cavities............................................. 1034.5. Measurement.Uncertainties......................................................................... 105

4.5.1. Bias.and.Precision.Errors................................................................. 1054.5.2. Velocity.Field.Uncertainties.in.Channel.Flow.................................. 1064.5.3. Measurement.Uncertainties.of.Entropy.Production......................... 1084.5.4. Entropy.Production.of.Free.Convection.in.Cavities......................... 109

References............................................................................................................... 109

Chapter 5 Entropy.Production.in.Microfluidic.Systems.................................... 111

5.1. Introduction.................................................................................................. 1115.2. Pressure-Driven.Flow.in.Microchannels..................................................... 112

5.2.1. Continuum.Equations.and.Slip.Boundary.Conditions...................... 1125.2.2. Case.Study.of.Exergy.Losses.in.Channel.Design............................. 113

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5.3. Applied.Electric.Field.in.Microchannels..................................................... 1175.3.1. Irreversibilities.with.a.Constant.Magnetic.Field............................... 1175.3.2. Case.Study.of.Channel.Design.at.Varying.Hartmann.Numbers...... 122

5.4. Micropatterned.Surfaces.with.Open.Microchannels................................... 1265.4.1. Fluid.Flow.Formulation.................................................................... 1265.4.2. Heat.Transfer.Formulation................................................................ 1315.4.3. Formulation.of.Entropy.Production.................................................. 1325.4.4. Case.Studies.of.Surface.Micropattern.Design.................................. 136

References............................................................................................................... 141

Chapter 6 Numerical.Error.Indicators.and.the.Second.Law............................. 143

6.1. Introduction.................................................................................................. 1436.2. Discretization.Errors.of.Numerical.Convection.Schemes........................... 145

6.2.1. Finite.Volume.Formulation............................................................... 1456.2.2. Central,.Upwind,.and.Exponential.Differencing.Schemes............... 1476.2.3. Case.Study.of.Nozzle.Flow.Analysis.and.Design............................ 152

6.3. Physical.Plausibility.of.Numerical.Results.................................................. 1576.3.1. Entropy.Correction.of.Numerical.Diffusion..................................... 1576.3.2. Case.Study.of.Shock.Capturing.in.a.Shock.Tube............................. 161

6.4. Entropy.Difference.in.Residual.Error.Indicators......................................... 1636.4.1. Formulation.of.Average.Entropy.Difference.................................... 1636.4.2. Case.Study.of.Error.Indicators.in.Supersonic.Flow......................... 165

References............................................................................................................... 173

Chapter 7 Numerical.Stability.and.the.Second.Law.......................................... 175

7.1. Introduction.................................................................................................. 1757.2. Stability.Norms............................................................................................ 1767.3. Entropy.Stability.of.Finite.Difference.Schemes.......................................... 180

7.3.1. Linear.Scalar.Advection................................................................... 1807.3.2. Nonlinear.Scalar.Advection.............................................................. 1897.3.3. Coupled.Nonlinear.Equations........................................................... 197

7.4. Stability.of.Shock.Capturing.Methods........................................................202References............................................................................................................... 210

Chapter 8 Entropy.Transport.with.Phase.Change.Heat.Transfer....................... 213

8.1. Introduction.................................................................................................. 2138.2. Entropy.Transport.Equations.for.Solidification.and.Melting...................... 2158.3. Heat.and.Entropy.Analogies.in.PhaseChange.Processes.............................220

8.3.1. Irreversibility.of.Interdendritic.Permeability...................................2208.3.2. Thermal.Recalescence.and.Dimensionless.Entropy.Ratio...............222

8.4. Numerical.Stability.of.Phase.Change.Computations...................................2278.4.1. Modeling.of.Two-Phase.Entropy.Production...................................2278.4.2. Iterative.Phase.Rules.and.the.Second.Law.......................................230

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8.4.3. Entropy.Correction.of.Numerical.Conductivity............................... 2328.4.4. Entropy.Condition.for.Temporal.Stability........................................2348.4.5. Case.Study.of.Melting.in.an.Enclosure............................................ 2378.4.6. Case.Study.of.Free.Convection.and.Solidification...........................240

8.5. Thermal.Control.of.Phase.Change.with.Inverse.Methods...........................2428.5.1. Formulation.of.an.Inverse.Method...................................................2428.5.2. Entropy.Correction.for.Numerical.Stability.....................................2448.5.3. Case.Study.with.Solidification.of.a.Pure.Material...........................246

8.6. Entropy.Production.with.Film.Condensation..............................................2508.6.1. Formulation.of.Heat.Transfer.and.Irreversibility.Distribution.........2508.6.2. Case.Study.of.Flat.Plate.Condensation.............................................256

References............................................................................................................... 258

Chapter 9 Entropy.Production.in.Turbulent.Flows............................................ 261

9.1. Introduction.................................................................................................. 2619.2. Reynolds.Averaged.Entropy.Transport.Equations....................................... 2629.3. Eddy.Viscosity.Models.of.Mean.Entropy.Production.................................2659.4. Turbulence.Modeling.with.the.Second.Law................................................2669.5. Measurement.of.Turbulent.Entropy.Production...........................................268

9.5.1. Formulation.of.Dissipation.Rate.......................................................2689.5.2. Large.Eddy.Particle.Image.Velocimetry.......................................... 2719.5.3. Case.Study.of.Turbulent.Channel.Flow............................................ 273

References...............................................................................................................284

Appendix................................................................................................................287

Nomenclature........................................................................................................299

Index.......................................................................................................................303

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Foreword

Various.aspects.of.the.First.and.Second.Laws.of.Thermodynamics.are.used.in.the.design.of.many.systems..They.are.typically.applied.to.components.and.not.in.a.truly.connected.fashion..As.the.title.implies,.this.book.has.been.written.and.organized.to.enable.the.design.of.energy-efficient.systems.of.fluid.systems..It.lays.out.the.theo-retical.methods.to.support.entropy-based.design..Chapters.1.through.3.lay.out.the.governing.equations.for.all.related.aspects.of.fluid.flow.with.an.emphasis.on.entropy.generation.and.energy.flow..The.different.sections.are.organized.to.support.compu-tation.of.the.flow.characteristics..Chapter.3.ends.with.case.studies.to.illustrate.the.use.of.the.preceding.methods.

A. critical. factor. in. all. design. processes. is. the. question. of. how. the. theories.compare.with.experimental.measurements..Chapter.4.starts.with.a.presentation.of.experimental.techniques.for.energy.systems..Then.there.are.case.studies.followed.by.consideration.of.the.uncertainties.in.such.measurements.

Chapter.5.is.an.example.application.of.the.theoretical.methods.to.microchannel.flows..Again,.it.has.case.studies.for.better.illustration.of.the.methods.

Chapters.6.and.7.address.the.critically.important.subject.of.potential.errors.in.the.computational.application.of.the.theoretical.methods..This.subject.is.typically.neglected,.but.even. the.exact.physical.equations.cannot.be.numerically.computed.with.complete.accuracy..The.discussions.address.numerical.convection,.numerical.diffusion,. linear. and. nonlinear. effects,. and. so. forth.. Again,. various. case. studies.clearly.show.the.effects.being.presented.

Chapters.8.and.9.end.the.book.with.theories.and.case.studies.of.entropy.transport..These.dynamic.considerations.are.a.critical.aspect.of.any.practical.design.problem...

It.is.my.belief.that.this.book.lays.out.the.theoretical.methods.for.efficient.design.of.energy.fluid.systems..It.presents.case.studies.for.easier.understanding.of.all.the.methods..A.very.important.contribution.is. the.consideration.of.both.experimental.and.computational.errors.

Dr. David J. Moorhouse

Director,.Multidisciplinary.Technologies.Center

Air.Force.Research.Laboratory

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Preface

The.word.“entropy”.is.derived.from.the.Greek.verb.to.chase,.escape,.or.rotate..It.is.generally. interpreted.in. three.semirelated.ways,.namely,.a.macroscopic.viewpoint.(classical. thermodynamics),. microscopic. viewpoint. (statistical. thermodynamics),.and.an.information.viewpoint.(information.theory)..At.a.philosophical.level,.some.believe.that. thermodynamic.entropy.can.be.interpreted.as.an.application.of. infor-mation.theory.to.a.particular.set.of.physical.phenomena..From.another.perspective,.entropy.is.often.described.as.a.type.of.clock,.as.it.is.the.only.quantity.in.the.physical.sciences. that. corresponds. to. a. particular. direction. for. time,. sometimes. called. an.arrow.of. time..As. systems.operate. in. time,. the.Second.Law.of.Thermodynamics.requires.that.entropy.of.an.isolated.system.can.only.increase.or.remain.constant,.but.never.decrease..For.open.systems,.entropy.production.must.be.nonnegative.

The.history.of.entropy.began.with.Lazare.Carnot,.who.in.1803.postulated.a.“loss.of.moment.activity”.in.any.machine.with.moving.parts,.due.to.energy.lost.by.friction..This.led.to.a.basic.concept.of.“transformation.energy”.or.entropy,.with.an.inference.that.perpetual.motion.was.impossible..Lazare’s.son,.Sadi.Carnot,.in.1824.published.Reflections on the Motive Power of Fire.. He. visualized. an. ideal. engine. where. a.“caloric”.(now.known.as.heat).converted.into.work.could.be.reinstated.by.reversing.the.motion.of.the.cycle,.a.concept.that.became.known.as.thermodynamic.reversibil-ity..Building.on.his.father’s.work,.Sadi.postulated.that.“some.caloric.is.always.lost,”.thus.setting.a.foundation.for. the.concept.of.available.energy.loss.through.entropy.production..In.the.1850s,.German.physicist.Rudolf.Clausius.gave.this.“lost.caloric”.a.mathematical.interpretation.and.set.forth.the.concept.of.a.thermodynamic.system,.whereby.in.any.irreversible.process,.a.small.amount.of.thermal.energy.is.dissipated.across.the.system.boundary..Afterward,.scientists.such.as.Ludwig.Boltzmann,.Wil-lard.Gibbs,.and.James.Maxwell.gave.entropy.a.statistical.basis.

Today. the. applications. of. entropy. are. widespread,. from. engineering. fluid.mechanics.and.thermodynamics,.to.information.and.coding.theory,.economics,.and.biology..Entropy.serves.as.a.key.parameter.in.achieving.the.upper.limits.of.perfor-mance.and.quality.in.many.engineering.technologies..As.future.technologies.press.toward.these.theoretical.limits,.entropy.and.the.Second.Law.will.have.an.increasingly.significant.role.of.importance..They.can.shed.new.light.on.various.flow.processes,.ranging.from.optimized.flow.configurations.in.an.aircraft.engine.to.highly.ordered.crystal.structures.(low.entropy).in.a.turbine.blade,.and.many.other.applications.

Entropy-based. design. (EBD). is. an. emerging. design. methodology. that. incor-porates.the.Second.Law.with.computational.fluid.dynamics.(CFD),.computational.physics.in.general,.and.experimental.techniques..This.book.provides.an.overview.of.EBD.and.applications.ranging.from.aerospace.to.microfluidics,.fluid.machinery,.and.others..It.builds.on.past.methods.like.exergy.analysis.and.entropy.generation.mini-mization.(EGM),.by.extending.those.methods.to.more.complex.configurations.(not.

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having.analytical.solutions),.other.applications.(such.as.Second.Law.compliance.in.CFD),.and.measurement.techniques.

Local. irreversibilities. in. fluids. engineering. systems. (such. as. friction. in. fluid.machinery). lead. to.reduced.system.efficiency..The.entropy.produced. in.fluid.flow.leads.to.pressure.losses.or.other.irreversible.conversion.of.kinetic.energy.into.inter-nal.energy..Past.methods.have.often.studied.these.losses.of.useful.energy.on.a.global.scale,.typically.through.a.single.loss.coefficient.(such.as.pressure.measurements.at.the.inlet.and.outlet.of.a.valve)..In.contrast,.this.book.outlines.new.advances,.showing.how.local.irreversibilities.can.be.tracked.in.complex.configurations,.both.numeri-cally.and.experimentally,.so.that.engineering.devices.can.be.redesigned.locally.to.improve.overall.performance.

An. example. is. EBD. with. CFD. applied. to. fluid. motion. through. a. turbine.. In.this. example,. flow. losses. arise. from. fluid. friction. along. the. blades,. viscous. mix-ing. in. the.blade.wakes.and.corner.or. tip.vortices,.as.well.as.other.flow.recircula-tions.with.the.channels.between.blades..The.regions.of.highest.entropy.production.at.the.inlet,.blade.inception,.and.wake.regions.identify.the.regions.where.the.most.substantial.design.improvements.can.be.made..Examples.of.possible.EBD.modifica-tions.might.include.changes.to.the.geometric.parameters.(blade.shape.and.angles,.height,.curvature.around.leading.edges),.cooling.holes.(design,.number,.location),.or.inflow.parameters.(cooling.mass.flow.rate,.temperature)..In.these.examples.and.oth-ers,.computational.and.experimental.techniques.are.needed.to.accurately.predict.the.entropy.transport.processes..This.book.focuses.on.development.of.these.techniques.for.EBD,.including.processes.with.turbulence,.phase.change,.microfluidic.transport,.and.other.complex.transport.phenomena.

Greg F. NatererJosé A. Camberos

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Acknowledgments

During.the.past.several.years,.numerous.colleagues.and.students.have.contributed.in.significant.ways.to.the.development.and.preparation.of.materials.in.this.book..The.authors.would.like.to.express.their.sincere.gratitude.for.this.valuable.input,.particu-larly.to.Kevin.Pope.(University.of.Ontario.Institute.of.Technology,.Oshawa,.Canada),.Olusola.Adeyinka.(Imperial.Oil,.Calgary,.Canada),.and.Emmanuel.Ogedengbe.and.Xili.Duan.(University.of.Manitoba,.Winnipeg,.Canada).

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Authors

Greg Naterer, Ph.D., is.a.professor.of.mechanical.engineering.and.a.Canada.research.chair.in.Advanced.Energy.Systems.at.the.University.of.Ontario.Institute.of.Technol-ogy.(UOIT),.Oshawa,.Canada..He.is.the.director.of.Research,.Graduate.Studies,.and.Development. in. the.Faculty.of.Engineering.and.Applied.Science..He.received.his.Ph.D..in.mechanical.engineering.from.the.University.of.Waterloo,.Canada,.in.1995..His.research.interests.involve.design.of.energy.systems,.hydrogen.technologies,.and.heat.transfer,.including.more.than.160.journal.and.conference.publications.in.these.fields.. Dr.. Naterer. is. currently. leading. an. international. research. team,. involving.Atomic.Energy.of.Canada,.Argonne.National.Laboratory,.and.different.universities.across.Ontario.and.abroad.to.build.a.copper–chlorine.cycle.for.producing.hydrogen.from.nuclear. energy..The.cycle. aims. to. combine. steam.with. intermediate. copper.and.chlorine.compounds.in.a.sequence.of.steps.to.split.water.into.hydrogen.and.oxy-gen..The.Cu-Cl.thermochemical.cycle.could.be.eventually.linked.with.nuclear.reac-tors.to.achieve.higher.efficiencies,.lower.environmental.impact,.and.lower.costs.of.hydrogen.production.than.any.other.conventional.technology..Dr..Naterer.authored.an.earlier.book.entitled.Heat Transfer in Single and Multiphase Systems.(CRC.Press,.2003)..He.has.codeveloped.two.patents.and.supervised.numerous.M.Sc..and.Ph.D..students,.as.well.as.research.assistants.and.postdoctoral.researchers..He.has.served.in.various.administrative.capacities.with.the.Canadian.Society.for.Mechanical.Engi-neering.(CSME),.American.Institute.of.Aeronautics.and.Astronautics.(AIAA),.and.American.Society.of.Mechanical.Engineers.(ASME)..He.is.a.fellow.of.CSME.and.an.associate.fellow.of.AIAA.

José Camberos, Ph.D.,. works. as. an. aerospace. engineer. for. the. U.S.. Air. Force.Research. Laboratory. at. Wright-Patterson. Air. Force. Base,. Dayton,. Ohio.. He. is.also. an. adjunct.professor. at. the.University.of.Dayton. and. the.Air.Force. Institute.of.Technology..He.received.his.Ph.D..in.aeronautical.and.astronautical.engineering.from.Stanford.University,.Stanford,.California..His.research.interests.include.high-.performance. computing,. numerical. analysis,. and. engineering. applications. of. the.Second.Law.of.Thermodynamics..Dr..Camberos.is.currently.working.in.the.Multi-disciplinary.Technologies.Center.at.the.Air.Force.Research.Laboratory.to.develop.systems-level.analysis,.design,.and.optimization.methods.based.on.entropy.produc-tion.as.a.unifying.element.to.quantify.and.improve.system.performance..He.is.also.leading. various. efforts. seeking. to. accelerate. computational. analysis. and. design,.ranging.from.reconfigurable.computing,.symbolic.computation,.and.advanced.meth-ods. in. the. design. of. experiments.. As. an. adjunct. professor,. he. has. also. mentored.several. M.Sc.. and. Ph.D.. students. in. aeronautics,. hypersonics,. and. computational.physics..He.is.a.member.of.Sigma.Xi,.American.Society.for.Engineering.Education.(ASEE),.and.an.associate.fellow.of.AIAA.

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1 Introduction

. Bac kg r o u n d

Effective thermal and fluid system design often requires a creative, iterative, and open-ended process to meet multifaceted objectives of an engineering system. It pro-vides concepts and specifications that will optimize the function, performance, and value of a system, for the mutual benefit of users and manufacturers. Some common tools for such design include computational fluid dynamics (CFD), computer-aided design (CAD), measurement techniques such as particle image velocimetry (PIV), and others. This book focuses on how entropy and the Second Law of Thermody-namics can enhance conventional design methods by providing an iterative meth-odology to reduce entropy production in a thermal system, thereby improving its energy efficiency.

Industrial design methodologies were first adopted widely in the late 1930s and early 1940s, with prominent industrial designers such as Raymond Loewy, Norman Bel Geddes, and Henry Dreyfuss. The importance of their methods has risen steadily since that time for various reasons. Economics has been a key factor because a manufacturer’s profitability depends on the product price in the marketplace and manufacturer’s cost to produce it. As manufactured products become a com-modity, cost savings are more difficult, and better industrial designs are needed to allow a product to gain higher profit margins. Also, good engineering designs can allow products to achieve certain attributes that are important for advertising and marketing purposes.

With increased worldwide awareness that the world’s fossil fuel resources are limited, major efforts have focused on the design of more efficient and environmen-tally sustainable energy devices and processes. Energy systems are often thoroughly scrutinized for possible design improvements. Past conventional technology has gen-erally detected energy losses on a system-wide or global scale, such as a single loss coefficient (i.e., valve loss coefficient). With the current state of this technology, the margins for improving the efficiency of existing devices can be relatively small. In this book, entropy-based design with local loss mapping is presented as a robust tool for reaching higher levels of system efficiency, thereby leading to energy savings in various industrial applications.

The fundamental principles governing the design of energy systems are Newton’s law of motion and the laws of thermodynamics. Newton’s Second Law of Motion and the First Law of Thermodynamics are the cornerstones on which virtually all energy systems are built today. The other laws have played a secondary support role. A limita-tion associated with the First Law of Thermodynamics is that it tracks only the quantity of energy. In contrast, the Second Law tracks “quality” of energy, or its work-producing potential. Thus, the Second Law has the unique advantage of offering a systematic tool

7262_C001.indd 1 1/19/08 11:25:40 AM

Entropy-BasedDesignandAnalysisofFluidsEngineeringSystems

for optimal energy usage and choice of technologies. The unique capabilities of the Second Law can be used to scrutinize flow irreversibilities locally, rather than globally. In this way, the problem regions can be clearly identified by the high entropy produc-tion rates, so designers can focus on those regions for improvements. A useful analogy is a sick patient telling a doctor that he or she is sick, without knowing the part of the body that is causing the ailment. Doctors can often use diagnostic tools to pinpoint the source. Similarly for a complex engineering system, large rates of entropy production within a device can identify problematic areas of concern because a commonly desired goal of devices is improving the efficiency through reduced entropy production. This goal is generally desired regardless of application, flow conditions, system parameters, and so on.

Local exergy, or the work potential of a device, can be more readily interpreted physically than entropy production because it contains the same dimensional units as energy. It can be related directly to economic indicators. For example, multiplying the local cost of electricity (per kilowatt hour) by exergy destroyed by moving fluid through a valve over a year can indicate a yearly expense of wasted energy therein. This expense can be interpreted directly in terms of lost revenue. Thus, an economic framework can be based on local entropy production rates or exergy losses in a fluids engineering system.

Furthermore, there exists a need for a standard metric from which the energy effi-ciency of all devices can be characterized. For example, fuel efficiency in a car is defined differently from that of a water heater’s efficiency, while still different than how a diffuser’s efficiency is defined, and so on. As a result, it is difficult for regula-tory and government agencies to identify a standard method for identifying the energy wasted by a given device. Entropy production gives a single, measurable quantity that is directly related to the efficiency of any device that transforms energy because it char-acterizes degradation of useful (mechanical) energy to less useful (internal) energy.

The utility of entropy and the Second Law have been widely documented in various disciplines, ranging from engineering fluid mechanics, to information and coding theory, economics, and biology. It will be emphasized frequently throughout this book how entropy serves as a key parameter in achieving the upper limits of performance and quality in many technologies. It can shed new light on various flow processes, ranging from optimized flow configurations in an aircraft engine to highly ordered crystal structures (low entropy) in a turbine blade, and other applica-tions (Bejan, 1996). It is likely not possible to find any other law of nature, wherein a proposed violation would bring more skepticism than violation of the Second Law of Thermodynamics.

Consider the implications of the Second Law in the thermal design of aircraft subsystems, involving work potential (Camberos, 2000a). Past authors have observed that there is no current systematic method for tracking work potential usage in the design of aircraft subsystems (Roth and Mavris, 2000). Exergy and entropy cal-culations can identify the loss of work potential within each subsystem and fluid flow process during an aircraft’s operation. These calculations can enable design-ers to identify key locations that incur the most significant losses. Moorhouse and Suchomel (2001) have discussed how flow exergy provides a unifying framework and set of metrics to more effectively analyze aircraft subsystems.

7262_C001.indd 2 1/19/08 11:25:40 AM

Introduction

Various analytical methods have been developed over the past several decades for Second Law analysis. Notable examples include (i) estimation of the theoretical ideal operating conditions of a proposed design (called exergy analysis, or EA) and (ii) minimization of the lost available work or entropy generation by design modi-fications (called entropy generation minimization, or EGM; Bejan, 1996). Exergy quantifies the capacity of an energy source to perform useful work. It is a measure of the maximum capacity of an energy system to perform useful work as it pro-ceeds to a specified final state in equilibrium with its surroundings. Exergy analysis focuses on closing the gap between maximum exergy and the actual work delivered by a device, through careful examination of the thermodynamic processes involved in a series of energy conversion steps (Dincer and Rosen, 2004). Subsequently, the exergy values at each point are used to evaluate Second Law efficiencies, which quantify the magnitude of irreversibilities (or exergy destruction) associated with the energy conversion process (Bejan, 1997; Rosen and Dincer, 2004). The method of EGM involves fluid mechanics, heat transfer, material constraints, and geometry, in order to obtain relationships between entropy generation and the optimal con-figuration. Typically, a functional expression for the entropy production in a process is derived (Poulikakos and Bejan, 1982; Zubair et al., 1987). Then the extremum of the derived expression that guarantees a minimum entropy generation is deter-mined by methods of calculus. Because analytical methods are often limited to simplified geometries, this book extends analytical EGM to numerical and experi-mental methods.

Opportunities for design optimization based on the Second Law can be enhanced through CFD as a design tool for complex problems and geometries. Entropy produc-tion can be obtained by postprocessing of the predicted flow fields (Sciubba, 1997). Many industrial problems in metallurgy, power generation, energy storage, aerody-namics, and other applications have been successfully solved by CFD. A designer can choose an optimum design from many possible alternatives at a remarkable speed using CFD. Combined EGM with CFD provides an emerging technology with prom-ising potential for design optimization of practical industrial problems.

For example, an application involving the design of air-cooled gas turbine blades was presented by Natalini and Sciubba (1999). The full Navier–Stokes equations of motion for turbulent viscous flow and the energy equations were solved with a finite element approach and a two-equation turbulence closure. By identifying the entropy generation rates corresponding to the fluid friction and heat transfer irreversibility, the authors determined which configurations had minimal thermodynamic loss in a turbine cascade. The computed flow field for pitched turbine blades (Kresta and Wood, 1993) can be postprocessed to identify regions of high local losses, thereby guiding engineers in local redesign of the blade profile to reduce such losses. Pre-dictions of entropy production have been used in various other applications such as free convection in inclined enclosures (Baytas, 2000), mixed convection in a verti-cal channel with transverse fin arrays (Cheng et al., 1994), laminar and turbulent flow through a smooth duct (Demirel, 1999; Sahin, 2000, 2002), flow in concen-tric cylinder annuli with relative rotation (Mahmud and Fraser, 2002), and diffusers (Adeyinka and Naterer, 2005). These studies are examples of how entropy produc-tion computations can successfully complement CFD technology.

7262_C001.indd 3 1/19/08 11:25:40 AM


Industrial flow problems usually involve turbulence. Numerical predictions of entropy production in a turbulent flow were given by Moore and Moore (1983). Moore’s work was the first documented effort to develop a numerical model for tur-bulent entropy production. The Moore model assumes that turbulent fluctuations of the heat flux and viscous dissipation in the positive definite entropy equation can be modeled by the addition of a turbulent conductivity and turbulent viscosity to the molecular conductivity and viscosity, respectively. It has been used to predict the mean local entropy production in a bent elbow (Moore and Moore, 1983), turbulent plane oscillating jet (Cervantes and Soloris, 2002), and a jet impinging on a wall (Drost and White, 1991). A finite volume method for predicting the mean viscous dissipation and entropy production in turbulent flows, based on the time-averaged turbulence equations, was described by Kramer-Bevan (1992).

In addition to the previous physical characteristics of entropy production, it can be interpreted alternatively in computational terms. Physical processes of viscous dissipation and heat transfer lead to entropy production. Past Second Law studies have shown how numerical procedures may also produce or destroy entropy, due to discretization errors, artificial dissipation, and nonphysical numerical results (Cox and Argrow, 1992; Naterer, 1999). Solutions of differen-tial equations that do not satisfy an “entropy condition” may be characterized by a lack of uniqueness, oscillations, and other unusual behavior (Adeyinka, 2002; Hughes et al., 1985; and others). Cox and Argrow (1992) computed local entropy production with a finite difference method for compressible flow. Jansen (1993) and Hauke (1995) applied an entropy-based stability analysis to turbulent flows. Jansen (1993) showed that the exact Navier–Stokes equations for compressible flow could lead to an entropy inequality, through a linear combination of equa-tions. The study determined what constraints the Second Law places on modeling of the averaged equations by linking entropy production to the solution variables. A major difficulty with numerical predictions can be the inability to ascertain error bounds. Solutions can be very sensitive to various parameters associated with the numerical algorithm (Naterer, 1999). This can make it difficult to judge the extent to which the computed results agree with reality. In numerical predic-tions of complex industrial flows, limited or no experimental data may be avail-able for validation purposes. In these cases, checking where predicted entropy production rates are positive (realistic) or negative (unrealistic) is a valuable tool for verification.

. g o v er n in g eq u at io n s o f fl u id fl o w a n d Heat tr a n sf er

1.2.1 Vec t o r a n d tenso r no t at io ns

In this book, conventional notations for vectors and tensors will be used. A vector will be denoted by boldface font or a vector hat. A unit vector is a vector of unit magnitude. For example, i and j refer to the unit vectors in the x and y coordinate directions, i.e., (1, 0) and (0, 1), respectively. The symbol | v | designates the magnitude

7262_C001.indd 4 1/19/08 11:25:41 AM

Introduction

of the indicated vector. When performing operations with matrices in this book, matrices are contracted when their individual entries are multiplied by each other and summed. For example, if

AA BB=

=

a a

a a

b b

b b11 12

21 22

11 12

21 22

;

(1.1)

then

AA BB: 11 11 12 12 21 21 22 22= + + +a b a b a b a b

(1.2)

Tensors are generalized notations for scalars (rank of zero), vectors (rank of 1), matrices (rank of 2), and so on. A tensor is denoted by a variable with subscripts. For example, aij represents the previously described matrix, where the range of subscripts is i = 1, 2 and j = 1, 2. When tensors use indices in this way, the notation is called indicial notation. The summation convention of tensors requires that repetition of an index in a term denotes a summation with respect to that index over its range. For example, in the previously cited case (dot product) involving two vectors,

u v u v u vi i = +1 1 2 2 (1.3)

The range of the index is a set of specified integer values, such as i = 1, 2 in the previ-ous equation. A dummy index refers to an index that is summed, whereas a free index is not summed. The rank of a tensor is increased for each index that is not repeated. For example, aij contains two nonrepeating indices, thereby indicating a tensor of rank 2 (i.e., matrix).

1.2.2 Ma ssa n d Mo Men t u Meq u at io ns

The governing equations of fluid flow and heat transfer can be expressed in either vector or tensor notations. For two-dimensional flows, the mass conservation equa-tion is given by

∂∂+ ∂∂+ ∂∂=ρ ρ ρ

t

u

x

v

y

( ) ( )0

(1.4)

For incompressible flows, this equation may be simplified wherein that the diver-gence of the velocity field (∇ ⋅ v) equals zero. The divergence of velocity may be interpreted as the net outflow from a control volume (fully occupied by fluid), which must equal zero at steady state, because any inflows are balanced by mass outflows.

The momentum equations represent a form of Newton’s law. Forces on a fluid element like pressure and shear forces balance the particle’s mass times its accelera-tion (i.e., total, or substantial derivative of velocity). The x-direction and y-direction

7262_C001.indd 5 1/19/08 11:25:45 AM


momentum equations can be expressed as

∂∂+ ∂∂

+ ∂∂

= ∂∂+∂∂+( ) ( ) ( )ρ ρ ρ σ τu

t

uu

x

vu

y x yFxx yx

bx

(1.5)

∂( )∂+∂( )∂

+∂( )∂

=∂∂+∂∂+

ρ ρ ρ τ σv

t

uv

x

vv

y x yFxy yy

by

(1.6)

where Fb refers to a body force.These equations cannot be solved in this form because there are more unknowns

(i.e., stresses, velocities, and pressure) than available equations. As a result, addi-tional relations called constituitive relations between the stresses and velocities are needed. In Newtonian fluids, the stresses are proportional to the rate of deformation (or strain rate). For incompressible flows of Newtonian fluids, we have the follow-ing two-dimensional constitutive relations for stresses in terms of the pressure and velocity fields:

σ µxx pu

x= - + ∂

∂2

(1.7)

σ µyy pv

y= - + ∂

∂2

(1.8)

τ µ τyx xy

u

y

v

x= ∂

∂+ ∂∂

=

(1.9)

Substituting these constitutive relations into the previous x-momentum equation and using continuity (mass conservation) to rewrite the left side,

ρ ρ ρ µ∂∂+ ∂

∂+ ∂

∂= - ∂∂+ ∂

∂+ ∂∂

u

tu

u

xv

u

y

p

x

u

x

u

y

2

2

2

2

+ Fbx

(1.10)

together with a similar y-momentum equation represents the two-dimensional Navier–Stokes equations. Analytical solutions of these equations are usually limited to simplified geometries because of the difficulties inherent in the nonlinear and coupled (with continuity equation) nature of the equations.

Fluid flow regions are generally classified as viscous or nearly inviscid regions. In a viscous region, such as a boundary layer, frictional forces are significant. A boundary layer refers to the thin diffusion layer near the surface of a solid body, where the fluid velocity decreases from its freestream value to zero at the wall over a short distance. In contrast to viscous regions, frictional forces are often small in comparison to fluid inertia in regions far from a surface or boundary layer. The Euler equations are a special form of the Navier–Stokes equations for frictionless (or inviscid) flow. An inviscid fluid refers to an idealized fluid with no viscosity. In this situation, the terms involving viscosity are absent from the governing equations.

7262_C001.indd 6 1/19/08 11:25:49 AM

Introduction

The fluid motion can be characterized as a potential flow, whereby the reduced gov-erning equations can be written in terms of a scalar potential function.

1.2.3 en er g ytr a nsp o r t eq u at io ns

In addition to the fluid flow equations, energy is another transported quantity of key importance in the analysis of thermal and fluid systems. The mechanical energy equations can be obtained by multiplying each ui momentum equation by ui (where i = 1, 2 for two-dimensional flows) and adding them together. Using the substantial derivative notation, we obtain

12

2 2D

Dtu v u

p

xv

p

yu

xu

yvxx yx( )+ = - ∂

∂- ∂∂+ ∂∂+∂∂+τ τ ∂∂∂+∂∂+ +

τ τxy yyx yx

vy

uF vF

(1.11)

Using the product rule and generalizing to a vector notation, the following mechani-cal energy equation is obtained:

12

( ) ( ) :2ρ τ τD

DtV pv p v( ) [ ] [ ]= - ∇ ⋅ - ∇ ⋅ + ∇ ⋅ ⋅ - ∇ + ⋅vv vv vv FFFF

(1.12)

where V u v= +2 2 refers to the total resultant magnitude of the velocity. The first term (left side) represents the rate of increase of kinetic energy of a fluid element with respect to time. On the right side, the second term gives the flow work done by pressure on the differential control volume to increase its kinetic energy. The third term represents an energy sink due to fluid compression in the mechanical energy equation, and it becomes zero for incompressible flows. The difference between the fourth and fifth terms on the right side gives the net fluid work done by viscous stresses to increase the kinetic energy of the fluid within the control volume. The lat-ter portion represents work lost through viscous dissipation, which is a degradation of mechanical energy into internal energy through viscous effects. This viscous dis-sipation is represented by t: ∇ v, which refers to the viscous stress tensor contracted with the velocity gradient.

For two-dimensional incompressible flows of a Newtonian fluid, it can be shown that the viscous dissipation term can be written as

τ µ µ: 22 2

∇ = ∂∂+ ∂∂

+ ∂

∂vv

u

x

v

y

u

yy

v

x+ ∂∂

≡

2

µΦ

(1.13)

where Φ refers to the positive-definite viscous dissipation function. This function is greater than or equal to zero. As a result, the conversion of mechanical energy into internal energy through viscous dissipation is an energy sink in the mechanical energy equation. Thus, mechanical energy is not conserved, but instead a portion of this energy is degraded and lost to internal energy through viscous dissipation.

7262_C001.indd 7 1/19/08 11:25:53 AM


It is degraded in the sense that a certain quality of energy is lost in the irreversible transformation, as internal energy normally has less ability than kinetic energy to perform useful work.

The conservation of total energy (internal plus mechanical energy) is called the First Law of Thermodynamics. Performing a total energy balance on a differential control volume within the fluid stream, it can be shown that the total energy equation can be written as

ρ τD

Dte V q pv v F v Sˆ ( )+

= -∇ ⋅ - ∇ ⋅ - ⋅ + ⋅ +1

22

(1.14)

where ê refers to internal energy and S is a source term. The rate of increase of total energy within the control volume equals the rate of energy addition by conduction, plus work done by pressure, viscous and external forces, plus internal energy gener-ated per unit volume ( S ).

The internal energy equation can be derived by subtracting the mechanical energy equation from the First Law (total energy equation). Performing this subtrac-tion and writing the results in a general vector form, we have

ρ τDeDt

q p v v Sˆ

:= -∇ ⋅ - ∇ ⋅ + ∇ +

(1.15)

where the fourth term (right side) refers to the viscous stress tensor contracted with the velocity gradient. It represents an internal energy source because it arises from the conversion of mechanical energy to internal energy through viscous dissipation. In the thermal energy equation, viscous dissipation represents an energy source, which corresponds to the energy sink previously observed in the mechanical energy equation. In other words, its magnitude is identical, but its sign changes in transpos-ing from the mechanical to internal energy equations.

. Mat HeMat ica l Pr o Per t ies o f en t r o Py a n d exer g y

Numerous past studies have examined the significance of exergy as a measure of work potential or maximum useful work (Boehm, 1989b, 1992). A common aspect in all of these analyses is the identification of exergy with useful work potential. For example, Szargut et al. (1988) define exergy as “the amount of work obtainable when some matter is brought to a state of thermodynamic equilibrium.” Similar definitions were documented by Bejan (1996) and Kotas (1985). Although engineers have accepted the capacity to do work as a measure of quality of energy, this does not invalidate another, less anthropomorphic approach. By conceptualizing “exergy” as a distance functional, one eliminates the need to introduce additional terms also found in the literature (e.g., “anergy,” “essergy,” etc.) or to fragment exergy into multiple forms as often done with energy. Concise and critical reviews of the origins and history of exergy have been reported by Bejan (1996), Haywood (1974), Kotas (1985), and Szargut et al. (1988).

7262_C001.indd 8 1/19/08 11:25:55 AM

Introduction

The previous section has outlined the governing equations for conserved vari-ables of mass, momentum, and total energy. In this section, entropy and the Sec-ond Law will be formulated, particularly fundamental properties associated with the nonconserved variables of entropy and exergy. Thermodynamics began as the science of heat, intended to provide extended mechanics that would account for a common experience, namely, that doing work on a body sometimes makes it hot-ter, and sometimes heating a body causes it to do work (Truesdell, 1985). Common experience shows that mechanical action does not always result in a mechanical response, so we need to add the concept of heating alongside the concept of working or power. The Second Law is often expressed in terms of “work potential” or exergy. The balance of exergy equation represents a synthesis of the First and Second Laws. Exergy places all thermodynamic processes in a given system on the same basis by providing a common reference and metric. This section examines the essence of the Second Law of Thermodynamics as a statement involving the existence of entropy, with particular mathematical properties, from which a corresponding statement for the existence of exergy follows. It will be shown that exergy represents an abstract, mathematical distance functional. The concept of exergy will be interpreted as a thermodynamic functional representing the distance of a given system from the state of equilibrium at a reference state.

1.3.1 c o n c aVit ypr o per t yo f en t r o py

The Second Law of Thermodynamics represents a natural foundation for thermo-physical processes. The concept of entropy, however, is often viewed as abstract. A fundamental feature of the Second Law reflects a concavity property of entropy (Camberos, 2000a). Given a set of thermodynamic variables, ξ and ζ, there exists a functional, entropy, S = S (ξ, ζ) such that S is a concave function of its arguments. This framework can be useful to unify various formulations of the Second Law, including the principle of nonnegative entropy generation itself (Lavenda, 1991).

Consider an example of a rigid material body at some temperature T immersed in a thermal reservoir at temperature T0 (e.g., a hot rock inside a cold room). Suppose T > T0 and we let the cooling process proceed from the initial time, t, to t0 when the body reaches thermodynamic equilibrium with its surroundings. The transfer of energy as the body cools equals

t

t

Qdt U U0

0∫ = -

(1.16)

where U0 = U (T0) at a final time t0 and U = U (T) at the initial time t. The variables Q and U refer to heat transfer rate and internal energy, respectively.

The Second Law of Thermodynamics requires that entropy is produced, but never destroyed, in an isolated system. Thus, Sgen ≥ 0 for an isolated system, where Sgen refers to the entropy generation. In the current example, the entropy flow associ-ated with heat transfer is –Q/T0, so the entropy balance equation is

S S ST

Q dtt

t

gen = - - ∫00

1 0

(1.17)

7262_C001.indd 9 1/19/08 11:25:57 AM

0 Entropy-BasedDesignandAnalysisofFluidsEngineeringSystems

where S = S(T) and S0 = S(T0). Substituting Equation 1.16 into Equation 1.17,

S S S

TU Ugen = - - -0

00

1( )

(1.18)

To write the change of energy in terms of temperature, we can use the definition of the specific heat (CV = ∂U/∂T). The entropy generated during the cooling process is then

S S S

C

TT TV

gen = - - -00

0( )

(1.19)

Using standard thermodynamic relations between the specific heat and entropy (CV/T = ∂S/∂T), the expression for entropy generation becomes

S S S

S

TT Tgen = - - ∂

∂-0

00( )

(1.20)

This expression indicates a concavity property of entropy as a function of T.To clarify the meaning of the concavity property, consider some arbitrary func-

tion F = F (X) such that F′′ < 0, where the inequality indicates that F is a concave function of its argument. Integration by parts requires

- - ′′ = - - ′ -∫ ( ) ( ) ( ) ( ) ( )(X X F X dX F X F X F X XX

X

1 2 1 2 21

2

XX1)

(1.21)

The result on the right-hand side has a geometric interpretation. Figure 1.1 illustrates the right side of the equation with a vertical line. Geometrically, we have

F X F X F X X X( ) ( ) ( )( )2 1 2 2 1 0- - ′ - ≥

(1.22)

where the equality holds if and only if X2 = X1. Comparing this result with Equation 1.20, it can be observed that positive entropy generation (the Second Law) is equivalent to asserting the concavity property of entropy when S = S(T).

Consider another example of a simple compressible substance, subject to both heat transfer, Q, and work, W, when relaxing to equilibrium with an environment at T0, P0, where P0 = P(T0, V0). Solving for the heat flow from an energy balance and writing the net compression/expansion work of the gas in terms of pressure and a volume difference,

Q dt U U P V V= - + -∫ ( ) ( )0 0 0

(1.23)

7262_C001.indd 10 1/19/08 11:26:02 AM

Introduction

The entropy balance then becomes

S S S

TU U

PT

V Vgen = - - - - -00

00

00

1( ) ( ).

(1.24)

Alternatively, by substituting the appropriate thermodynamic relations and using Sgen ≥ 0,

S S

ST

T TSV

V V00

00

0 0- - ∂∂

- - ∂∂

- ≥( ) ( ) .

(1.25)

The inequality asserts the concavity of entropy as a function of T and V. Equality holds if and only if (T, V) = (T0, V0).

Exergy represents the maximum work potential when bringing the system to equilibrium with its surroundings. In this example, it is given by

X T S S C T T P V VO O V O O O= - - - - -( ) ( ) ( ).

(1.26)

Standard thermodynamic relations provide

C

T

S

TV = ∂∂

P

T

S

V= ∂∂

(1.27)

f ig u r e. Downward concave function (entropy).

S(T

)

S(T0) – S(T) – S 0(T0 – T )

T T0

S(T0)

7262_C001.indd 11 1/19/08 11:26:08 AM


Multiplying the entropy inequality in Equation 1.24 leads to

T S S U U P V V0 0 0 0 0 0( ) ( ) ( ) ,- - - - - ≥

(1.28)

where C T TV ( )0 - can be interchanged with ( )U U0 - . Identifying the left side as exergy and taking the time rate of change,

X T S C T P VV= - + +0 0 . (1.29)

Also, from the entropy balance equation for this problem,

S S

ST

TSV

V= + ∂∂

+ ∂∂gen .

(1.30)

Substituting Equation 1.30 into Equation 1.29 and replacing terms defined by

∂ ∂ =S U T/ /1 and ∂ ∂ =S V P T/ / yields

X

TT

C T P TPT

V T SV- - - -

= -1 0

0 0 0 gen..

(1.31)

From the definition of exergy, it can be shown that the following thermodynamic relations hold:

∂∂= -

∂∂= -

XV

P TPT

XT

TT

CV0 001; .

(1.32)

Because the entropy generation is nonnegative, the previous relations yield

X

XT

TXV

V- ∂∂

- ∂∂

≤ 0.

(1.33)

This result asserts the mathematical property of convexity for X = X (T, V). Thus, the concavity of entropy is equivalent to the convexity of exergy.

Figure 1.2 shows an example of exergy as a convex function of temperature. A geometric complementary relation exists between entropy and exergy, as shown by the concavity inequality for entropy and the line segment that defines exergy (see Figure 1.3). Exergy has an absolute minimum at the point of equilibrium. The tan-gent slope at this point coincides with the horizontal axis of zero exergy. The straight vertical line from an arbitrary initial state in Figure 1.2 represents the corresponding distance to equilibrium. The distance to equilibrium is represented equally well by the vertical line shown in Figure 1.1 (concavity of entropy) or the horizontal line in Figure 1.3 (geometric representation of exergy).

7262_C001.indd 12 1/19/08 11:26:15 AM

Introduction

X (T

)

X‚ X (T0) = 0

T0 T

X(T )

f ig u r e. Exergy function as a convex function of T.

S (T

)

S0 – S

X CV (T0 – T )

CVT CVT0

S (CVT0)

f ig u r e. Geometric representation of entropy.

7262_C001.indd 13 1/19/08 11:26:18 AM


1.3.2 d ist a n c efu n c t io n a l w it h respec t t o eq u il ibr iu Mco n d it io ns

In addition to its convexity property, exergy may be interpreted as a thermodynamic metric or distance functional. Define a metric, x, based on the Hessian of entropy (second-order tensor of derivatives with respect to temperature and volume) and the following inner product,

( ) ( )x x x S xT

xx, ≡ ⋅ - ⋅0 (1.34)

where

( )S

S

T

S

T V

S

T V

S

V

xx 0

0

2

2

2

2 2

2

=

∂∂

∂∂ ∂

∂∂ ∂

∂∂

(1.35)

represents the Hessian. The superscript T refers to matrix transpose and

x T VT ≡ ,( )

(1.36)

represents an algebraic vector of the corresponding thermodynamic variables. The inner product of (x, x) results in nonnegative values, as guaranteed by the concavity property of entropy. A general mathematical metric can be defined by the following distance functional:

|| || ( )x x x x- ≡ D ,D /

01 2

(1.37)

where Dx = x - x0.To clarify the importance of these equations with respect to exergy, consider the

construction of the norm || ||x , which requires evaluation of the second-derivative terms of the Hessian. Thermodynamic relations for a simple compressible substance (Bejan, 1996) give

∂∂= -

2

2 2

S

T

C

TV

(1.38)

∂∂ ∂

= - -2

2

1S

T V TP T

κκ β( )

(1.39)

∂∂

= - - +

2

2 2

21 2S

V T

P TP

C

C T

C VV

P

Vκκ β

(1.40)

where

β ≡ ∂

∂1V

V

T (1.41)

7262_C001.indd 14 1/19/08 11:26:25 AM

Introduction

and

κ ≡ - ∂∂

1V

V

P (1.42)

represent the volumetric coefficients of thermal expansion and isothermal compress-ibility of the gas, respectively. Using these definitions to calculate the Hessian yields the following inner product:

( )x xC T

H

THV

C V

VV

P, = - +

12

0

2

0

2

0κκ β

(1.43)

where H = CVT + P0V.Consider two cases: an ideal gas and an incompressible substance. For an ideal gas,

κ βP T= =1 1

(1.44)

and the inner product simplifies to

( )x x C

T

T

P V

C T

C P V

C T VVV

P

V

, =- +

2

0

02 2

02

02

0 0 (1.45)

Simplifying further with the ideal gas equation of state,

PV RT= ˆ (1.46)

together with the following relations:

CR

V = -

ˆ

γ 1

CR

P = -γγ

ˆ

1

γ ≡ C

CP

V (1.47)

yields

( )

ˆˆx x

R T

TR

V

V, =

-+γ 1

2

0

2

0 (1.48)

where R= mR/M and m, M, and R are the mass, molecular weight, and universal gas constant, respectively. The subscript O denotes reference conditions for pressure and temperature. Replacing (U, V) with the corresponding differences ( )U U V V- , -0 0 yields the following square of a true mathematical distance functional,

|| || ˆ

ˆx xR T

TR

VV0

2

2

0

2

011 1- =

--

+ -

γ

(1.49)

7262_C001.indd 15 1/19/08 11:26:34 AM


Following the same procedure for an incompressible substance, where κ β, → 0, and C C CP V, → , yields

|| ||x x CT

T02

2

0

1- = -

(1.50)

This result of a mathematical distance functional can be directly related with exergy. Consider a system near equilibrium conditions and expand the entropy func-tion in a Taylor’s series. Using the previous definition of xT = (T, V) and neglecting higher-order terms,

S SS

TT T

S

VV V x x ST

xx≈ + ∂∂

- + ∂∂

- + - ⋅00

00

0 0 012( ) ( ) ( ) ⋅⋅ -( )x x0

(1.51)

Using the definition of exergy from the previous section, entropy derivatives in terms of exergy and the previous result for the distance functional, it can be shown that

X T x x≈ -1

2 0 02|| ||

(1.52)

This formulation of exergy as a distance functional with respect to equilibrium con-ditions provides a more systematic and mathematically rigorous interpretation than various definitions of “work potential” in undergraduate textbooks. The previous results show that exergy represents a physical measure of the distance from equilib-rium conditions for a system at some arbitrary state.

The convexity and distance functional properties of exergy have been presented here to aid understanding of exergy. The concept of exergy has been interpreted through a connection between a system and its environment. Standard textbooks often introduce and discuss “availability” or exergy in the context of “a system’s potential to do work in a reversible manner.” Many modern texts (such as Cengel and Boles, 1989; Müller, 1985) also introduce a number of work terms (reversible work, available work, etc.) in an effort to clarify and expand on the subject. However, this can lead to more confusion and cluttering of terminology. This situation was observed more than 30 years ago (Haywood, 1974a,b). Second Law analyses have found well-deserved attention (Szargut et al., 1988), but the cluttering of terminology and obscurity in the definitions often remain. By providing fundamental mathemati-cal properties of exergy as a state variable, this section has provided a valuable alter-native interpretation.

The Second Law has deep and significant implications for engineering systems. As future machines become increasingly complex and sophisticated in their abil-ity to transform energy into various forms, exergy and the Second Law will have an increasingly important role in prescribing their upper limits of performance. Since the industrial revolution, the Second Law served only a secondary role by pre-scribing what the real physical world allowed. Complex machines of the future will require a more interconnected relationship, as they press toward the maximum limits

7262_C001.indd 16 1/19/08 11:26:38 AM

Introduction

of engineering design, precisely where the true power of the Second Law becomes evident. Therefore, it is essential that fundamental properties of entropy and exergy as state variables are well understood. Systematically developing the foundations of the Second Law from the essence of entropy as a concave function of state variables, we can advance that concept, together with a simplicity that will make it possible for future engineers and scientists to achieve what we can now only imagine.

. g o v er n in g eq u at io n s o f en t r o Py a n d t Hesec o n d law

The First and Second Laws are physical principles governing all thermophysical processes, and the addition of constitutive relations describes the response of various classes of materials (Truesdell, 1984, 1985). As discussed in the previous section, a general axiom of thermodynamics postulates the existence of a concave thermo-dynamic variable called entropy. The Second Law then stipulates that the rate of entropy generation must be nonnegative in all thermophysical processes, that is,

genS ≥ .0 The mathematical property of concavity implies certain restrictions on the constitutive relations for any material body. This section will use this property to develop the governing equations for entropy and the Second Law.

1.4.1 c l o sed syst eM

For a closed system, let xk represent independent variables in the constitutive func-tional relation, such that U = U (xk), W = W (xk), etc. Then the mathematical expres-sion for the Second Law can be written as

SS

Sk

- ∂∂- =∑ ξξ gen 0

(1.53)

For S S U V= ,( ), the Second Law becomes

S

S

UU

S

VV S- ∂

∂- ∂∂

- =gen 0

(1.54)

Simplifying this result by using thermodynamic relations for the derivatives (CV/T = ∂S/∂T, P/T = ∂S/∂T, and CV = ∂U/∂T) and the First Law,

S

Q W

T

P

TV S- + - - =gen 0

(1.55)

Relating the work term and third term yields

S

Q

TS- - =gen 0

(1.56)

7262_C001.indd 17 1/19/08 11:26:43 AM


which represents the entropy balance for a closed system from classical thermodynamics. This result is well known, although the previous derivation has shown that it follows from the concavity property of entropy.

High-quality energy refers to energy from which a great amount of useful work can be extracted, so exergy is used to refer to the work potential of that energy. Lower quality energy like internal energy can produce less work and therefore reflects also lower exergy. Thus, exergy quantifies a qualitative aspect of energy. In standard practice, to derive an equation representing the balance of exergy, one typically considers a closed system at some uniform arbitrary state, (P, T), rela-tive to ambient conditions at (P0, T0). To measure the distance of a system from the reference or so-called “dead state,” imagine a reversible process whereby the system relaxes to thermodynamic equilibrium with the surroundings. The energy balance equation simplifies to

U U Uin out- = D

(1.57)

where the total change in energy is

D = - = ∫U U t U t U dt

t

t

( ) ( )2 1

0

(1.58)

A closed system relaxes to equilibrium with its surroundings through work and heat transfer. Integrating the balance of energy equation over time,

t

t

t

t

Q dt W dt U0 0

∫ ∫+ = D

(1.59)

with the integral limits defined at an initial time when the system is at (P, T) and the final time when the system has reached equilibrium with the surroundings at (P0, T0). To replace the heat interaction term with a state variable, one can use the following definition of entropy:

S dtT

Q dt T S S Q dt∫ ∫ ∫= ⇒ - =1

00 0( )

(1.60)

For the work term, the energy quality directly relates to the useful work extracted. It is the maximum amount of work done during a thermodynamically reversible pro-cess. For a simple compressible substance,

W dt PV dt P P Vdt P

W

∫ ∫ ∫= - = - - -

( )0 0

useful

VV dt∫

(1.61)

The first term on the right-hand side defines the maximum “useful work” available, and the second term represents the work done by the ambient pressure acting on a

7262_C001.indd 18 1/19/08 11:26:47 AM

Introduction

moving boundary. Substituting the previous two equations into the energy balance,

T S S W P V V U U0 0 0 0 0( ) ( )- - - - = -useful

(1.62)

where the subscripts indicate the value at reference conditions. Solving for the useful work term gives

W T S S U U P V Vuseful = - - - - -0 0 0 0 0( ) ( ) ( )

(1.63)

which is equivalent to the exergy defined earlier as

X T S S U U P V V≡ - - - - -0 0 0 0 0( ) ( ) ( )

(1.64)

Dividing by the total mass gives the specific exergy (in other words, exergy per unit mass):

φ ≡ - - - - -T s s u u P v v0 0 0 0 0( ) ( ) ( )

(1.65)

Generalizing to include kinetic and gravitational potential energy requires only that we replace u with e, which is the total specific energy given by e u V gz= + +1

22 .

Typically, V0 = 0 and z0 = 0 at the reference state.With the exergy defined in this manner, one can study the change in exergy when

the state of a system changes. As a system undergoes a process from one thermo-dynamic state to another, a corresponding change in exergy occurs. Combining the First and Second Laws as expressed by the energy and entropy balance equations for a compressible substance of fixed mass leads to an exergy balance equation. In integral form, the First and Second Laws become

L Q dt W dt U U1 2 1: + = -∫ ∫

(1.66)

and

LQ

Tdt S S S2 2 1: + = -∫ gen

(1.67)

Combining by taking L T L1 0 2- gives

Q dt W dt TQ

Tdt T S U U T S S∫ ∫ ∫+ - - = - - -0 0 2 1 0 2 1gen ( )

(1.68)

Collecting terms and replacing the right-hand side with equivalent terms using the definition of exergy gives

1 00 2 1 0 2-

+ + - - = -∫ ∫T

TQ dt W dt P V V T S X( ) gen XX1

(1.69)

7262_C001.indd 19 1/19/08 11:26:55 AM


Other than a difference in sign on the work term, Equation 1.69 is a typical result found in textbooks on elementary thermodynamics (Cengel and Boles, 1989). If we identify the first term on the left side as “exergy transfer due to heat interaction” in the same sense that we identify entropy transfer due to heat interactions, and the second term (in brackets) as the “exergy transfer due to work interaction,” then Equation 1.69 reduces to the following result:

I X XX - = Ddes (1.70)

where Ix is the exergy current due to work and heat transfer and Xdes = T0 Sgen is the exergy destruction (called the Gouy–Stodola identity).

Alternatively, it is useful to interpret the quantity expressed by Xdes as the dis-tance from which the system approaches thermodynamic equilibrium with its envi-ronment. Recognizing the Second Law through the increase of entropy principle, it is required that

T S0

0

0

0gen

Real World

Ideal World

Impossible

>=<

(1.71)

Because we have associated exergy as equivalent to a measure of work potential, this term can be described as “exergy degeneration” or a loss of potential work due to real-world, irreversible effects. Entropy generation has a corresponding destruction of exergy:

X T Sdes gen= 0 (1.72)

A system in the real world undergoes spontaneously a process that brings it closer to thermodynamic equilibrium with its surroundings.

1.4.2 o pen syst eM

During an unsteady process where a substance goes from an initial (inlet; subscript “in”) to a final (exit; subscript “out”) state, the quality of energy changes, and a cor-responding change occurs in its thermodynamic distance from equilibrium. Com-bining the First and Second Laws as expressed in the energy and entropy balance equations for an unsteady process, one may obtain the balance equation for exergy. This derivation is commonly provided in undergraduate thermodynamics textbooks. The First Law for a control volume can be expressed as

Q W m h V gz m h V gzk + + + +

- + +

∑ ∑

12

12

2 2

in =∑

out

ECV

(1.73)

7262_C001.indd 20 1/19/08 11:26:58 AM

Introduction

where m, h, and Ecv refer to the mass flow rate, enthalpy and time rate of change of exergy in the control volume, respectively.

The corresponding form of the Second Law is

Q

Tms ms S Sk

k

+ - + =∑∑ ∑ in out

gen CV

(1.74)

Combining these equations yields

1 0 2

2

12

12

- + + + +( )

- + +

∑ ∑T

TQ W m h V gz

m h V

k

in

ggz T Sd

dtE T S( ) - = -∑ out

gen CV0 0 ( )

(1.75)

If we define a specific “flow exergy,” y, in the same sense that enthalpy represents a “flow energy,” then the exergy balance equation simplifies to

1 0

0 0- + + + - -∑∑ T

TQ W P V m m Tk ( ) ψ ψin out gen

S X=∑ CV

(1.76)

where

ψ φ= + -( )P P v0 (1.77)

and f is the specific exergy, (e - e0) + P0 (v - v0) - T0 (S - S0). Identifying the first two terms on the left side of Equation 1.76 as the transfer of exergy due to work and heat transfer, respectively, and the third and fourth terms as exergy transfer due to mass flow, the exergy balance equation for a control volume reduces to

in out gen CV X X T S X- - =0 (1.78)

where “in” and “out” terms represent the flow of exergy into and out of the control volume.

In addition to its role in determining the direction of natural processes and a criterion for thermodynamic equilibrium, the Second Law can also characterize the efficiency of engineering devices (Bejan, 1996). Carnot (1960, English translation from French by R.H. Thurston) conceived and developed the Second Law to account for the performance and limits of heat engines. Isentropic efficiency characterizes the performance of various engineering devices, such as turbines and compressors. In the context of exergy, the Second Law defines a more general measure of perfor-mance that applies not only to turbines and compressors, but heat exchangers, mix-ing processes, and other devices. A measure of performance for any engineering device should compare its efficiency, relative to the efficiency of an ideal device (no irreversible losses) operating under the same conditions. This measure of performance is called the “Second Law efficiency” or effectiveness, which can be

7262_C001.indd 21 1/19/08 11:27:03 AM


defined as follows,

ηIIdes

supp

≡ -1

X

X (1.79)

where the subscripts “des” and “supp” refer to destroyed and supplied, respectively.Past literature has interpreted this measure of performance as a “rational effi-

ciency” of a process or device (Müller and Ruggeri, 1998). Essentially, the effective-ness of any process equals the fractional change in the exergy relative to the exergy supplied. The concept of effectiveness applies to any thermophysical process, includ-ing heat engines, refrigerators, heat exchangers, mixing, throttling, and so forth. It is always bounded between zero and one.

The Second Law of Thermodynamics will: (i) determine the direction of change for spontaneous, natural processes; (ii) establish criteria for equilibrium in thermo-dynamic systems; and (iii) provide the theoretical limits for the performance of engi-neering systems and processes. Items (i) and (ii) identify the role of the Second Law as a limiter in abstracting the differences in response of different materials via the constitutive relations. Item (iii) identifies the role of the Second Law enumerated by the concept of effectiveness, as a limiter to indicate how a system relaxes to equilib-rium conditions with its surroundings while producing or consuming work.

. f o r Mu l at io n o f en t r o Py Pr o d u c t io n a n d exer g y dest r u c t io n

In the previous section, formulations of entropy transport and the Second Law were developed. In those equations, entropy production and exergy destruction were key parameters that characterized the efficiency of the thermal system or device. In this section, detailed expressions for these parameters will be developed, from which design methodologies can be established to reduce and minimize entropy produc-tion, thereby optimizing system performance.

1.5.1 c l o sed syst eM

From Section 1.3.1, for a closed system exchanging energy with its surroundings through work and heat transfer, the exergy balance equation can be expressed as

1 00 0-

+ + - =T

TQ W P V T S X( )

. . .

gen

(1.80)

Substitution for the heat flow and work term (relating compression/expansion work to pressure and change of volume) leads to

1 00 0-

+ - + - =T

TU PV PV P V T S X( )

. . . . . .

gen

(1.81)

Using thermodynamic relations for the exergy gradients leads to the following simi-lar result as Equation 1.33,

- = - ∂

∂- ∂∂

T S XX

UU

X

VV0 gen

(1.82)

7262_C001.indd 22 1/19/08 11:27:07 AM

Introduction

which is similar to results derived previously for the convexity property of exergy in Section 1.3.1. The convexity property of exergy is intrinsically linked with the Second Law.

1.5.2 l in ea r ad Vec t io n eq u at io n (w it h o u t dif f u s io n )

Entropy is transported throughout a problem domain through advection of scalar quantities like fluid momentum and internal energy. For example, scalar transport of fluid momentum leads to frictional irreversibilities, while transport of internal energy involves convective heat transfer and thermal irreversibilities. In this section, the exergy balance equation is developed with respect to scalar transport of a general scalar quantity, h(x,t). The governing equation for the one-dimensional scalar advec-tion equation without diffusion is given by

∂∂+ ∂∂=η

t

f

x0

(1.83)

which represents pure advection and f(h) equals the “flux of h.”According to the entropy concavity principle, the corresponding balance of

entropy is given by

∂∂- ′ ∂∂≥S

tS

t

η0

(1.84)

Substituting for the second term using Equation 1.83 and applying the chain rule,

∂∂+ ′ ′ ∂

∂≥S

tS f

xη

0

(1.85)

The one-dimensional form of the entropy transport equation can also be expressed as

genS

S

t

F

x= ∂∂+ ∂∂

(1.86)

where F represents the “transfer of entropy with h.” It is a term arising from pure convective transport of h. Subtracting Equation 1.86 from Equation 1.85 and using the chain rule for ∂F/∂x gives

( ' ' ')F S f

x- ∂

∂≥η 0

(1.87)

The strict equality must be enforced to avoid violation of the Second Law, so a compatibility condition, ′ = ′ ′F S f , is obtained as a constitutive restriction required by the Second Law. This result implies

∂∂+ ∂∂=S

t

F

x0

(1.88)

and it follows that Sgen = 0. This result is well known that reversible processes have zero entropy generation, although the previous derivation shows an additional requirement of compatibility between derivatives of entropy and its flux, F.

7262_C001.indd 23 1/19/08 11:27:12 AM


The corresponding balance of exergy equation is given by

∂∂- ′ ∂

∂≤X

tX

tη

0.

(1.89)

Substituting for the second term using Equation 1.83 and applying the chain rule,

∂∂+ ′ ′ ∂

∂≥X

tX f

x

η0

(1.90)

When the balance of exergy is written in an analogous form as entropy transport,

genX

X

t

G

x= ∂∂+ ∂∂

(1.91)

where G represents the “transfer of exergy with h.” It is a term resulting from the purely convective transfer of h across boundaries. Subtracting Equation 1.91 from Equation 1.90 and using the chain rule for ∂G/∂x gives

( ' ' ')G X f

x- ∂

∂≤η 0

(1.92)

In this case, the strict equality to satisfy the Second Law leads to an exergy com-patibility condition, ′ = ′ ′G X f , which is a constitutive restriction required by the Second Law. Also, it leads to

∂∂+ ∂∂=X

t

G

x0

(1.93)

and it follows that Xdes = 0. In the next section, the previous procedure will be extended to scalar advection, including diffusion.

1.5.3 l in ea r ad Vec t io n eq u at io n (w it h dif f u s io n )

In this section, a similar procedure will be used to derive the exergy destruction rate corresponding to scalar advection with diffusion. The governing equation for one-dimensional advection with diffusion is

∂∂+ ∂∂= ∂∂

η ηt

f

xD

x

2

2 (1.94)

where F = chand c equals a constant advection velocity. The variable D refers to a diffusion coefficient. The corresponding balance of exergy is given by

∂∂- ′ ∂

∂≤X

tX

t

η0

(1.95)

7262_C001.indd 24 1/19/08 11:27:19 AM

Introduction

Substituting for the second term using Equation 1.94 gives

∂∂+ ′ ∂

∂- ′ ∂

∂≤X

tX

f

xDX

x

2

20

η

(1.96)

Applying the compatibility condition and the chain rule,

∂∂+ ∂∂- ∂∂

′ ∂∂

+ ∂

∂ ′′X

t

G

xD

xX

xD

xX

η η 2

dees

X

≤ 0

(1.97)

where G = cX. The exergy destruction term is labeled because it is the only term that must be nonnegative. To preclude any violation of the Second Law, the strict equality must be enforced because the magnitude of all terms on the left side is not known beforehand. Because X is convex in h, then X″ < 0. Hence, we arrive at two expres-sions for the Second Law corresponding to scalar advection with diffusion:

desX D

xX= ∂

∂ ′′

2η

(1.98)

and

desX Dx

Xx

X

t

F

x= ∂∂

′ ∂∂

- ∂∂+ ∂∂

η

(1.99)

When imposing the principle of nonnegative exergy destruction, the first expression represents a constitutive restriction on the diffusion parameter: D ≥ 0 . The second expression represents the true exergy balance equation for this process. The third term contains the effects of the diffusive flux (diffusive transport of h such as fluid friction or heat conduction).

1.5.4 n aVier –st o keseq u at io ns

Since the Euler equations represent inviscid fluid motion, they are limiting cases of the Navier–Stokes equations, which describe the dynamic motion of a viscous, heat-conducting fluid. The Navier–Stokes equations can be expressed in the following tensor form,

∂∂+∂∂=ρ ρ

t

V

xj

j

0

(1.100)

∂∂+ ∂∂

+ - - =E

t xEV PV V q

jj j ji i j[ ]τ 0

(1.101)

7262_C001.indd 25 1/19/08 11:27:24 AM


∂∂+ ∂∂

+ - - =E

t xEV PV V q

jj j ji i j[ ]τ 0

(1.102)

where Vj refers to the velocity component in the j-coordinate direction and where E u V= +ρ ρ1

22 represents the total energy (internal plus kinetic energy).

These equations are underdetermined, because they contain more unknowns than equations. Consequently, additional information is required. The constitutive relations provide the additional closure information. Typically, these include the ideal gas law, P RT= ,ρ the assumption of a thermally perfect gas (cvdepends only on temperature, T), and Fourier’s relation for heat conduction,

q kT

xjj

= - ∂∂

(1.103)

Also, the following constitutive relations will be used for the viscous stress tensor of a Newtonian fluid,

τ µ λ δjij

i

i

j

j

jji

V

x

V

x

V

x=

∂∂+ ∂∂+∂∂

(1.104)

where d ij is the Kronecker delta function. The entropy transport equation associated with processes modeled by the Navier–Stokes equations is

genS

S

t

F

x x

q

Tj

j j

j= ∂∂+∂∂+ ∂∂

(1.105)

where S s= ρ and F sVj j= ρ . Because one more unknown has been added (specific entropy, s), another constitutive relation is needed, namely, the functional relation between entropy and the other field variables. This relation must satisfy the concav-ity property of entropy. For an ideal gas, the entropy functional (from thermodynam-ics; written in nondimensional form) is

s T T( ) ln lnρ

γρ, =

--1

1 (1.106)

where g is the ratio of specific heats.Because the state variables include mass, momentum, and total energy, it is conve-

nient to define an algebraic state vector, q V V V ET ≡ , , , , ρ ρ ρ ρ ρ1 2 3 , so that the entire set of conservation equations reduces to

∂∂+ ∂∂=q

t

f

xk

k

0

(1.107)

7262_C001.indd 26 1/19/08 11:27:31 AM

Introduction

where f is an algebraic flux vector. The functional relation for the entropy is S = S(q), and the concavity property is expressed as

S SS

qq q0

0

0 0- - ∂∂⋅ - ≥( )

(1.108)

where equality holds if and only if q = q0. The transient generalization of Equation 1.108 is the following extension of Equation 1.53 for the state variables:

∂∂- ∂∂⋅ ∂∂≥S

t

S

q

q

t0

(1.109)

Replacing ∂q/∂t and rearranging terms yields the following entropy generation rate:

genS

T

V

x

q

T

T

xji i

j

j

j

= ∂∂- ∂

∂τ

2

(1.110)

This result may be obtained by other means (Müller, 1985), but the approach here aims to emphasize the intrinsic connection between entropy concavity and the entropy generation equation in the Second Law of Thermodynamics.

Given the Fourier relation and the formula for the viscous stress tensor, the Second Law requires that

k ≥ ≥ + ≥0 0 02

3µ λ µ

(1.111)

These results are well known, and they have been documented in past literature dealing with thermodynamics and kinetic theory (Bird, 1976, 1994; Chapman and Cowling, 1990; Müller and Ruggeri, 1998). The origin of the inequalities arrives from the mathematical property of entropy concavity, as a function of the field vari-ables. Two expressions were obtained for the entropy generation: one that places restrictions on the constitutive relations; the other represents the entropy transport equation, Equation 1.105. The corresponding exergy destruction can be obtained by the Gouy–Stodola theorem. It may also be obtained directly by defining exergy from the concavity of entropy and then constructing the appropriate balance equation.

Consider the one-dimensional Navier–Stokes equations with the flux vectors separated into convective and dissipative parts as follows:

∂∂+ ∂∂+ ∂∂=q

t

f

x

f

x

v

0

(1.112)

where

f

V q

V

x

q kT

x

v = -- +

= ∂∂

= - ∂∂

043

ττ

τ µ

(1.113)

7262_C001.indd 27 1/19/08 11:27:36 AM


From Equation 1.106 and S s= ρ , the row vector of entropy derivatives is

S s

VT

VT Tq, , ,= + -

--

2

2 11γ

γ (1.114)

Then the exergy can be written as

X q T S S S q qq( ) [ ( )]≡ - - , ⋅ -0 0 00

(1.115)

where the subscript comma notation refers to differentiation.The exergy balance equation and exergy destruction rate can be obtained by

starting with the convexity relation for exergy in Section 1.3.1, and substituting for the time derivatives to give

∂∂- , ⋅ ∂

∂= ∂∂+ , ⋅ ∂

∂+ ∂∂

X

tX

q

t

X

tX

f

x

f

xq q

v

(1.116)

where X T S Sq q q, = , - , .0 0( ) Invoking the chain rule and the corresponding compat-

ibility condition leads to

X

fS

q

q

xG

q

x

G

xq q, ⋅∂⋅ ∂∂= , ⋅ ∂

∂= ∂∂

(1.117)

Also, note that

Xf

xT S

f

xT S

f

xq

v

q

v

q

v

, ⋅ ∂∂= , ⋅ ∂

∂- , ⋅ ∂

∂0 00

(1.118)

and

T S f x T sT

xV

qv

0 0 00

0 10 1

0

, ⋅ = --, ,

⋅ ∂∂

-- +

γγ

ττ qq

xq V

= ∂∂

-( )τ

(1.119)

In addition, the following equation can be derived:

S

f

x x

q

T T

V

x

q

T

T

xq

v

, ⋅ ∂∂= ∂∂ -

∂∂+ ∂

∂τ

2

(1.120)

Substituting these results into Equation 1.116 leads to

∂∂+ ∂∂- ∂∂+ ∂∂

-

+ ∂X

tSU

Vx x

TT

qTT

τ τ1 0 0 VVx

TT

qTx∂

- ∂∂=0

20

exergy destruction

(1.121)

7262_C001.indd 28 1/19/08 11:27:43 AM

Introduction

This identifies the exergy destruction term, and by comparison with Equation 1.110, note that des gen

X T S= 0 . The formula for the exergy destruction becomes

desXX

t

G

x

V

x x

T

T= - ∂

∂+ ∂∂

+ ∂∂- ∂∂

-

τ1 0 qq

(1.122)

Because the equations of fluid flow presume local thermodynamic equilibrium, there is no inconsistency when applying classical thermodynamic principles as represented by entropy concavity and exergy convexity. These principles are mathematical prop-erties, not limited to thermodynamics.

To apply the Second Law of Thermodynamics for availability analyses in prac-tice requires the balance of exergy equation and a functional formula for exergy. The construction of the entropy and exergy balance equations has been derived with-out specifying an entropy formula, except for the case of an ideal gas. To obtain the proper formula for S S= ( )ξ , general optimization principles can be applied. For example, Jaynes’ Maximum Entropy Principle (Jaynes, 1991; Levine and Tribus, 1979) is based on a generalization of the Second Law, when applied to constrained equilibria. Kapur and Kesavan (1992) provide a comprehensive and detailed proce-dure for generalized entropy optimization principles. If the domain of the dependent variable x is known, then the Maximum Entropy Principle obtains the proper form of entropy for a probability distribution function that quantifies fluctuations in that variable about its mean value. For example, if ξ ∈ ,∞[ )0 , then the MaxEnt principle prescribes S = lnξ . If the dependent variable u ∈ -∞,∞( ), then MaxEnt prescribes

S u= - /2 22σ , where2 12σ = is set without loss of generality. The Second Law, in essence, provides a way to (i) obtain a formula for entropy, and (ii) construct the bal-ance equation for entropy (Liu, 1972; Müller, 1967).

The mathematical property of entropy concavity has served multiple purposes, including restricting the types of constitutive relations allowed for modeling of real-world processes. The Second Law is a powerful concept that determines how physi-cal processes can be modeled, so that mathematical models reflect physical reality. This chapter has developed formulae for the balance of entropy and exergy, as required to enforce the restrictions prescribed by the Second Law. The advantage of using exergy balances (instead of entropy) is that they provide a concept that unifies the First and Second Laws into a single principle. This unified approach provides the basis for constructing a single metric across the spectrum of possible thermophysical systems and processes.

r ef er en c es

Adeyinka, O.B. 2002. Entropy and Second Law Optimization in Computational Thermo- fluids with Laser Based Measurements. M.Sc. thesis, University of Manitoba.

Adeyinka, O.B. and G.F. Naterer. 2005. Entropy based metric for component level energy man-agement: Application to diffuser performance. Int. J. Energ. Res., 29(11): 1007–1024.

Baytas, A.C. 2000. Entropy generation for natural convection in an inclined porous cavity. Int. J. Heat Mass Transfer, 43: 2089–2099.

7262_C001.indd 29 1/19/08 11:27:49 AM


Bejan, A. 1996. Entropy Generation Minimization: The Method of Thermodynamic Optimi-zation of Finite-Time Systems and Finite-Time Processes. CRC Press, Boca Raton, FL.

Bejan, A. 1997. Advanced Engineering Thermodynamics. 2nd ed., John Wiley & Sons, New York.

Bird, G.A. 1976. Molecular Gas Dynamics. Oxford University Press, New York.Bird, G.A. 1994. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford

University Press, New York.Boehm, R.F., Ed., 1989. Second Law Analysis — Industrial and Environmental Applications.

Proceedings of the Winter Annual Meeting of the American Society of Mechanical Engineers, AES 26. ASME San Francisco CA.

Boehm, R.F., Ed., 1992. Thermodynamics and the design, analysis, and improvement of energy systems. Proceedings of the Winter Annual Meeting of the American Society of Mechanical Engineers, AES 27. ASME, San Francisco, CA.

Camberos, J.A. 2000a. An alternative interpretation of work potential in thermophysical processes. AIAA J. Thermophysics Heat Transfer, 14(2): 177–185.

Camberos, J.A. 2000b. Non-linear time–step constraints based on the Second Law of Thermodynamics. AIAA J. Thermophysics Heat Transfer, 14(3): 231–244.

Carnot, S. 1960. Reflections on the Motive Power of Fire and on Machines Fitted to Develop That Power (English translation from French by R.H. Thurston, 1824). Dover Pub-lications, New York.

Cengel, Y.A. and M.A. Boles. 1989. Thermodynamics, An Egineering Approach. McGraw-Hill, New York.

Cervantes, J. and F. Soloris. 2002. Entropy generation in a plane oscillating jet. Int. J. Heat Mass Transfer, 45: 3125–3129.

Chapman, S. and T.G. Cowling. 1990. The Mathematical Theory of Non-Uniform Gases. Cambridge University Press (reprint; originally published in 1939), London.

Cheng, C.H., Ma, W.P., and W.H. Huang. 1994. Numerical predictions of entropy generation for mixed convective flows in a vertical channel with transverse fin arrays. Int. J. Heat Mass Transfer, 21: 519–530.

Cox, R.A. and B.M. Argrow. 1992. Entropy production in finite-difference schemes. AIAA J., 31(1): 210–211.

Demirel, Y. 1999. Irreversibility profiles in a circular couette flow of temperature dependence materials. Int. Commn. Heat Mass Transfer, 26: 75–83.

Dincer, I. and M. Rosen. 2004. Exergy as a driver for achieving sustainability. Int. J. Green Energ., 1(1): 1–19.

Drost, M.K. and M.D. White. 1991. Numerical predictions of local entropy generation in an impinging jet. ASME J. Heat Transfer, 113: 823–829.

Hauke, G.A. 1995. A Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation for the K-Epsilon Model. Ph.D. thesis, Stan-ford University, Stanford, CA.

Haywood, R.A. 1974. A critical review of the theorems of thermodynamic availability, with concise formulations. Part 1. J. Mechanical Eng. Sci., 16(3): 160–173.

Hughes, T.J.R., Mallet, M., and L.P. Franca. 1985. Entropy Stable Finite Element Methods for Compressible Fluids: Application to High Mach Number Flows with Shocks. Finite Element Methods for Nonlinear Problem, Europe-US Symposium. Trondheim, Norway, 762–773.

Jansen, K.E. 1993. The Role of Entropy in Turbulence and Stabilized Finite Element Meth-ods. Ph.D. thesis, Stanford University, Stanford, CA.

Jaynes, E.T. 1991. Probability Theory as Logic, in Maximum-Entropy and Bayesian Meth-ods. P.F. Fougére, Ed., Kluwer, Dordrecht, 1–16.

7262_C001.indd 30 1/19/08 11:27:49 AM

Introduction

Kapur, J.N. and H.K. Kesavan. 1992. Entropy Optimization Principles with Applications. Academic Press, San Diego, CA.

Kotas, T.J. 1985. The Exergy Method of Thermal Plant Analysis. Butterworth & Co., London.Kramer-Bevan, J.S. 1992. A Tool for Analysing Fluid Flow Losses. M.Sc. thesis, University

of Waterloo, Ontario, Canada.Kresta, S.M. and P.E. Wood. 1993. The flow field produced by a pitched blade turbine: char-

acterization of the turbulence and estimation of the dissipation rate. Chem. Eng. Sci., 48: 1761–1774.

Lavenda, B.H. 1991. Statistical Physics: A Probabilistic Approach. John Wiley & Sons, New York.

Levine, R.D. and M. Tribus. 1979. The Maximum Entropy Formalism. MIT Press, Cambridge, MA.

Liu, I–Shih. 1972. Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rational Mech. Anal., 46(2): 131–148.

Mahmud, S. and R.A. Fraser. 2002. Analysis of entropy generation inside concentric cylinder annuli with relative rotation. Int. J. Thermal Sci., 42: 513–521.

Moore, J. and J.G. Moore. 1983. Entropy Production Rates from Viscous Flow Calculations Part II. Flow in a Rectangular Elbow. American Society of Mechanical Engineers Gas Turbine Conference, ASME Paper 83-GT-71, Phoenix.

Moorhouse, D.J. and C.F. Suchomel. 2001. Exergy Methods Applied to the Hypersonic Vehicle Challenge. 2nd AIAA Plasmadynamics and Laser Conference, 4th Weakly Ionized Gases Workshops. AIAA Paper 2001-3063, Anaheim, CA.

Müller, I. 1967. On the entropy inequality. Arch. Rational Mech. Anal., 26(2): 118–141.Müller, I. 1985. Thermodynamics. Pitman Publishing, Marshfield, MA.Müller, I. and T. Ruggeri. 1998. Rational Extended Thermodynamics. Springer–Verlag,

Heidelberg.Natalini, G. and E. Sciubba. 1999. Minimization of the Local Rates of Entropy Production

in the Design of Air-Cooled Gas Turbine Blades. ASME J. Engineering Gas Turbine Power, 121: 466–475.

Naterer, G.F. 1999. Constructing an entropy-stable upwind scheme for compressible fluid flow computations. AIAA J., 37(3): 303–312.

Poulikakos, D. and A. Bejan. 1982. A fin geometry for minimum entropy generation in forced convection. Trans. Am. Soc. Mech. Eng., 104: 616–623.

Rosen, M.A. and I. Dincer. 2004. Effect of varying dead-state properties on energy and exergy analyses of thermal systems. Int. J. Thermal Sci., 43(2): 121–133.

Roth, B. and D. Mavris. 2000. A Method for Propulsion Technology Impact Evaluation Via Thermodynamic Work Potential. AIAA/USAF/NASA/ISSMO Symposium on Mul-tidisciplinary Analysis and Optimization, AIAA Paper 2000-4854, Long Beach, CA.

Sahin, A.Z. 2000. Entropy generation in turbulent liquid flow through a smooth duct sub-jected to constant wall temperature. Int. J. Heat Mass Transfer, 43: 1469–1478.

Sahin, A.Z. 2002. Entropy generation and pumping power in a turbulent fluid flow through a smooth pipe subjected to constant heat flux. Exergy Int. J., 2: 314–321.

Sciubba, E. 1997. Calculating entropy with CFD. Mech. Eng., 119(10): 86–88.Szargut, J., Morris, D.R., and F.R. Steward. 1988. Exergy Analysis of Thermal, Chemical,

and Metallurgical Processes. Hemisphere, New York.Truesdell, C. 1984. Rational Thermodynamics. 2nd ed. Springer-Verlag, Heidelberg.Truesdell, C. 1985. The Elements of Continuum Mechanics. 2nd printing. Springer-

Verlag, Heidelberg.Zubair, S.M., Kadaba, P.V., and R.B. Evans. 1987. Second Law-based thermoeconomic

optimization of two-phase heat exchangers. J. Heat Transfer, 109: 287–294.

7262_C001.indd 31 1/19/08 11:27:50 AM

33

2 Statistical and Numerical Formulations of the Second Law

2.1 In t r o d u c t Io n

The foundations of the Second Law of Thermodynamics are based on statistical thermodynamics of intermolecular exchange. However, its impact extends from the microscopic scale to the largest scales of engineering systems and, more generally, the environment and the earth. Dincer (2001) presented a detailed study that outlines the key role of exergy in environmental sustainability of energy systems. The envi-ronmental impact of waste emissions and power generation systems can be effec-tively characterized by methods of exergy analysis (Rosen and Dincer, 1997).

At the microscopic level, a close relationship exists between the concepts of entropy and probability, the most well known of which is associated with Ludwig Boltzmann. The concavity property of entropy is directly related to a given prob-ability distribution function (PDF) for a fluid. In this chapter, statistical formulations of the Second Law and the Clasius–Duhem (C-D) inequality will be described. The C-D inequality represents the irreversible increase of entropy required by the Second Law of Thermodynamics, and it is a supplementary equation in fluid mechanics. By relating entropy directly to a PDF, one can show that a nonequilibrium version of the entropy function (and also a modified C-D inequality) can be obtained. These probability functions will be outlined in this chapter. Some of the concave entropy functions obtained for the nonequilibrium functions will be shown to be less than or equal to the entropy associated with the equilibrium value, in accordance with the Second Law.

Entropy and probability are intrinsically related. However, no general agree-ment exists among scientists as to what this relation means or even exactly what is the relationship. Jaynes asserts that probability is a “logic of science.” In this way, probability theory (as logic) may be applied to any field of science. Posing ques-tions or problems from one scientific field in terms of concepts and principles from another can prove fruitful, if properly directed. This chapter attempts to frame some fundamental questions regarding the statistical aspects of fluid motion, in terms of the logic of probability. Some of the issues include establishing criteria for numeri-cal techniques based on physical principles, within the logical framework of prob-ability theory, as well as deriving practical mathematical expressions and formulae for implementing the results. Because kinetic theory utilizes many concepts and principles of probability and statistics, there is a wealth of ideas to gain from this

7262_C002.indd 33 1/19/08 11:31:24 AM

34 Entropy-Based Design and Analysis of Fluids Engineering Systems

field, as well as inspiration. This chapter will derive macroscopic rules, principles, and formulae by statistical averaging, rather than constructing constitutive relations based on one particular microscopic model or process. Unlike moment methods that suffer from closure problems, this chapter will assume given constitutive relations (which become constraints on the moments) that fully specify a distribution of prob-ability in the molecular velocity variable. The well-known conservation laws of fluid mechanics can then be obtained by taking subsequent moments of the Boltzmann equation, with a given PDF and set of collisional invariants that include molecular mass, momentum, and energy.

This chapter will also present a given probability distribution obtained with Jaynes’s maximum entropy principle. This distribution is then modified to accom-modate velocity and temperature gradients in a gas. It will be rewritten to highlight Gauss’s error law (Lavenda, 1991). Some interesting conclusions follow when the macroscopic entropy is obtained by taking the moment of - ln( )pdf . A function will be obtained that essentially represents entropy associated with the PDF. This function is different for equilibrium (i.e., Euler equations) and nonequilibrium (i.e., Navier–Stokes) conservation laws. This difference will be evident from the entropy associated with each kind of PDF.

2.2 c o n ser vat Io n Law s a s Mo Men t s o f t h eBo Lt zMa n n eq u at Io n

The dynamics of an ideal, monatomic, dilute gas in the absence of external forces are theoretically governed by the following Boltzmann equation (neglecting body forces):

∂∂+ ∂

∂= ∂

∂

( ) ( ) ( )ng

tv

ng

x

ng

tkk coll

(2.1)

where n is the number density and g is the PDF for the molecular velocity vk in an inertial frame. The function g is not a “velocity distribution function” because it is the probability that is distributed, not the velocity. Hence, gdvk equals the prob-ability that the molecular velocity lies between vk and vk + dvk. The repeated index k denotes a sum, and the right-hand side represents the time-rate of change in g, due to molecular collisions.

Moment equations can be generated by multiplying the Boltzmann equation by any function of molecular velocity, Q(vi), and integrating over velocity space as follows:

∂∂

< > + ∂∂

< > = Dt

n Qx

n v Q Qk

k( ) ( ) [ ] (2.2)

The first operator in Equation 2.2 is the expectation

< > = ,

-∞

∞

-∞

∞

-∞

∞

∫ ∫ ∫Q Qg dv dv dv 1 2 3

(2.3)

7262_C002.indd 34 1/19/08 11:31:27 AM

Statistical and Numerical Formulations of the Second Law 35

and the second operator is the collision integral given by

D = ∂

∂

-∞

∞

-∞

∞

-∞

∞

∫ ∫ ∫[ ]( )

Q Qng

tdv dv dv

coll1 2 33. (2.4)

Moments of the collision integral are identically zero when the arbitrary func-tion of molecular velocity, Q(vi), is one of the five collisional invariants, ψ ( )v Q m v vi

INVi≡ = , , / 1 22 , where m is the molecular mass and v2 represents the

square of the velocity magnitude. This general result holds for any distribution func-tion g and for any molecular interaction law. Taking moments of the Boltzmann equation, given the set of collisional invariants, yields the following conservation laws of gas dynamics:

∂∂

< > +∂∂

< > =t

nx

n vk

k( ) ( )ψ ψ 0 (2.5)

Expanding Equation 2.5 using each of the collisional invariants in turn gives a set of equations for the conservation of mass, momentum, and energy.

By introducing a set of relative velocity components, C v ui i i= - , where u vi i= < > is the expected value of the corresponding velocity variable, the following results are obtained for the central moment stress tensor, thermodynamic pressure, viscous stress tensor, internal energy, and heat flux vector, respectively, as follows:

σ ρij i jC C= < > (2.6)

p kk= 1

3σ (2.7)

τ σ δ δij ij ij ijp

i j

i j= - + =

≠=

;( )

( )

0

1 (2.8)

e Cint = < > 1

22 (2.9)

q C C C C C Ci i= < > = + +ρ 1

22 2

12

22

32; (2.10)

where C C C C212

22

32= + + .

The conservation laws for gas dynamics can then be written in the following familiar form:

∂∂+∂∂=Q

t

F

xk

k

0

(2.11)

7262_C002.indd 35 1/19/08 11:31:35 AM


where the algebraic state and flux vectors are, respectively,

Q

u

u

u

E

F

u

u u

k

k

k

=

=+

ρρρρρ

ρρ σ1

2

3

1 kk

k k

k k

k ki i k

u u

u u

u E u q

1

2 2

3 3

ρ σρ σ

ρ σ

++

+ +

(2.12)

The internal energy, eint, and heat flux vector, qi, are given for a monatomic gas. When the gas has a different internal structure, the procedure may be modified to accommodate the intermolecular degrees of freedom. A common approach will assume that all internal molecular energy modes exist in equilibrium, both internally and with translational degrees of freedom.

The total internal energy, eint, can be expressed in terms of the temperature, T, by the following equilibrium relation:

e RTint = = ,ξ ξ σ

2 22 (2.13)

where x equals the number of degrees of freedom, R is the gas constant, and σ 2 = RT is the variance that specifies the second central moment for some PDF, g g v uk k= | ,( )σ 2 , to be determined. In this notation, it is understood that the probability function g is conditional on knowing the parameters uk and s2, so that it is fully specified. The fluid velocity is the first moment, whereas the variance (temperature) is the second central moment. When the first and second moments of a probability distribution are known and the variable exists in the range ( )-∞,∞ , then the equilibrium or maxi-mum entropy distribution is a central probability distribution (Kapur and Kesavan, 1992). Also, when the off-diagonal correlation coefficients are zero, it is known as a normal or Gaussian PDF. The additional internal energy is introduced through the parameter x. For a monatomic gas, x = 3, whereas for a diatomic gas with two addi-tional degrees of freedom (due to molecular rotation), x = 5. Diatomic gas molecules have a dumbbell structure, so the energy associated with axial rotation is negligible.

2.3 ext en d ed Pr o Ba BILIt y dIst r IBu t Io n s

To account for the amount of energy carried by a particle with a certain internal structure, the kinetic energy mv2 2/ must be replaced by ( )1

22mv + ε , where ε is the

additional internal energy per particle. The collisional invariants are then

ψ ε= , , + m mv mvi

12

2 (2.14)

To include the effects of internal structure, one can use the mathematical expression for the probability distribution. The additional degrees of freedom can be expressed by defining the variables ω k, with k = ,1 2 for the two rotational degrees of freedom,

7262_C002.indd 36 1/19/08 11:31:44 AM


corresponding to a diatomic molecule with a linear structure. The expectation opera-tor is then also modified to include integration over ωk ∈ -∞,+∞ ,( ) with an expecta-tion < >ω k equal to zero. One can therefore set ε ω= 1

22m with ω ω ω2

12

22= + in

Equation 2.14 and use a multivariate PDF, g g v ui i i= , | , .( )ω σ 2

Assuming Equation 2.1 continues to hold for the extended distribution function, g v ui i i( ), | ,ω σ 2 , and the collision integral on the right side is interpreted properly, then an additional integral over ω k is required to generate Equation 2.2 as moments of Equation 2.1. The quantities in Equation 2.14 must continue to be conserved in a collision. Consequently, Equation 2.4 becomes zero and Equation 2.5 remains unchanged. Evaluating the left side of Equation 2.5 for the five different quantities in Equation 2.14 gives identical results to those obtained for the monatomic gas, for all quantities that contain polynomials in vi alone. This can be shown because inte-gration over the ω k variables can be taken independently from integration over vi. Therefore, the conservation laws for mass and momentum are recovered as written.

The same observation also applies to the first term in the quantity 12

2 2m v( ),+ω , and the conservation of energy can be written as

∂∂

< > + < >

+∂∂

< > + <t

vx

v vk

kρ ρ ω ρ ρ12

12

12

12

2 2 2 vvkω 2 0>

= (2.15)

Substituting the central moments, Equation 2.6 to Equation 2.10 into Equation 2.15 and using the previous results for the monatomic gas gives

∂∂

+∂∂

+ + =t

Ex

u E u qk

k ki i k( ) ( )ρ ρ σ 0 (2.16)

where E e uint≡ + ,12

2

e Cint = < + >1

22 2( )ω (2.17)

and

q C C Ci i i= < > + < >ρ ρ ω1

212

2 2

(2.18)

The conservation law Equation 2.11 continues to hold, provided Equation 2.9 is replaced by Equation 2.17, and Equation 2.10 by Equation 2.18. Therefore, the con-servation law Equation 2.11 can be used with the state and flux vectors as they occur in Equation 2.12, provided definitions in Equation 2.17 and Equation 2.18 are used when a state of equilibrium exists between the internal modes and the translational degrees of freedom (Vincenti and Kruger, 1965).

Since the conservation laws in Equation 2.11 can be developed for a general fluid through phenomenological principles alone, the set is more general than implied by a kinetic theory derivation, because they are also the conservation equations for fluid dynamics. For an ideal gas flow, the kinetic theory approach is necessary in that it shows that Equation 2.11 is valid for any degree of translational nonequilibrium,

7262_C002.indd 37 1/19/08 11:31:53 AM


i.e., any PDF for a translational velocity. If equilibrium conditions are assumed, then the Maxwell–Boltzmann PDF (MB-PDF), gMB, can be used, and the conservation laws are the Euler equations because the viscous stresses and heat flux are zero. If instead a Chapman–Enskog (CE-PDF) distribution gCE is chosen, then the set of moment equations can be interpreted as the Navier–Stokes equations because the corresponding expressions describe the dynamics of a viscous, heat-conducting fluid. Note that a PDF can be used in Equation 2.11. As long as g is fully specified, the set becomes closed. If g remains general, then there is a closure problem when using a moment method with kinetic theory because t ij and qi are unknown quantities in the equations. The conservation equations in Equation 2.11 are not the Navier–Stokes equations, until one introduces gCE or some other PDF that incorporates terms to account for the viscous, heat-conducting effects.

In the conservation law, Equation 2.11, specifying Fi completely in three dimen-sions requires the evaluation of 15 quantities. But the task is simplified for a finite volume together with Gauss’s divergence theorem. Equation 2.11 can be written in the following integral form:

∂∂

+ =∫ ∫tQ dV F dS

V Sn 0 (2.19)

where S encloses the volume V, and Fn is the projection of Fi onto the unit outward normal for the surface element dS. If V is a rectangular volume in Cartesian coor-dinates, then only five quantities need to be calculated for each planar surface, pro-vided Fn can be evaluated directly. Using the notation of Equation 2.5, the state and flux vectors can be generated from moments as follows:

Q n F n vn n= < > = < >ψ ψ (2.20)

where vn is the molecular velocity component normal to the planar surface. In Equation 2.20, the scalar quantity Q is transported across a fixed surface by vn, thus creating a flux in that quantity. The five fluxes defined in Equation 2.20 are total fluxes. In the general case, they contain both the inviscid (Euler) fluxes, as well as the nonequilibrium effects due to viscous stresses and heat conduction. Expressions for the viscous stresses and heat conduction terms can be obtained readily from clas-sical fluid dynamic theory (Vincenti and Kruger, 1965). Together with the ideal gas equation of state, these nonequilibrium terms complete the constitutive relations for the field equations. They will serve to establish the nonequilibrium PDFs described in the following section.

2.4 seLec t ed Mu Lt Iva r Iat ePr o Ba BILIt y d Ist r IBu t Io n fu n c t Io n s

This section summarizes four important probability distributions: the MB-PDF, the central distribution PDF, the CE-PDF, and the skew-normal PDF, which have been analyzed by Camberos (1997a,b) in terms of the Second Law. The notation in this section is standard in probability theory.

7262_C002.indd 38 1/19/08 11:31:55 AM


2.4.1 Ma xw el l –Bo l t zMa n n Pr o Ba Bil it yDist r iBu t io n Fu n c t io n

First consider the MB-PDF. This equilibrium distribution function is a special case of the central probability distribution. It is fully specified by the first moment and the second central moment (a temperature parameter), s2, known as the variance. Consider a monatomic gas for simplicity. The equilibrium probability distribution for a set of ξ = 3 variables is represented by the MB-PDF,

g z CC

z C zMB

oo T

o( )( )

exp| = | | - ⋅ ⋅ - /

/-

1 1 2

3 21

212π (2.21)

where z = (z1,z2,z3) is the set of standardized velocity variables written relative to expectation values such that z v uk k k≡ - / .( ) σ The matrix Co contains the central moments. For an equilibrium distribution, it is given by

C Io = ×3 3 (2.22)

where I3 3× is the identity matrix, sized appropriately for ξ = 3 variables. The matrix in Equation 2.22 has no off-diagonal terms, so the multivariate probability distribu-tion could have been expressed as the product of three univariate central probability distributions in each variable. However, the form given in Equation 2.22 will be retained for consistent notation, when compared with PDFs in following sections.

2.4.2 c en t r a l Dist r iBu t io n Pr o Ba Bil it yDist r iBu t io n Fu n c t io n

The second function selected is the central distribution PDF. The central distribution (CD) for the number of variables sufficient to describe the molecular model of interest is

g(z barz, C)

The second function selected is the central distribution (CD) PDF. The CD for the number of variables sufficient to describe the molecular model of interest is gCD.

It has the same form as the right side of Equation 2.21 for the MB-PDF, but now the more general covariant matrix C contains all of the central moments. Compo-nents of C are given by

c rx x x x

ij iji i

i

j j

j

= ≡-

-

σ σ

(2.23)

which are known in probability theory as the correlation coefficients. Note that rii = ,1 because σ jj j jx x2 2= < - > .( ) The MB-PDF is a special case of Equation 2.23, whereby all of the mixed central moments are zero. Taking the expectation value of ψ using gMB gives the following algebraic vector of macroscopic state variables:

Q n n g v u dx dx dxMB MB

i i= < > = | ,-∞

∞

-∞

∞

∫ ∫ψ ψ σ( )21 2 3

(2.24)

7262_C002.indd 39 1/19/08 11:32:02 AM


which is given in Equation 2.12 with a total specific energy E u= + ,12

2 23( )σ where ui are the fluid velocity components in Cartesian coordinates.

The algebraic vector for the flux of particles, F, across a volume boundary is given by the expectation value of the state function multiplied by the velocity com-ponents normal to the boundaries:

F n vk

EEk

MB= < >ψ (2.25)

On evaluating the expectation values for each component, with v z uk k k k= + ,σ one finds that Equation 2.25 produces the familiar inviscid flux vectors for the Euler equations, as noted by the superscript. If, instead, expectation values are taken by the central PDF with nonzero correlation coefficients, one obtains the same state vector, but

F n vk k

CD CD= < >ψ (2.26)

which now contains the viscous terms due to velocity gradients. These fluxes could be used, for example, to represent the Navier–Stokes equations.

2.4.3 c h a PMa n –ensk o g Pr o Ba Bil it yDist r iBu t io n Fu n c t io n

Next, consider the CE-PDF. To accommodate heat-conduction effects, the equilib-rium MB-PDF or the CD-PDF must be modified. A well-known approach to mod-ify the MB-PDF is to apply the perturbation technique employed by Chapman and Enskog (1939). This technique yields a pseudo-PDF that incorporates the effects of velocity and temperature gradients when deviations from equilibrium are not too severe. One can write the so-called Chapman–Enskog pseudoprobability distribu-tion in different ways. To maintain consistency with the notation used in this chapter, the CE-PDF is written as

g z q T z T z q z zCE T( ) ( ), | , , = - ⋅ ⋅ + ⋅ -

σ 2 31

12

15

gMB

(2.27)

where

z

z

z

z

z

z

z

z

= ≡

1

2

3

3

13

23

33

(2.28)

The effects of heat conduction are included in the following parameter:

q q q q qq

kk= , , , ≡ ,( ˆ ˆ ˆ ) ˆ

1 2 3 3ρσ (2.29)

which are called skewness coefficients in probability theory because of their effects on the PDF.

7262_C002.indd 40 1/19/08 11:32:07 AM


Define the following nondimensional stress matrix

ττ τ ττ τ ττ τ τ

τ≡

11 12 13

21 22 23

31 31 33

; ij == τ ji

(2.30)

This accounts for the effects of velocity gradients, because we define the dimension-less viscous stresses by

tkiki≡ .τρσ 2

(2.31)

Evaluating the expectation value in Equation 2.24 using gCE instead of the equilib-rium MB-PDF gives the same state vector, but now the flux vectors in Equation 2.25 will include both the viscous and heat-conducting terms. Hence, the full Navier–Stokes equations are represented by Equation 2.11, when gCE is used together with the constitutive relations for tki and qk.

2.4.4 s kew -n o r Ma l Pr o Ba Bil it yDist r iBu t io n Fu n c t io n

Finally, with a skew-normal PDF, we can choose any probability distribution when generating the moment equations. One may consider alternatives to the CE-PDF, as long as the appropriate field equations are obtained. One approach is to use the PDF directly and a probabilistic approach when constructing it. To incorporate the effects of viscosity, which lead to second-order moments with nonzero correlation coefficients, a multivariate central probability distribution is sufficient. This is the maximum entropy probability distribution, when the first and second moments are specified. However, to include the effects of heat conduction, one needs to specify the third-order central moments. If only these are specified and no others, then a resulting maximum entropy analysis becomes infeasible since the exponential func-tion cannot be normalized when third-order powers are included. The only recourse is the Chapman–Enskog results, with skewness coefficients in the construction of the PDF, which will be a modification of the central distribution. This leads to a skew-normal pseudo-PDF (SN-PDF). In this PDF, a nonzero third-order moment is obtained, although the symmetry of a multivariate central distribution is retained. The skew-normal distribution is defined by variable tii from Equation 2.31.

g z C z z g z CSN CD( ) ( )| , = - ⋅ -

|θ θ113

3 (2.32)

where i iiz z tˆ = / - .1 The skewness coefficients θ θ θ θ= , ,( )1 2 3 must be determined from the moment constraints, thereby leading to the full Navier–Stokes fluxes. The PDF on the right side of Equation 2.32 is the multivariate CD-PDF with correlation coefficients given by Equation 2.23.

7262_C002.indd 41 1/19/08 11:32:11 AM


The moments n vkSN< >ψ produce the Navier–Stokes fluxes, which are obtained

by specifying gSN with the following covariant matrix:

C

r r r

r r r

r

=11 1

212 1 2 13 1 3

21 1 2 22 22

23 2 3

σ σ σ σ σσ σ σ σ σ

331 3 1 32 2 3 33 32σ σ σ σ σr r

.

(2.33)

where σ τk kk2 1≡ - and

r r

i j

i j

ji ij

ij

ii jj

= ==

-- -

≠ .

1

1 1

ττ τ( )( )

(2.34)

The structure of the covariant matrix implies that the molecular translational modes are not at equilibrium with the fluid temperature.

To obtain the correct energy fluxes, the skewness coefficients qk must be related to the heat-conduction terms. Hence, the third-order moments are equated to the expectation of the flux of thermal energy, relative to fluid velocity. The resulting constraints provide a unique solution, but the solution gives a complicated set of equations for the three unknown parameters, q1, q2, q3. A more convenient form of the result is

θ θ θ θ= , , = ⋅

-( )

ˆ

ˆ

ˆ1 2 3

1

1

2

3

K

q

q

q

(2.35)

where kq expressions were previously defined in Equation 2.29. Also, K can be expressed by

K

r r r

r r≡

σσ

σ

1

2

3

11 12 13

12

0 0

0 0

0 0222 23

13 23 33

1

2

3

0 0

0 0

0 0

r

r r r

a

a

a

(2.36)

where

a ri

k

ik k==∑

1

3

2 2σ (2.37)

7262_C002.indd 42 1/19/08 11:32:17 AM


and u,t ij and qi are given by local conditions and constitutive relations. The skew-normal distribution is then fully specified by the parameters qk and covariance matrix C. This then yields the full Navier–Stokes fluxes after evaluating the moments, n vk< >ψ , with g gSN= .

2.5 c o n ca v een t r o Py fu n c t Io n s

From elementary thermodynamics, the specific entropy for an ideal gas with con-stant specific heats can be written in the following nondimensional form:

s TEQ =-

′ - ′11γ

ρln ln (2.38)

where ′T and ′ρ are the nondimensionalized temperature and mass density ratios, with respect to some reference state. As a function of the fluid state variables, the specific entropy is concave. The concavity property of entropy can be interpreted through proper probability distributions. The probabilistic approach to statistical physics developed by Lavenda (1991) asserts that “the connection between entropy and probability is through a law of error for extensive thermodynamic variables and Boltzmann’s principle is a consequence of it.” This “law of error” is realized as an inequality expressing the concavity property of entropy. Concavity is directly related to the logarithm of a probability distribution as follows:

- = - - ′ - +ln ( ) ( ) ( ) ( )( )g x s x s x s x x x constant (2.39)

where the prime notation (v′) denotes a derivative.Physical phenomena that are characterized by a probability density, g(x), for

some relevant variable, x, can be examined in terms of Gauss’s principle, where one identifies the average value, x , with the most probable value. The function g(x) is a general, unknown probability density, not limited to conditions at equilibrium. The concavity property of entropy requires that

s x s x s x x x( ) ( ) ( )( )- - ′ - > 0 (2.40)

The inequality defines a strictly concave function. In thermodynamics, the average value x is uniquely determined at the equilibrium state. Note that Equation 2.40 does not assert that the entropy, s(x), of the nonequilibrium state is always less than the entropy of the equilibrium state, s x( ). A key feature of Equation 2.40 lies in the realization that the entropy function is a constrained maximum, where the derivative ′s x( ) has the role of a Lagrange multiplier for the corresponding constraint, obtained

directly as a function of the conserved state variable or variables. Thus, the entropy tends to increase only when the state variable differs from its value at equilibrium. Taking the average of Equation 2.40 gives s x s x( ) ( )- > ,0 which implies a principle of nondecreasing entropy.

To obtain the entropy associated with a PDF, the expectation value of the nega-tive of the natural logarithm is taken as follows:

s g= <- >ln (2.41)

7262_C002.indd 43 1/19/08 11:32:26 AM


where the functional dependence is given with respect to a set of standardized vari-ables. This allows one to define

ϕ = ⋅ ⋅ + | |-1

21z C z CT ln (2.42)

which is “-1n g” with the constant omitted, such that

s s gPDF EQ CD

- = <- >ln (2.43)

where sEQ is the thermodynamic (equilibrium) entropy. The expectation value is taken relative to the standardized variables, so the right side of Equation 2.43 will be either negative for a nonequilibrium PDF or zero for an equilibrium PDF. With the MB-PDF, a constant is obtained, although it does not change the implications.

This analysis can be extended for the Maxwell–Boltzmann entropy. For the MB-PDF,

ϕ MB z C z CT

o o= ⋅ ⋅ + | |-12

1 ln (2.44)

where Co is the identity matrix so | |= .Co 1 Taking the expectation value yields

s s z z z z zMB EQ TMB- =< > = < ⋅ >= < + + >=ϕ 1

212

321

222

32 (2.45)

As anticipated, the entropy associated with the MB-PDF is equivalent to the thermo-dynamic entropy, within a constant.

For the Chapman–Enskog entropy, with the CE-PDF,

ϕ ϕ εCE MB= + +ln 1 (2.46)

where ε contains the nondimensional parameters associated with velocity and tem-perature gradients, as expressed in Equation 2.27. The expectation value of the sec-ond term in Equation 2.46 cannot be evaluated analytically. One possibility is to expand the term, assuming that ε << 1. But this gives a result indicating that the nonequilibrium entropy is greater than the equilibrium entropy, which is not possible in the context of physical theory (see Figure 2.1). Thus, for the CE-PDF,

s s s sCE EQ MB EQ- = - = 3

2 (2.47)

The entropy associated with the equilibrium Euler equations is the same as the function associated with the nonequilibrium Navier–Stokes equations.

Correspondingly, for the central distribution entropy, with the following central distribution PDF,

ϕ CD z C z CT= ⋅ ⋅ + | |-1

21 ln (2.48)

7262_C002.indd 44 1/19/08 11:32:33 AM


Taking the expectation value yields

< > = + | |ϕ CD C

32

ln (2.49)

where ln | |C is the logarithm of the determinant of the covariance matrix. For sim-plicity, one can assume only t11 0≠ . For this case, the constraint is t t t22 33 11 2= = - / . Then

s s t tCD EQ- = + - - / 3

212

1 1 211 112ln ( )( ) (2.50)

Under certain conditions restricting possible values of the nondimensional shear stress t11, the entropy associated with the CD-PDF is always less than the equilibrium value, as shown in Figure 2.2. The range implies that - < <2 111t , which includes values typically found in actual physical processes, such as shock waves.

Finally, for the skew-normal entropy, the SN-PDF is

ϕ ϕ εSN CD= + +ln 1 (2.51)

f Ig u r e2.1 Chapman–Enskog expectation value for entropy difference relative to equilib-rium entropy.

0.4

SCE–S0

t11

0.2

0.0

0.0

0.0

0.5

0.5

1.0

1.0

–1.0

–1.0

–0.5

–0.5

q

7262_C002.indd 45 1/19/08 11:32:39 AM


where ε contains the parameters associated with thermal and velocity gradients. This presents the same difficulty as the CE-PDF, because this term cannot be evalu-ated analytically. Using the same analysis mentioned previously, an expression can be obtained by assuming that ε << 1. But again for some combination of values of t11 and

1q , the result is nonphysical. For the range typically found in practice ( ),- < <1 011t the entropy associated with the SN-PDF is less than the equilibrium value (see Figure 2.3). Due to this uncertainty, it is assumed that

s s s sSN EQ CD EQ- = - (2.52)

In the next section, these results will be used to develop a statistical formulation of the Second Law.

2.6 st at Ist Ica Lfo r Mu Lat Io n o f t h esec o n d Law

Performing an entropy balance over a differential control volume for inviscid flows (Euler equations), it can be shown that

genk

kS

s

t

u s

x = ∂

∂+ ∂∂

( ) ( )ρ ρ (2.53)

f Ig u r e2.2 Central distribution function expectation value for entropy difference relative to thermodynamic entropy.

SCD–S0

t11

0.0

00.5

1

–1.5

–1.0

–0.5

–2.0

–2

–1

0.0

0.5

1.0

–1.0

–0.5

q

7262_C002.indd 46 1/19/08 11:32:44 AM


where the first term on the right side is the time-rate-of-change of entropy in a differ-ential control volume. For processes modeled by the Euler equations, which are gener-ated as moments of the Boltzmann equation with the MB-PDF, the entropy generation is zero. For the Navier–Stokes equations, it was shown in the previous chapter that

genS

TT T

T = ∇ ⋅∇ +κ µ

2F (2.54)

where F is the viscous dissipation function (strictly nonnegative). For processes gov-erned by the Navier–Stokes equations, entropy generation is always nonnegative, as the coefficient of thermal conductivity, k, and viscosity, m, is nonnegative. Motion of a viscous, heat-conducting fluid will yield a net production of entropy.

A statistical approach for deriving the entropy production equation is to extend the technique for conservation laws, as moments of the Boltzmann equation, to the Second Law. The entropy generation rate can be expressed as

gen

kkS

tn

xn v = ∂

∂< > + ∂

∂< >( ) ( )ϕ ϕ (2.55)

f Ig u r e2.3 Skew-normal expectation value for entropy difference relative to thermody-namic entropy.

t11

0.0

0.5

1.0

–0.5

SSN–S0

2

0

–1.0

–2

0

1

–1

–2

q

7262_C002.indd 47 1/19/08 11:32:48 AM


which is consistent with kinetic theory, because the collision integral for the quan-tity represented by ϕ = - ln g is nonzero. It remains to determine under what con-ditions the entropy generation rate is nonnegative, to satisfy the Second Law of Thermodynamics. For the MB and CE probability distributions, this is already established because both yield the same functional expression for the entropy. For the CD and SN distributions, the expression for entropy in these distribu-tions differs from the equilibrium entropy, so extra terms appear when the entropy generation rate is evaluated using Equation 2.55 with either of these distribu-tions. However, it is evident that a nonequilibrium version of the Clausius-Duhem inequality emerges.

The expression in Equation 2.55 may be considered a generalized version of the standard Clasius-Duhem expression for the entropy production rate. The equa-tion will contain additional terms due to the nonequilibrium effects of velocity and temperature gradients. As expected, these effects appear not only in the constitutive relations for the stress tensor, but also in the entropy function itself. This satisfies the requirement of the Second Law of Thermodynamics, as the nonequilibrium entropy is less than its corresponding equilibrium value. But the result is different from the result obtained with the Chapman–Enskog formalism, when constructing the pseu-doprobability density (a perturbation) for the Navier–Stokes equations.

2.7 n u Mer Ica Lfo r Mu Lat Io n o f t h esec o n d Law

The previous sections have developed statistical formulations of entropy and the Second Law. From this basis and the governing equations developed previously for the Second Law, numerical solutions of the governing equations can be determined. This section develops a numerical formulation of the Second Law. Many types of numerical methods exist for the solution of the Navier–Stokes equations, such as finite differ-ences, elements, volumes, and so forth. This section uses a particular method (called a CVFEM; control volume-based finite element method) to illustrate how discretization of the Second Law can be accomplished. Similar procedures can be readily developed with other methods, by postprocessing of the computed velocity and temperature fields to determine the entropy production rates throughout the flow field.

2.7.1 Disc r et izat io n o Ft he Pr o Bl eMDo Ma in

A typical two-dimensional domain in Figure 2.4 is subdivided into linear, quadri-lateral finite elements. The grid is collocated, so that the velocity components, pres-sure, and temperature are obtained at nodes located at every element corner. The finite element discretization uses a local ( )s,τ coordinate system that defines the shape functions and other element properties. A local numbering scheme, ranging from 1 to 4, is used within each element, so that the finite element equations can be devel-oped locally and independently of the mesh configuration. Following a conventional assembly procedure (Schneider, 1988) for finite elements, the local nodal equations are assembled into the global system of equations involving global nodes. The conservation principle is applied over an “effective” control volume defined by all subvolumes from elements surrounding a particular node in the mesh. Each element is subdivided into four subcontrol volumes (SCVs), with internal SCV boundaries

7262_C002.indd 48 1/19/08 11:32:50 AM


coincident with the local coordinate surfaces defined by s = 0 and t = 0 (see Figure 2.4; note origin of axes at the center of an element).

Transported quantities across the edges (surfaces) of a control volume are approximated from values at the midpoint of a subsurface, called the integration point, where “ip” refers to the integration point in Figure 2.4. Interpolation within each element yields

x N xi

i i==∑

1

4

(2.56)

y N yi

i i==∑

1

4

(2.57)

φ ==∑i

i iN1

4

F (2.58)

where xi, yi, and Fi are nodal values of the spatial coordinates and f, respectively. For quadrilateral, isoparametric elements, the linear shape functions, Ni, are given by

N s t114

1 1= + +( )( )

(2.59)

N s t2

14

1 1= - +( )( ) (2.60)

N s t3

14

1 1= - -( )( ) (2.61)

N s t4

14

1 1= + -( )( ) (2.62)

f Ig u r e2.4 Schematic of a finite element and control volume discretization.

Local Node Number

2

Node

SCV 2

SCV 3

SCV 1

SCV 4

(SS2) s

ip 2 ip 1

ip 3

Finite Element

Subvolume

Control Volume

4

ip 4 t

1

7262_C002.indd 49 1/19/08 11:32:57 AM


The spatial derivatives of the scalar function are evaluated according to

∂∂= ∂

∂=∑φx

N

xi

ii

1

4

F (2.63)

∂∂= ∂

∂=∑φy

N

yi

ii

1

4

F (2.64)

To obtain the x and y derivatives of the shape functions in Equation 2.63 and Equation 2.64, the chain rule for partial derivatives is applied as follows:

∂∂= ∂∂∂∂+ ∂∂∂∂

N

s

N

x

x

s

N

y

y

si i i (2.65)

∂∂= ∂∂∂∂+ ∂∂∂∂

Nt

Nx

xt

Ny

yt

i i i (2.66)

Solving for the x and y derivatives,

∂∂∂∂

=

∂∂

-∂∂

-∂∂

∂∂

Nx

Ny

J

yt

ys

xt

xs

i

i

1

∂∂∂∂

Ns

Nt

i

i

(2.67)

where J is the determinant of the Jacobian of transformation given by

Jxs

yt

ys

xt

= ∂∂∂∂- ∂∂∂∂

(2.68)

The derivatives of the global coordinates with respect to local coordinates in Equation 2.67 are obtained from the x and y nodal values as follows:

∂∂= ∂

∂=∑x

s

N

sx

i

ii

1

4

(2.69)

∂∂= ∂

∂=∑x

tNt

xi

ii

1

4

(2.70)

∂∂= ∂

∂=∑y

s

N

sy

i

ii

1

4

(2.71)

∂∂= ∂

∂=∑y

tNt

yi

ii

1

4

(2.72)

7262_C002.indd 50 1/19/08 11:33:04 AM


The local derivatives of the shape functions required in Equation 2.72 can be found by differentiating Equation 2.59 through Equation 2.62. In the upcoming approxi-mation of transient and source terms, the area of a two-dimensional control volume bounded by a specific range of s and t is given by

dA Jdsdt= (2.73)

2.7.2 Disc r et izat io n o Ft he co nse r vat io n eq u at io ns

The discrete conservation equations are obtained by integrating the differential equations over a discrete control volume (or two-dimensional area, encompassed by a surface, S). Using the Gaussian theorem, the standard form of the integral equation for a conserved quantity, f, can be expressed as

A S S AtdA dn dn PdA∫ ∫ ∫ ∫∂

∂+ ⋅ - ∇ ⋅ =( )

( ) ( )ρφ ρ φ φv Γ (2.74)

The term on the right side refers to a source term, where v and dn refer to the veloc-ity and differential normal vector at the surface, respectively. Equation 2.74 applies to each control volume, as well as the solution domain as a whole. To discretize Equation 2.74 in two dimensions, a particular finite element illustrated in Figure 2.4 is considered. Let the variable Q represent the flow of f across the edge of an ele-ment. The flows consist of a diffusive component and convective component. The integral forms of the components are given by the second and third terms on the left side of Equation 2.74, convective and diffusive.

The first and second subscripts on S and Q will denote the subsurface and nodal point numbers, respectively. The subscripts e1 and e2 will refer to flows into the control volume through the surfaces, which lie on the exterior edge of the element. Therefore, the equation governing the conservation of f over SCV1 (subquadrant 1 of element in Figure 2.4) can be written as

Q Q Q Q PdV

tdVe e

scv scv11 4 1 11 2 1

1 1, , , ,+ + + + = ∂

∂∫ ∫ ρφ

(2.75)

To complete the discretization of the integral conservation equation, the surface and volume integrals need to be approximated. For example, the diffusive compo-nent of Q4,1 is approximated by

Q dnN

xy

N

Sj

xi ip

y4 1

1

4

4 44 1

,=

= ∇ ⋅ = ∂∂| D - ∂

,∫ ∑( )Γ Γ Γφ ii ip

jyx

∂| D

4 4 F

(2.76)

where the gradient functions have been evaluated using the shape functions. Note that the surface integral in Equation 2.76 is approximated by the product of the flux evalu-ated at the surface integration point and the length of the surface (see Figure 2.4).

7262_C002.indd 51 1/19/08 11:33:08 AM


Similarly, the convective component is given by

Q dn v x u yS

ip ip ip i4 1 4 4 4 4 4 4

4 1

, = ⋅ = D - D,∫ ( )ρ φ ρ φ ρ φv pp

(2.77)

where the lower case variables, f ip and uip, denote the integration point values. Also, Dx4 and Dy4 are respective changes in the x and y directions, as subsurface 4 is tra-versed in a counterclockwise direction.

Some common schemes for obtaining the integration point variable, f ip, in terms of the nodal values are the upwinding differencing scheme (UDS), central differenc-ing scheme (CDS), exponential differencing scheme (EDS), and a physical influ-ence scheme (PINS) (Naterer, 1999; Schneider, 1988). UDS uses an upstream value to approximate the scalar at the integration point. CDS uses a linear interpolation between adjacent nodes, and EDS is a “hybrid” scheme, which obtains a smooth transition from the CDS scheme, for low Peclet numbers (Pe u xi i= D /ρ Γ), to the UDS scheme for high Peclet numbers (Patankar, 1980). The value taken for the convected variable is determined based on an interpolated value at the nodal points, related by integration point coefficients. PINS predicts the integration point value of the scalar by a local approximation of the governing equation at the integration point. Thus, each integration point equation becomes an approximation to the appropriate partial differential equation, including all physical influences on the upwind value.

For the transient term, a lumped mass approximation is adopted. The approach assumes that f is uniformly equal to the nodal value over the whole control volume. The transient term is represented in the following form:

∂∂

=-D∫+

tdV J

tscv

n n

1

11

1ρφ ρ ( )F F

(2.78)

where the superscripts n and 0 refer to the new and old time levels, respectively, and J denotes the area of SCV1. Finally, for a given source-type term, such as the pressure gradient, body force, or the contribution from viscous stress terms in the momentum equations, the source term is evaluated as

scv

P dV P J1

12

12∫ = | , (2.79)

where a midpoint integration has been used in evaluating P at ( )12

12, . The two-

dimensional domain considered in this discretization is assumed to have a unit depth normal to the plane of interest. Thus, the volume and area integrals reduce to area and line integrals, respectively. Each of the aforementioned components of the dis-crete equation can be assembled into Equation 2.74 and represented in a final matrix form. If the contributions of the four separate control volume equations are taken together, the algebraic equations can be written as

[ ] [ ] A A a B Bt d a t pφφ φφ φφ φ φφ φ+ + = + (2.80)

7262_C002.indd 52 1/19/08 11:33:12 AM


where the superscripts f on A denote a f equation and a multiplier of a f variable. The superscripts t, a, d, and p refer to the transient, advection, diffusion, and source, respectively. The square brackets denote a 4 × 4 matrix, and the braces denote a column vector. A row in the matrix represents the SCV contribution to the corre-sponding control volume equation, and the column indicates the integration point (or nodal point) of the variable multiplied. In the CVFEM, the element stiffness matrix is generated from a control volume formulation.

2.7.3 Disc r et izat io n o Ft he sec o n Dlaw

The entropy production rate can be calculated numerically from the entropy trans-port equation (Cheng et al., 1994; Merriam, 1988). Alternatively, after simplifying and using the Gibbs equation, which relates entropy to the temperature, pressure, mass, density, and internal energy, an alternative positive definite form of the entropy production equation can be obtained (Bejan, 1996; Naterer and Camberos, 2001). This positive definite form was presented in previous sections, and it will be further developed in this section with a numerical formulation and CVFEM.

For the numerical discretization of entropy production, let Vj denote the volume associated with node j, so the integral form of the Second Law can be written as

V

S

tF q dsj

j

S j

∂∂+ ⋅ ≥∫ ( ) .

0

(2.81)

where q represents the vector of conserved variables.

The entropy flux (second term) results from the contribution of four different SCVs within four different elements sharing node j. The resulting equation for the effective control volume surrounding node j becomes

VS

tFj

j

i

i∂∂+ D ≥

=∑

1

8

0

(2.82)

where the summation (over node i) refers to 8 integration points of the effective control volume.

Two alternative temporal discretizations are considered in this formulation of the Second Law. The first way is a semidiscrete approach, whereby the entropy time derivative is transformed by the chain rule, i.e.,

∂∂= ∂∂|∂∂

S

t

S q

q

q

tj

jj( )

(2.83)

Now, substitute expressions of the conservation equations in to ∂ ∂q tj / and place the

resulting form of ∂ ∂S tj / into Equation 2.82, thereby giving

- ∂∂| D + D ≥

= =∑ ∑S q

qf Fj

i

i

i

i( )

1

8

1

8

0

(2.84)

In this approach, no temporal differencing is applied.

7262_C002.indd 53 1/19/08 11:33:18 AM


The second approach uses a backward-difference in time to evaluate the entropy storage term. For two-dimensional quadrilateral elements, the Second Law becomes

VS S

tFj

jn

jn

i

i+

=

-D

+ D ≥∑1

1

8

0

(2.85)

where the superscript n + 1 denotes evaluation at the current time step and the super-script n refers to the previous time step. Equation 2.85 is the fully discrete form of the Second Law.

For an implicit time advance, the semidiscrete and fully discrete entropy produc-tion rates become

( )s jn

n

j

i

i n

i

iPS

qf F

++

=

, +

=

= - ∂∂| D + D∑ ∑1

1

1

81

1

8

,, + ≥n 1 0

(2.86)

and

( )s jI

jjn

jn

i

i nP VS S

tF =

-D

+ D ≥+

=

, +∑1

1

8

1 0

(2.87)

respectively. In Equation 2.86 and Equation 2.87, the notation expressing the func-tional dependence of S on

q, S q( )

, has been dropped for brevity. For a variable, h,

in the range of n n≤ ≤ +η 1, the relationship between Sn+1 and Sn can be found from a Taylor’s expansion as follows:

S S q q

qq qj

njn

jn

jn

jn= - -( ) ∂∂ + ... + -+

,+

, ,+1

11

11

41

444

1,

+( ) ∂∂

j

njn

qS

+ -( ) ∂∂ + ... + -( ) ∂,

+, ,

+,

12 1

11

14

14q q

qq qj

nj

nj

nj

n

∂∂

q

Sj4

2

η

(2.88)

The transport equation for the conserved variables can be written similarly as Equation 2.85 using an implicit time advance as follows:

jn

jn

j i

i nq qt

Vf+

=

, +- + D D ≥∑1

1

8

1 0

(2.89)

The value of ( )jn

jnq q+ -1 can be replaced in Equation 2.88 by using Equation 2.89.

The resulting form of Sjn may then be substituted into Equation 2.87 to obtain an

expression for ( )s jIP as follows:

( )s jI

n

j

i

i n

i

i nPS

qf F = - ∂

∂| D + D

+

=

, +

=

, +∑ ∑1

1

8

1

1

8

11

-D

-( ) ∂∂ + ... + -,+

, ,+

,V

tq q

qq qj

jn

jn

jn

jn

2 11

11

41

4(( ) ∂∂

≥

qSj

4

2

0η

(2.90)

7262_C002.indd 54 1/19/08 11:33:28 AM


Substituting the expression for the semidiscrete equation into Equation 2.90 using Equation 2.87, it can be shown that

( ) ( )s jI

s jn j

k l

kl k lP PV

th = -

D+

= =∑∑1

1

4

1

4

2α α (2.91)

where hkl denotes the entries of the Hessian matrix, H S q= ∂ /∂2 2 , and αk = ( )k j

nk jnq q,

+,-1

. Note that hkl depends on the entropy functions, S q( )

and F q( )

, and the problem parameters under consideration.

The Hessian matrix entries (second derivatives of entropy) can be derived ana-lytically for specific cases such as compressible flows, subject to the Navier–Stokes equations of motion. The entries require derivatives of entropy, with respect to the vector of conserved variables. For example, the entropy derivative with respect to

q

for one-dimensional compressible flows can be written as (Camberos, 1995; Merriam, 1988; Naterer, 1999)

S s c

u

Pc

u

Pcq v v, = + - + -

, - - , γ γ ρ γ ρ ρ( ) ( )1

212

vv P

( )γ -

1 (2.92)

Physically, the result S dqq, represents the cumulative effect of changes in all con-

served quantities on the entropy change. Because H is convex (negative definite), then the quadratic form given by the double sum in the semidiscrete entropy produc-tion becomes negative for all ( )α α α α1 2 3 4, , , . As a result,

( ) ( )s j

Is j

nP P ≥ +1 (2.93)

Thus, the fully discrete entropy production rate for a numerical scheme is equal to or greater than the semidiscrete entropy production. From Equation 2.91, the effects of entropy production due to temporal and spatial discretization may be separated from each other. In the upcoming chapters, numerical simulations of entropy production that utilize this result and formulation will be presented.

r ef er en c es

Bejan, A. 1996. Entropy Generation Minimization: The Method of Thermodynamic Optimiza-tion of Finite-Time Systems and Finite-Time Processes. CRC Press, Boca Raton, FL.

Camberos, J.A. 1994. Probabilistic Approach to the Computational Simulation of Gasdy-namic Processes. Doctoral dissertation, Department of Aeronautics and Astronautics (SUDA AR No. 668), Stanford University, Stanford, CA, 102–105.

Camberos, J.A. 1997a. Comparison of Split-Fluxes Generated from Selected Probability Distributions (preprint). AIAA Paper 97-2095.

Camberos, J.A. 1997b. Comparison of Selected Probability Distributions for Gas Dynamic Simulations Inspired by Kinetic Theory. AIAA Paper 97-0340.

7262_C002.indd 55 1/19/08 11:33:35 AM


Camberos, J.A. 1999. Nonlinear Time-Step Constraints Based on the Second Law of Ther-modynamics. 37th Aerospace Sciences Meeting and Exhibit. AIAA Paper 99-0558, Reno, NV.

Chapman, S. and T.G. Cowling. 1960. Mathematical Theory of Non-Uniform Gases. Cambridge University Press, London. (Reprint of 1939 original.)

Chapman, S. and T.G. Cowling. 1939. The Mathematical Theory of Non-Uniform Gases. University Press, Cambridge, U.K. (Reprint Edition 1990.)

Cheng, C.H., Ma, W.P., and W.H. Huang. 1994. Numerical predictions of entropy generation for mixed convective flows in a vertical channel with transverse fin arrays. Int. J. Heat Mass Transfer, 21: 519–530.

Dincer, I. 2001. Exergy and the environment: A global perspective. Int. J. Global Energy Issue, 15(3/4): 363–374.

Kapur, J.N. and H.K. Kesavan. 1992. Entropy Optimization Principles with Applications. Academic Press, New York.

Lavenda, B.H. 1991. Statistical Physics. John Wiley & Sons, New York.Merriam, M.L. 1988. An Entropy-Based Approach to Nonlinear Stability. Ph.D. thesis,

Stanford University, Stanford, CA.Naterer, G.F. 1999. Constructing an entropy-stable upwind scheme for compressible fluid

flow computations. AIAA J., 37(3): 303–312.Naterer, G.F. and J.A. Camberos. 2001. The Role of Entropy and the Second Law in Com-

putational Thermofluids. AIAA 35th Thermophysics Conf. AIAA Paper 2001-2758, June 11–14. Anaheim, CA.

Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere, New York, 79–101.

Rosen, M.A. and I. Dincer. 1997. On exergy and environmental impact. Int. J. Energy Res., 21(7): 643–654.

Schneider, G.E. 1988. Elliptic systems: finite-element method 1, in W.J. Minkowycz, E.M. Sparrow, G.E. Schneider, and R.H. Pletcher, Eds., Handbook of Numerical Heat Transfer. Wiley Interscience, New York, chap. 10.

Vincenti, W.G. and C.H. Kruger, Jr. 1965. Introduction to Physical Gas Dynamics. John Wiley & Sons, New York.

7262_C002.indd 56 1/19/08 11:33:36 AM

57

3 Predicted Irreversibilities of Incompressible Flows


Entropy production is a key parameter in determining the maximum theoretical lim-its of energy efficiency of engineering devices. Inefficiencies in a fluids engineer-ing system arise from thermal, friction, and other thermodynamic irreversibilities. The Second Law of Thermodynamics can provide a systematic way of establishing optimal performance in these systems. For example, the Carnot cycle efficiency is based on processes that require the least amount of heat input, to deliver the max-imum power output by giving ideal performance without irreversibilities. Actual heat engines are often compared against this Carnot limit. The rate of entropy gen-eration, or any related measure of efficiency based on the Second Law, such as the Second Law efficiency, can be used to quantify the magnitude of irreversibilities in thermofluid applications. Power-generation devices (such as steam turbines) deliver maximum power output, and power-consuming devices (i.e., compressors, pumps) consume the least power when the rate of entropy generation is minimized.

In this chapter, dissipative energy losses will be characterized through local rates of entropy production, which could be alternatively expressed in terms of exergy destruction. Past studies have shown the importance of such exergy tracking in vari-ous industrial applications, such as fluid machinery, transportation (Dincer et al., 2004), cogeneration district energy systems (Rosen et al., 2004), and others. Many other past efforts have been devoted to the design of highly efficient energy devices and systems. Consequently, these devices have been thoroughly scrutinized for pos-sible design improvements. With the current state of this technology, the margins of increasing such performance further are often relatively small. This chapter dis-cusses how the Second Law can offer new ways of reaching the upper limits of tech-nological performance, based on local predictions of thermofluid irreversibilities. Past studies have described various analytical and empirical techniques for entropy-based optimization of engineering systems, most notably the method of entropy gen-eration minimization (Bejan, 1996). Some typical examples include extended fins in forced convection (Poulikakos and Bejan, 1982) and two-phase heat exchangers (Zubair et al., 1987). An analytical approach typically involves the derivation of a functional expression for the entropy generation in a process. The extrema of the functional expression, which characterize the minimum entropy production, are then determined by analytical methods of calculus.

The rate of entropy generation is a derived quantity, which is predicted from postprocessing of the temperature and velocity distributions. For complex flow configurations, this typically requires numerical methods like computational fluid

7262_C003.indd 57 1/19/08 11:37:22 AM


dynamics (CFD). In this chapter, it will be shown that an entropy-based local loss analysis remains consistent with the usual global loss distribution given by tradi-tional loss correlations. A component-level design methodology can be enhanced by the capability of identifying the source and specific location of highest entropy production. This approach can be more valuable than examining global losses like a duct’s end-to-end pressure loss, because the desired overall performance can be improved by redesigning a component locally. Unlike past methods involving global coefficients characterizing the overall performance, this chapter discusses a new entropy-based metric for characterizing local losses of available energy. Also, meth-ods are developed to predict upper bounds on entropy, thereby allowing designers to use the Second law to develop systems with higher performance and efficiency.

3.2 En t r o py tr a n spo r t Eq u at Io n f o r In c o mpr EssIbl Efl o w s

In tensor notation, the conservation form of the general scalar transport equation in multidimensions can be written as

∂∂+ ∂∂

= - ∂∂

∂∂

+( )

( )ρφ ρ φ φ

φt xu

x xS

jj

j j

Γ

(3.1)

where j = 1,2,3 and f is a general scalar quantity or dependent variable, such as tem-perature, velocity, or concentration transported throughout the flow field by diffu-sion or convection. The terms on the left side of Equation 3.1 represent the transient storage and convective flux. The first term on the right side is the diffusive flux. The last term represents production or sources of f in the volume. In the modeling of Equation 3.1, Γ and Sφ are generalized variables representing the diffusion coef-ficient and source terms, respectively.

The conservation equations involve equalities, whereas the Second Law involves an inequality. As discussed in previous chapters, the entropy transport equation can be written as

∂∂+ ∂∂≡ ≥S

t

F

xP

i

is 0

(3.2)

where sP is the entropy production rate and S = rs represents the entropy per unit volume. The component of the entropy flux in the xi direction, Fi, may be expressed in terms of the velocity component and heat flux in that direction, vi and qi, as follows:

F v s

qTi i

i= +ρ . (3.3)

The specific entropy, s, in the flux term of Equation 3.2 can be obtained from the Gibbs equation as follows:

dsT

dep

Td= +1

2ρρ

(3.4)

7262_C003.indd 58 1/19/08 11:37:27 AM

Predicted Irreversibilities of Incompressible Flows 59

where e is the internal energy per unit mass, r represents density, and p is pressure. Integration of the Gibbs equation leads to

D = +∫ ∫s cdT

T

p

Td

T

T

vr

s

r

s

ρ

ρ

ρρ

2

(3.5)

where the subscripts r and s denote a specified initial (or reference) state and the current state, respectively. The variable cv represents the specific heat, which will be assumed to be constant (formulation is limited to liquid flows or incompressible gas flows over small to moderate temperature differences).

For an incompressible fluid, Equation 3.5 becomes

D = - =

s s s c lnT

Tr vs

r

(3.6)

For an ideal gas,

s c lnT

TRln sv

s

r

s

rr=

-

+ρ

ρ

(3.7)

Substituting the ideal gas law into Equation 3.7,

s c lnp p

s c lnp

svs r

s rr v r= /

/+ =

+

∗

∗

( )( )ρ ρ ρg g

(3.8)

where g is the ratio of specific heats.When combined with the Gibbs equation, the entropy transport equation provides

a way of calculating the local entropy generation for an open system. As discussed in the previous chapter, an alternative way of formulating sP is (Bejan, 1996)

si

ij i

jP

k

T

T

x T

u

x = ∂

∂+ ∂

∂≥

2

2

0τ

(3.9)

where k is the thermal conductivity and τ ij is the viscous stress arising from velocity gradients in the fluid motion,

τ m δiji

j

j

i

k

kij

u

x

u

x

u

x= ∂

∂+∂∂

- ∂∂

23

(3.10)

In Equation 3.10, m and δ ij refer to the dynamic viscosity and Kronecker delta, respectively. The last divergence term in Equation 3.10 will vanish under the assump-tion of flow incompressibility.

7262_C003.indd 59 1/19/08 11:37:34 AM


In Equation 3.9, Fourier’s law has been used to represent heat conduction. Also, a Newtonian fluid was assumed for the viscous stress term. Based on these mod-els, Equation 3.9 becomes a positive definite expression for the entropy genera-tion rate, as it represents a sum of squared terms. Temperature, T, is expressed in absolute (Kelvin) units. The positive definite equation applies to both compressible and incompressible Newtonian fluids. In Equation 3.9, the first term on the right side represents entropy generation due to heat transfer across a finite temperature difference, whereas the second term represents the local entropy generation due to viscous dissipation (i.e., conversion of kinetic energy into internal energy through fluid friction).

The vector form of the positive-definite equation for entropy production can be expressed as follows:

sP

k T T

T T = ∇ ⋅∇ + ≥( )

20

mF (3.11)

where F is the viscous dissipation function, which involves velocity gradients in the fluid motion. In Equation 3.11 the first term represents entropy generation due to heat transfer across temperature gradients in the fluid. The second term is the local entropy generation due to viscous dissipation. For turbulent flows, the effec-tive thermal conductivity can be approximated by the sum of the molecular eddy conductivities, whereas effective viscosity is the sum of the molecular viscosity and eddy diffusivity. An upcoming chapter will focus on detailed modeling of entropy transport in turbulent flows.

For a nearly isothermal process, the thermal contribution to entropy generation is neglected. The resulting form of the equation, representing the viscous dissipation contribution alone to flow loss, is given by the second term on the right side of the previous equation, that is,

sPT

u

y

v

x

u

x

v

y = ∂

∂+ ∂∂

+ ∂

∂+ ∂∂

m2 2 2

2

≥ 0

(3.12)

where the expression in square brackets is the viscous dissipation function, F. This result is directly related to the mechanical power needed to transport fluid through a system. Unlike velocity or temperature, the measurement of entropy cannot be performed directly. However, the previous equation can be used as an indirect way of characterizing the flow irreversibility. For example, the entropy produced by fric-tion irreversibility can be estimated by measured gradients of velocity. In upcom-ing chapters, these velocity gradients will be obtained through postprocessing of experimental velocity data or numerical results obtained from a CFD solution of the Navier–Stokes equations.

7262_C003.indd 60 1/19/08 11:37:37 AM


3.3 f o r mu l at Io n o f lo ss co Ef f Ic IEn t s IntEr ms o f En t r o py pr o d u c t Io n

Conventional loss parameters, such as a global head loss or pressure recovery coef-ficient, typically cannot identify the specific locations and sources of flow losses within a fluid system. This section presents a formulation that allows local irre-versibilities to be scrutinized and converted to local distributions of the loss coeffi-cient. In this way, a designer could use local loss mapping to detect locations of high entropy production (or flow irreversibility), thereby allowing local design changes of geometrical or other parameters to improve system efficiency. It will be shown that local rates of entropy production for an incompressible flow can be converted to local loss parameters, thereby leading to a more generalized approach to loss analysis.

3.3.1 En t r o pypr o d u c t io n in BEr n o u l l i’sEq u at io n

Consider incompressible viscous flow through a streamtube. A streamtube refers to a three-dimensional tube that encompasses a fluid streamline within a channel or other flow configuration. The Bernoulli equation identifies the head loss along this flow path as follows:

pgz V

pgz V hl

1

11 1

2 2

22 2

212

12ρ ρ

+ + = + + +

(3.13)

where hl is the head loss and the subscripts 1 and 2 refer to different points along the streamline. Also, p, g, z, and V refer to pressure, gravitational acceleration, elevation, and total velocity, respectively. It can be shown that Bernoulli’s equation represents an inte-grated form of the following differential mechanical energy equation (Naterer, 2002):

ρ τ τD

DtV p

12

2

= - ⋅∇ + ∇ ⋅ ⋅ - : ∇ + ⋅v v v F vb

(3.14)

where D/Dt, τ , Fb, and v refer to the total (substantial) derivative, shear stress tensor, body force, and fluid velocity vector, respectively. The colon symbol (:) represents matrix contraction between the shear stress and velocity gradient matrices (yielding the viscous dissipation function). The temporal portion of the substantial derivative on the left side vanishes for steady-state conditions.

The previous equation requires that the net convection of kinetic energy (first term) balances the sum of flow work (second term), net work of viscous stresses (third term), plus the net work done by body forces to increase kinetic energy (fifth term), minus the viscous dissipation (fourth term). Rewriting the gravitational body force term, integrating over a streamtube control volume, V, and expressing the vec-tor gradient in the streamwise direction, s, it can be shown that

V V V

Vs

Vp

gz dV dV d∫ ∫ ∫∂∂

+ +

= ∇ ⋅ ⋅ - : ∇ρρ

τ τ12

2 v v VV .

(3.15)

7262_C003.indd 61 1/19/08 11:37:40 AM


The net viscous work term (first term on right side) is the work done by viscous stresses in the fluid element against the surroundings to change the kinetic energy of the fluid.

Consider a control volume, A(ds), of finite width in the cross-stream direction and differential length in the streamwise direction (see Figure 3.1). Integrating over this control volume and assuming a uniform mass flow rate through the streamtube encompassing the control volume, it can be shown that

m

md V

Pgz

mdV

R R V V1

221

21∫ ∫ ∫+ +

= ∇ ⋅ ⋅ -

ρτ τv :: ∇

vdV (3.16)

where Rm is a reference global mass flow rate. The last term on the right side refers to viscous dissipation within the control volume. It represents a loss term in Equa-tion 3.16, which can be directly related to the entropy generation, based on Equation 3.18. Performing that substitution and comparing to Bernoulli’s equation, the head loss becomes

hm

TP dVlR V

s= ∫1

(3.17)

Alternatively, this result can be expressed in terms of the local rate of exergy destruc-tion, dX , due to friction irreversibilities of viscous dissipation at ambient tempera-ture, T0,

hm

XT

TdVl

R Vd= ∫1

0

(3.18)

This result represents a valuable local alternative to conventional global loss charac-terization. It can be observed that the available energy loss within a fluid element is a local volumetric phenomenon involving exergy destruction.

y

1 21ds

A

∆s

2

f Ig u r E3.1 Flow losses along a streamtube in recirculating flow.

7262_C003.indd 62 1/19/08 11:37:45 AM


In contrast to other methods characterizing the flow losses through global empiri-cal coefficients, an entropy-based approach allows local tracking of flow losses, because V can be taken as an arbitrarily located discrete volume. Entropy production encompasses all flow irreversibilities (thermal, friction, chemical, and so forth), unlike other variables such as pressure, which are commonly used in loss analysis. Reduced entropy production is a common objective in fluids engineering systems while changes of individual flow variables are generally problem dependent. For example, higher pressure losses with added baffles may be helpful to increase heat transfer rates in a heat exchanger, but reduced pressure losses are needed in pipe flows, as they entail lower pumping input power. Thus, tracking local pressure changes does not gener-ally identify the problem areas. On the other hand, lower entropy production rates are desired in both cases, and they provide a more robust and common design objective.

3.3.2 l o ssco Ef f ic iEn t sin a pl a n Edif f u sEr

Consider an example of incompressible viscous flow through a diffuser of unit depth, as shown in Figure 3.2. Assuming a uniform velocity profile between the outlet (subscript 2) and the inlet (subscript 1) of the duct, a balance of total energy, E, over the entire duct gives

dE

dtm e p gz V Q m e p= + + + + - +

1 1 1 1 1

22 2

12

υ υυ2 2 221

2+ + -

gz V W (3.19)

where m, e, v, Q, and W refer to the mass flow rate (constant throughout the stream-tube), internal energy (per unit mass), specific volume (per unit mass), heat trans-fer, and boundary work, respectively. For steady-state conditions without boundary work, Equation 3.19 becomes

pgz V

pgz V

m e e Q1

11 1

2 2

22 2

2 2 112

12ρ ρ

+ + = + + + - -

( )mm

. (3.20)

Also, applying an entropy balance to a differential section in Figure 3.2 and using the Gibbs equation,

ˆ ˆdQ Tmde

Tp

dv

TT dPi s

= +

-

(3.21)

Out

W

1

z

Q

2In

dQ

f Ig u r E3.2 Streamtube for diffuser analysis.

7262_C003.indd 63 1/19/08 11:37:50 AM


where the notation of dQ is used rather than dQ, since Q represents a process, not a

thermodynamic state variable. Summing over n sections throughout the entire duct from the inlet (i = 1) to the exit (i = n),

Q m e e T dPi

n

i s i= - -=

,∑( ) ˆ2 1

1 (3.22)

By comparing Equations 3.13, 3.20, and 3.22,

hm e e Q

m mT dPl i s i

i

n

= - - = ≥=∑

( ) ˆ

,2 1

1

10

(3.23)

This result confirms that the head loss in Bernoulli’s equation is a measure of irre-versibility, which represents a loss of mechanical energy per unit mass of the flowing fluid. It represents the irreversible dissipation of kinetic energy into internal energy of the fluid.

As discussed previously, current design technology usually detects a loss of use-ful energy on a global scale using a single loss parameter, such as a valve loss coef-ficient. The previous results suggest that flow losses can be tracked locally based on the entropy production rate. A measure of the overall loss in the bulk fluid entails a summation of local entropy production rates in fluid elements centered on a stream-line through the domain. In this way, entropy generation can be used as an alterna-tive metric of flow loss in fluid systems. The information provided to the designer by this entropy-based metric can be more valuable than global data characterizing the end-to-end flow loss. Unlike the conventional loss characterization with a global head loss or pressure recovery coefficient characterizing an entire device, local loss characterization with the entropy-based metric allows the designer to identify the source and specific location of head losses.

3.3.3 c a sEst u d yo f ch a n n El a n d dif f u sEr dEsig n

In this section, results will be presented to link entropy generation with conventional loss parameters for channel and diffuser flow problems. The case study considers an incompressible viscous flow between two horizontal plates with a length of L. The plates are spaced 2w apart (y-direction). If the plates are very wide, the fully devel-oped velocity profile does not change in the z-direction, so that

u uy

wc= -

1

2

(3.24)

where the centerline velocity, uc, can be expressed in terms of the channel pressure drop, Dp, and average (mean) velocity, u , as follows:

uw p

Luc =

- D =

2

232m (3.25)

7262_C003.indd 64 1/19/08 11:37:56 AM


The total head loss over the entire channel, hl, can be obtained from the inte-grated frictional irreversibility of entropy production as follows:

hm

TP dVm

u

ydV

u

wlV V

c

S= = ∂

∂

=∫ ∫ -

1 1 42

2

4

m m

ww

w

y DLdy∫ 2 (3.26)

which gives

hu L

wlc= 2

2

mρ

(3.27)

Using the previous expression for uc,

hp p

l =-1 2

ρ (3.28)

Thus, the entropy-based formulation of head loss for channel flow reduces to the expected result of D /p ρ , which is the required head loss between two wide horizontal flat plates. Alternatively, the friction factor and the mechanical energy loss can be related to entropy production for the channel flow as follows:

fWT

mLuP dV

Vs= ∫2

2 (3.29)

By using sP as a metric of evaluation, the equivalent friction factor becomes a product of the local entropy generation integrated over the domain and a constant, based on averaged values of the flow variables. Alternatively, exergy is defined as the work potential that can be extracted from an energy source. The exergy destroyed in a process reflects the extent to which the operation of an actual system departs from the theoretical limit of the ideal system. This departure is proportional to the entropy generation.

For the purpose of extending this analysis to more complex geometric con-figurations, a validation study with a numerical simulation was performed. Using a control-volume-based finite element method (CVFEM) (Naterer, 2002), valida-tion of the numerical model was carried out through comparisons with the previous analytical solution of incompressible, viscous flow between two horizontal plates (representing a channel flow). The fully developed velocity profile does not change in the streamwise direction. The analytical solution of the velocity distribution is differentiated to find the spatial variation of entropy production in the channel. The predicted results of velocity and local entropy production, for water flow at 290 K with an average velocity of 0.0504 m/s through a duct with a width of 0.02 m, are illustrated in Figure 3.3a and Figure 3.3b, respectively. The numerical and analytical results show close agreement. The entropy production rate is maximum near the wall

7262_C003.indd 65 1/19/08 11:38:00 AM


due to viscous effects. It becomes zero at the center of the channel, where the cross-stream velocity gradient is zero.

To link the entropy generation with traditional loss parameters, the local entropy generation is integrated over the domain and related to the friction factor based on Equation 3.29 as follows:

P f mLu

wTs =2

2 (3.30)

f Ig u r E3.3 Channel flow results: (a) velocity and (b) entropy production.

90

Loca

l Ent

ropy

Gen

erat

ion

(W/m

3 K)

0.000 0.005 0.010 0.015 0.020 0.025y (m)

(b)

AnalyticComputed80

70

50

60

40

30

20

10

0

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

Velo

city

(m/s

)

0.000 0.005 0.010 0.015 0.020 0.025y (m)

(a)

AnalyticComputed

7262_C003.indd 66 1/19/08 11:38:03 AM


where f m L, , , and T are the friction factor, mass flow rate, plate length, and the average temperature of the fluid, respectively. When Equation 3.30 is integrated over the volume, it gives the total entropy produced in viscous laminar flow between the inlet and exit locations of any two-dimensional expansion section of the same inlet width.

For laminar flow between parallel plates, it is known that f ReD= /48 , so Equation 3.30 becomes

PRes

D

=12 mLu

wT

2

(3.31)

Equation 3.31 shows how the total entropy generated in a fully developed flow between parallel plates can be evaluated based on geometrical and flow data. An entropy-based formulation used for optimization purposes could be benchmarked against this value for validation. The entropy production results for u = .0 0504 m/s, L = 15 cm, and w = 2 cm with water at 290 K from Equation 3.31 and the numeri-cal formulation are 8 522 10 7. × /- W K and 8 519 10 7. × /- W K, respectively. This close agreement provides useful validation of the formulation.

Consider another example of gas flow through a subsonic diffuser, which is widely encountered in aerospace and other applications. In this particular configuration, an incoming flow experiences an area expansion. Flow losses arising from an area expansion lead to reduced diffuser effectiveness. The flow characteristics are highly dependent on the area expansion ratio and the Reynolds number. Entropy production can be used as a basis of correlating this optimal flow configuration, with respect to different flow conditions and system parameters. The two-dimensional geometry of the expansion section in the numerical simulation is shown in Figure 3.4. A fully developed velocity profile is prescribed at the inlet, and a Neumann condition is applied at the outlet. For a given outlet-to-inlet area ratio, the velocity fields were computed for different expansion angles. Reynolds numbers of 301 and 602 were investigated with an area ratio kept at 1.5. The flow remained unstalled until θ = 10o and θ ≅ 7o for Reynolds numbers of 301 and 602, respectively. When the expansion angle increases, the boundary layer separates from the top wall. A recirculation cell is formed in a similar way as flow past a backward facing step. This separation arises from the unfavorable pressure gradient introduced by the expansion.

The predicted velocity distribution (Re = 602) and the corresponding entropy generation contours (values multiplied by 105) at an expansion angle of 60o are shown in Figure 3.5a and Figure 3.5b, respectively. For a Reynolds number of 301,

= 0

= 0

H = 1 cm

2H Vin

f Ig u r E3.4 Schematic of a plane diffuser.

7262_C003.indd 67 1/19/08 11:38:10 AM


the flow reattaches at approximately 8.5 step heights from the expansion. Figure 3.5b illustrates the flow irreversibilities, as characterized by the contours of local entropy production. For comparison purposes, three major regions of entropy generation and loss distribution are evident. The first region is the channel leading into the expan-sion. The second region is the diverging region, which starts at the inlet and ends where the diverging section joins the wider channel. The third region continues from that point to the outlet. Three subregions of importance are identified in the diverg-ing section: a recirculation and reattachment region close to the top wall, a region of separation close to sharp corners at the beginning of the expansion, and flow along the bottom wall. Entropy production in the recirculation or attachment zone is not predominant, because the flow is relatively slow in that region and the velocity gradients are small. The entropy production is high, close to the separation region. It diminishes when the flow decelerates to fill the larger channel. The flow near the bottom of the expansion also exhibits high entropy generation, due to the wall shear effects. The entropy generation map provides a useful way of detecting the detailed structure of the mechanical energy loss in the expansion section.

The variation of total entropy generation in the unstalled flow regime for the diverging section is shown in Figure 3.6. It is interesting to observe that the loss decreases in the region with less expansion, until approximately 3 5. o for Re = 301 and 3 0. o at Re = 602. Higher angles cause an increase in the mechanical energy loss, due to greater channel length. The trend also confirms the dependence of flow losses on Reynolds number, when the flow is laminar. In Figure 3.6, an optimal angle is predicted at the point of minimized entropy production. This entropy-based approach provides a useful alternative and systematic way of establishing the opti-mized geometrical configuration. The same approach of summing local entropy pro-duction rates can be applied to any fluids engineering device. Since this methodology entails tracking of local losses throughout an individual device, rather than global Second Law analysis, the approach provides a valuable component-level energy man-agement tool.

Figure 3.7 illustrates results for loss characterization over a larger range of expansion angles. The values have been normalized by Equation 3.31 (divided by the

f Ig u r E3.5 (a) Velocities and (b) entropy production contours (*105 W/m3K) for Re = 602 with a 60o expansion.

(a)

0.13075 0.39134

11/31 0/8222

0.13075 78222

0.26105

0.26105

(b)

7262_C003.indd 68 1/19/08 11:38:14 AM


volume). The resulting parameter after normalization is directly proportional to the loss coefficient. As observed with previous studies, an optimal angle yielding the least flow losses in the expansion section exists. This optimum corresponds to narrow angles (approximately 3.5o for Re = 301 and 3.0o at Re = 602) and unstalled flow conditions.

Based on these results, a new entropy-based metric that locally characterizes the pressure recovery factor (or any other global loss parameter) is defined as follows:

h θ= - ,

,1

P

Ps

s ref (3.32)

0.58

0.53

0.51

0.49

0.47

0.45

Nor

mal

ized

Tot

al E

ntro

py G

ener

atio

n

0 2 4 6 8Expansion Angle, Degrees

Re = 301Re = 602

f Ig u r E3.6 Total normalized entropy production in the unstalled region.

0.90

0.60

0.30

0.00

Nor

mal

ized

Tot

al E

ntro

py P

rodu

ctio

n

0.0 20.0 40.0 60.0 80.0Expansion Angle, Degrees

f Ig u r E3.7 Flow loss characterization in the unstalled region.

7262_C003.indd 69 1/19/08 11:38:17 AM


where Ps,θ is the entropy production for a diffuser with an expansion angle of q. Also, Ps ref, represents a loss that would occur in the absence of the expansion sec-tion, given by Equation 3.31. An interesting observation from Figure 3.7 is noticed for expansion angles higher than 10o, whereby small changes in the expansion angle cause a larger loss. In addition to providing a physically based measure of loss, the entropy-based approach also leads to added insight into the specific location of flow losses and the flow structures leading to those losses.

3.4 u ppEr En t r o py bo u n d s In cl o sEd syst Ems

Upper bounds of system performance provide a useful design parameter for ensuring that maximum system capabilities are not exceeded. For example, upper bounds on cooling capabilities of a heat pipe can ensure that maximum operating temperatures are not exceeded during convective cooling of a microelectronic assembly. Various other examples arise in thermal design of aerospace, manufacturing, automotive, and other applications (Naterer, 2002). In this section, a method of establishing upper entropy bounds for convection problems is developed. These bounds involve both friction and thermal irreversibilities arising during convective heat transfer within an enclosure.

Various past methods have been developed for establishing upper bounds in ther-mal systems. Martins and da Gama (2000) developed an upper bound for solutions of coupled heat conduction and radiative heat transfer problems, subject to nonlinear boundary conditions. An auxiliary function was used to establish upper bounds, while confirming that the Laplacian of the temperature field satisfied certain inequality con-straints. In heat conduction problems, an upper bound for thermal shape factors was derived for two-dimensional layers by Hassani et al. (1993). Upper bounds for conduc-tion contact resistances were developed by Bobeth and Diener (1982). These bounds are functions of a two-point correlation function of the local contact resistance. Upper bounds for random arrangements of circular contact spots of equal size can be pre-dicted by variational principles with stochastically varying local contact resistances.

In convection problems, upper pressure bounds were developed for establishing criteria in thermal destabilization of wall shear flows (Mikic, 1998). These bounds involved admissible system perturbations, which lead to the onset of turbulence and upper bounds for the wall sublayer scales in fully developed turbulent flows. Entropy transport characterizes the dissipation of kinetic energy in these layers (Adeyinka and Naterer, 2005). For laminar separated flows, upper bounds have been derived to predict laminar instabilities of self-sustaining oscillations and vortex shedding (Mikic, 1998).

This section focuses on upper entropy bounds for problems involving internal forced convection in enclosures or tanks (Lui and Naterer, 2007). Numerous past stud-ies have been conducted on forced convection with internal confined flows (such as Eames and Norton, 1998; Homan and Soo, 1998; Naterer, 2001; Sinai, 1985). Entropy bounds can provide useful new insight regarding the dynamics of flow mechanisms in these problems. For example, entropy production characterizes the mixing, flow struc-tures, and magnitude of frictional dissipation in a tank. As a result, it can provide use-ful guidance for effective design and control of internal flows. It can establish optimal conditions to reduce input power during fluid mixing, while transferring fixed rates of heat transfer to a fluid. The objective of this section is to develop analytical expressions

7262_C003.indd 70 1/19/08 11:38:19 AM


for the upper entropy bounds and correlate these limits with different geometrical con-figurations. In this way, design changes could reduce input power required by energy conversion devices. Alternatively, changes of design parameters to alter entropy bounds can enhance mixing in chemical processing applications.

3.4.1 u ppEr Bo u n d so f th Er ma l ir r Ev Er s iBil it y

Consider three-dimensional convective heat transfer in a bounded domain, V, gov-erned by the following equations of fluid motion and energy transport:

∂∂+ ⋅∇ = - ∇ + ∇

v

tv v

pv

ρν 2 (3.33)

∂∂+ ⋅∇ = ∇T

tv T T a 2

(3.34)

No-slip conditions (v = 0) and Neumann boundary conditions are applied along the

walls of a closed domain, that is,

∂∂=T

n0

(3.35)

where n refers to the unit outward normal direction on the surface, Ω, which encom-passes the volume of the problem domain, V. .

A “temperature excess” is the difference between the actual temperature at some position, T, and the average initial temperature. It is defined as follows:

τ = -| |

=∫TV

T t dVV

10( ) (3.36)

Both actual and average temperatures satisfy the governing equations, so

∂∂+ ⋅∇ = ∇τ τ a τ

tv 2 (3.37)

subject to Neumann boundary conditions.It can be shown that t has a zero average at all times. Entropy production aris-

ing from heat transfer over a finite temperature difference involves a squared tem-perature and squared temperature gradient (as part of the thermal irreversibility), so the previous equations will be squared. Multiplying Equation 3.37 by t and then integrating over V,

V V VtdV v dV dV∫ ∫ ∫∂

∂+ ⋅∇( ) = ∇τ τ τ τ a τ τ 2

(3.38)

7262_C003.indd 71 1/19/08 11:38:25 AM


which can be rewritten to give

12

2 2∂∂

= - | ∇ |∫ ∫tdV dV

V Vτ a τ (3.39)

From Poincare’s inequality (Zeidler, 1985) for any function, s, with first deriva-tives that are square integrable, there is some positive constant c < 1 such that

V V V V

dV dVc

dV dV∫ ∫ ∫ ∫+ | ∇ | ≤ | ∇ | +

σ σ σ σ2 2 2

21

(3.40)

Using this inequality, Equation 3.39 becomes

∂∂

= - ∇ ≤ -∫ ∫ ∫tdV dV C dVτ a τ a τ2 2 22 2

v v v (3.41)

because t has a zero average at all times, where C = c/(1 - c). Integrating this result,

V

CtdV C e∫ ≤ -τ a20

2 (3.42)

where

C t dV02 0= =∫ τ ( )

v (3.43)

This result indicates that the squared temperature excess decreases exponentially over time. This result will be used to show that the temperature excess gradient (needed in the thermal irreversibility of entropy production) also decreases exponentially.

Consider a practical example of a closed domain (such as a tank) with fluid containing a fixed initial amount of total energy. For example, consider a magnetic stirrer for thermochemical processing to generate uniform mixtures in a rectangular tank (see Figure 3.8). In this example, uniformly distributed magnitudes of entropy production and high total entropy are desired to maximize mixing. It is worthwhile to consider how different geometrical configurations affect the steady-state entropy field in the tank, particularly the upper bound of total entropy, so maximum mixing could be achieved with minimal power input to the system. After the mixing stops, the velocity field approaches zero in the steady state ( v→ 0 as t→∞). Also, the first derivatives of velocity approach zero at the steady state, when the fluid motion stops.

The mixing process within the enclosure decays over time. Numerous past stud-ies have considered time decay of diffusive mixing. For example, for diffusive mixing associated with oscillation of a plane wall, the velocity gradient changes exponen-tially with time (White, 1974). This oscillation is similar to the problem considered here, whereby a mixer suddenly stops and the fluid velocities decrease over time.

7262_C003.indd 72 1/19/08 11:38:30 AM


Thus, consider that the velocity gradients decrease exponentially over time in this cur-rent problem (note: other decay trends, such as decays bounded by a constant multiple of t-2, would yield analogous results). For exponential decay, consider that F ≤ -d e d t

12

for positive constants d1, d2, and T t Tmin( )= ≥0 (minimum temperature), where F refers to the viscous dissipation function.

The total entropy production includes thermal irreversibilities (due to heat trans-fer) and friction irreversibilities (due to viscous dissipation). First, consider the ther-mal irreversibilities, which involve squared temperatures and squared temperature gradients. Applying integration by parts to calculate the time derivative of the squared temperature gradient,

d

dtdV

n td

tV V∫ ∫ ∫| ∇ | = ∂∂∂∂

- ∇ ∂∂

τ τ τ τ τ2 22 2Ω

Ω

dV (3.44)

The right side satisfies the following inequality:

RHS dV dV v

V V V≤ - ∇ + ∇ + | | | ∇ |∫ ∫ ∫2

12 2 2 2 2a τ a τa

τ( ) ( ) 22 dV (3.45)

Since the velocity magnitude approaches zero in the steady state and it remains bounded below a throughout the time period t ≥ F,

RHS dV dVV V

≤ - ∇ + | ∇ |∫ ∫a τ a τ( )2 2 2 (3.46)

For the first term on the right side, the following inequality can be used:

V V V VdV dV dV dV∫ ∫ ∫ ∫| ∇ | = - ∇ ≤ + ∇τ τ τ τ τ2 2 2 2 21

212

( ) (3.47)

f Ig u r E3.8 Schematic of a tank mixing problem.

L

Volume ofTank (V)

Surface ofTank (Ω)

NonuniformInitial T(x, y, z, t)

Velocity Decayafter the MixerStops (Time F)

Fluid Mixer in Heated Tank

7262_C003.indd 73 1/19/08 11:38:35 AM


Combining this result with the previous right side,

d

dtdV C e dV

V

Ct

V∫ ∫| ∇ | ≤ - | ∇ |-τ a a τa20

2 2 (3.48)

Integrating this inequality over time from t = F to some other arbitrary time, t, yields

V

t F

VT dV e T t F dV

C

C∫ ∫| ∇ | ≤ | ∇ = | +-

- -2 2 0

1 2a ( ) ( ) -- - - +

e eC t t F FC2 2a a ( )

(3.49)

This result will establish bounds and an exponential decay of the thermal irre-versibility of entropy production. The entropy transport equation can be written as

ρ ρ m∂∂+ ⋅∇ = ∇ +s

tv s k

T

T T 2 F

(3.50)

The total entropy within the domain, V, is given by

S t s x t dVV

( ) ( , )= ∫ ρ (3.51)

This total (spatially integrated) entropy within the domain becomes a function of time only, and its derivative (with respect to time) is greater than or equal to zero, that is,

dS

dtk

T

TdV

TdV

V V= | ∇ | + ≥∫ ∫

2

20m F

(3.52)

This result represents an integrated form of the Second Law. Using the previous result of the bounded thermal irreversibility, it follows that the total entropy is bounded according to S(t) ≤ M, where

M S Fk

TT t F dV

C e

Cmin V

C F

= + | ∇ = | +

∫-

( ) ( )a

a

22 0

2

2 + | | -md V e

T d

d F

min

1

2

2

(3.53)

Also, the total derivative of entropy on the left side of Equation 3.52 approaches zero in the steady state, i.e., lim ( )

tS t

→∞′ = 0. In a case with a constant initial temperature

throughout the domain, the temperature will remain at that same constant value for

all times, and thus ∇ ≡T 0 and C0 0= . The upper bound of the total entropy simpli-fies considerably for this case, that is,

M S Fd V e

T d

d F

min

= + | | -( )

m 1

2

2

(3.54)

7262_C003.indd 74 1/19/08 11:38:43 AM


In Equation 3.53, the analysis indicates that C should be made as large as pos-sible, in order to minimize the value of M. Fluids having small values of k and m and large values of a lead to maximized upper bounds on the total entropy, defined through M. The geometrical configuration can be modified to maximize the value of the constant C. From Equation 3.41, this constant satisfies C w dV w dV

V V∫ ∫≤ | ∇ |2 2 ,

where w is any function whose normal derivative vanishes at the boundary and satisfies ∫v wdv = 0.

From a variational formulation of the eigenvalue problem, it can be shown that C is given by the smallest eigenvalue of the following problem,

∇ + = , =2 0 0w w

dw

dnonλ Ω (3.55)

This problem has a solution of l = 0, and w is any nonzero constant function. The solution will be the second eigenvalue of the eigenvalue problem. Of all domains with a fixed volume, it is known that a disk in two dimensions and a sphere in three dimensions yield the largest value of C (Weinberger, 1956). In the next section, these results and analysis will be used to determine the optimal aspect ratio to minimize entropy bounds associated with mixing in the tank.

3.4.2 o pt ima l aspEc t rat io o f uppEr En t r o pyBo u n d s

Consider the problem of determining the optimal aspect ratio of the rectangular mixing tank (Figure 3.8), which gives the largest possible value of C. Let the cross- sectional dimensions of the rectangle be L and L-1. A basis for the space of functions on the rectangle with the left corner at the origin is

B

x

LyL

x

LyL

x

Ly= , , , ,cos cos cos cos cos cos

π π π π π π22 LL

x

LyL, , cos cos

2π π

(3.56)

These functions are also eigenfunctions with the following corresponding eigenvalues:

E =2

22 2

2

22 2

2

22 2

2

22π π π π π π π π

LL

LL

LL

LL, , , ,+ +4

44 2,,

(3.57)

Consequently, the minimum eigenvalue is given by min( )π π2 2 2 2L L- , . The goal is to find the value of L that maximizes this eigenvalue. In other words, find

EL

LL

max max min= ,

>

0

2

22 2π π (3.58)

It can be shown that the maximum occurs at L = 1, which gives a square. The maxi-mum value of C becomes π 2 9 87≈ . .

7262_C003.indd 75 1/19/08 11:38:48 AM


Consider a disk of unit area (radius of π - /1 2). The eigenvalues are given by ( )ζ πmn

2 , where J′n(zmn) = 0. In other words, zmn is the mth zero of the Bessel func-tion derivative J′n. The smallest value of these zeros is approximately 1.8412, which is the first zero of J′1. This leads to C ≈ .10 65. Hence, a disk yields a larger value of C than a square with the same area. In three dimensions, a cube is the domain which yields the eigenvalueπ 2, as it gives the largest value of C among all rectangular sol-ids of unit volume. The sphere of unit volume gives a value of C ≈ .11 26. This result suggests that disks and spheres are better domains than rectangles and rectangular solids for minimizing the upper bound of total entropy in the steady state.

As an example of a Dirichlet case, consider a one-dimensional problem where the temperatures at the left and right boundaries (x = 0 and x = 1) are fixed at T0 and T1, respectively, with T0 > T1. Suppose that the initial temperature satisfies T t T T( ) min( )= ≤ ,0 0 1 . Heat transfer is initiated and after some time, the temperature approaches a linear profile, with a temperature of T0 on the left side and T1 on the right boundary. This means that |∂T/∂x|2 will approach a positive constant. In this case, the previous analysis yields

′ ≥ ∂∂∫S t k

T

T

xdx( )

12

2

0

1

(3.59)

The entropy derivative is larger than a positive constant for all times. The total entropy grows unboundedly over time, because the heat supply from the left bound-ary provides a continued entropy flow into the domain over time. This boundary condition is more suitable for mixing problems if a goal is to maximum mixing and entropy.

3.4.3 c a sEst u d yo f mixin g ta n kdEsig n

Consider another example of fluid mixing in a tank, but with nonuniform initial pro-files of velocity and temperature. Nondimensional numerical results were obtained by numerical integration with a spectral method and integration over the spatial domain to give the net entropy production (Lui and Naterer, 2007). Spectral methods seek solutions of the form ∑ =j j j

n a1 φ for certain basis functions denoted byf j. Con-sider a rectangular domain, V, bounded by [-L-1, L-1] × [-L, L] with the following nonuniform initial temperature and velocity distributions:

T t T L xy

LL ( ) ( )= = + - -

0 1 10

2 22

g

(3.60)

v

e

L L L

Lx yL

nt

=+

-

-

-

π π π2

122 2

2cos sinLL

L Lxy

L

, sin( ) cosπ π2

2 (3.61)

where T0, g, and h are positive constants. Note that the velocity field satisfies conser-vation of mass (zero divergence), decays exponentially, and vanishes at the bound-ary of the domain. After normalizing with x x L= / and y y L= / , the new domain

7262_C003.indd 76 1/19/08 11:38:57 AM


becomes a square bounded by [- 1,1] × [- 1,1]. In the numerical simulations, a Che-byshev spectral collocation method was used to solve the temperature equation (Lui and Naterer, 2007).

Fluid motion is initially generated by the fluid mixer, but then the mixer within the tank is turned off, and the fluid velocity approaches zero after a period of time. An example of a magnetic stirrer for chemical mixing is illustrated in Figure 3.8. The upper bound of total entropy has practical significance because it characterizes the effectiveness of mixing and the system input power required to achieve certain levels of mixing. This mixing is dependent on the initial tem-perature and velocity profiles, which characterize the fluid motion and heat trans-fer leading to entropy production during the mixing process. The upper entropy bounds can be predicted without solving the detailed transient equations in the tank with CFD. Analytical results for the upper entropy bound are developed independently of this transient motion, although they depend on the initial level of fluid mixing in the tank.

A simulation was performed with mixing of methane inside the tank. Based on a nonuniform initial temperature with T0 = 873 K, g = 10, h = 0.0155, L = 3, and a time step of 0.00065 s, the predicted results are shown in Figure 3.9 for 17, 19, and 21 Chebyshev modes of a spectral method, respectively. In Figure 3.10, 10 Chebyshev modes were used with a time step of 0.28 s. The initial temperature is fixed at 373 K, and the total entropy at equilibrium is plotted for various aspect ratios of the rectangle. It can be observed that the total entropy increases at larger values of L. In Figure 3.11, the total entropy at equilibrium is minimized for a square (L = 1). The total entropy at equilibrium increases at higher aspect ratios, namely, the higher ratio of surface

172119

0.025

0.02

0.015

0.01

0.005

0

Tota

l Ent

ropy

(J/K

)

0 100 200 300 400 500 600 700Time (sec)

f Ig u r E3.9 Total entropy for methane for L = 3.

7262_C003.indd 77 1/19/08 11:38:58 AM


0 2000 4000 6000 8000 10000 12000 14000

9

8

7

6

5

4

3

2

1

0

Tota

l Ent

ropy

(J/K

)

Time (sec)

× 10–3

L = 3

L = 2.5

L = 2

L = 1.5

L = 1

f Ig u r E3.10 Effects of tank width on total entropy change for methane.

0 0.5 1 1.5 2 2.5 3L(m)

1

2

3

4

5

6

7

8

9× 10–3

Tota

l Ent

ropy

at E

quili

briu

m (J

/K)

f Ig u r E3.11 Total entropy at equilibrium (373 K) at varying tank widths.

7262_C003.indd 78 1/19/08 11:39:00 AM


area to volume for the domain. Frictional irreversibilities along the wall constitute a major component of the total entropy production, so larger surface areas lead to higher entropy production and total entropy at equilibrium. The results in this case study pro-vide examples of how entropy bounds can give useful insight for better understanding of energy conversion, mixing, and fluid dynamics in tanks and enclosures.

3.5 c a sEst u d y o f au t o mo t Iv Efu El cEl l dEsIg n

The next case study in this chapter involves entropy-based design of fuel cells (Naterer and Tokarz, 2006). It will be shown that entropy and the Second Law provide a valu-able design tool for achieving higher efficiency of fuel cells. This includes friction and thermochemical irreversibilities of gas flow through the fuel cell channels.

3.5.1 El Ec t r o c h Emic a l ir r Ev Er s iBil it iEsin a po r o u sEl Ec t r o d E

In a fuel cell, entropy production of irreversible chemical reactions, diffusion, and ohmic heating leads to voltage losses (or polarization). During operation of a solid oxide fuel cell (SOFC), fuel and oxidant are continuously supplied to the anode and cathode, respectively (see Figure 3.12). Oxygen molecules combine with free elec-trons from the external circuit to produce negative oxygen ions (O-), which migrate through the electrolyte and generate ohmic heating with entropy production. Hydro-gen molecules diffuse simultaneously through the anode. They combine with the oxygen ions and liberate electrons, while producing H2O and heat. As a result, free electrons flow through the external circuit as an electrical current. They return to the fuel cell at the cathode, where they combine with oxygen molecules to again produce oxygen ions. A proton exchange membrane fuel cell (PEMFC) operates in a similar manner, except that hydrogen ions flow to the cathode where water molecules are produced. The chemical balances for an SOFC and PEMFC are given by

SOFC: H g O H O g e2 2 2( ) ( )+ ⇒ += - ( )anode (3.62)

12 2 2O g e O( ) + ⇒- =

( )cathode (3.63)

PEMFC: H g H aq e2 2 2( ) ( )⇒ ++ - ( )anode (3.64)

12 2 22 2O g H aq e H O l( ) ( ) ( )+ + ⇒+ -

( )cathode (3.65)

Irreversibilities within a fuel cell lead to voltage losses and lost power to aux-iliary devices like blowers to sustain cyclical operation of the fuel cell. The Nernst equation gives the ideal (reversible) performance of a fuel cell, in terms of the ideal standard potential, E0, ideal equilibrium potential, E, and nonstandard product or reactant temperatures and pressures. For a PEMFC, the maximum theoretical volt-age is given by

E ERT

F

P

P

RT

FH

H O

= +

+

0

2 42

2

ln llnP

PO2

0

(3.66)

7262_C003.indd 79 1/19/08 11:39:05 AM


where P0 refers to the standard pressure (1 atm) and other variables are defined in the “Nomenclature.” Voltage losses are often characterized by the polarization, h. The total polarization is the potential difference, DE, between the reversible voltage and the cell voltage when current flows through the circuit, that is,

h = = -DE E Erev current (3.67)

The reversible voltage computed at the wall (subscript w) and bulk value (subscript b) can be expressed in terms of the ideal standard potential for the chemical reaction, E0, that is,

E E

RT

FCw H

I= + ( )0

2 2ln (3.68)

E E

RT

FCb H= + ( )0

2 2ln (3.69)

Current FlowEnd Plate

AnodeElectrolyte MatrixCathodeSeparator Plate

Oxidant FlowAnode

Solid Wall (Bipolar Plate)

Porous Electrode Interface

Fuel Flow

Repeating Unit

y = h

y

x

Gas Flow

v(x, 0) = Suction Velocity

(a)

f Ig u r E3.12 Schematic of a PEMFC (a) stack, channel flow and (b) operating components.

Electric Current

Flow e– (+) (–) Bipolar Plate

Gas Channel

H2 H+ O2

Anode

Electrolyte

Cathode0.5O2 + H+ + 2e– H2O

Gas Channel

Bipolar Plate

7262_C003.indd 80 1/19/08 11:39:11 AM


The difference between these voltages is called the concentration loss or concentra-tion polarization. It represents the difference of the reversible potential computed from the gas concentration at the wall and bulk value.

The concentration polarization at the anode (subscript conc,a) can be written as (Naterer and Tokarz, 2006, and references therein)

hδ δconc a

a

H

a

HHIRT

F

p pp

RT, ln exp= - - -

⋅

22 2

2 222

2

2F

l

D pi pH a

H eff aHI⋅ ⋅

δ

( )

(3.70)

The electrochemical polarization is related to entropy production, Ps, according to

h = ⋅T P

Fs

2 (3.71)

Thus, entropy production due to concentration irreversibilities within the anode can be written as

P Rp p

pRT

Fs conca

H

a

HHI

, ln exp= - ⋅ - -

⋅

δ δ2 2

2 2⋅⋅ ⋅

δH a

H eff aHI

l

D pi p2

2

2( )

(3.72)

This result was derived for the anode of a PEMFC. An analogous result for the cath-ode polarization can be derived with the same procedure:

PR RTl

FD pi

Rc

c effIs,conco

= - -

+

21

81

2

In( )

TTl

FD pic

c effIH4

2( ) o

(3.73)

In addition to these concentration irreversibilities, the total entropy production within the electrodes includes activation and ohmic irreversibilities. These losses can be written as (Naterer and Tokarz, 2006)

PR

n

i

i

i

i

R

n

i

is acte c c e a

, ln ln= + +

+4

14

0

2

02

0

++ +

i

i a

2

02

1 (3.74)

PF C D

Tl n

i

is ohmicH m

m d L, ln= - -

+21

2

(3.75)

The expression for the activation polarization of an SOFC is identical to this result, except that i0c is replaced by 2i0c (first term on right side) and i0a is replaced by 2i0c (second term on the right side). These changes occur due to the different half-cell reactions.

The voltage losses become independent of the limiting current density of the anode, i0a, for a small anode thickness in a PEMFC (or a small cathode thickness in

7262_C003.indd 81 1/19/08 11:39:16 AM


an SOFC). The anode concentration polarization diminishes significantly when the anode thickness becomes much smaller than the cathode thickness. For PEMFCs, the following TAP-TCS model (Thin Anode for PEMFC, or Thin Cathode for SOFC approximation) neglects the anode concentration polarization. After combining all irreversibilities,

PR

n

i

i

i

i

R

n

i

i

i

ise a a e c

= + +

+ +4

14

0

2

02

0

2

ln ln002

2

12

1

2

c

H m

m d L

F C D

Tl n

i

i

R

+

- -

-

- ln

lnn, ,

18

18

2

-

+RTl

FD pi

RTl

FD pc

c eff OI

c

c eff O22I

i

(3.76)

where ne and i0c refer to the number of moles of electrons produced per half-cell reaction and the cathode exchange current density, respectively. On the right side, the five terms represent the activation irreversibility (first and second terms; anode plus cathode), ohmic irreversibility (third term), and concentration irreversibility (fourth term), respectively. For an SOFC, a similar entropy production equation as the PEMFC result can be derived, except the derivation involves different half-cell reactions, so pa, dH2O, pH2, la, Da(eff), Dc(eff), and pO2 in a PEMFC are replaced by pc, dO2, pO2, lc, DO2(eff), Da(eff), and pH2, respectively, for an SOFC.

3.5.2 f o r mu l at io n o f ch a n n El fl o w ir r Ev Er s iBil it iEs

To calculate the total entropy production within a fuel cell stack, additional friction and thermal irreversibilities within the gas channels must be formulated. Consider gas flow through a uniform fuel channel involving either hydrogen in the anode-side channel or oxygen in the cathode-side channel (see Figure 3.12). The steady-state gas motion is bounded between a solid wall (bipolar plate) and a porous wall (gas- diffusion layer), where suction flow occurs due to the permeable interface. As a result, the fuel and oxidant concentrations will decrease along the channel, and the bulk gas velocity increases to conserve mass. In the following simplified integral analysis, the gas concentration will be assumed as uniform across the channel, but varying in the streamwise direction.

Assuming a parabolic variation of velocity across the channel,

u x y A x y B x y G x( , ) ( ) ( ) ( )= + +2 (3.77)

A no-slip condition is applied along the upper wall (y = h), while a slip condition of u x u x( , ) ( )0 = x is applied at the bottom wall (see Figure 3.12), where x represents the slip coefficient. This coefficient depends on the permeability of the porous electrode, and it generally varies between 0.1 and 1. Furthermore, the mean velocity is defined as

u xh

u x y dyh

( ) ( , )= ∫1

0 (3.78)

7262_C003.indd 82 1/19/08 11:39:19 AM


Applying the boundary conditions and integrating the resulting profile to estab-lish the mean velocity gives the unknown coefficients (A, B, and G), thereby yielding

u

u

y

h

y

h= - + - +( ) ( )3 6 2 3 2

2

2x x x (3.79)

For two-dimensional laminar gas flow under steady-state conditions through a fuel channel, the reduced mass and x-momentum equations are given by

∂∂+ ∂∂=( ) ( )ρ ρu

x

v

y0 (3.80)

ρ mu

u

xv

u

y

p

x

u

y

∂∂+ ∂∂

= - ∂∂+ ∂∂

2

2

(3.81)

where the density can be expressed as the product of gas concentration, C(x), and molecular weight, M. Although the gas density varies due to gas concentration changes, it remains an incompressible flow in terms of the Mach number. Inte-grating the continuity equation across the channel and applying the slip boundary condition,

h

d Cu

dxCV x

( )( )= 0 (3.82)

where V0(x) is the suction velocity at the base of the fuel channel, defined by

VnF

i x

C x01=

( )( )

(3.83)

The current density at a particular x-position is related to the gas concentration by the following Tafel equation:

i x

i

C x

C

F

RT

( ) ( )exp

0 0

=

ga h (3.84)

Nondimensionalizing the velocity ( u u u= / 0) and gas concentration ( C C C= / 0), the integrated continuity equation becomes

d Cu

dxK C K

i

nFhC u

F

RT

( ); exp

+ = =

1 1

0

0 0

0g a h (3.85)

7262_C003.indd 83 1/19/08 11:39:27 AM


Substituting the u-velocity profile and solving the continuity equation, subject to appropriate boundary conditions along the top and bottom walls,

v

V

y

h

y

h

y

h0

3

3

2

22 3 3 2 1= - - - - -( ) ( )x x x (3.86)

Then, substituting the velocity profiles into the mass and momentum equations, assuming a constant pressure gradient in the streamwise direction and integrating the momentum equation across the channel,

K

d Cu

dxK K C u K u2

2

3 4 5 0( ) + + + =g (3.87)

where

K2

2215

15

65

= - +x x (3.88)

KP

MC u30 0

2= (3.89)

Ki

nFhC u

F

RT40

0 0

=

x a hexp (3.90)

KMh C u5 2

0 0

12 112

= -x

m (3.91)

Solving the coupled mass and momentum equations with a series solution, sub-ject to inlet conditions of u( )0 1= and C( )0 1= , yields the following linearized channel flow approximation (LCF model),

C x C K x KK K K K K

K( ) ( );= - = - - -

0 6 61 2 3 4 5

2

12

(3.92)

u x u K x KK K K K K

K( ) ( );= + = - - -

0 7 71 2 3 4 5

2

1 (3.93)

In the LCF model, higher-order terms have been neglected in a series solution for short channels, microfuel cells, or small values of x (near the inlet section of a fuel cell channel). LCF refers to a linearized approximation of channel flow profiles, including the gas concentration and velocity profiles.

The purpose of these simplifications is to allow closed-form approximations for entropy production, particularly so net irreversibilities can be analytically

7262_C003.indd 84 1/19/08 11:39:34 AM


minimized to illustrate how entropy-based design can be used as a predictive tool to improve the fuel cell performance. The combined friction and thermal irrevers-ibilities within the fuel channel can be expressed as

PT

u

y

k

T

T

ys''' = ∂

∂+ ∂

∂

m2

2

2

(3.94)

Substituting the differentiated velocity profile, integrating across the channel, and expressing the total entropy production due to friction within a channel of length L, per unit depth,

Pu L

hTh hs = - + - +4

3 3 3 6 42

2 2m x x x( ) (3.95)

Alternatively, this result can be written in terms of the channel Reynolds number, Reh. Combining all irreversibilities yields the following total entropy production within the fuel cell. It can be shown that (Naterer and Tokarz, 2006)

PRT

n F

i

iK x

F

RTse a

= -

-2

11 0

06sinh ( ) expg a h

+ -

-2

11 0

06

RT

n F

i

iK x

F

RTe c

sinh ( ) expg a h

- - -

RT

F

p

p

p

pa

HI

H

a

HI

H21

2 2 2 2

lnδ δ -

exp ( ) exp

,

RT

F

l

Di K x

F

RTH a

a eff212

0 6

δ a hg

- - -RT

F

RTl i K x

FDc

c41

1

80 6ln( )g

,,

exp(

eff OI

c

p

F

RT

RTl i K x

2

110 6a h

+ - ))

exp,

g a h8

2FD p

F

RTc eff OI

-- - -

+21 1

20

6

F C D

Tl n

i

iK x

F

RTH m

m d L

ln ( ) expg a h

+ - + - +4

3 3 3 6 42

2 2m x x xu L

hTh h( )

(3.96)

In the TAP-TCS model, the concentration polarization (third term on the right side) becomes negligible. In the following section, numerical predictions will be studied to outline how entropy-based design can provide a valuable tool for improving fuel cell performance.

3.5.3 pr o t o n Exc h a n g EmEmBr a n Efu El cEl l (pEmf c )a n d so l id oxid Ef u El cEl l (sof c )dEsig n

In this section, numerical results for PEMFCs and SOFCs will be presented (Naterer and Tokarz, 2006). Problem parameters are presented in Table 3.1 and adopted from Chan and Xia (2002), Chen et al. (2004), Ghadamian and Saboohi (2004), and Kim et al. (1999). Entropy production involves electrochemical reactions at the electrode surfaces, mass transfer ohmic heating, and frictional losses within the fuel channel.

7262_C003.indd 85 1/19/08 11:39:37 AM


The overall efficiency of a fuel cell system depends on other losses, such as gas blow-ers, pumps, electrical losses (DC power conversion to AC), electrolysis, fuel storage, and others. Unlike conventional methods of characterizing fuel cell losses that use overpotential or polarization curves, the current entropy-based method provides a use-ful alternative by encompassing all losses of available energy. It strives toward the upper limits of performance imposed by the Second Law. Entropy production provides a useful parameter for systematic optimization of design parameters in fuel cells.

Irreversibilities and inefficiencies are important factors in evaluating feasibility of energy conversion processes. This is particularly evident when comparing fuel cells against other possible methods of future power generation, such as advanced diesel engines for automobiles. For example, automotive PEMFCs can consume 10% or more power to drive pumps, blowers, heaters, and controllers (Bossel, 2003). DC power is converted to voltage-adjusted DC or frequency-modulated AC, and the electrical efficiency of the electric drive train can be about 90%. The process of generating hydrogen for fuel cells has numerous irreversibilities. For electrolysis, an overall efficiency of 70% for power plant to hydrogen production can be achieved.

t a bl E3.1o peratingconditionsandproblemparameters

protonExchangemembranefuelcell

Channel temperature, T (K) 353.15 Inlet gas velocity, u0 (m/s) 0.7Inlet gas pressure, P0 (atm) 2.0Inlet gas concentration, C0 (mol/m3) 69.0Exchange current density, i0 (A/m2) 0.00001Activation overpotential, h (V) 0.3Reaction order, g 0.5Electrons transferred in reaction, n 4 Charge-transfer coefficient, a 2.0Molar weight, M (kg/mol) 0.032 Viscosity, m (kg/ms) 0.00002 Channel height, h (m) 0.001 Standard equilibrium potential, E0 (V) 1.167 Pressure gradient (Pa/m) 250.0Slip coefficient, x 0.1

solidoxidefuelcell

Operating temperature, T (oC) 750 Operating pressure, p (atm) 2.0Electrolyte resistance, Ri (Ωcm2) 0.092 Concentration resistance, Rconc (Ωcm2) 0.297 RT/4F 0.02204 Exchange current density, i0 (A /cm2) 0.113 Effective diffusion coefficient, Da,eff (cm2/s) 0.166 Cathode thickness, lc (m) 0.00005 Average pore radius, (mm) 0.5 Electrolyte thickness, le (mm) 40.0 Anode thickness, la (mm) 750.0 Partial pressure ratio, ph2/ph2o 32.352

7262_C003.indd 86 1/19/08 11:39:38 AM


Furthermore, hydrogen gas needs to be compressed (about 90% efficient) or lique-fied (65% efficiency) for transportation purposes. Hydrogen gas can be delivered to filling stations by pipelines, which takes about 10% energy (higher heating value, or HHV) for gaseous hydrogen, or about 6% for liquid hydrogen. Also, about 3% energy is needed to transfer gaseous hydrogen from a large storage tank into a car’s tank. When combined with the efficiency for conversion of electricity in fuel cells, Bossel (2003) has reported a “power plant to wheel efficiency” of about 22% for typical operating conditions in a PEMFC, compared with advanced diesel (25%) and hybrid electric with SOFC range extension (33%). Although promising advances like thermochemical hydrogen production will significantly improve the power plant to wheel efficiency, the improvement of fuel cell efficiency will continue to be an important issue for their widespread adoption in the transportation sector. An entropy-based design provides a more powerful design tool for this purpose of improving fuel cell efficiencies.

In a PEMFC, hydrogen fuel is consumed at the electrode surface when it reacts, releases electrons, and creates hydrogen ions (or protons). Electrons produced at the anode pass through an electrical circuit to the cathode, while protons diffuse through the electrolyte. The oxygen concentration decreases due to chemical reactions along the electrode surface. In Figure 3.13, the predicted results with the LCF model show close agreement with past data reported by Chen et al. (2004). Due to fuel consumption in the x-direction of the channel, the gas density decreases. From requirements of mass conservation, the gas velocity then increases. A slip-flow boundary condition is applied along the anode or channel interface, which affects the magnitude of gas velocity. Along this interface, entropy production arises from friction and viscous dissipation of kinetic energy to internal energy within the gas stream. As a result, added blower power is con-sumed from the cell output voltage, to overcome pressure losses created by gas friction. This energy conversion differs from electrochemical irreversibilities characterized by

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.000 0.002 0.004 0.006 0.008 0.010x(m)

Co

ncen

trat

ion

(C),

Velo

city

(u)

C (Chen et al. 2004)

C (LCF Model)

u (Chen et al. 2004)

u (LCF Model)

(P = 400 Pa/m, h = 1 mm, Proton Exchange Membrane Fuel Cell

= 0.1)

f Ig u r E3.13 Velocity and concentration profiles in the fuel channel.

7262_C003.indd 87 1/19/08 11:39:39 AM


conventional polarization methods. An entropy-based design can compare all irrevers-ibilities directly against each other, in terms of their lost work potential.

In the LCF model, a slip velocity and slip coefficient (x) were applied in bound-ary conditions at the porous electrode interface, when predicting the gas velocity profile in the fuel channel. In Figure 3.14, entropy production is shown at varying slip coefficients and channel heights, with a minimum point in each case at a slip coefficient of x = 0.5. For a fixed mass flow rate through a fuel channel, the gas velocity and entropy production increase with smaller channel heights. At low slip coefficients, additional friction at the wall yields higher entropy production. On the other hand, higher slip coefficients affect the suction flow through the porous inter-face. The momentum balance alters the velocity profile and near-wall velocity gra-dient along the top wall, as well as skewing of the velocity profile in the lateral (y) direction. The entropy production rises, and the optimal point is reached midway, when net viscous dissipation within the channel is minimized.

Figure 3.15 illustrates the combined ohmic, concentration, and activation irre-versibilities within the electrode. The total entropy production increases at higher interface surface resistances, R(i) (units of kΩ/cm2), due to higher ohmic heating. Electrode materials with higher electrical conductivity could reduce these ohmic losses. Entropy production rises rapidly at low current densities, due to high activa-tion losses. Activation losses also contribute to higher entropy production at larger current densities. Some possible ways of reducing these irrversibilities include dif-ferent materials, higher reactant concentrations (possibly using oxygen instead of air), or higher operating temperatures. Furthermore, larger electrode surface rough-nesses would increase the effective surface area, while increasing the exchange cur-rent density and reducing overall entropy production.

Figure 3.16 illustrates close agreement between predicted results with the cur-rent TAP-TCS model and past data. Voltage losses increase at higher electrode

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Slip Coefficient ( )

Entr

opy P

rodu

ctio

n (W

/m3 K

*106 )

h = 0.001 m

h = 0.002 m

h = 0.003 m

PEMFC (T = 353 K, P = 1 atm)

f Ig u r E3.14 Entropy production at varying slip coefficients in a PEMFC.

7262_C003.indd 88 1/19/08 11:39:40 AM


surface resistances, due to additional ohmic heating. The TAP-TCS voltage losses were calculated based on entropy production, rather than empirical polarization methods documented by Ghadamian and Saboohi (2004). The results provide use-ful validation of the current predictive model of entropy production. As mentioned previously, an entropy-based approach provides a useful alternative for characterizing voltage losses, as it encompasses all types of irreversible losses within the fuel cell. For example, power consumed by the fuel and air blowers, due to channel entropy produc-tion, comes at the expense of output voltage generated by the fuel cell. Thus, channel

0

50

100

150

200

250

0 100 200 300 400 500 600 700 800Current Density (mA/cm2)

Entr

opy P

rodu

ctio

n (J/

kg K

)

PEMFC (T = 373 K, P = 1 atm)

R(i) = 0.00003

R(i) = 0.00006

R(i) = 0.00009

f Ig u r E3.15 PEMFC entropy production.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 200 400 600 800Current Density (mA/cm2)

Vol

tage

(V)

TAP/TCS ModelGhadamian, Saboohi (R = 0.00001) TAP/TCS ModelGhadamian, Saboohi (R = 0.0005)

PEMFC (T = 373 K, P = 1 atm, iL = 800 mA)

f Ig u r E3.16 PEMFC voltage profile.

7262_C003.indd 89 1/19/08 11:39:43 AM


flow, ohmic heating, diffusion, and concentration losses are irreversibile losses that can all be characterized consistently according to their rates of entropy production. Their exergy losses will reduce the overall efficiency of the fuel cell. Unlike past methods characterizing system losses through a “polarization” or “overpotential,” the entropy-based approach can provide a more robust way of calculating all losses of available energy including frictional, thermal, electrochemical, and so forth.

Previous figures have investigated PEMFCs, whereas Figure 3.17 illustrates results for SOFCs. Voltage losses were calculated based on the entropy formulation and compared successfully against measured data in Figure 3.17. The results of volt-age losses were derived from the predicted entropy production, which could also be expressed in terms of exergy destruction, after multiplying by the operating temper-ature of the fuel cell. This provides another useful parameter for design purposes, as exergy losses have equivalent units of power. Thus, power lost to irreversibilities could be calculated directly, or converted to economic losses after multiplying by the local cost of electricity per kilowatt hour of operation of the fuel cell.

3.6 c a sEst u d y o f fl u Id mac h In Er y dEsIg n

This last case study applies the method of entropy-based design to loss coefficients and analysis of power generation from fluid machinery (particularly turbines in this case study). The mechanical power generated by steam, gas, or wind turbines is highly dependent on the shape of blades, velocity field, and other factors (Leclerc et al., 1999). The turbine power output is related to the change of kinetic energy, internal energy, and heat transfer rate from the system encompassing the turbine. Consider a control volume including a turbine, with the inlet, outlet, and other boundaries suf-ficiently far from the turbine to permit uniform conditions along those boundaries

f Ig u r E3.17 Voltage profile (SOFC; T = 750°C).

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 Current Density (A/cm2)

Volta

ge (V

) Experimental (T = 750°C)

Predicted (Kim, 1999)

Predicted (TAP/TCS Model)

i0s = 4.95 A/cm2, i0 = 0.113 A/cm2

7262_C003.indd 90 1/19/08 11:39:44 AM


(see Figure 3.18). For example, a water turbine submerged below a free surface of water is considered. A standard undergraduate thermodynamic analysis would yield the following energy balance for the control volume:

dE

dtm e p gz V Q W m e= + + +

+ - - + 1 1 1 1 1

22 2

12

υ pp gz Vυ2 2 221

2+ +

(3.97)

where m1, e, υ , Q, and W refer to mass flow rate (constant throughout streamtube), internal energy (per unit mass), specific volume (per unit mass), heat transfer, and boundary work, respectively. The left side of the equation becomes zero under steady-state conditions.

Applying an entropy balance to a differential section in Figure 3.18 and using the Gibbs equation,

dQ Tmde

Tp

d

TTdSgen

= +

-υ

(3.98)

In this equation, the heat transfer differential represents a process, not a thermo-dynamic property or state variable. Summing over n sections throughout the entire channel from the inlet (i = 1) to the exit (i = n), it can be shown that

Q m e e T dSi

i

n

gen i= - -=∑( ) ,2 1

1 (3.99)

Substituting this result into the energy balance, the following result is obtained for incompressible flows:

W m gz V gz V T dSi

i

n

gen i= + - -( ) -=∑

12 1 1

21 2

2

1, (3.100)

W

Out

V2

A2

21

dQ

V1

A1

In

f Ig u r E3.18 Schematic of a flow channel for turbine analysis.

7262_C003.indd 91 1/19/08 11:39:50 AM


This result suggests that the power output is maximized when the net entropy pro-duction over the control volume encompassing the turbine is minimized. Thus, a key objective is to minimize the energy availability lost on the right side of the equa-tion from entropy production due to viscous mixing, flow separation, and other flow irreversibilities. This goal can be achieved through design modifications (such as modifications of the blade shape, gap spacing, thickness, and so forth), using CFD or experimental techniques like particle image velocimetry (PIV), which would provide whole-field data for local entropy production rates (Adeyinka and Naterer, 2004). These methods of calculating and measuring whole-field distributions of entropy production will be presented in upcoming chapters.

r Ef Er En c Es

Adeyinka, O.B. and G.F. Naterer. 2004. Numerical and Experimental PIV/PLIF Studies of Entropy Production in Natural Convection. AIAA 42nd Aerospace Sciences Meeting and Exhibit. Jan. 5–8, Reno, NV.

Adeyinka, O.B. and G.F. Naterer. 2005. Modeling of entropy production in turbulent flows. ASME J. Fluid Eng., 126(6): 893–899.

Bejan, A. 1996. Entropy Generation Minimization: The Method of Thermodynamic Optimiza-tion of Finite-Time Systems and Finite-Time Processes. CRC Press, Boca Raton, FL.

Bobeth M. and G. Diener. 1982. Upper bounds for the effective thermal contact resistance between bodies with rough surfaces. Int. J. Heat Mass Transfer, 25(8): 1231–1238.

Bossel, U. 2003. Efficiency of Hydrogen Fuel Cell, Diesel-SOFC-Hybrid and Battery Electric Vehicles. European Fuel Cell Forum (October 20). Morgenacherstrasse, Germany.

Cengel, Y.A. and M.A. Boles. 2002. Thermodynamics: An Engineering Approach. McGraw-Hill, New York.

Chan, S.H. and Z.T. Xia. 2002. Polarization effects in electrolyte/electrode-supported solid oxide fuel cells. J. Appl. Electrochem., 32: 339–347.

Chen, F., Wen, Y.Z., Chu, H.S., Yan, W.M., and C.Y. Soong. 2004. Convenient two-dimen-sional model for design of fuel channels for proton exchange membrane fuel cells. J. Power Sources, 128: 125–134.

Dincer, I., Hussain, M.M., and I. Al-Zaharnah. 2004. Energy and exergy utilization in trans-portation sector of Saudi Arabia. Appl. Thermal Eng., 24(4): 525–538.

Eames, P.C. and B. Norton. 1998. The effect of tank geometry on thermally stratified sensible heat storage subject to low Reynolds number flows. Int. J. Heat Mass Transfer, 41(14): 2131–2142.

Ghadamian, H. and Y. Saboohi. 2004. Quantitative analysis of irreversibilities causes voltage drop in fuel cell (simulation and modeling). Electrochimica Acta, 50: 699–704.

Hassani, A.V., Hollands, K.G.T., and G.D. Raithby. 1993. A close upper bound for the conduc-tion shape factor of a uniform thickness, 2D layer. Int. J. Heat Mass Transfer, 36(12): 3155–3158.

Homan, K.O. and S.L. Soo. 1998. Laminar flow efficiency of stratified chilled-water storage tanks. Int. J. Heat Fluid Flow, 19(1): 69–78.

Kim, J.W., Virkar, A.V., Fung, K.Z., Mehta, K., and S.C. Singhal. 1999. Polarization effects in intermediate temperature, anode-supported solid oxide fuel cells. J. Electrochem. Soc., 146: 69–78.

Leclerc, C., Masson, C., Ammara, I., and I. Paraschivoiu. 1999. Turbulence Modeling of the Flow around Horizontal Axis Wind Turbines, Wind Engineering. Multi-Science Publishing, Essex, U.K., 279–294.

7262_C003.indd 92 1/19/08 11:39:51 AM


Lui, S.H. and G.F. Naterer. 2007. Upper entropy bounds for transient forced convection. Heat Mass Transfer, 43: 295–308.

Martins, R. and S. da Gama. 2000. An upper bound estimate for a class of conduction heat transfer problems with nonlinear boundary conditions. Int. Commun. Heat Mass Transfer, 27(7): 955–964.

Mikic, B.B. 1998. On destabilization of shear flows: concept of admissible system perturba-tions. Int. Commun. Heat Mass Transfer, 15: 799–811.

Naterer, G.F. 2001. Establishing heat-entropy analogies for interface tracking in phase change heat transfer with fluid flow. Int. J. Heat Mass Transfer, 44(15): 2903–2916.

Naterer, G.F. 2002. Heat Transfer in Single and Multiphase Systems. CRC Press, Boca Raton, FL.

Naterer, G.F. and C.D. Tokarz. 2006. Entropy based design of fuel cells. ASME J. Fuel Cell Sci. Technol., 3(2): 165–174.

Poulikakos, D. and Bejan, A. 1987. 1982 (Nov.). Fin geometry for minimum entropy genera-tion in forced convection. ASME J. Heat Transfer, 104: 616–623.

Rosen, M.A., Le, M.N., and I. Dincer. 2004. Exergetic analysis of cogeneration-based district energy systems. IMechE-Part A: J. Power Energy, 218(6): 369–376.

Sinai, Y.L. 1985. Fundamental sloshing frequencies of stratified two-fluid systems in closed prismatic tanks. Int. J. Heat Fluid Flow, 6: 142–144.

Weinberger, H.F. 1956. An isoperimetric inequality for the n-dimensional free membrane problem. J. Rational Mech. Anal., 5: 633–636.

White, F.M. 1974. Viscous Fluid Flow. McGraw-Hill, New York.Zeidler, E. 1985. Nonlinear Functional Analysis and Its Applications. 2A. Springer-

Verlag, Heidelberg.Zubair, S.M., Kadaba, P.V., and Evans, R.B. 1987. Second Law-based theroeconomic optimi-

zation of two-phast heat exchangers. ASME J. Heat Transfer, 109(2): 287–294.

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95

4 Measured Irreversibilities of Incompressible Flows


This chapter presents experimental methods to enable measurements leading to the spatial distribution of entropy production within a flow field. Measured entropy produc-tion provides a valuable diagnostic tool from which economic impact of exergy losses (losses of work potential) could be determined. Rosen and Dincer (2003) have devel-oped exergoeconomic methods to assess economic impact of exergy losses in various industrial systems, such as power plants operating on various fuels and thermal energy storage systems (Dincer and Rosen, 2000). Linking exergy losses directly with finan-cial losses is a powerful tool for driving changes within energy systems to reduce losses of useful work, which would be otherwise underestimated without understanding their economic impact on system viability.

In this chapter, the experimental techniques will focus on the combined use of particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF). PIV measures the spatial distribution of fluid velocity, whereas PLIF is used to acquire temperature data in a flow field. The combined PIV-PLIF method offers certain advantages over standard methods of anemometry for experimental stud-ies of entropy production. Previous methods, limited by single-point measurement techniques, can only measure single-point entropy production or averaged entropy production over a finite volume. On the other hand, PIV-PLIF methods provide whole-field methods, while allowing nonintrusive and time-varying measurements of the instantaneous velocity and temperature distributions within a flow field. This chapter presents a detailed description of methods to collect physical data on the detailed structure of entropy production throughout a flow field. The PIV and PLIF techniques provide multipoint instantaneous data, so they enable measured data for local variations of the entropy production rates. In this chapter, the experimental techniques will give whole-field measurements of entropy production with these nonintrusive, optical methods.

4.2 ExpEr ImEn t a l tEc h n Iq u Es o f Ir r Ev Er sIbIl It y mEa su r EmEn t

4.2.1 Vel o c it yFiel d Mea su r eMen t

Whole-field velocity data are needed before the local entropy production rates can be determined. PIV is a widely used experimental method based on light scattering by small particles in a flow fluid, which are illuminated by two laser light pulses at very short intervals. The scattered light has the same frequency as incident laser light at

7262_C004.indd 95 1/19/08 11:41:38 AM


low wavelengths. In contrast, laser-induced fluorescence (LIF) does not result from a scattering process, but rather an absorption and wavelength conversion process. The light emitted by molecules and atoms in a de-excitation process, induced by absorption of a photon of higher energy (from a laser source), is red-shifted to lon-ger wavelengths. These combined features of PIV-PLIF allow synchronization of measurement techniques for both thermal and friction irreversibility measurements, without duplication of hardware.

The optical configuration for a typical PIV-PLIF setup consists of a light source, light sheet optics, fluorescent dye for PLIF, processor with software, and tracer par-ticles for PIV and CCD or CMOS cameras (see Figure 4.1). In two-dimensional PIV, the pulsed laser illuminates a planar cross section in the center of the flow region of interest, parallel to the flow and perpendicular to the camera. The camera captures the image of the illuminated particles in successive frames at each instant when the light sheet is pulsed. The two successive images are processed, subdivided into small interrogation regions, and matched based on a correlation analysis to determine the displacement of a group of particles, elapsed time, and the local fluid velocity. Denoting M as the magnification of the camera, the velocity is given by a first-order estimate as follows:

UM s

t= DD

(4.1)

where Ds is a displacement vector in the image plane and Dt is the pulse time interval (Willert and Gharib, 1991).

Interrogation analysis is an important element of the PIV technique. The spatial velocity distribution is obtained over a regular grid of small subregions using statisti-cal methods. The recorded image frame is divided into small areas, called interroga-tion areas. Correlation-based techniques are used within each interrogation region to produce a vector representing the average particle displacement. Autocorrelation and cross-correlation techniques are used for high particle density image analysis, whereas other methods like particle tracking and particle pairing are limited to relatively

Base of Plexiglass Flow Chamber

Camera

PIV Seeding Particles

Test Apparatus

Pulsed Laser Optics

Light Sheet

IncomingFlow

f Ig u r E4.1 Schematic of particle image velocimetry configuration.

7262_C004.indd 96 1/19/08 11:41:41 AM

Measured Irreversibilities of Incompressible Flows 97

low-density images. A high-density image arises when the number of particles is between 7 and 10 image pairs per interrogation area. In autocorrelation, an inter-rogation area is correlated with itself. In this section, the analysis will correlate an interrogation area with a second area, which is offset in the mean flow direction. The cross-correlation employed within each interrogation area allows a precise determina-tion of the direction of displacement to give instantaneous values of both components of fluid velocity in two dimensions. Westerweel (1997) and Adrian (1991) provide addi-tional details regarding interrogation analysis in PIV methods.

The PIV resolution becomes more important for high Reynolds number experi-ments that attempt to resolve small-scale variations embedded with in a large-scale motion. Such scenarios exist in turbulence measurements and cases where small-scale flow structures around large objects must be resolved. Two key velocity resolu-tion issues arise in these types of problems (FlowMap, 1998), namely, (1) the dynamic velocity range, which relates to the ability to resolve very small velocity displacements between particle image pairs, and (2) the dynamic spatial range, which relates to the size of the smallest velocity structure that can be resolved in the flow field.

The dynamic spatial range is defined as the field of view in the object space, divided by the smallest resolvable spatial variation (Adrian, 1997). This range coin-cides with the number of independent vectors obtained from the interrogation analy-sis (without overlapping). The smallest-length scale that can be resolved is given by

λminI pN d

M=

(4.2)

where Lo is the physical dimension of the field of view in the x direction, LI is the corresponding pixel dimension of the camera, N is the number of interrogation areas, and dp is the pixel pitch of the CCD array.

For a 32 × 32 pixel interrogation area, each flow field is resolved to a factor of approximately 32 in the field of view. This dynamic spatial range would be low for turbulence measurements. A decrease in the resolved length scale, λmin, would require the reduction of the view area size to a fixed number of interrogation cells. Higher resolution can be achieved by a higher magnification of the measurement area, such as extension rings between the lens and the camera. However, higher mag-nification of the image may lead to higher velocity bias errors. Better modifications include a higher resolution CCD (higher number of pixels) or higher format record-ing media with physical dimensions on the order of 1 cm. The dynamic range is the ratio of the maximum velocity to the minimum velocity resolvable by a particular PIV system. The minimum resolvable velocity occurs in the order of the root mean square (rms) error, when determining the displacement of the particle image.

4.2.2 t eMper at u r eFiel d Mea su r eMen t

Spatial variations of temperature within the flow field are needed to determine the thermal irreversibilities of entropy production, and they can be determined from PLIF. In PLIF, molecules and atoms of a fluorescent dye are excited to a higher electronic energy state, by pulsed laser absorption and fluorescence. The local fluo-rescence intensity, I, varies with intensity of excitation light, Ie; concentration of the

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fluorescent dye, C; quantum efficiency as a function of temperature, f; and the molar absorptivity, e, as follows:

I fI C Te= ε φ( ) (4.3)

where f is a factor corresponding to the optical setup. For a known concentration and excitation energy, the quantum energy decreases at higher temperatures. This depen-dence constitutes the basis for PLIF temperature measurements. The temperature is determined as follows:

T TI

fI Crefe

− = DDε φ (4.4)

Thus, quantitative analysis is based on temperature calibration images that correlate the variation of intensity of the image with the local temperature and laser energy.

The first step in the PLIF calibration procedure is to find the optimum concentra-tion resulting in the maximum temperature resolution with low absorption phenom-ena. The corresponding absorption, A, can be calculated from

A e l CRhod= − η (4.5)

where hRhod is the extinction coefficient of Rhodamine B in water and l is the optical path length. The experimental procedure would involve running a series of trials at a fixed energy level to determine the optimum concentration at which the temperature resolution is maximum, while maintaining linearity between the gray level and tem-perature. The measurement precision of a particular concentration value is indicated by the slope of the curve obtained in the preliminary experiment. Typically, the temperature resolution approaches an asymptotic minimum at an optimum concen-tration, and then it increases thereafter.

Signal processing consists of a final translation of the recorded images to tem-peratures via the calibration maps. The final calibration relates the response of every pixel of the CCD camera to varying temperature, laser energy levels, and concen-tration. The temperature at discrete locations in an actual measurement region is determined from

T TI I

refref− =

−β

(4.6)

where Iref is the intensity of the fluorescent signal at the reference temperature, Tref . The denominator is statistically determined during calibration. The wavelength of the fluorescence emitted from PLIF is longer than the wavelength of the reflected laser light, thereby making simultaneous measurements of both velocity and tem-perature possible. An optical filter can be attached to the front of the camera for the fluorescent image to cut off reflected light from the PIV particles.

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The camera and image capturing systems detect particle images and fluorescent images successively at two different instants. Postprocessing of the velocity and tem-perature measurements will allow the estimation of entropy production. In addi-tion, the temperatures can be resampled with spatial resolution of the PIV vectors, ensuring maximum correlation between the thermal and friction irreversibilities. Experimental correlations between velocity and temperature will provide useful data for turbulent entropy transport modeling in upcoming chapters.

4.2.3 po st pr o c ess in g Fo r en t r o pypr o d u c t io n Mea su r eMen t

Unlike velocity or temperature, the measurement of entropy cannot be performed directly. But the entropy production equation can be used in an indirect way to characterize the flow irreversibility. The measured velocities and temperatures are extracted over a discrete grid in the PIV software. The velocity and temperature fluids at grid position (i,j) are denoted by u(i,j), v(i,j), and T(i,j), respectively. From Section 3.2, a positive definite expression for entropy production rate was derived in terms of a sum of squared terms representing the frictional irreversibility (viscous dissipation) and thermal irreversibility (due to heat transfer). Discretizing that result for two-dimensional flows yields the following expression for entropy production in terms of measured velocity and temperature gradients, centered about the point (i,j):

sPk

T i j

T i j T i j

x

k

T i =

,+ , − − ,

D

+

( )( ) ( )

(2

21 1,,

, + − , −D

+,

j

T i j T i j

y

T i j

u i

)( ) ( )

( )(

2

2

2

1 1

µ ,, + − , −D

+ + , − − ,D

j u i j

y

v i j v i j

x

1 1 1 1) ( ) ( ) ( )

++,

+ , − − ,D

+

,2

1 1 2 2µ

T i j

u i j u i j

x

v i j

( )( ) ( ) ( ++ − , −

D

1 1) ( )v i j

y

(4.7)

where Dx and Dy refer to the grid spacing in the x and y directions.When calculating the previous derivatives of velocity, errors can occur as fol-

lows: (1) bias error associated with the displacement measurement, and (2) a propa-gated uncertainty due to spatial differentiation of the velocity field. For a smaller grid size, the bias error decreases. The bias error associated with the fast Fourier transform-based cross-correlation algorithm in commercial PIV software can been minimized by a subpixel resolution of the PIV images. The entropy production algo-rithm contains multiple products of velocity derivatives. Hence, it is imperative to minimize the error associated with the determination of spatial derivatives. Two approaches can be taken in this regard. A twice-differentiable empirical function could be fitted to the data. The spatial derivative is then obtained directly through the differential of the empirical function. This approach requires an elaborate, often

7262_C004.indd 99 1/19/08 11:41:47 AM


difficult, interpolation routine for multidimensional output of PIV. Another approach is a local piecewise smoothing of the experimental data followed by the applica-tion of forward differences, central differences, or a Richardson central difference scheme over an adaptive window to calculate the derivatives.

Smoothing or filtering of experimental data reduces the noise in terms of experi-mental scatter, and it performs a least-squares approximation through a path that minimizes error for all data points in the field. In commercial PIV software, an average filter can usually be implemented in the form of a top-hat Gaussian filter with uniform weighting. The size of vectors in the neighborhood of a position (i,j) is specified by odd numbers, m and n. The filter calculates an average of vectors in a rectangular domain of size m × n surrounding a vector. The average value is substi-tuted for all entries in the initial matrix. The average can then be calculated by the following formula:

u x ymn

u ii x

x

j y

y

n

n

n

n

( ) (, == −

+

= −

+

−

−

−

−

∑ ∑1

12

12

12

12

,, j)

(4.8)

In addition to the average filter, a spline fit based on a second-order polynomial least-squares algorithm can also be used for data smoothing. Smoothing algorithms mitigate against error in the calculation of derivatives and resulting entropy produc-tion. They provide better approximations to an actual flow loss distribution. The interpolation of smooth curves or surfaces should be limited to flow structures pres-ent in the raw data from which they were obtained.

4.3 c a sEst u d y o f mag n Et Ic st Ir r In g ta n k dEsIg n

This section applies the previous techniques to a case study involving measured entropy production of fluid mixing induced by a magnetic stirrer in a cuvette cube. In this case study, a PIV camera views a magnetic stirrer commonly used in chemi-cal processing laboratories (see Figure 4.2). The rotational speed of the stirrer is

Camera

Illuminated Planeof Measurement

Light Sheet

Magnetic Stirrer

f Ig u r E4.2 Schematic of magnetic stirrer.

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90 r/min, which allows the camera, running at 30 Hz, to resolve each rotation with 20 image frames. Other problem parameters are summarized as follows:

Cube side length: 60 mmCamera: 30 HzLaser: Double-pulsed Nd:YAG laser at 10 mJ per pulseLight sheet entering the cuvette that is approximately 5 mm below the free surfaceSeeding: 50-mm polyamide particlesBackground: Ambient light used to capture the magnet stirrer in the images

Using Dantec Flow Map software, Figure 4.3 illustrates the measured velocity field within the plane of the light sheet used to illuminate the particles. Based on this velocity field, entropy production rates are determined and plotted in Figure 4.4. The regions of high mixing yield the highest rates of entropy production. In this example, the practical application of a magnetic stirrer involves mixing of chemicals to pro-vide uniform mixtures. As a result, uniformly distributed magnitudes of entropy production would be desired to maximize mixing, rather than minimal entropy pro-duction in other applications like fluid machinery or power generation. Based on the

••••

••

f Ig u r E4.3 Measured velocity field.

7262_C004.indd 101 1/19/08 11:41:52 AM


measured entropy production rates, the impeller could be redesigned to extend the diffusive effects induced by mixing.

By summing the local entropy production measurements, the results provide a useful basis from which the energy efficiency of fluids engineering devices can be effectively characterized. Using the First Law of Thermodynamics, the thermal efficiency of a heat exchanger is defined differently from a water heater’s efficiency, and still different from a diffuser’s efficiency (in terms of pressure), and so on. Due to such inconsistencies, difficulty arises when trying to establish a standard way of identifying a device’s energy wastefulness. Unlike methods based on the First Law, local or summed entropy production rates can provide a single, measurable quantity that is directly related to the efficiency of any energy-consuming or energy-producing device. The magnetic stirrer example in this section represents a single application where entropy production measurements can provide useful insight for design purposes. The practical utility of the method can be extended to numerous other applications, such as aerospace, automotive, power generation, turbomachin-ery, sprays, combustion, indoor ventilation, processing industries, and others.

f Ig u r E4.4 Surface profiles of entropy production.

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4.4 c a sEst u d y o f nat u r a l co n v Ec t Io n In cav It IEs

In this second case study, the measurement technique will be applied to natural convection in an enclosure. Free convection in enclosures occurs in various practical applications. Some examples include cooling of microelectronic assemblies, heat transfer between panes of glass in double-pane windows, solar collectors, and gas-filled cavities surrounding a nuclear reactor core. Although the physical processes of free convection have been widely documented in the literature, fewer studies have considered the related significance of entropy and the Second Low. For example, con-vective cooling within a microelectronic assembly entails free convection, whereas pressure losses occur with forced convection of air past internal components. In this instance, each unit of entropy produced leads to a corresponding unit of heat flow which is desired to be removed, but is not removed due to entropy production. This entropy production leads to pressure losses when kinetic energy is dissipated to internal energy, which works against the desired objective of component cooling.

Consider two-dimensional free convection within a square enclosure. The experi-mental setup involves PLIF for measuring temperatures within the test cell, as well as PIV for velocity measurements. An experimental study was conducted by Adeyinka and Naterer (2005) to measure entropy production in a 39 × 29-mm test cell. The cell depth of 59 mm was designed to minimize three-dimensional variations of thermal and flow fields along the plane of symmetry. Values of temperature at discrete locations in the measurement domain were obtained from the method of PLIF. In the commercial PLIF software, statistical averages are available to establish whole-field statistics of the LIF data. Further details regarding the experimental setup are described by Adeyinka and Naterer (2005).

The cavity is illuminated from above at the vertical plane of symmetry by an Nd:YAG pulsed laser. A CCD camera captures the sequence of image maps. The temperatures are recorded after their steady-state conditions are reached in both velocity and temperature fields. The Rayleigh number is controlled by adjusting fluid temperatures into the aluminum heat exchanger side walls. The PIV images are postprocessed by a fast Fourier transform based on a cross-correlation scheme in the Dantec Flow Map software. The PLIF images are resampled by a calibration map with a spatial resolution corresponding to the velocity map. As discussed in previous sections, the measured velocity vectors are displayed by the PIV software over a dis-crete grid. Using the velocity measurements and PLIF temperature measurements, the conversion algorithm for determining entropy production is then applied.

The PLIF measurements are used for temperatures in the expression for entropy production in Equation 4.7. For this buoyancy-driven problem, the tem-perature field varies spatially, thereby affecting the frictional entropy production in Equation 4.7. The nonintrusive method of PIV is used for whole-field measure-ments of velocity, which are then postprocessed by spatial differencing to yield local rates of entropy generation. In the experimental studies, the working fluid was water (Pr = 8.06). The left hot and right cold walls were maintained at 20 and 10 oC, respec-tively, thereby yielding a Rayleigh number of 5.35 × 106. The measured velocities indicate that a single clockwise recirculation cell developed with highest velocities near the side walls. The fluid velocities diminish rapidly at locations farther from

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the wall. The velocities become too small for PIV vectors to be displayed in the central region of the cavity. The U-velocity results and V-velocity results along the vertical and horizontal midplanes, respectively, were also obtained. In each case, the velocities were nondimensionalized with respect to the maximum velocity, while the spatial coordinate was nondimensionalized with respect to the cavity width.

Close agreement between predicted and measured results was established. The measured velocity field is slightly skewed to the right side of the cavity, so some discrepancy between predicted and measured results was observed near the right wall. The numerical simulation assumes a perfectly insulated boundary on both horizontal walls of the cavity, which leads to complete symmetry without skewing of the velocity field. The experimental apparatus closely approaches this idealization, but any slight heat gains through the horizontal boundaries could potentially lead to asymmetry of the buoyancy-driven flow. Velocity measurements were obtained within 1 mm from the wall. In view of their importance in subsequent spatial dif-ferencing for entropy production at the wall, additional measurements were obtained by resolving the velocity field closer to the wall.

Surface plots of U-velocity values across the entire cavity were also obtained. The maximum horizontal velocity occurs near the top corner of the cold wall. Unlike fluid flow of air at Pr = 0.71, where the maximum U-velocity is closer to the hot wall in the top corner of the cavity, the predicted and measured results for water (Pr = 8.06) exhibit a maximum magnitude closer to the top corner of the cold wall. Buoyancy-induced acceleration of fluid up the hot wall leads to an adverse pressure gradient and velocity change, when the fluid is redirected horizontally near that corner. This momentum exchange involves a balance between fluid inertia and forces imparted by pressure, friction, and fluid buoyancy. The frictional resistance of the fluid along the wall increases, when the momentum diffusion rate exceeds the rate of heat dif-fusion (Pr > 1). This affects the overall momentum balance on the fluid, thereby altering pressure gradients near the top corners of the cavity and changing the trends of maximum fluid velocity for air (Pr < 1) and water (Pr > 1). Also, the distance of this maximum velocity point from the wall changes at different Prandtl numbers. Similarity solutions of free convection along a vertical wall confirm that the point of maximum velocity moves closer to the wall at higher Prandtl numbers (Naterer, 2002).

Postprocessing of the measured velocity results yields the spatial variation of entropy production throughout the cavity. The peak values occur at the vertical walls, corresponding to the locations of largest spatial gradients of velocity. Away from these points, entropy production decreases sharply to approximately zero close to the wall, which corresponds to the local maximum and zero gradient of V-velocity near the wall. Beyond this local maximum of velocity, entropy production increases to a local maximum and decreases back to nearly zero in the central region of the enclosure. The illustrated results have been normalized, with respect to a reference entropy production, Ps (ref), at the local maximum. The entropy production reaches a minimum value in the center of the cavity, where the stagnation point of the recir-culation cell is observed.

Near-wall measurements of V-velocity and entropy production in the midregion of the cavity at the cold wall were also obtained. The measured maximum U and V

7262_C004.indd 104 1/19/08 11:41:54 AM


components of velocity were 0.611 and 1.69 mm/s, respectively, for this particular study. The predicted maximum U and V components of velocity are 0.632 and 1.89 mm/s, respectively. The close agreement between predicted and measured velocities near the wall are important because near-wall spatial gradients of velocity are needed for the entropy production calculations. Although PIV technology is limited by camera resolu-tion and particle tracing of small-scale structures near the wall, the current experimental study successfully measured velocity and derived entropy production at very close prox-imity to the wall. A resolution of 0.2 mm was achieved in the wall region, which pro-vided good near-wall accuracy that becomes particularly important for turbulent flows.

Measured oscillations of entropy production can be reduced through filtering of the velocity data. In the experimental study, a 3 × 3 average filter was used for smoothing of the raw velocity vectors, before calculating the entropy production. Previous PIV studies (Luff et al., 1999) have shown that filtering does not introduce additional error into the measured velocity, but it serves to mitigate uncertainty by averaging velocities at surrounding grid points. The measured results illustrate the benefit of filtering, particularly for the near-wall raw data points and removing ran-dom uncertainty in the measured velocity gradients. This measurement procedure for entropy production provides a useful diagnostic tool for identifying the local flow losses, so that energy conversion devices can be redesigned locally around regions of highest entropy production.

4.5 mEa su r EmEn t un c Er t a In t IEs

4.5.1 Bia sa n d pr ec is io n er r o r s

Uncertainty analysis involves systematic procedures of calculating error estimates for experimental data (Coleman and Steele, 1995). Measurement errors of entropy produc-tion arise from various sources. They can be broadly classified as bias errors and pre-cision (or random) errors. Bias errors remain constant during a set of measurements. They are often estimated from calibration procedures or past experience. This section will assess both bias and precision errors in the entropy production measurements.

Elemental bias errors arise from calibration procedures or curve-fitting of cali-brated data. Also, “fossilized” bias errors arise when measuring and tabulating ther-mophysical properties. Although such errors are usually less than ±1%, Coleman and Steele (1989) describe cases involving higher levels of fossilized bias errors. Moffat (1988) defines a “conceptual bias,” which includes a residual uncertainty due to vari-ability arising in the true definition of the measured variable. For example, if point measurements are used to approximate bulk temperatures at the inlet and exit of a duct, then the difference between these temperatures and the bulk mean temperature contributes to a conceptual bias error, because point measurements cannot fully cap-ture the spatially averaged bulk value.

In contrast to bias errors, precision errors appear through scattering of mea-sured data. Such errors are affected by the measurement system (i.e., repeatability, resolution) or spatial and temporal variations of the measured quantity. Also, the procedure itself may lead to precision errors arising from variations in operating conditions. If an error can be estimated statistically, then it is usually considered to

7262_C004.indd 105 1/19/08 11:41:55 AM


be a precision error. Otherwise, it is generally assumed to be a bias error. Anticipated precision errors are often used to guide experimental designs and procedures, in view of collecting data within a desired range of measurement uncertainty.

Gui et al. (2001) outline precision errors and other PIV measurement uncertainties in a towing tank experiment. Precision errors are reduced by increasing the number of measurement samples. Alekseeva and Navon (2002) found temperature uncertain-ties based on first- and second-order adjoint equations. An adjoint formulation of an inverse heat transfer problem leads to uncertainty indicators for the correspond-ing direct problem. Hessian maximum eigenvalues from the second-order adjoint equations can be used to evaluate the uncertainty indicators (Alekseeva and Navon, 2002). Pelletier et al. (2003) show how sensitivity equations provide key information regarding which parameters most affect the flow response. Measurement uncertain-ties of flow parameters depending on input data errors (such as initial and boundary conditions) can be effectively calculated with adjoint equations. Alekseeva and Navon (2003) use adjoint temperatures to calculate the transfer of uncertainties from such input data. Propagated uncertainties (Kline and McClintock, 1953) are often classified according to zero-order or higher-order uncertainties. In the former case, all parameters affecting the measurements are assumed to be fixed, except for the procedure of the experiment. Thus, data scattering arises from instrumentation reso-lution alone. In the latter case (higher-order uncertainty), control of the experimental operating conditions is considered, so factors such as time are included. The degree of variability of operating conditions can be expressed by the standard deviation.

Measurement uncertainties of primary variables (such as fluid velocity) with various experimental techniques have been widely reported previously, i.e., Kline (1985), Lassahn (1985), Moffat (1982), and others. Postprocessing of measured data, such as measured vorticity from postprocessed PIV data, entails additional uncer-tainties in the conversion algorithm. Conventional error indicators (AIAA Standard, 1995) can be extended to the scalar variable of entropy production. In this case, bias errors must be specifically correlated with sensitivity coefficients of the mea-sured entropy production. Equation 4.7 expressed the measured entropy production as a postprocessed variable. Before assessing the experimental uncertainties in this method, the first step is assessing the uncertainties of measured velocities.

4.5.2 Vel o c it yFiel d un c er t a in t iesin ch a n n el Fl o w

PIV incurs certain errors from statistical correlations in the interrogation areas when determining the fluid velocities. For example, consider the problem of laminar chan-nel flow where the average fluid velocity, U, for an interrogation area at any instant is measured by the following equation:

UsL

tLo

I

= DD

(4.9)

where Dt is the time interval between laser pulses, Ds is the particle displacement from the correlation algorithm, Lo is the width of the camera view in the object plane, and LI is the width of the digital image.

7262_C004.indd 106 1/19/08 11:41:56 AM


The total error, ε, in a measured quantity is a sum of the bias component, B, and a precision component, P. The bias error of the measured velocity is related to the elementary bias errors based on the sensitivity coefficients, that is,

B B B B Bu s s t t Lo Lo L LI I

2 2 2 2 2 2 2 2 2= + + +η η η ηD D D D (4.10)

where the sensitivity coefficients with respect to an arbitrary variable, c , is given by

η χχ = ∂ ∂u / (4.11)

The elementary bias limits (t, Ds) are usually specified by the manufacturer. The width of the camera view in the object plane, Lo, depends on distances and configu-rations related to the experimental setup, so the bias limit for Lo is determined from calibration procedures, not manufacturer’s specifications. In this calibration, the physical dimensions and spatial resolution of the camera view in the measurement plane are determined. Then the width of the digital image can be determined by the number of pixels corresponding to these dimensions. The width of the camera view in the object plane and bias limit for Lo are determined. Then, the uncertainty associ-ated with this bias limit can be reduced with a more refined procedure for measuring Lo. The PIV image pairs are cross-correlated with an interrogation window, which yields a value of Ds in the centerline. Combining the contributions of each bias error and the sensitivity coefficient, the velocity error can then be determined.

The precision error (P) of an average value, χ , measured from N samples is given by

P

t

N= σ (4.12)

where t is the confidence coefficient, t equals 2 for a 95% confidence level, and s is the standard deviation of the sample of N images. The standard deviation is defined as follows:

σ χ χ=−

−=∑1

11

2

Nk

N

k( ) (4.13)

where the average quantity is defined by the following equation:

χ χ==∑1

1N

k

N

k (4.14)

Typical values of the standard deviation along the centerline and the near-wall region of the channel can be determined from the procedure, thereby yielding the precision limits and resulting total uncertainty of the measured velocity.

7262_C004.indd 107 1/19/08 11:42:01 AM


4.5.3 Mea su r eMen t un c er t a in t ieso Fen t r o pypr o d u c t io n

Based on the previous velocity results, the errors of measured entropy production can be estimated. A data reduction equation for entropy production of laminar chan-nel flow is approximated by

sP

T

u

y

k

T

T

y = D

D+ D

D

µ2 2

(4.15)

The total uncertainties (B + P) for the U (velocity), T (temperature), and y (position) variables are

U U

T T

y y

i i U

i i T

i i y

i

i

i

= ±= ±= ±

εεε

The uncertainty in DU is obtained as follows:

ε θ ε θ εD , + , + , − , −= ± ′ + ′u u i u i u i u i( ) ( )1 1

21 1

2 (4.16)

where

u ii

u

u, −′ = ∂ D

∂1θ( )

(4.17)

Note that q¢u,i = -1 = -1 and q¢u,i = -1 = 1 or vice versa. The uncertainty of DT is calculated in the same manner as Equation 4.16 and Equation 4.17, except that the velocity component, U, is replaced by temperature, T. Similarly,

ε θ ε θ εD , + , + , − , −= ± ′ + ′y y i y i y i y i( ) ( )1 1

21 1

2 (4.18)

where

y ii

y

y, −′ = ∂ D

∂1θ( )

(4.19)

Neglecting the error in reported thermophysical properties, the data reduction equa-tion for entropy production leads to

ε η ε η ε η ε η εP T T u u y y T Ts

2 2 2 2 2 2 2 2 2= + + +D D D D D D (4.20)

Based on this equation and the previous procedure of calculating individual uncer-tainties, the experimental uncertainty of entropy production can be determined. The measured uncertainties represent a maximum error bound within the 95% confi-dence interval.

7262_C004.indd 108 1/19/08 11:42:06 AM


4.5.4 en t r o pypr o d u c t io n o FFr eeco n Vec t io n in caVit ies

Consider another example involving measurement uncertainties of free convection experiments described in Section 4.4. A similar procedure is adopted for the bias and precision errors. However, certain differences exist due to variations of temperature within the enclosure. Unlike the previous channel flow problem, irreversibilities in this problem vary spatially due to both velocity and temperature variations across the flow field. For this problem, the bias error of the measured velocity is related to the elementary bias errors and sensitivity coefficients as follows:

B B B B Bu s s t t L L L Lo o I I

2 2 2 2 2 2 2 2 2= + + +D D D Dη η η η (4.21)

where the same definition of sensitivity coefficients is used, i.e., η χχ = ∂ / ∂U . By com-bining the contributions from each source of bias and the sensitivity coefficient, a full-scale velocity bias error is obtained. Similarly as described previously, the precision error (P) of an average value, χ , is measured from N samples. The data reduction equation for friction irreversibility of entropy production in this problem then becomes

PT

u

y

v

x

u

xsy x x=

DD

+ DD

+

DD

µ2 2 2

++ DD

2v

yx

(4.22)

The same definitions are applied from the previous problem, including the total uncertainties for the U, T, y, DU, and Dy variables. The total uncertainty of entropy production becomes

ε η ε η ε η ε η εP T T U U V V y ys

2 2 2 2 2 2 2 2 2= + + +D D D D D D (4.23)

Then, the total uncertainty of measured entropy production can be determined. The reader is referred to past studies by Adeyinka and Naterer (1995), which provide detailed examples of measurement uncertainties of entropy production in various applications.

r Ef Er En c Es

Adeyinka, O.B. and G.F. Naterer. 2005. Particle image velocimetry based measurement of entropy production with free convection heat transfer. ASME J. Heat Transfer, 127(6): 615–624.

Adrian, R.J. 1991. Particle-imaging techniques for experimental fluid mechanics. Annu. Rev. Fluid Mechanics, 23: 261–304.

Adrian, R.J. 1997. Dynamic ranges of velocity and spatial resolution of particle image velo-cimetry. Measurement Sci. Technol., 8: 1393–1398.

AIAA-Standard-S017-1995, Assessment of Experimental Uncertainty with Application to Wind Tunnel Testing. American Institute of Aeronautics and Astronautics. Washington, D.C.

Alekseeva, A.K. and M.I. Navon. 2002. On estimation of temperature uncertainty using the second order adjoint algorithm. Int. J. Computational Fluid Dynamics, 16(2): 113–117.

7262_C004.indd 109 1/19/08 11:42:10 AM


Alekseeva, A.K. and M.I. Navon. 2003. Calculation of uncertainty propagation using adjoint equations. Int. J. Computational Fluid Dynamics, 17(4): 283–288.

Coleman, H.W. and W.G. Steele. 1989. Experimentation and Uncertainty Analysis for Engi-neers. John Wiley & Sons, New York.

Coleman, H.W. and W.G. Steele. 1995. Engineering application of experimental uncertainty analysis. AIAA J., 33: 1888–1896.

FlowMap Particle Image Velocimetry Instrumentation: Installation and User Guide. Dantec Dynamics, Demark, 1998.

Gui, L., Longo, J., and F. Stern. 2001. Towing tank PIV measurement system, data and uncer-tainty assessment for DTMB Model 5512. Exp. Fluids, 31: 336–346.

Kline, S.J. 1985. The purpose of uncertainty analysis. ASME J. Fluids Eng., 107: 153–160.Kline, S.J. and F.A. McClintock. 1953. Describing uncertainties in single-sample experi-

ments. Mechanical Eng., 75: 3–8.Lassahn, G.D. 1985. Uncertainty definition. ASME J. Fluids Eng., 107: 179.Luff, J.D., Drouillard, T., Rompage, A.M., Linne, M.A., and J.R. Hertzberg. 1999. Experi-

mental uncertainties associated with particle image velocimetry (PIV) based vortic-ity algorithms. Exp. Fluids, 26: 36–54.

Moffat, R.J. 1982. Contributions to the theory of single-sample uncertainty analysis. ASME J. Fluids Eng., 104: 250–260.

Moffat, R.J. 1988. Describing the uncertainties in experimental results. Exp. Thermal Fluid Sci., 1: 3–17.

Naterer, G.F. 2002. Heat Transfer in Single and Multiphase Systems. CRC Press, Boca Raton, FL.

Pelletier, D., Turgeon, E., Lacasse, D., and J. Borggaard. 2003. Adaptivity, sensitivity and uncertainty: Toward standards of good practice in computational fluid dynamics. AIAA J., 41(10): 1925–1933.

Rosen, M.A. and I. Dincer. 2003. Exergoeconomic analysis of power plants operating on various fuels. Appl. Thermal Eng., 23(6): 643–658.

Westerweel, J. 1997. Fundamentals of digital particle image velocimetry. Measurement Sci. Technol., 8: 1379–1392.

Willert, C.E. and M. Gharib. 1991. Digital particle image velocimetry. Exp. Fluids, 10: 181–193.

7262_C004.indd 110 1/19/08 11:42:11 AM

111

5 Entropy Production in Microfluidic Systems


Microfluidic irreversibilities of friction, heat transfer, and electrodynamic transport have significance in the achievement of various technological goals involving micro- and nanoenergy systems, for biodevices, micropower sources, and other applications (Gad-el-Hak, 1999). Microelectromechanical systems (MEMS) have promising applications to aerodynamics, drag reduction, and slow control. For example, embed-ded surface microchannels can take advantage of local slip-flow behavior to reduce wall friction and entropy production of external flows (Naterer, 2004). In these appli-cations, pressure losses arising from flow irreversibilities affect the power consump-tion and performance of microsystems. This chapter examines how entropy and the Second Law have importance in the design and optimization of microdevices.

Fluid flow through microchannels has been studied extensively by many authors (Cho et al., 2001; Ng and Tan, 2004; Zhao et al., 2001), including experimental, numerical, and theoretical studies. Ng and Tan (2004) developed a three-dimen-sional finite volume model of fluid motion within rectangular microchannels based on the Navier–Stokes equations, including an electric double layer (EDL) along the walls. Electromagnetic effects of the EDL can be modeled as a type of body force and source term in the momentum equation. Cho et al. (2003) developed a condition for electrowetting on dielectric (EWOD) in microfluidic motion through parallel-plate channels.

In past studies, some conflicting opinions have arisen with regard to the appar-ent viscosity of fluids in microchannels. This debate involves whether the apparent (required or measured) viscosity of a microchannel flow equals the bulk viscosity at large distances away from the wall. For thin films, Israelachvili (1986) reported values of apparent viscosity that were much larger than the bulk viscosity. Migun and Prokhorenko (1987) reported that the apparent viscosity increases for capillar-ies smaller than a micron in diameter. However, other researchers (Anderson and Quinn, 1972) have reported that the apparent and bulk viscosities are nearly equal for flows in capillaries.

The bulk viscosity is generally determined from classical thermodynamics, whereby curve fits of measured data to interpolation polynomials are used to estimate the variations of viscosity with temperature and pressure. Using meth-ods of statistical thermodynamics, a velocity or temperature distribution func-tion can be used to include the effects of intermolecular interactions (Ferziger and Kaper, 1972). Avsec (2003) applied statistical methods to include translational, rotational, vibrational, and electron excitation effects on property evaluation

7262_C005.indd 111 1/19/08 11:44:13 AM


for liquids. Gad-el-Hak (1999) included a Knudsen number (Kn) dependence in the velocity distribution function.

Pfahler et al. (1990, 1991) measured the friction factor of liquids (isopro-pyl alcohol, silicon oil) and gases (nitrogen, helium) flowing through microchan-nels etched in silicon. The liquid flow rate was measured as a function of channel size, pressure drop, and type of fluid. It was observed that the fluid’s apparent vis-cosity was smaller than values predicted for macroscale flows. Although some speculation regarding rarefaction and thermal effects were noted, the authors “do not have a satisfactory theoretical explanation for the phenomena observed” (Pfahler et al., 1991). A trend of decreasing friction factor at lower Reynolds numbers was observed, although macroscale theory predicts a constant friction fac-tor for laminar flow.

Entropy production includes both frictional and thermal irreversibilities, which lead to pressure losses in microchannel flows. Additional irreversibilities of phase transition, electromagnetic transport, and radiative heat transfer have been reported previously (Bejan, 1996; Naterer, 2001). Camberos (2003) predicted the electro-magnetic irreversibilities in compressible flows with computational fluid dynamics (CFD) applications of improved aircraft design. This chapter develops models of entropy production including such irreversibilities, but focuses on applications to microdevices. The main objectives of this chapter involve showing how entropy and the Second Low provide key insight regarding fluid friction, pressure losses, and energy conversion in microdevices.

5.2 Pr essu r e-d r Iv en Fl o w In MIc r o c h a n n el s

5.2.1 Co n t in u u mEq u at io ns a n d sl ipBo u n d a r yCo n d it io ns

Microchannel flows can be subdivided into different flow regimes. Depending on the Knudsen number (Kn), different methods of CFD are needed. The ratio of the mean free path of the fluid to a characteristic length scale of the problem is called the Knudsen number. The flow regimes include the continuum flow 0 10 3≤ ≤( )-Kn , slip flow 10 103 1- -≤ ≤( )Kn , transition flow 10 101 1- -≤ ≤( )Kn , and free-molecule regimes 101 ≤ ≤ ∞( )Kn . At 10-3 < Kn < 10-1, the flow lies within the slip-flow regime where the continuum-based equations (Navier–Stokes equations) and a slip bound-ary condition are applied. In this regime of fluid motion, a linear relation between applied stress and strain rate is used for Newtonian fluids.

Consider the following Navier–Stokes equations for gas flow in the slip-flow continuum regime,

∂∂+ ∂∂+ ∂∂=( ) ( ) ( )ρ ρ ρ

t

u

x

v

y0

(5.1)

∂∂+ ∇ ⋅ = - ∂

∂+ ∇ ⋅ ∇( )

( ) ( )ρ ρ µu

tvu

p

xu

(5.2)

∂∂+ ∇ ⋅ = - ∂

∂+ ∇ ⋅ ∇( )

( ) ( )ρ ρ µv

tvv

p

yv

(5.3)

7262_C005.indd 112 1/19/08 11:44:19 AM

Entropy Production in Microfluidic Systems 113

subject to slip boundary conditions at the walls of the microchannel. The symbols, ρ and µ denote the density and dynamic viscosity of the fluid, respectively. These properties will be assumed uniform throughout the channel. To calculate the tem-perature field,

p RT= ρ (5.4)

which represents the ideal gas equation of state.Maxwell’s first-order slip velocity (Maxwell, 1879) will be used for boundary con-

ditions at the walls of the microchannel. This boundary model incorporates two coef-ficients involving velocity and temperature gradients at the wall, i.e.,

u uu

y

T

xgas wall

wall wall

- = ∂∂

+ ∂∂

ξ ξ1 2

(5.5)

where ξσσ

ξ µρ

λ1 1

2 3

4= - =λ and

Tgas. It is unusual to observe streamwise temperature

gradients affecting the slip velocity at the wall. A higher component of thermally induced wall slip occurs at smaller gas densities, as the internal energy of gas mol-ecules has greater impact on a near-wall region of intermolecular interactions when fewer molecules occupy the region. These mechanisms of near-wall energy exchange can be characterized through the entropy production rate. The frictional dissipa-tion of kinetic energy to internal energy, including near-wall friction associated with velocity slip, occurs at a rate given by the entropy production rate. When this entropy production rate is multiplied by temperature, it becomes the local rate of exergy destruction, ′′′Xd , per unit volume.

Using the Gibbs equation, it can be shown that the rate of exergy destruction for near-isothermal microchannel flows can be written directly in terms of the velocity gradients as follows:

Xu

y

v

x

u

x

v

yd′′′ =∂∂+ ∂∂

+ ∂

∂+ ∂∂

µ2 2

2

2

(5.6)

Thus, wall slip affects the velocity profile and resulting exergy destruction. The frictional dissipation of kinetic energy leads to pressure losses in microchannels, which depend on cross-stream velocity and streamwise temperature gradients, according to Equation 5.5.

5.2.2 Ca sEst u d yo f ExEr g ylo ssEsin Ch a n n El dEsig n

A numerical study of microfluidic entropy production was conducted by Ogedengbe et al. (2006), with a finite volume discretization of the continuum governing equations. In this section, numerical results of nitrogen gas flow through microchannels (from that study) are examined at varying mass flow rates, pressure ratios, Reynolds numbers, and channel aspect ratios. Figure 5.1 shows a schematic of the microchannel flow configu-ration, and Table 5.1 outlines the problem parameters and thermophysical properties

7262_C005.indd 113 1/19/08 11:44:24 AM


of the gas. Sample results of slip-flow velocity profiles are illustrated in Figure 5.2. It can be observed that the wall velocity increases at lower momentum accommodation coefficients, due to a higher resulting slip coefficient (note: x2 = 0 in Figure 5.2).

In Figure 5.3, the cross-stream profile of the entropy production rate (per unit volume) is illustrated at varying pressure ratios in the fully developed section of the channel. The pressure ratio refers to the inlet pressure divided by the outlet pressure. The area covered under each curve represents the entropy production per unit area at a specific location, x*. It can be observed that the entropy production rate rises when the pressure ratio falls from 2.7 to 1.34. The lowest entropy production occurs at the pressure ratio of 3.0. The diffusion layer grows faster at lower pressure gradients, thereby leading to higher entropy production. The minimum entropy production rate (per unit volume) occurs at the midpoint of the channel, due to the zero transverse velocity gradient at that position.

In Figure 5.4, the predicted entropy production rate (per unit area) and varying slip coefficients (ζ2) in the streamwise direction (x* = x/L) are illustrated. At each x* position, the entropy production is calculated based on an integrated profile across the channel. The inlet velocity profile is uniform, so a developing flow region leads to higher exergy destruction (per unit area) in the x-direction, when the diffusion layer propagates inward to the core of the microchannel. It was confirmed that the velocity profile reaches a fully developed condition upstream of the outlet. As a result, the entropy production rate (per unit area) reaches a peak value near the outlet and remains constant thereafter to the outlet. The pressure declines along the micro-channel, so temperature decreases according to the ideal gas law. As a result, the slip coefficient rises in the x-direction.

Uniform Inlet Velocity

Rectangular Microchannel

Fully-DevelopedVelocity Profile(Near Outlet)

Outlet

Wall Slip(Boundary Condition)

FIg u r e5.1 Schematic of microchannel flow problem.

t a bl e5.1ProblemParametersandFluidProperties

Length (μm) 1560, 2560, 3560, 4560, and 5560Height (μm) 1.0Dynamic viscosity (Ns/m2) 0.000016Gas constant (J/kg K) 296.8Outlet pressure (kPa) 100.8Pressure ratio (Pin / Pout) 1.34, 2.70, 3.00, and 3.34Reynolds number (Re) 0.001, 0.002, and 0.003

7262_C005.indd 114 1/19/08 11:44:26 AM


1.0

0.8

0.6

y/H

0.4

σ = 1.0

Kn = 0.0579Re = 9.9 × 10–4

Nitrogen (Pout = 100.8 kPa)Pin/Pout = 3.001/ε = 1,560

σ = 0.6

σ = 0.2

σ = 0.8

σ = 0.4

No-Slip

0.2

0.00.0 0.4 0.8

u/Uinlet

1.2 1.6

FIg u r e5.2 Predicted velocity profiles at varying slip coefficients.

0

100

200

300

400

500

600

0.0 0.2 0.4 0.6 0.8 1.0 y*

Entr

opy P

rodu

ctio

n Ra

te (p

er U

nit V

olum

e), W

/m3 K P(in)/P(out) = 1.34

P(in)/P(out) = 2.701

P(in)/P(out) = 3.00

P(in)/P(out) = 3.34

Nitrogen (Pout = 100.8 kPa) Kn = 0.0579 ε = 19 × 10–4

Re = 19 × 10–4

P s

FIg u r e5.3 Predicted entropy production rate (per unit volume) at varying pressure ratios.

7262_C005.indd 115 1/19/08 11:44:31 AM


The total entropy production rate within the microchannel is calculated based on the sum of individual rates from all control volumes within the domain. This includes control volumes in both developing and fully developed sections of the microchannel. In Figure 5.5, the total entropy production rate (per unit area) is illustrated at varying Reynolds numbers and channel aspect ratios, ε (note: fixed pressure ratio of 2.7).

6.1E – 05 0.0 0.2 0.4 0.6

Entropy Production Slip Coefficient

x* 0.8 1.0

1.18E – 08

1.19E – 08

1.19E – 08

1.20E – 08

1.20E – 08

1.21E – 08

1.21E – 08

6.1E – 05

6.1E – 05

6.1E – 05

Entr

opy P

rodu

ctio

n Ps

˝ (W

/m2 K)

6.1E – 05

6.1E – 05

6.1E – 05

6.1E – 05

6.2E – 05

6.2E – 05

6.2E – 05 Nitrogen (Pout = 100,800 Pa)Kn = 0.0579 ε = 19 × 10–4

Re = 19 × 10–4

Pin/Pout = 19 × 10–4

Velo

city

Slip

Coe

ffici

ent (

2)

FIg u r e5.4 Streamwise exergy destruction rate (per unit area) with a varying slip coefficient.

Re = 0.00099Re = 0.0019Re = 0.00282

Pin/Pout = 3.34

1,0000.E + 00

4.0E – 07

Entr

opy P

rodu

ctio

n Ra

te (p

er U

nit A

rea)

, W/m

2 K

6.0E – 07

1.2E – 06

16E – 06

2,000 3,0001/ε

4,000 5,000 6,000

P s

FIg u r e5.5 Changes of entropy production rate (per unit area) at varying Reynolds numbers.

7262_C005.indd 116 1/19/08 11:44:34 AM


The entropy production rate (per unit area) increases linearly with 1/ε, because the microchannel volume increases when 1/ε rises. Also, it increases at higher Reynolds numbers, because the mass flow increases and fluid friction rises. Because entropy production characterizes the frictional dissipation of kinetic energy and resulting pressure losses, the current results show that entropy production can serve as a key parameter to improve energy efficiency of microsystems.

5.3 a PPl Ied el ec t r Ic FIel d In MIc r o c h a n n el s

5.3.1 ir r Ev Er s iBil it iEsw it h a Co ns t a n t mag n Et iCfiEl d

Surface and electromagnetic forces are key differences that distinguish microchan-nel flows from fluid motion in large-scale channels. A common method of flow control in microchannels involves electromagnetic forces that are exerted on the fluid. Manipulating different charge patterns along the walls of a microchannel will affect the speed and direction of electrokinetic flow. A charged surface of a micro-channel can attract ions of the opposite charge in the surrounding fluid. The result-ing spatial gradient of ions can lead to an EDL. This EDL contains an immobile inner layer and an outer layer, which can be affected by an external electric field. The EDL reduces the liquid velocity and affects the frictional losses. For example, the friction coefficient increases when the ionic concentration of an aqueous solution decreases. In this section, spatial changes of the electromagnetic forces on the fluid will be considered when predicting entropy production of microchannel flows.

Consider a nonpolarized thermomagnetic field that is exerted on a steady-state fully developed flow in a rectangular microchannel (see Figure 5.6). The separation between the walls, 2b, is assumed to be much larger than the distance of 2a. An electromagnetic wave is polarized if the electric field vibrates in only one direction.

y

b

u(z)

Wall

Inner Layer(Immobile)

Diffusive Layer (Mobile)

Bulk Solution

Positive Ion Negative Ion Neutral Molecule

x z

a

E i

q B

FIg u r e5.6 Schematic of an applied electric field in a microchannel.

7262_C005.indd 117 1/19/08 11:44:37 AM


The electromagnetic waves transmitted through the microchannel are nonpolarized, when the applied electric field is assumed to vibrate in many directions, simulta-neously. Polarization is a phenomenon associated with transverse electromagnetic waves. Longitudinal waves, such as sound, are nonpolarized. Ordinary light is another example of nonpolarized electromagnetic waves, because the electric field vibrates in multiple directions at the same time. The nonpolarized waves traveling in the y-direction of the microchannel are a superposition of many waves. For each wave, the electric field is perpendicular to the y-axis, and the angle it makes with the x-axis varies for different waves. For polarized waves, the angle that the electric field makes with the x-axis would be unique.

The general form of the momentum equation for electrohydrodynamic flow is

ρ ρ µ∂∂+ ⋅∇ = -∇ + ∇ ⋅ ∇ + ×

v

tv v p v i B( ) (5.7)

where the last term represents the electromagnetic force. The variables i and B refer to the current density and magnetic field strength, respectively. The following reduced form of the momentum equation is approximated for steady-state microchannel flow at small Reynolds numbers (see Figure 5.6):

0 = -∇ + ∇ ⋅ ∇ + ×p v i B( )µ

(5.8)

Consider fluid velocity, magnetic field, and current density fields that are mutually orthogonal. The net force exerted by the magnetic field on the fluid is perpendicular to the direction of the fluid velocity. The applied electric field is nonpolarized, and the cross-product of the electromagnetic source term in Equation 5.8 is simplified to give the following reduced form of the x-momentum equation:

02

2= - + +dp

dx

d u

dzi By zµ (5.9)

The terms represent pressure, viscous, and electromagnetic forces on the liquid. Using Ohm’s law to express the current density in terms of fluid velocity yields

µ σd u

dzB u

dp

dxe z

2

22+ = (5.10)

where σ e and Bz refer to the electrical conductivity and magnetic field strength (z-direction), respectively. For fully developed flow in the microchannel, the pressure gradient becomes constant and independent of the magnetic field strength.

In terms of the Hartmann number, M (where M aBz e= σ µ/ ), Equation 5.10 becomes

µ µd u

dz

M

au

dp

dx

2

2

2

2-

=

(5.11)

7262_C005.indd 118 1/19/08 11:44:42 AM


Applying no-slip boundary conditions at z = a and z = -a (note: origin of coordinates at the midplane of Figure 5.6), the analytical solution of Equation 5.11 becomes

u za dp dx

M

Mz a

M( )

( / ) cosh( / )cosh

= -

2

21

µ

(5.12)

After nondimensionalizing the z-coordinate (z* = z/a) and velocity (u* = u/ub where ub refers to the mean velocity), it can be shown that

u

M M M MzM M M

**cosh cosh( )

cosh sinh= -

- (5.13)

Using a large Hartmann number approximation (LHA model), the velocity becomes

u eM z* ( )*= - -1 1 (5.14)

Using similar assumptions for the energy equation, the reduced form of the energy equation becomes

kd T

dz

du

dz

iy

e

2

2

2 2

0+ + =µσ

(5.15)

Solving this equation subject to the boundary conditions of a uniform wall heat flux, on

θ

µ( )

( ) ( ) (cosh* *

* *

zk T T

uq

C z C Mzw

bw= - =

- + -2

2 22 1 8 ccosh )/ cosh cosh

sinh sinh

*M M Mz M

C C M M

+ -+ +

2 2

4 8 22 22M

(5.16)

where C = M(K – 1) cosh M – K sinh M and K represent a nondimensional load factor (ratio of the applied electric field strength to the product of the mean velocity and magnetic field strength).

It can be readily verified that this result is symmetrical about the midplane of the microchannel. In the case of a constant wall temperature, the thermal boundary conditions are given by

θ( )± =1 0 (5.17)

Solving the internal energy equation subject to these boundary conditions, it can be shown that

θ =-

× - + -

M

M M M

Cz

C

MM

2

2

22

21

2

( cosh sinh )

( ) (cosh co* ssh *) (cosh cosh *)Mz M Mz+ -

14

2 2

(5.18)

This result shows that both the Hartmann number and load factor affect the tempera-ture profile.

7262_C005.indd 119 1/19/08 11:44:48 AM


Under the assumptions outlined previously, the corresponding reduced form of the exergy transport equation can be expressed as (Camberos, 2002):

DX

Dt zk

T

T

T

z

u

zXd= ∂

∂-∂∂

+ ∂∂-1 0 ( )τ

(5.19)

where DX/Dt refers to the total convective derivative of exergy, X. The rate of exergy destruction, Xd, can be represented by the reference temperature, T0, multi-plied by the local entropy production rate, Ps. Thus, entropy production is needed to calculate the exergy destroyed by friction and thermal and electromagnetic irreversibilities.

The entropy production rate can be expressed as a sum of positive-definite terms corresponding to friction, thermal, and electromagnetic irreversibilities, that is,

Pk T T

T T TE v B E v Bs

e= ∇ ⋅∇ + + + × ⋅ + ×2

µ σΦ( ) ( ) (5.20)

The terms on the right side represent a sum of squared terms, so the entropy pro-duction is positive, thereby complying with the Second Law of Thermodynamics. Based on the previous assumptions in the fluid flow and heat transfer formulations, it can be shown that the reduced form of the entropy production equation can be written as

Pk

T

dT

dz T

du

dz

i

Tsy

e

= +

+2

2 2 2µσ (5.21)

Define the following nondimensional entropy production, Ps*, and wall temperature:

P

P

k ass*

/=

2 (5.22)

θµw

w

b

kT

u=

2 (5.23)

Using these variables, the nondimensional entropy production equation becomes

Pd dz du dz i

sw w

y** * *( / )

( )

( / )=

++

+++

θθ θ θ θ θ θ

2

2

2 2

ww (5.24)

In an upcoming case study, a detailed analysis of this entropy production (particu-larly the electrohydrodynamic term) will be presented.

7262_C005.indd 120 1/19/08 11:44:54 AM


Substituting the velocity and temperature profiles, the nondimensional entropy production at the wall becomes

Pq M M

M M Ms ww

w,*

* sinhcosh sinh

= +-

2

2

2 21

θ θww w

M K+2 2

θ (5.25)

This expression for the entropy production rate can be simplified for large Hartmann numbers. For example, the exponential e-M becomes less than about 0.03% of eM for values of the Hartmann number above 3. In this case, the entropy production at the wall becomes

Pq u

kT

M

M

M Ks w

w

w

b

w w,*

*

( )=

+

-+

2

2

22

4

2

2

1θµ

θ

22

θw

(5.26)

This result can be rearranged in terms of the Reynolds number (Re), magnetic Prandtl number (Prm), and thermomagnetic number (N), respectively, as follows,

P

q mN Ks w

w

w,* Re (= ′

+ +-14 16

12 2

3 5

2

8 2ρβ µ θ

)) Pr Rem2 2

(5.27)

where

Re = ρµu ab

(5.28)

Prme e= µµ σρ

(5.29)

NH

kTe

e w

=2

σ

(5.30)

and me and He refer to the magnetic permeability and electric field strength, respec-tively. Also, b and qw are the microchannel aspect ratio (b/a) and wall heat flux per unit length, respectively.

Differentiating the previous result with respect to the Reynolds number and set-ting the result equal to zero yields the following optimal Reynolds number:

ReL opt

B

N K,

/

/

.[ ( ) ]=

+0 5741

1 5

2 6 1 10β

(5.31)

where the duty parameter, B, is given by

Bq m

m w

= ′ 2 2

5

ρµ θPr

(5.32)

7262_C005.indd 121 1/19/08 11:45:00 AM


The irreversibility distribution ratio (or entropy generation number), Ns, is defined as the ratio of the actual entropy generation to the minimum rate of entropy genera-tion, i.e.,

NP

Pss

s

=

,min

(5.33)

After substituting the previous expression for ReL,opt, it can be shown that

NsL

L opt

L

L opt

=

+

-15

45

8 2Re

Re

Re

Re, ,

(5.34)

The first term on the right side outlines the rate of change due to the thermal irrevers-ibility, whereas the second term includes the combined friction and electromagnetic irreversibilities. The result suggests that the entropy generation number changes faster at low Reynolds numbers (below ReL,opt), when the thermal irreversibility is the largest portion of the total irreversibility.

5.3.2 Ca sEst u d yo f Ch a n n El dEsig n at va r yin g ha r t ma n n nu mBEr s

Using the previous formulation, the section will present a case study with entropy production in a microchannel (Naterer and Adeyinka, 2005). Predicted results will be compared against computational simulations with a finite element volume formu-lation. Predicted results will be compared against past data reported by Bejan (1996), Salas and coworkers (1999), and Adeyinka and Naterer (2004). The predicted results will be presented in terms of nondimensional variables described in previous sec-tions. These nondimensional variables include the cross-stream coordinate (z* = z/a), velocity (u* = u/ub), Hartmann number (M), temperature (θ), Reynolds number (Re), load factor (K), and duty parameter (B). In Figure 5.7 and Figure 5.8, the predicted nondimensional velocity and temperature fields are shown at varying Hartmann numbers. It can be observed that the temperature and near-wall temperature gradi-ent decrease at lower Hartmann numbers. Also, they decrease at larger values of the load factors. The electromagnetic resistance of fluid motion decreases at larger load factors, when the magnetic field strength decreases. As a result, the fluid velocity increases in the denominator of the nondimensional temperature, thereby reducing the temperature of the fluid. As expected, the temperature decreases when the wall heating rate is reduced.

In Figure 5.9, the rate of exergy destruction is illustrated for a microchannel half-width of 43 mm, load factor of 0.5, Hartmann number of 20, and varying mag-netic field strengths. It can be observed that exergy destruction decreases at lower magnetic field strengths, due to smaller electromagnetic irreversibilities. The exergy destruction reaches a peak value at the wall. Then it decreases to a local minimum and rises to a uniform nonzero value in the core of the microchannel. The exergy

7262_C005.indd 122 1/19/08 11:45:02 AM


destruction increases for wider microchannels because added surface area increases the friction irreversibilities. Also, the electromagnetic irreversilibity decreases at higher load factors due to a lower magnetic field strength. As a result, the higher load factor of K = 1.0 in Figure 5.10 yields a monotonically decreasing exergy destruc-tion toward zero in the midplane of the microchannel, without a local minimum and rising trend observed in Figure 5.9. The results in Figure 5.9 and Figure 5.10 indi-cate the friction irreversibility is highest near the walls, whereas the electromagnetic

0.900.0

0.6

0.4

0.2

u/u b

0.8

1.0

1.2

0.91 0.92 0.93 0.94 0.95z/a

0.96 0.97 0.98 0.99 1.00

Tillack, Morley (M = 40)LHA Model (M = 40)Tillack, Morley (M = 100)LHA Model (M = 100)

FIg u r e5.7 Predicted velocity profile at varying Hartmann numbers.

0.0 0.0

0.6

0.4

0.2 Non

dim

ensio

nal T

empe

ratu

re

0.8

1.0

1.2

0.2 0.4 0.6 0.8 z/a

1.0

q* = 1; M = 2

q* = 2; M = 4 q* = 2; M = 2 q* = 1; M = 4

K = 0.5 Constant Wall Heat Flux

FIg u r e5.8 Nondimensional temperature at various heating rates.

7262_C005.indd 123 1/19/08 11:45:04 AM


irreversibility is dominant within the core of the microchannel. If the fluid friction is sufficiently small at low Reynolds numbers and the Hartmann number is sufficiently large, the maximum exergy destruction may occur at the midplane. Unlike classi-cal problems involving convective heat transfer without electromagnetic forces, the point of maximum exergy destruction may not be located at the wall. In this case, local loss coefficients would be better represented in terms of local exergy destruc-tion, rather than friction coefficients at the wall, which may not best reflect the most relevant location of dissipative losses in electrohydrodynamic flows.

0.0 0

300

200

100

X des

t

400

500

600

0.1 0.2 0.3 0.4 1 – z/a

1.5

Predicted (B = 8 kV/m)

M = 20 (Salas, 1999)LHA Model (M = 20; B = 15 kV/m) Predicted (B = 11 kV/m)

K = 0.5a = 43 m

FIg u r e5.9 Exergy destruction at varying magnetic field strengths.

0.00 0

400

200

X des

t

600

800

1,000

0.05 0.10 0.15 0.20 0.25 0.30 0.35 1 – z/a

0.40

Predicted (a = 30 m)

M = 20 (Salas, 1999)LHA Model (a = 43 m; M = 20)Predicted (a = 35 m)

K = 1.0

FIg u r e5.10 Exergy destruction at varying channel widths.

7262_C005.indd 124 1/19/08 11:45:07 AM


In Figure 5.11, the optimal Reynolds number, ReL,opt, which minimizes the net exergy destruction, is plotted at varying duty parameters, aspect ratios, and flow configurations. The optimum for electrokinetic flow is lower than other configura-tions, and it increases at higher duty parameters and smaller aspect ratios. This opti-mum involves a balance, which minimizes the net exergy destruction arising from combined effects of friction and thermomagnetic irreversibilities. The friction irre-versibility is reduced with a smaller surface area and net friction, but higher, larger irreversibilities occur due to a higher temperature (between fluid and wall) needed to transfer a specified heat flow (q´) over a smaller area. However, the thermal irrevers-ibility decreases with a larger surface area, because a smaller temperature difference between the fluid and wall is needed to transfer the fixed heat flow. This reduction comes at the expense of higher friction irreversibilities, when a larger surface area (i.e., larger Reynolds number) contributes to added surface friction. Furthermore, the electromagnetic irreversibility increases at higher Reynolds numbers, because Ohm’s law implies that this irreversibility is proportional to the velocity squared. These trends contribute to the physical mechanisms that minimize the net rate of exergy destruction at ReL,opt in Figure 5.11.

Operating at other conditions below or above ReL,opt implies that additional elec-tric input power is needed to transfer fixed rates of mass and heat flow through the microchannel. Additional power is needed to offset higher internal irreversibilities, which dissipate kinetic energy into internal energy, rather than transferring power for mass transport. In the case of electro-osmotic flow, additional input power is needed to generate sufficient charge distributions along the wall for the specified mass flow rate. Other methods of microfluidic flow control, such as pressure or ther-mocapillary driven flow, would also entail wasted power input to overcome system irreversibilities.

The previous results of entropy production (or exergy destruction) have practical significance in electrokinetic flow control in microchannels. Exergy losses charac-terize the friction, pressure losses, and kinetic energy dissipated to internal energy within the microchannel. As a result, they have an important role in the performance

1 1.0E – 01

1.0E + 01 1.0E + 02 1.0E + 03 1.0E + 04 1.0E + 05 1.0E + 06 1.0E + 07

1.0E + 00

ReL,

opt

1.0E + 08

10 B

100

Film Condensation (Adeyinka, 2004)

Electrokinetic Flow (Aspect Ratio = 5) Electrokinetic Flow (Aspect Ratio = 10) Electrokinetic Flow (Aspect Ratio = 25)

Turbulent Boundary Layer (Bejan, 1996)

FIg u r e5.11 Optimal Reynolds number at varying magnetic field strengths.

7262_C005.indd 125 1/19/08 11:45:08 AM


of microdevices. For example, consider cooling of an electronic assembly with microchannels. Each unit of exergy destroyed corresponds to an amount of internal energy, which could have been removed by convective cooling, but was not removed due to energy dissipated by the thermomagnetic irreversibilities. Additional power input is needed to offset these irreversibilities. As future microdevice technologies become more complex in terms of energy conversion between various subsystems, the spatial tracking of entropy production throughout these networks will become an increasingly valuable tool in reaching the highest levels of performance and device efficiency.

5.4 MIc r o Pat t er n ed su r Fac es w It h oPen MIc r o c h a n n el s

5.4.1 f l u id fl o w fo r mu l at io n

Controlled surface roughness has importance in various fluids engineering appli-cations. Surface roughness affects the boundary layer formation in aerodynamics of aircraft, vehicles, and so forth. Extended surfaces (fins), modified surface pro-files, and other passive techniques of heat transfer enhancement are commonly used in industrial heat exchangers. Random microscale features of a surface are often modeled as a lumped or overall surface roughness. Recently, advances in microma-chining fabrication can allow surface profiles to be carefully designed for various purposes. In this section, the effects of embedded surface microchannels on bound-ary layer flow and convective heat transfer will be examined. It will be shown that Entropy-Based Surface Microprofiling (EBSM) enables drag reduction and lower entropy production of convective heat transfer, due to slip-flow conditions within the embedded microchannels. Using EBSM, the power consumption to transfer speci-fied rates of mass and heat flow across a surface can be reduced.

Consider external flow past a flat surface with embedded open microchan-nels (see Figure 5.12 and Figure 5.13). This flow configuration closely resembles a flat plate boundary layer flow, with a Blasius similarity solution for streamwise changes of flow variables. But open microchannels are aligned parallel to the incom-ing flow along the surface, with micron or submicron scale depth. Unlike random surface roughness, the well-controlled profiles of these embedded microchannels

u∞ T∞

yx

Tw

q˝

L

W

Diverging Microchannels

Slip-Flow Region No-Slip Region

u∞ T∞

Wns Ws

d

L

FIg u r e5.12 Schematic of embedded microchannels.

7262_C005.indd 126 1/19/08 11:45:10 AM


can allow geometrical optimization for drag reduction and heat transfer enhancement. A basic geometry of flat plate flow is considered in this section, although, the tech-nique can be readily extended to more complex geometries.

In the x-z direction (i.e., side view within a microchannel), a Couette-type flow is encountered. A Couette flow refers to a one-dimensional shear layer enclosed by fixed velocities at both edges of the layer. For example, a linear variation of velocity occurs between a moving plate and a stationary wall below the plate. In the case of the open microchannel, a nonzero Blasius velocity and slip velocity are encountered at the top and base of the microchannel. Diffusion-dominated transport of momen-tum in the z-direction yields a similar linear profile between both edge velocities. The slope of this profile decreases in the x-direction, as the top velocity changes in the slip-flow pattern of the boundary layer development.

Considering a front view of the plate in the z-y direction at a fixed x-location, the cross-stream flow is neglected, and the z- and y-velocities will be assumed to be negligible, relative to the streamwise (x-direction) velocity component. Transi-tion between slip-flow and no-slip regimes occurs at the top corners of the embedded microchannels. Consider a slip-flow embedded microchannel, with a Knudsen num-ber and characteristic length based on both depth and width. At the top corners, the near-wall slip-flow profile approaches no-slip behavior before reaching the top edge of the microchannel. This transition occurs because the local Knudsen num-ber decreases at the top edges of the microchannel. This transition may produce a small submicron semicircular type of zone of influence at the top corners. Little or no experimental data has been reported in the technical literature regarding such slip-flow variations arising from this transition. Past studies have mainly reported slip-flow coefficients based on measured mass-flow rate slopes against various pres-sure differences in closed microchannels. The current transition regime and mixed Knudsen numbers would not arise in those cases. Also, past measurements yield a single (net) coefficient, without spatial variations across the microchannel. In this method, a single coefficient simulates a spatial variation and transition regime in a similar fashion.

No-SlipRegion

No-SlipRegion

No-SlipRegion

No-SlipRegion

f(η, a)

Slip- Flow Region

h(y) = (Ltanθ1cotθ1)ηa

Tangent at Profile Base y

L

Slip-Flow Region

(a) (b)

x

y

h(y) = ytanθ1 θ1 x

L

θ1

θ2

θ2

δ(x) = xcotθ1

δ = Lcotθ2

FIg u r e5.13 Top view of microchannels with (a) converging and (b) diverging profiles.

7262_C005.indd 127 1/19/08 11:45:11 AM


Analytical solutions of laminar boundary layer flow can be determined from the method of similarity selection. The predicted velocity field, u, from this method for flat plate boundary layer flow can be expressed in terms of a similarity variable, h, freestream velocity, u∞, and a stream function derivative as follows:

u

u∞

= ′f ( )η (5.35)

where

η

ν= ∞y

u

x (5.36)

Transforming the two-dimensional, steady, laminar boundary layer equations with this similarity variable, the governing equation for mass and momentum transport within the boundary layer becomes the following well-known Blasius equation,

′′′ + ′′ =f f f( ) ( ) ( )η η η1

20 (5.37)

This nonlinear ordinary differential equation will be solved by a Runge-Kutta method, subject to boundary conditions of f ′ (∞) = 1 and f (0) = 0. The no-slip condi-tion at the wall is f ′ (0) = 0.

The analytical solution can be obtained from successive integrations as follows:

uuu

f

f d* 0

( )

exp /2

= = ′ =-

∞

∫∫η

ηηη

0dd

f d d

η

η ηη

exp /20

-

∫∫

∞ˆ

0

(5.38)

It can be shown that the functional forms of the momentum and thermal energy equations are equivalent, so a similar procedure yields the following result for the nondimensional temperature within the boundary layer:

θ ηη

η

( )

exp /20

=--

=-

∞

∫T T

T T

f dw

w

Pr ˆ

00

0

η

η

η

η η

∫∫∫ -

∞

d

f d d

exp /20

Pr ˆ

(5.39)

When the Prandtl number of the fluid is close to 1 (common fluids such as air), the pre-vious expressions for nondimensional velocity and temperature become identical.

7262_C005.indd 128 1/19/08 11:45:15 AM


For the case of a slip-flow condition at the wall (within an embedded micro-channel), the boundary condition involves the nonzero wall velocity and spatial gra-dient of velocity (second derivative of the stream function at the wall), so that

′ = ′′f f(0) (0)1K (5.40)

where

K Kn Rex x11/22= -

σσ

(5.41)

In Equation 5.41, Knx and Rex are the local Knudsen and Reynolds numbers, respec-tively. The boundary condition implies that wall slip increases with higher velocity gradients at the wall. For no-slip conditions, f ′′ (0) = 0.3321. It can be shown that f ′′ (0) varies with K1 for the slip-flow problem, that is,

′′ =+

fK

(0)1.39

4.185 0.96 11 11.

(5.42)

After the third-order Blasius equation is solved, subject to the slip-flow bound-ary condition, the resulting stream function, f(h), can be numerically differentiated to yield the velocity field and wall shear stress, τw, distributions, i.e.,

τ ρ µw = ′′∞

1/2

(0)1 2 3 2

1 2

/ /

/

u

xf

(5.43)

Rearranging this result in terms of the local Reynolds number,

τρ

wxu

Re∞

-= ′′ ⋅2

f (0) 1/2

(5.44)

Unlike macroscale systems with a no-slip condition at the wall, the slip-flow conditions and in a microchannel can lead to lower shear stresses along the walls. These trends have been investigated previously for gases (Martin and Boyd, 2001) and liquids (Choi et al., 2002). Choi et al. (2002) have reported higher water flow rates induced by a different surface coating along a microchannel wall, thereby leading to a variation of slip velocity and shear stress along the wall. Such slip-flow effects increase when the channel height decreases and the wall shear stress increases. The percentage reduction of wall shear stress due to the slip-flow condi-tion, as compared to the no-slip solution, can be determined from the previous simi-larity solutions. The result follows from the difference of 100% (0.3321 - f ′′ (0)),

7262_C005.indd 129 1/19/08 11:45:19 AM


where f ′′ (0) refers to the result from the slip-flow solution. The value of 0.3321 corresponds to the classical Blasius solution with a no-slip velocity at the wall. Martin and Boyd (2001) have examined the validity of key assumptions adopted in this similarity solution. At all locations except the leading edge, the stream-wise velocity gradient is assumed to be much smaller than the cross-stream (y-direction) gradient.

In the slip-flow condition, the Knudsen number, Kn, characterizes the degree of rarefaction of fluid motion. As discussed previously, the continuum assumption of fluid flow is considered valid when Kn ≤ 10-3, whereas free molecular flow occurs when Kn ≥ 10. Between these two limits, the slip-flow regime exists within the range of 10-3 ≤ Kn ≤ 10, and a transition region occurs for 10-3 ≤ Kn ≤ 1. A similar condition of temperature discontinuity exists for the thermal problem, but “no-jump” (thermal problem) replaces the condition of “no-slip” (flow prob-lem). The boundary between the slip-flow and transition regimes is problem- and geometry-dependent.

The principles underlying the no-slip, no-jump conditions for velocity and temperature require that there cannot be any finite discontinuities of velocity and temperature at the wall. Such discontinuities would entail infinite velocity and temperature gradients, thereby leading to infinite viscous stresses and heat fluxes. Based on continuum theory, the no-slip, no-jump conditions require an infinitely high number of collisions between the fluid and solid surface. In practice, such assumptions lead to reasonably accurate predictions, provided that Kn < 0.001 for gases. For flows at higher Knudsen numbers, the mean free path of molecules is no longer sufficiently small relative to a characteristic length of the micron or sub-micron device (such as the microchannel height). Slip-flow conditions entail direct momentum exchange of intermolecular interactions near the wall. The probability of a fluid molecule striking another fluid molecule within an embedded microchannel, rather than a wall, decreases at higher Knudsen numbers. A molecule may reflect from several walls before colliding with another fluid molecule traveling in the prin-ciple flow direction. Some molecules reflect specularly, and others reflect diffusely from the surface of the walls. Thus, a portion of momentum of incident molecules is lost to the wall, while the remaining portion is retained by the reflected molecules. The tangential momentum accommodation coefficient is used to represent the frac-tion of incident molecules that is reflected diffusely. This coefficient typically var-ies between 0.2 and 0.8, and it depends on the fluid properties, solid wall, and the surface finish.

For an idealized smooth wall, the incident angle exactly matches the reflected angle of impacting molecules. The molecules conserve tangential momentum, thereby not exerting shear on the wall. This process of specular reflection leads to perfect slip at the wall. But for an actual wall with surface roughness, the molecules reflect at some random angle, which is uncorrelated with their incident angle. Per-fectly diffuse reflection requires zero tangential momentum for the reflected fluid molecules to be balanced by a finite slip velocity, to account for the shear stress trans-mitted to the wall. A near-wall force balance requires that the difference between the slip velocity and wall velocity balances the product of mean free path and velocity gradient perpendicular to the wall. This balance will be applied as the slip-flow

7262_C005.indd 130 1/19/08 11:45:19 AM


boundary condition in the previous similarity solution of boundary layer flow. Slip occurs only when the mean velocity of molecules changes significantly over a dis-tance of about one mean free path.

5.4.2 h Eat tr a nsf Er fo r mu l at io n

For convective heat transfer analysis in the thermal boundary layer, the reduced form of the governing energy equation is simplified as follows for the current problem:

ρ ρc uT

xc v

T

yk

T

yp p

∂∂+ ∂

∂= ∂∂

2

2 (5.45)

subject to the following boundary equations at the edge of the boundary layer and wall, respectively,

T x y T( , )→∞ = ∞

(5.46)

- ∂∂= ′′k

T

yqw

0 (5.47)

In this problem, the wall heat flux is specified, and the wall temperature, Tw(x), is unknown. When the Prandtl number of the fluid is close to 1 (common fluids such as air), the functional form of the boundary layer equations for velocity and tempera-ture become analogous. When the velocity solution is obtained from the momentum equations, it can be modeled as a known coefficient when solving the temperature boundary layer equation.

Define the following variable as the wall temperature difference:

θw wx T x T( ) ( )= - ∞ (5.48)

After solving the energy equation, subject to the boundary conditions and evaluating the result at y = 0 for the wall temperature difference, it can be shown that (Kays and Crawford, 1980)

θ ξw x wxk

q x( ).

[ ( / ) ]/ / /= ′′ -- - -0 62311 3 1 2 3 4 2Pr Re //3

0d

x

ξ∫

(5.49)

A thermal boundary condition with a constant heat flux has been applied at the wall.In the problem configuration, the y-direction is perpendicular to the wall. Thus,

both the solid side (y → 0-) and fluid side of the wall (y → 0+) are assumed to have an equivalent heat flux, ′′qw,, passing through an infinitesimal control volume along the wall (y = 0), due to conservation of energy. After the boundary layer equation is solved, subject to the constant heat flux condition at the wall, the wall temperature can be obtained as a function of x, ′′qw, and T∞. In the no-slip (no-jump) case, the

7262_C005.indd 131 1/19/08 11:45:24 AM


spatial temperature profiles on both sides of the wall must meet at a common wall temperature. Each profile exhibits a certain slope at the wall to match the Fourier heat flux corresponding to the specified flux of the boundary condition.

But in the slip-flow case, these profiles are assumed to be offset at the wall, due to a temperature jump condition. In addition, the near-wall slopes of temperature are assumed to match the corresponding slopes of the equivalent no-slip case, when the same speci-fied flux condition is applied at the wall. More specifically, ′′qw, is specified and T(x, y → 0+) is solved in the slip-flow problem. Then T(x, y → 0-) can be obtained from a thermal jump condition, while simultaneously matching the required Fourier heat flux at the solid side of the wall (y → 0-). The same constant wall flux can be obtained in both no-slip and slip-flow cases, provided that the near-wall temperature slopes are equivalent.

Due to the similarity between molecular diffusion of heat and momentum near the wall at fluid Prandtl numbers close to 1, the magnitude of temperature jump at the wall can be approximated with the momentum accommodation coefficient. But the previ-ous convective heat transfer analysis does not need this accommodation coefficient in the solution procedure, when a specified flux boundary condition is used at the wall. It is only needed if spatial temperature variations within the wall are required, or a con-jugate (conduction and convection) analysis is required to find the wall heat flux.

The local convection coefficient can be evaluated based on the result from the temperature field, thereby leading to the following expression for the local Nusselt number (Kays and Crawford, 1980):

Nux x= θ0

1 3 1 2Pr Re/ / (5.50)

where q0 = 0.453 for the current case of a specified flux boundary condition. It can be shown that the same Nusselt number is obtained for the case of a constant wall tem-perature, except that the leading coefficient becomes q0 = 0.3321. This heat transfer coefficient is about 36% lower than the value at the same point on the plate with a constant wall flux. But the average Nusselt number and average qw are only about 2% lower than the case of a constant wall heat flux at x = L.

5.4.3 f o r mu l at io n o f En t r o pypr o d u Ct io n

The total entropy production over a plate of length L and a width of W consists of a thermal irreversibility and a friction irreversibility, which can be expressed in inte-gral form as follows:

Sq

Tgen =′′

+∞

∞

∞∫ ∫ ∫

2

0 0 0

W L W

w

dxdy

h

u

Tdxdyτ

00

L

∫ (5.51)

The previous correlations for the convection coefficient (based on the Nusselt number) and wall shear stress will be substituted into this equation. In this section, the entropy production will be analyzed for the following three cases: (i) diverging and converging embedded microchannels; (ii) unspecified (exponential) profile of microchannels; and (iii) unspecified cross-stream variation of the microchannel geo-metrical profile.

7262_C005.indd 132 1/19/08 11:45:26 AM


c ase(i):divergingandconvergingMicrochannels,laminarFlow,u niformwallheatFlux

For diverging (or converging) microchannels, the predicted entropy production involves a parallel section and varying slip areas. Using the previous expressions for the heat transfer coefficient and wall shear stress, it can be shown that the total entropy production over the plate becomes

Sgen

2

= ′′

′∞ ∞∫q

kT

ReReνθ

2

2 20

1 2

0un

Lx

s

/

( )ddyd dydx

W dx

nsx

W ns L

ReRe

ReRe

+′

+ -

∫ ∫0

2

0

1 2

0 0

/

( )θ

WW

s

x

s

L

nf

d

∫

∫

+

′′∞

∞

u

T Re

Re2µ0

1 2

0( )/

yydf

dydx

W ds

xx

W nWs L

ReRe

ReRe

+′′+ -

∫ ∫0

2

01 2

0

0( )/

ss

S Sf h

∫

+ + , ,

gen gen

(5.52)

where the subscripts s and ns refer to slip-flow and no-slip regions, respectively. The number of microchannels is denoted by n and other geometrical parameters are illustrated in Figure 5.12. The latter two integrals, S fgen, and S hgen, , refer to the wall friction irreversibility difference and wall thermal irreversibility difference due to slip minus no-slip conditions, that is,

Su

nhx

s

L

gen

2q,

/

= ′′

′∞ ∞∫ν

θ

2

2 20

1 2

2Re

ReRe

(( ) ( )

/

0 0

1 2

0-′

∫ Re

Rex

nsxdyd

θ

δ

(5.53)

S nf

fs

x

L

gen

2

, /

( )=

′′- ′′∞

∞∫u

T Re

Reµ2

0

01 2

ffdydns

xx

( )/

01 2

0 ReRe

∫

δ

(5.54)

It can be shown that the total entropy production over the plate becomes a sum of entropy production rates for parallel microchannels (subscript p) and irreversibility dif-ference integrals for diverging and converging microchannels ( S fgen,

plus S hgen,), i.e.,

S S S Sp h fgen gen gen gen= + +, , , (5.55)

The second and third terms in Equation 5.52 represent a parallel microchannel term in Equation 5.55. The result in Equation 5.55 applies to diverging microchannels. An analogous result is obtained for converging microchannels, after subtracting the latter two terms (rather than adding the terms). Without the latter two terms, the result represents the entropy generation over a surface with interspersed parallel microchannels.

c ase(ii):unspecified(exponential)Profile,laminarFlow,u niformwallheatFlux

Consider another case where the best geometrical profile of embedded micro-channels is unknown (or unspecified). The profile is characterized by an unknown

7262_C005.indd 133 1/19/08 11:45:32 AM


function, h(y), which is determined based on minimization of entropy production along the microprofiled plate. Define a nondimensional profile variable as follows:

ηθ

η= ≤ ≤y

L cot;

2

0 1 (5.56)

The varying distance between the microchannel base and edge point (see Figure 5.13) becomes

h L= ( tan cot )θ θ η1 2 (5.57)

An exponentially varying profile shape is defined by

h L a= ( tan cot )θ θ η1 2 (5.58)

From this definition, it is required that h(0) = 0 and h(1) = Ltan(q1) cot(q2).In terms of these variables, the thermal irreversibility difference integral becomes

Su

nhs

gen

2

,

/

/ ( )= ′′

′

-′∞ ∞

q

kT

νθ

1 2

2 1 22

10

1θθ

θ ηns h

L

L x dxd( )

cot /

0 0

1

21 2

∫ ∫ (5.59)

Performing the integrations with the varying geometrical profile,

Sq

kT unKhgen,

.. cot= ′

-∞ ∞

2

2 11 11

20 9212ν θ

33 25 2

23 2

11 2

a L+

cot tan/ / /θ θ Re (5.60)

Similarly, the friction irreversibility integral becomes

S n f ff sgen

5/2

,

/ /

( ( )=

′′ - ′′∞

∞

u

T

ρ µ1 2 1 2

2 0 nnsh

L

L x dxd( )) cot /00

1

21 2∫ ∫ -θ η

(5.61)

which yields

Su

T Kfgen, .

.

. .=

+-∞

∞

µρ

2

11 11

5 56

4 185 0 961.. cot

cot tan/ /

3282

22

3 22

1 21

× -+

na

θθ θ

ReL

3 2/

(5.62)

The same result is obtained as the previous case (embedded linearly converging and diverging microchannels), except that the factor 2/5 is replaced by 2/(3a + 2) in the thermal irreversibility integral. Also, 2/3 is replaced by 2/(a + 2) in the friction

7262_C005.indd 134 1/19/08 11:45:38 AM


irreversibility integral. When a value of a = 1 (linear profile) is substituted into these expressions, the generalized results match the special geometrical case for linearly converging and diverging microchannels.

The previous irreversibility difference integrals are combined into the total entropy production over the plate (per unit width). Then, the optimal shape profile of the embed-ded microchannels can be obtained by differentiating that expression with respect to the profile parameter, a, and setting the result equal to zero. For a diverging profile,

63 2

22

02 2

A

a

B

a( ) ( )+++

=

(5.63)

where

Aq

u TnK= ′

∞ ∞

2

2 11 11 5 2

230 921

νρ

θ. cot tan. / /221

1 2θ ReL/

(5.64)

Bu

T K=

+

-

∞

∞

µρ

2

11 11

5 564 185 0 96

1 328.

. ..

.

n Lcot tan/ / /3 22

1 21

3 2θ θ Re

(5.65)

For a converging profile, a minus sign is placed before each expression for the coefficients A and B. Solving the previous algebraic equation for the optimal profile coefficient,

aA B

A B A B A B A B=+

- - + + - + +(2

3 93 3 9 3 3 32( ) ( )( ) ))

(5.66)

When substituted into the profile distribution for h(y), the resulting shape of the embedded microchannels minimizes the entropy production over the plate.

c ase(iii):unspecifiedcross-streamProfilevariation,laminarFlow,u niformwallheatFlux

Leaving out the integration of total entropy production in the y-direction, the mini-mization of entropy production yields a detailed variation of microchannel pro-file in that direction. The thermal and friction irreversibility difference integrals become

Sq

kTK f ahgen,

. /. ( ( ,= ′

-∞

2

2 11 11 3 20 461 1 η)) cot tan )/ / /3 2

23 2

11 2θ θ ReL-

(5.67)

Su

T Kfgen, .

.. .

.=

+

-∞

∞

2

11 11

2 784 185 0 96

0µ

6664 1 1 2 1 22

1 21

1

-( ( , ) cot tan )/ / /f a Lη θ θ Re //2

(5.68)

7262_C005.indd 135 1/19/08 11:45:43 AM


Combining these irreversibility integrals with the portions arising from parallel embedded microchannels, the total entropy production over the plate becomes

Sq

kTK Kgen =

′

+ +∞

2

2 11 114 016 0 461 0 461. . ..

111 11 3 2 3 2

23 2

11. / / /( ( , ) cot tan )-

-

f a

L

η θ θ

Re 11 22

11 11

2 784 185 0 96

0 664/.

. ..

.+

+

+∞

∞

u

T K

µ

++

-

-2 78

4 185 0 960 664 1

11 11

1.. .

. (.

/

Kf 22 1 2

21 2

11 2( , ) cot tan )/ / /a Lη θ θ

Re

(5.69)

The first, second, fourth, and fifth terms represent the irreversibility contributions from the parallel microchannel profile. The remaining third and sixth terms, involv-ing the trigonometric factors, represent the contributions arising from profile correc-tions (due to deviations of the profile width in the streamwise direction).

5.4.4 Ca sEst u d iEso f su r faCEmiCr o pat t Er n dEsig n

In this section, EBSM results illustrate how the method can provide an effective design tool for reducing drag and entropy production in external flows along a flat plate. Numer-ical results for air (300 K) are considered. In Figure 5.14, the change of optimized pro-file parameter, a, at varying Reynolds numbers, wall heat fluxes, and slip coefficients, is presented. Geometrical and surface parameters are shown in the figure. Each profile parameter minimizes the combined entropy production of thermal and friction irrevers-ibilities under each set of flow conditions. This parameter affects the relative proportion of surface area containing slip-flow and no-slip conditions. The friction irreversibility

0.0

1.0

2.0

3.0

4.0

5.0

6.0

800 1,200 1,600 2,000 2,400 2,800 3,200Reynolds Number (ReL)

Opt

imal

Pro

file P

aram

eter

(a)

q' = 14 W/m (K1 = 0.1)

q' = 14 W/m (K1 = 0.5)

q' = 16 W/m (K1 = 0.1)

q' = 16 W/m (K1 = 0.5)

A ir (300 K) n = 1,200

1 = 0.1 2 = 0.8

d/W = 1.0E – 06 (Diverging Microchannels)

θ θ

FIg u r e5.14 Change of optimized profile parameter with Reynolds number.

7262_C005.indd 136 1/19/08 11:45:46 AM


increases for larger surface areas exposed to no-slip conditions, due to added surface friction. But the thermal irreversibility decreases, because a smaller temperature differ-ence (between the wall and surrounding fluid) is needed to transfer a fixed rate of heat flow, q . Slip-flow conditions within the embedded microchannels lead to lower fric-tion irreversibility, but only over a certain range of conditions, because they contribute simultaneously to additional surface area with friction.

Slip-flow conditions affect the momentum exchange of intermolecular interac-tions near the wall. The probability of a fluid molecule striking another molecule within an embedded microchannel, rather than a wall, decreases at higher Knudsen numbers. From results obtained in this section, the Knudsen number varies between about 0.02 and 0.07. These values fall within 0.001 < Kn < 0.1, which represents the range governed by the Navier–Stokes equations with slip-flow boundary conditions (Gad-el-Hak, 1999).

In Figure 5.14, the optimized surface profile parameter decreases at higher Reynolds numbers. At a fixed freestream velocity, the surface area increases at higher Reynolds numbers. Also, smaller profile parameters lead to a decreasing slip-flow area. At higher Reynolds numbers, the minimal entropy production moves to lower values of the profile parameter. The friction irreversibility rises earlier at those lower values, due to larger surface area. Also, Figure 5.14 shows that the profile param-eter increases at higher wall heat fluxes. More slip-flow area is needed to overcome added thermal irreversibility at those higher heat fluxes.

Air flow at 300 K past a surface with 2800 parallel microchannels and a surface heat transfer rate of 50 W/m is considered in Figure 5.15. The figure shows predicted trends of entropy production over a range of Reynolds numbers. The benchmark solu-tion refers to the asymptotic no-slip limit, when correlations for the Blasius similarity solution can be integrated directly to yield the net entropy production. This case with-out microchannels represents the classical boundary layer flow and convective heat transfer from a flat nonprofiled surface. It can be observed that the current numerical

0.01

0.10

1.00

10.00

100.00

1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06ReL

S gen

(W/m

3 K)

Numerical (K1 = 0)Benchmark (No-Slip Limit)Experiment (K1 = 0; Czarske et al.)Microchannels (K1 = 1, d/W = 0.00001)Microchannels (K1 = 2, d/W = 0.00001)Microchannels (K1 = 2, d/W = 0.00002)

(Air, 300 K, q´ = 50 W/m, n = 2,800)

FIg u r e5.15 Reduced entropy production with embedded surface microchannels.

7262_C005.indd 137 1/19/08 11:45:47 AM


slip-flow formulation approaches this benchmark solution properly in the no-slip limit, when the slip coefficient becomes K1 = 0. The close agreement between pre-dicted and benchmark results provides useful validation of the numerical modeling.

Experimental data have been reported by Czarske et al. (2002) regarding wall shear stresses in the friction irreversibility portion of total entropy production. This data represent measured changes of skin friction coefficients at varying Reynolds numbers in the no-slip limit case. In Figure 5.15, this measured data (filled circle markers) is used for comparisons against the numerical modeling (solid line) and benchmark data (open circle markers) in the no-slip limit case. Close agreement is reached in these comparisons, thereby providing additional useful evidence regard-ing the current model’s reliability.

The entropy production increases at low Reynolds numbers, when the smaller surface area leads to a high thermal irreversibility. When the surface area decreases, a larger temperature difference (between the wall and surrounding fluid) is needed to transfer a fixed rate of heat transfer from the wall, q . On the other hand, friction irreversibilities increase at higher Reynolds numbers, due to more friction over a larger surface area. Thus, an optimal Reynolds number occurs at a certain interme-diate range, where the entropy production rate is minimized. The predicted results show that the embedded microchannels allow lower values than the minimum entropy production without microchannels, due to slip-flow conditions within the microchannels. As a result, the adaptive microprofiling provides a useful technique of reducing entropy production in external flows. In Figure 5.15, this entropy produc-tion decreases at higher slip coefficients and shallower microchannels. Drag reduc-tion occurs at the higher slip coefficients, whereas less microchannel depth reduces the friction irreversbility, due to less overall surface area.

In Figure 5.15, it can be observed that the plate without embedded microchannels exhibits the lowest entropy production up to the critical Reynolds number. But this trend changes appreciably at larger Reynolds numbers. When the plate length and surface area become larger, the thermal irreversibility decreases, and added area leads to greater surface friction. The resulting entropy production becomes lower for cases with microchannels, because the added friction irreversibility is more notice-ably reduced by slip-flow conditions when the surface area increases. The beneficial impact of drag reduction by slip-flow conditions is not noticeable at lower Reynolds numbers, as thermal irreversibilities constitute a larger portion of the total entropy production. Additional surface area of embedded microchannels appears to raise friction irreversibilities more than frictional reduction by slip-flow conditions.

When analyzing the external flow conditions in these problems, the Reynolds number is characterized by the streamwise coordinate, x, and plate length, L. All geometrical and external flow parameters were selected so that the Reynolds number remains below the point of transition to turbulence at ReL = 5 × 105. The formula-tion could be extended to external turbulent flows, provided that turbulence equa-tions are supplied for the convective heat transfer and wall friction correlations. The open microchannel flow depends on the microchannel depth (or hydraulic diam-eter), so similarities exist with closed microchannel flows. According to Sharp and Adrian (2004), who performed measurements of pressure drops in microtubes; they confirmed that transition to turbulence occurs at Reynolds numbers of about 1800.

7262_C005.indd 138 1/19/08 11:45:48 AM


Microtube transition to turbulence has similarities with rectangular channels of the same hydraulic diameter. The transition to turbulence was estimated by the authors when pressure drops exceeded macroscopic Poiseuille flow results for laminar flow resistance. Also, micro-PIV measurements of mean velocity and root mean square (rms) velocity fluctuations at the centerline were monitored at the transition point. Experimental uncertainties of ±1% systematic and ±2.5% rms random errors were reported by Sharp and Adrian (2004).

For microchannel depths and external flow velocities considered in this sec-tion, the Reynolds numbers are well below the transition point of 1800. The velocity required in the Reynolds number is best represented by the velocity at the top of the open microchannel (not the freestream velocity). Because this corresponds to the base of the boundary layer in external flow, it is approximately equal to the slip-flow veloc-ity at the wall. At low slip coefficients, this becomes much smaller than the freestream velocity. For example, the similarity solution of f' (0) suggests that the wall velocity is about 0.02% of the freestream velocity at K1 = 0.3. This produces much lower esti-mates of the microchannel Reynolds number, as compared with the freestream veloc-ity. Thus, the open microchannel flow is assumed to be fully laminar.

In Figure 5.16, the ratio of actual entropy production to the minimum entropy production (called the entropy generation number, Ns) is plotted at varying length ratios (L/Lopt) and expansion angles of the microchannels. Linearly converging micro-channels and airflow at 300 K are considered. Other problem parameters are depicted in Figure 5.16. For small surface areas (low values of L/Lopt), the net entropy produc-tion occurs mainly from the thermal irreversibility. The varying expansion angles have minor effects on Ns at low values of L/Lopt, as those characteristics mainly affect the friction irreversibilities. On the other hand, the slip-flow friction irreversibilities rise faster than the no-slip case for all the expansion angles in Figure 5.16. For a specified surface length, the entropy production increases faster at smaller base and exit expansion angles, relative to the corresponding minimum entropy production, which decreases with added slip-flow area.

1

10

100

1E – 03 1E – 02 1E – 01 1E + 00 1E + 01 1E + 02 1E + 03L/Lopt

NS

Plate; without MicrochannelsExpansion Angles: 0.1, 0.4 (rad)Expansion Angles: 0.1, 0.9 (rad)Expansion Angles: 0.3, 0.4 (rad)Expansion Angles: 0.3, 0.9 (rad)

Air (300 K)q´ = 10 W/md/W = 1.0E – 06K1 = 0.1(Converging Microchannels)

FIg u r e5.16 Comparison of predicted entropy generation number with benchmark result.

7262_C005.indd 139 1/19/08 11:45:50 AM


Unlike previous cases with diverging microchannels, the predicted results in Figure 5.17 consider converging microchannels. The microchannel converges into the central parallel section, rather than expanding outward from it. On the verti-cal axis of Figure 5.17, the slip-flow area decreases at lower values of the profile parameter. For example, because the geometrical configuration represents con-verging microchannels, the slip-flow area at a = -0.7 exceeds the slip-flow area at a = -0.6. In Figure 5.17, the profile parameter decreases at higher slip coefficients. A higher slip coefficient overcomes the added friction irreversibility of less slip-flow area. Also, the profile parameter decreases at lower Reynolds numbers, which also entails reduced friction irreversibilities with a smaller surface area.

The corners of the embedded microchannels represent a transition connecting the no-slip regime (above microchannel; Kn < 0.001) and slip-flow regime (within a microchannel; 0.001 < Kn < 0.1). When calculating the local Knudsen number, the corresponding length scale must accommodate both microchannel depth and width, or a hydraulic diameter-based length. Otherwise, no-slip conditions could be errone-ously predicted near the corners. For example, a wide microchannel with a submi-cron or nanoscale depth could produce an unrealistically small Knudsen number if the width alone were used. It is expected that the local Knudsen number decreases below 0.001 and moves into the no-slip regime at some point near the top corner. This arises with diminished effects of side walls on the intermolecular interactions near the corners. This transition to no-slip conditions is assumed to produce a small submicron semicylindrical type zone of influence at the top corners. This zone pen-etrates mainly into the open microchannel, as fully no-slip conditions are expected outside of the microchannels.

The previous results have demonstrated a new method of surface-embed-ded microchannels for reducing wall friction, while simultaneously improving heat transfer effectiveness by reducing the overall entropy production. It is shown that local slip-flow conditions within the surface microgrooves can reduce the net

–1.0

–0.9

–0.8

–0.7

–0.6

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Slip Coefficient (K1)

Opt

imiz

ed P

rofil

e Par

amet

er (a

) q´ = 100 W/m (Re = 200,000)

q´ = 100 W/m (Re = 400,000)

q´ = 200 W/m (Re = 200,000)

q´ = 200 W/m (Re = 400,000)

A ir (300 K) n = 2,400 U = 40 m/s

2 = 0.9 d/W = 2.0E – 06 (Converging Microchannels)

θ

FIg u r e5.17 Sensitivity to slip coefficient (converging microchannels).

7262_C005.indd 140 1/19/08 11:45:51 AM


entropy generation below minimum values achieved without such embedded micro-channels. Entropy production is reduced when less kinetic energy is dissipated to internal energy, due to the slip-flow surface behavior. The method of surface micro-profiling takes advantage of optimally placed slip-flow regions interspersed in the cross-stream direction across the surface.

In contrast to other conventional methods that optimize macroscale parameters of surfaces, such as the width or aspect ratio of plates in external flow, this section has optimized the microscale features of a surface. The optimal spacing between microchannels and microchannel aspect ratios were predicted with a newly devel-oped technique called EBSM. These conditions establish the most effective com-promise between friction and heat transfer irreversibilities. It was shown that embedded open microchannels within a surface can sufficiently reduce wall fric-tion through slip-flow conditions, to overcome added friction from the larger sur-face area of these added microchannels. Similar enhancements of added thermal effectiveness can be achieved with the new technique, thereby offering a useful alternative over conventional methods of heat transfer enhancement, such as baf-fles, fins, and spiraling.

r eFer en c es

Adeyinka, O.B. and G.F. Naterer. 2004. Optimization correlation for entropy production and energy availability in film condensation. Int. Commun. Heat Mass Transfer, 31(4): 513–524.

Anderson, J.L. and J.A. Quinn. 1972. Ionic mobility in microcapillaries. Faraday Trans. I, 68: 744–748.

Avsec, J. 2003. Calculation of equilibrium and nonequilibrium thermophysical properties by means of statistical mechanics. J. Tech. Phys., 44: 1–17.

Bejan, A. 1996. Entropy Generation Minimization. CRC Press, Boca Raton, FL.Camberos, J.A. 2002. On the Construction of Exergy Balance Equations for Availability

Analyses. AIAA/ASME 8th Joint Thermophysics Conference, AIAA Paper 2002-2880 (24–27 June). St. Louis, MO.

Camberos, J.A. 2003. Quantifying Irreversible Losses for Magnetohydrodynamic (MHD) Flow Simulation. AIAA 36th Thermophysics Conference, AIAA Paper 2003-3647 (23–26 June). Orlando, FL.

Cho, S.K., Moon, H., and C.J. Kim. 2003. Creating, transporting, cutting and merging liquid droplets by electrowetting based actuation for digital microfluidic circuits. J. Micro-electromech. Syst., 12: 70–80.

Choi, C.H., Westin, K.J.A., and K.S. Breuer. 2002. To Slip or Not to Slip — Water Flows in Hydrophilic and Hydrophobic Microchannels. International Mechanical Engineer-ing Conference and Exposition. Proceedings of IMECE 2002-33707 (Nov. 13–16). New Orleans, LA.

Czarske, J., Buttner, L., Razik, T., Muller, H., Dopheide, D., Becker, S., and F. Durst. 2002. Spatial Resolved Velocity Measurements of Shear Flows with a Novel Differential Doppler Velocity Profile Sensor. 11th International Symposium on Applications of Laser Techniques to Fluid Mechanics (July 8–11). Lisbon, Portugal.

Ferziger, J. and H.G. Kaper. 1972. Mathematical Theory of Transport Processes in Gases. North-Holland, London.

Gad-el-Hak, M. 1999. The fluid mechanics of microdevices — the freeman scholar lecture. ASME J. Fluids Eng., 121(March): 5–33.

7262_C005.indd 141 1/19/08 11:45:52 AM


Israelachvili, J.N. 1986. Measurement of the viscosity of liquids in very thin films. J. Colloid Interf. Sci., 11(1): 263–271.

Kays, W. M. and M.E. Crawford. 1980. Convective Heat and Mass Transfer. McGraw-Hill, New York.

Martin, M.J. and I.D. Boyd. Blasius Boundary Layer Solution with Flip Flow Conditions. Rarefied Gas Dynamics 22nd International Symposium. Sydney, Australia, July 9–14, 2000.

Maxwell, J.C. 1879. On stresses in rarefied gases arising from inequalities of temperature. Phelps Trans. R. Soc., 170: 231–256.

Migun, N.P. and P.P. Prokhorenko. 1987. Measurement of the viscosity of polar liquids in microcapillaries. Colloid J. USSR, 49(5): 894–897.

Naterer, G.F. 2001. Establishing heat-entropy analogies for interface tracking in phase change heat transfer with fluid flow. Int. J. Heat Mass Transfer, 44(15): 2903–2916.

Naterer, G.F. 2004. Adaptive surface micro-profiling for microfluidic energy conversion. AIAA J. Thermophys. Heat Transfer, 18(4): 494–501.

Naterer, G.F. and O.B. Adeyinka. 2005. Microfluidic energy loss in a non-polarized thermo-magnetic field. Int. Journal Heat Mass Transfer, 48: 3945–3956.

Naterer, G.F. and J.A. Camberos. 2003. Entropy and the second law in fluid flow and heat transfer simulation. AIAA J. Thermophys. Heat Transfer, 17(3): 360–371.

Ng, E.Y.K. and S.T. Tan. 2004. Computation of three-dimensional developing pressure-driven liquid flow in a microchannel with EDL effect. Numerical Heat Transfer: Part A: Appl., 45(10): 1013–1027.

Ogedengbe, E.O.B., Naterer, G.F., and M.A. Rosen. 2006. Slip flow irreversibility of dissipa-tive kinetic energy and internal energy exchange in microchannels. J. Micromechan-ics Microengineering, 16: 2167–2176.

Pfahler, J., Harley, J., and H. Bau. 1990. Liquid transport in micron and submicron channels. Sensors Actuators, A21–A23: 431–434.

Pfahler, J., Harley, J., Bau, H., and J.N. Zemel. 1991. Gas and liquid flow in small channels. Symp. Microelectromech. Sensors Actuators Syst., 32: 49–60.

Salas, H., Cuevas, S., and M.L. de Haro. 1999. Entropy generation analysis of magnetohydro-dynamic induction devices. J. Phys. D: Appl. Phys., 32: 2605–2608.

Sharp, K.V. and R.J. Adrian. 2004. Transition from laminar to turbulent flow in liquid filled microtubes. Exp. Fluids, 36(5): 741–747.

Zhao, B., Moore, J.S., and D.J. Beebe. 2001. Surface-directed liquid flow inside micro-channels. Science, 291(5506): 1023–1026.

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143

6 Numerical Error Indicators and the Second Law


As discussed in previous chapters, entropy provides a valuable design tool for ana-lyzing performance of fluid engineering systems and designing alternatives that improve energy efficiency. This book has been primarily focusing on local entropy tracking within individual components of a system, while other authors such as Rosen and Dincer (1999) have developed comprehensive methods of overall exergy analyses of a system. This chapter focuses on the role of entropy and the Second Law of Thermodynamics in numerical simulations, particularly involving error indicators for computational fluid dynamics (CFD). Entropy indicates the degree of molecular chaos or randomization, and this disorder can be interpreted in a physical sense (a traditional view), as well as a computational sense (a more recent view). The tradi-tional view may be traced back to pioneering developments by the German math-ematical physicist, Rudolf Clausius, in 1850, on the importance of entropy in steam engine performance. Computational modeling of entropy has arisen more recently with the advent of digital computers. It relates entropy and the Second Law with dis-cretization errors (Naterer and Schneider, 1987), artificial dissipation (Hughes et al., 1986), and nonphysical numerical results (Majda and Osher, 1979). This chapter will focus on numerical errors, whereas the following chapter will describe the role of entropy and the Second Law in solution uniqueness and numerical stability of CFD simulations.

In early pioneering work, Lax (1971) implemented a discrete entropy equation to identify physically relevant and unique solutions in finite difference compressible flow simulations. Harten (1983) then symmetrized the governing equations through a change of variables (entropy gradient variables) to improve the performance of iterative algebraic solvers. Merriam (1987) has shown that satisfaction of the Sec-ond Law is sufficient, in some cases, to ensure stability of compressible flow com-putations. This numerical stability also suggested that entropy could serve as an effective error indicator and criterion for solution convergence. Camberos (1998) showed that entropy provides an effective measure of residual error and convergence because of its physical significance with a full functional dependence on all fluid state variables. Naterer and Schneider (1994) have demonstrated that solution errors and nonphysical phenomena, such as numerical oscillations, coincide with a discrete violation of the Second Law. In this way, solution accuracy and entropy production are closely related in a numerical sense. Conventional error indicators with Taylor

7262_C006.indd 143 1/19/08 3:42:31 PM


series expansions can only indicate the limiting behavior of numerical errors, when the grid spacing approaches zero. A mathematical Taylor series analysis generally lacks a physical interpretation. It usually cannot identify errors for coarse grids. It also cannot identify nonphysical aspects of the numerical results. Entropy and the Second Law provide a useful alternative because they establish a physical basis that relates physical plausibility of results, numerical accuracy, convergence, reliability, and stability of simulations, all within the scope of the Second Law.

MacCormack’s second-order time-split scheme (MacCormack and Baldwin, 1975) and the ARC2D and ARC3D codes of Pulliam and Steger (1980) were key pio-neering developments of Navier–Stokes solvers for three-dimensional viscous com-pressible flows. Subsequent advances were made for boundary layer and shock wave interactions, unstructured grids (finite elements; Lohner et al., 1984), and conserva-tion-based methods (finite volumes; Karki and Patankar, 1989). However, much effort and difficulty arose from general error analysis and robustness of the numerical codes. Solutions were sensitive to time steps, grid spacing, and empirical constants in the schemes. Rigorous order accuracy often could not be established. Complicated prob-lems required specialized “tuning” of coefficients, but the tuning would be altered for each new problem. More grid points and faster computers could achieve more accurate solutions, but they could not necessarily bring a more robust or stable algorithm.

As many numerical methods lacked a unified approach to error analysis, subse-quent developments implemented entropy and the Second Law for this purpose in CFD codes for viscous compressible flows. Hughes and coworkers (1986) developed finite element schemes that satisfied the Second Law in a global sense. However, numerical oscillations may still occur in individual elements because the Second Law was not enforced at a local level. Merriam (1987) presented a general methodol-ogy for satisfying the entropy inequality on a cell-by-cell basis. A general method of analysis, rather than a specific finite element or finite volume scheme, was presented. Entropy-based corrections for error reduction were later implemented for compress-ible flows (Naterer and Schneider, 1994) and duct flows (Nellis and Smith, 1997).

Finite volume methods are widely used for compressible flow simulations because of their capabilities, conservation properties, and physically based discreti-zation (Patankar, 1980). The discrete equations are obtained by integration of the governing equations over discrete control volumes. Approximations of the convec-tion and diffusion terms are usually made at the midpoint of the volume surface (integration point). Unlike the conservation equations, entropy is governed by an inequality (Second Law). This chapter focuses on how entropy and the Second Law can effectively characterize the accuracy and numerical errors inherent in discrete modeling of the conservation equations, such as convective upwind schemes (next section). Also, this chapter will present a novel entropy-based approach for calculat-ing the residual error in steady-state problems. The technique calculates the differ-ence in entropy (averaged over the computational domain) as a metric to analyze solution convergence. Several steady-state calculations of viscous compressible flow fields will be presented, together with an averaged metric based on entropy. This metric is the difference in the averaged entropy from one time step to the next step. Convergence is reached when the entropy-based residual is reduced by several orders of magnitude. The attractive feature of an entropy-based residual is that it provides

7262_C006.indd 144 1/19/08 3:42:32 PM

Numerical Error Indicators and the Second Law 145

a global measure of changes in the numerical solution, and it implicitly depends on all problem variables simultaneously (mass, momentum, and energy). Typically, conventional error residuals and convergence indicators depend on a single-state variable or certain metrics that are limited to a selected set of problems. In contrast, the universality of entropy does not suffer from these limitations. Thus, it makes an ideal candidate as a robust error indicator and criterion for convergence.

6.2 d Isc r et Izat Io n er r o r s o f nu mer Ica l c o n v ec t Io n sc h emes

6.2.1 Fin it eVo l u meFo r mu l at io n

This section presents a procedure that applies entropy principles to a numerical scheme that satisfies the Second Law for a component of the overall formulation, namely, the convection scheme. The governing equations for viscous compressible flow and heat transfer are the Navier–Stokes equations. These equations have been presented in earlier chapters, but they will be rewritten here in a generalized trans-port form for a subsequent entropy analysis. Define a vector of conserved quantities, q, and a corresponding flux, f, with an advective component, fa, and a pressure (p) and diffusive component, fd.

q f=

=

ρρρρ

ρ ρρ ρρ ρ

u

v

e

u v

uu uv

vu vvand

(ρρ ρ

τ ττ

e p u e p v

p

pxx xy

yx

+ +

-- +

-) ( )

0 0

+++ - + -

ττ τ τ τ

yy

xx xy x xy yy yu v j u v j

(6.1)

The heat flux vector, j, in Equation 6.1, can be related to temperature, T, by Fourier’s law. For each conserved quantity, there exists a corresponding transported scalar, f . For example, x-momentum is conserved (q2 = ru), and the scalar f = u is transported by the flow in the momentum equation. The governing equations can be written in the following conservation form or a nonconservation transport form (excluding continuity):

∂∂+ ∇ ⋅ + ∇ ⋅ =q

f ft

a d 0 ( )Conservation Form (6.2)

ρ φ ρ φ∂∂+ ⋅∇ +∇ ⋅ =

tdv f 0 ( )Nonconservation Form (6.3)

The components of the stress tensor, t, in Equation 6.1 are

τ µ µxx

u

x

u

x

v

y= ∂

∂- ∂

∂+ ∂∂

223

(6.4)

7262_C006.indd 145 1/19/08 3:42:36 PM


τ µ τxy yx

u

y

v

x= ∂

∂+ ∂∂

=

(6.5)

τ µ µyy

v

y

u

x

v

y= ∂

∂- ∂

∂+ ∂∂

223

(6.6)

where m is the dynamic viscosity. In addition, the ideal gas law, p = r RT, and the relations g = cp/cv and R = cp - cv, where cp and cv refer to specific heats, allow cal-culations of pressure as follows:

p e u v= - - -

( )γ ρ1

12

12

2 2 (6.7)

Consider a numerical discretization with a problem domain subdivided into finite volumes and elements. In one dimension, Figure 6.1a illustrates the grid structure, whereas Figure 6.1b shows the appropriate schematic definitions for two dimensions. In one dimension, a control volume is defined by the two adjacent half-elements surround-ing each node, and the integration points (ip) are located at the control volume surfaces. The integration point resides at the element midpoint. Linear interpolation functions are used within each element to represent the variation of dependent scalars, in terms of nodal variables. Integrating Equation 6.2 over a control volume and time step,

V V t

t t

St t dV t dV dAd∫ ∫ ∫ ∫+ ∆ - + ⋅ =

+∆

q q f n( ) ( ) t 0 (6.8)

i–1 i–1/2 i+1/2 i i+1

Control Volume (a)

Integration Points at Control Volume Surfaces

Element

Local Node Number 2

Finite Element

(b)

SubvolumeNode

1

Control Volume Upwind Difference

Upstream Point

Flow Direction

SCV 1

iP

SCV 2

SCV 44

f Ig u r e6.1 (a) One-dimensional, and (b) two-dimensional schematic of a finite volume.

7262_C006.indd 146 1/19/08 3:42:43 PM


The discrete equations can be obtained by integration over a specific time interval t t tn n≤ ≤ +1 and one-dimensional volume x x xi i- / + /≤ ≤1 2 1 2. The discretized conserva-tion and nonconservation (transport) forms of the governing equations, Equation 6.2 and Equation 6.3, respectively, then become

q q fi

nin

i

t

++ /-

∆+

11 22 1 2

1

0-∆

=

-∆

- /

+

fi

i

in

in

x

t

( )Conservation Form

φ φ ++∆

- ∂∂

=∑1

xm

xso

i ip

ip ip ipip

φ φΓ uurces ( )Nonconservation Form

(6.9)

(6.10)

where m refers to the mass flow rate, Γ represents a general diffusion coefficient, and “sources” refers to the remaining source-type terms in the governing equation. Conventional methods of interpolation are used to approximate the transient, dif-fusion, and source terms. To specify a well-posed algebraic system, the advection terms in Equation 6.10 at the integration points need to be related to nodal variables. This requires modeling for the transport of a scalar quantity, f, across a control vol-ume surface, or integration point, such as f = u in the momentum equation.

6.2.2 Cen t r a l ,upw in d ,a n d expo n en t ia l diFFer en Cin g SChe meS

Various methods can be used in numerical schemes to estimate an integration point value such as φi+ /1 2. For example, the approximation φ φi i+ / =1 2 represents an upwind differencing scheme (UDS). In two-dimensional problems, an analogous procedure is the skew upwind differencing scheme (SUDS), which uses the local flow direction to determine the appropriate upstream location for the scalar vari-able approximation. Without the influence of pressure forces on the integration point velocity, there can be a nonphysical decoupling between pressure and veloc-ity. For example, if a large pressure gradient in a flow field has no direct influence on the integration point velocity, it could lead to direct violation of the Second Law. In contrast to UDS, the central differencing scheme (CDS) uses linear interpola-tion between adjacent nodal values to find the integration point variable, that is, φ φ φi i i+ / += + /1 2 1 2( ) . In CDS, the convective flux dependence on downstream vari-ables may have nonphysical trends when the Peclet number is high and upstream convection influences are dominant.

Convection models may use some combination of adjacent nodal values for the integration point approximations. For example, hybrid schemes such as the exponen-tial differencing scheme (EDS) provide the correct balance between UDS and CDS influences, based on the local grid Peclet number (Pe u xi i= ∆ /ρ Γ) (Minkowycz et al., 1988). EDS obtains a smooth transition from CDS for Pe→ 0 to UDS for Pe→∞. Neglecting transient, pressure, and source terms in Equation 6.3 and solving the resulting equation subject to specified values of f at the nodes yield the EDS solution. Evaluating f at the integration point with this EDS solution,

7262_C006.indd 147 1/19/08 3:42:51 PM


φ α φ α φi i i+ / += +

+ -

1 2 1

12

12

(6.11)

where

α = - --

≈+

/1

2 11 5

2 2

2

( )e

e

Pe

Pe

Pe

Pe

(6.12)

The latter approximation is an alternative to reduce the computational expense of frequent exponential calculations (Minkowycz et al., 1988, see Chapter 7 by Raithby, G.D. and Schneider, G.E.). This scheme has first-order accuracy, in terms of the Taylor series truncation error. Higher-order schemes, such as quadratic upstream interpolation for convection kinetics (QUICK) (Leonard, 1979), reduce the discreti-zation errors through quadratic interpolation for integration point values. The finite element differential scheme (FIELDS) solves an approximation to the governing equations at the integration point to incorporate the local fluid physics, such as local pressure and source term effects (Schneider and Raw, 1987). It will be useful to determine whether FIELDS, UDS, and other schemes comply with the requirements of the Second Law at a local (control volume) level. Such schemes will be denoted as “entropy-stable” schemes. Solutions that obey the Second Law will exhibit proper physical characteristics.

A flow field governed only by the conservation laws in Equation 6.2, but not nec-essarily the discrete form of the Second Law, could display unusual physical behav-ior. For example, it is highly improbable that heated fluid elements could become sufficiently organized to independently produce a cold fast fluid stream that converts internal energy to kinetic energy. Although possible through the First Law, the sta-tistical probability of observing this process is extremely small, according to the Second Law. The Second Law states that entropy, which is a property of matter that measures the degree of disorder at the microscopic level, can be produced, but never destroyed in an isolated system. These observations also apply to numerical compu-tations, wherein numerical approximations should not produce nonphysical results that violate the Second Law.

The Second Law of Thermodynamics is written below in a form similar to Equation 6.2:

P Ss t x= + ≥, ,F 0 (6.13)

where the subscript notation with a comma refers to differentiation. For example, the subscript “x” refers to a partial derivative with respect to x in one dimension, or the divergence operator in multidimensions. Also, sP refers to the entropy produc-tion rate and S(q) and F(q) represent the thermodynamic entropy and entropy flux, respectively, that is,

S s= ρ (6.14)

F us= ρ (6.15)

7262_C006.indd 148 1/19/08 3:42:56 PM


and s represents the specific entropy. For an ideal gas,

s c logp p

v=//

0

0( )ρ ρ γ

(6.16)

where the subscript 0 refers to values at a specified initial state. In Equation 6.13, the equality refers to reversible processes, and the inequality refers to irreversible processes.

The entropy and entropy flux must obey two important mathematical properties:

S convexity, <qq 0 ( )

S compatibility, , ,=q q qf F ( )

The convexity condition requires irreversible processes to produce entropy. It ensures that entropy is bounded from above, because S,qq must be a negative definite matrix. The entropy distribution typically reaches a maximum value at thermal and mechanical equilibrium. In the compatibility criterion, F,q represents the entropy flux derivative matrix (a second-order tensor) with a vector component in each of the three coordinate directions. Also, f,q is a third-order tensor that denotes a derivative of four fluxes in three directions with respect to four conservation vari-ables. The compatibility condition guarantees the existence of an entropy flux satisfying the Second Law, whenever an entropy conservation principle holds for reversible processes.

For a discrete volume, the Second Law can be expressed as

s

in

in

i i

iP

S S

t x = -

∆+ -

∆≥

++ / - /

11 2 1 2 0

F F (6.17)

After the solution of the conservation equations is obtained, an additional step is required to find q(x,t) from the nodal and integration point values so that S(q) and F(q) can be properly integrated. An approach that does not violate the Second Law during this reconstruction step is needed. In this way, if a negative entropy production rate arises in the numerical analysis, it can be attributed to the discretized conserva-tion equations, rather than the entropy inequality, Equation 6.17. Thus, assume that q = qi within the control volume, where the subscript i refers to node i. This assumption meets the previous requirement because a piecewise constant distribution maximizes the entropy within each control volume with respect to the choice of qi. A fundamen-tal result of thermodynamics states that for all processes at a constant total volume and energy, the entropy increases or remains constant. Thus, when a system reaches thermal and mechanical equilibrium, then its entropy must be a maximum. Because the state transition from q(x,t) to qi is physically irreversible (in practice), the entropy contained within an isolated control volume must increase and achieve a maximum value at the equilibrium state, q(x,t) = qi. It will be approximated that q is piecewise constant, at its integration point value, along each control surface.

7262_C006.indd 149 1/19/08 3:43:00 PM


The entropy is expanded with a truncated Taylor series as follows:

S S S t t S t ti i

nt

ntt

n= + - + -, ,( ) ( )( )12

2η (6.18)

where t tn ≤ ≤η . Similarly, defining q qi i≤ ≤ + /ζ 1 2 and expanding the entropy flux about an integration point,

F F F Fi i i i i+ / , + / , + /= + - + -1 2 1 2 1 2

12q qqq q q q( ) ( )(ζ ii )2 (6.19)

If q is scalar, then the squared term in Equation 6.19 represents a scalar multiplica-tion. Otherwise, when q is a vector, then the term is evaluated by the product of the vector and its transpose. In a similar fashion, we can expand the entropy flux about the other integration point.

F F F Fi i i i i i- / , - / , - /= + - + -1 2 1 2 1

12q qqq q q q( ) ( )(ζ 22

2) (6.20)

Substituting these relations into the expression for the entropy production,

s t

i iP S F

xS = + -

∆

+, ,

+ / - /,q

q q1 2 1 2 12 ttt

i i i it Fx

∆ + - - -∆

,

+ / - /qq

q q q q( ) ( )1 22

1 22

(6.21)

Using the compatibility condition, and simplifying the first term in Equation 6.21,

s t

i iP S

x = + -

∆

+, , ,

+ / - /q qq f

q q1 2 1 2 122

1 22

1 22

S t Fxtt

i i i i, ,

+ / - /∆ + - - -∆

qq

q q q q( ) ( )

(6.22)

This equation expresses the entropy production rate in terms of several problem vari-ables, so it is difficult to implement or verify the positive definite character of each indi-vidual term. Simplifying the expression, in the first term with another Taylor series,

q q q qi i x i i xx i ix x x x+ / , - / , + /= + - + -1 2 1 2 1 2

212

( ) ( ) (6.23)

Writing another similar expansion about x xi= - /1 2 and substituting the results into Equation 6.22,

s t x ttP S S t = + + ∆ ≥, , , ,q q f( )

12

0 (6.24)

The row vector S,q in Equation 6.24 represents a rate of change of entropy with respect to the conserved quantity. If the conservation equations are solved in an exact

7262_C006.indd 150 1/19/08 3:43:08 PM


fashion, then the first term in Equation 6.24 vanishes, and the second term remains positive to satisfy the entropy inequality. It should be noted that Equation 6.24 remains valid at both the control volume level (where the overall conservation equa-tions are applied) and the integration point level, where modeling of convective terms, such as φ φi i+ / =1 2 (UDS), are made. In the latter case, violation of the inequality in Equation 6.24 may lead to nonphysical errors such as false diffusion (in the case of UDS) or oscillations (in the case of CDS). The subgrid modeling of convection at the integration points should be governed by the same entropy requirements as the overall conservation equation.

The bracketed component of the previous entropy inequality contains the scalar conservation equation. Discretizing that equation by standard differencing techniques,

L a

ta u

x

n ni i

i

( )q + = -∆

+

-∆

++ /δ ρ φ φ ρ φ φ

1

1

21 2

//

+ -

∆ /

+ -+ / +

2 2

23

1 24

1aP P

xai i

i

iΓ φ φii i

ix+ / +

∆

1 2

2

φ (6.25)

In this form, the exact equation, L(q) = 0, Equation 6.3, is replaced by a discrete approximation, L( ) ,q + =δ 0 where L() refers to the differential operator (left side) in Equation 6.3 and δ refers to discretization errors at xi+ /1 2. It is known that δ → 0 as the grid and time step are refined. The conventional models for integration point approximations can be extracted from Equation 6.25 as follows:

1. CDS for a a a1 2 3 0= = = =δ and a4 0≠ .

2. UDS for a a a1 3 4 0= = = =δ and a2 0≠ .

3. EDS for a a1 3 0= = =δ , a Pe2 = /α , and a4 1 2= - /( )α .

Figure 6.2 shows these coefficients and their dependence on Pe. As Pe→∞, it can be shown from Equation 6.25 that | | | |a a4 2<< . Thus, the UDS upstream values dominate the downstream influences, even though a2 becomes small.

0.1 0.01 0.001 1 10 100 1000 Peclet Number

–0.6

–0.4

–0.2

0

0.2

0.6

0.4

Upw

ind

Coeffi

cien

ts

a2 (CDS) a4 (CDS)a2 (EDS) a4 (EDS)a2 (UDS) a4 (UDS)

f Ig u r e6.2 Conventional upwind coefficients.

7262_C006.indd 151 1/19/08 3:43:22 PM


The following case study aims to determine whether these coefficients violate the entropy inequality in Equation 6.24. The first term after the equality in Equation 6.24 in a product between each component of the row vector S,q and a conservation (or transport) equation represented by each component of a column vector, i.e., S,q1 (continuity) + S,q2 (momentum transport equation) + S,q3 (energy transport equa-tion). The numerical entropy production of each individual contribution should be high enough to prevent an overall negative sum and a violation of Equation 6.24. The effect of a particular transport equation can be isolated within Equation 6.24 to determine its impact on entropy stability of the numerical method.

6.2.3 Ca SeSt u d yo Fno zzl eFl o w an a lySiSa n d deSig n

This case study examines the entropy stability of convection schemes for converging- diverging nozzle and channel flows (see Figure 6.3). Numerical simulations were con-ducted with a control volume-based finite element formulation of the viscous com-pressible flow equations (Naterer, 1999). Figure 6.4 shows the steady-state density, velocity, pressure, and temperature distributions, respectively, for the three example cases. For the fully subsonic flow, the flow accelerates in the converging section until a point of maximum velocity and minimum pressure is reached at the throat. Then the flow decelerates in the diverging section until it reaches the prescribed outlet condition. However, in the supersonic flow example, the maximum Mach number (Ma = 2.2) occurs at the outlet because the flow continually accelerates through the diverging section of the nozzle to meet the prescribed outlet condition.

The example involves compressible air flow through a converging–diverging nozzle with a specified cross-sectional area, A(x):

A x A A Ax

xth e th( ) ( )= + - - , ≤

2

15

5

(6.26)

A x A A Ax

xth e th( ) ( )= + - - , ≥

2

51 5

(6.27)

where 0 10≤ ≤x m[ ] and Ath and Ae represent the throat and exit areas (Ae = 2.035 Ath; see Figure 6.3). Gas properties for air include g = 1.4, cp = 1004[J/kgK], and R = 287 [J/kgK]. In this problem, the stagnation pressure (P0 = 93.75 [kPa]), temperature

ShockWave

NozzleInlet

Pe

5.0 5.0

Mai , Pi , Ti

Converging-DivergingNozzle

f Ig u r e6.3 Schematic of converging-diverging nozzle.

7262_C006.indd 152 1/19/08 3:43:26 PM


(T0 = 300 [K]), and inlet Mach number (Mai = 0.3) are specified. The inlet conditions are used as initial conditions throughout the problem domain. No-slip conditions are defined along the nozzle walls. Numerical simulations were conducted with a time-marching scheme from the initial conditions to the final steady-state solution.

Three different back pressures are specified, and these three values pro-duce the following results in the diverging section of the nozzle: (1) subsonic flow (Pe = 80 [kPa]); (2) mixed flow with a shock wave (Pe = 50 [kPa]); and (3) supersonic flow (Pe = 8.8 [kPa]). If the back pressure is not low enough to induce sonic condi-tions at the throat, then the flow remains subsonic throughout the nozzle (Case 1). As the back pressure is further reduced, the throat eventually becomes sonic, and the mass flux through the nozzle reaches a maximum value. If the back pressure is reduced below this critical condition, then the throat remains choked at the sonic value, and a normal shock wave occurs in the diverging section to meet the outlet condition (Case 2). The design pressure ratio is achieved when the back pressure is further reduced until the diverging flow is entirely supersonic (Case 3). This design condition is often used for efficient operation of a rocket exhaust.

For accelerating flow in a converging nozzle, the row vector S,q in Equation 6.24 refers to a rate of change of entropy with respect to a conserved quantity. For exam-ple, if the x-momentum (q2 = ru) increases by an incremental amount, dq2, when the flow accelerates through a converging nozzle, then the term S,q2dq2 represents the associated entropy increase. The term Sqdq would represent the cumulative effect

0 0.00

0.25

0.50

0.75

1.00

1.25

1.50

1 2 3 4 5 6 7 8 9 10 Position (m)

Den

sity (

kgm

3 )

Subsonic Outlet

Supersonic Outlet Flow with Shock

0 0.00

100.00

200.00

300.00

400.00

500.00

600.00

1 2 3 4 5 6 7 8 9 10 Position (m)

Velo

city

(m/s

)

Subsonic Outlet


0 0.00

20.00

40.00

60.00

80.00

100.00

120.00

1 2 3 4 5 6 7 8 9 10

Pres

sure

(kpa

)

Subsonic Outlet

Supersonic Outlet

Flow with Shock

Position (m)

0 100.00

150.00

200.00

250.00

300.00

350.00

450.00

400.00

1 2 3 4 5 6 7 8 9 10Position (m)

Tem

pera

ture

(kg)

Subsonic Outlet


(a) Density

(b) Velocity (d) Temperture

(c) Pressure

f Ig u r e6.4 Profiles of (a) density; (b) velocity; (c) pressure; and (d) temperature.

7262_C006.indd 153 1/19/08 3:43:28 PM


of changes in all conserved quantities on the entropy change. From Equations 6.1, 6.7, 6.14, and 6.16, the entropy derivative becomes

S s c

u

Pc

u

Pcv v v, = + - + -

, - - ,q γ γ ρ γ ρ ρ( ) ( )1

212 (( )γ -

1P

(6.28)

The components of S,q are illustrated in Figure 6.5 for the case of air with g = 1.4, T = 300[K], r = 1.16[kg/m3], cv = 717.4[J/kgK], and P0 = 101[kPa]. In Equation 6.26 for one-dimensional flows from left to right, the sign of S,q1 may be positive or nega-tive, but S q, ≤2

0 and S q, ≥30. These properties can be observed in Figure 6.5. In

physical terms, this implies that the entropy would decrease with a positive change, dq2. In Figure 6.5, it can be observed that | |S q, 2

for supersonic flow is larger than the corresponding subsonic value at a fixed pressure.

For subsonic converging-diverging nozzle flow, from Equation 6.28, numerical entropy production associated with discretization of the momentum equation is

2 21 21

2( ) ( )S

c u

Pa u

u u

xx

v i i

i

, ,+ /=

- - -∆ /

qf

γ ρρ

+-

∆ /+

- ++ / + + /aP P

xa

u ui i

i

i i3

1 24

1 1 2

2

2Γ

uu

xi

i∆

2

(6.29)

This equation indicates a set of necessary constraints on conventional upwind schemes to satisfy the entropy inequality. For example, consider an accelerating flow in a subsonic converging-diverging nozzle (i.e., ∂ ∂ >u x/ 0), upstream of the throat. If the flow remains subsonic, then it will decelerate upstream of the throat due to the upcoming area expansion in the duct. As a result, the concavity of the velocity profile changes in the region upstream of the throat, from concave upward to concave down-ward. Under these conditions with CDS, UDS, SUDS, or EDS ( a3 0= ), it can be observed that S x, , <qf 0 in Equation 6.29 and the entropy inequality in Equation 6.24 is violated. In practice, viscous terms are negligible outside the boundary layer in

3 4 5 6 7 1 2 –2

–1

0

1

2

3

4

Pressure (atm)

Alternating Signs

Positive–Definite

Negative–Definite

Subsonic (Ma=1.5)

Sq1/1000[J/kgK]Sq2 [m/sK]Sq3*1000 [1/K]

Sq1/1000[J/kgK]Sq2 [m/sK]Sq3*1000 [1/K]

Subsonic (Ma=0.5)

Entr

opy D

eriv

ativ

e

f Ig u r e6.5 Rate of entropy change.

7262_C006.indd 154 1/19/08 3:43:36 PM


this case, but nevertheless, Equation 6.29 still remains negative for accelerating flows. Thus, the upwinding must be modified to prevent negative entropy produc-tion. For example, the schemes may apply the approximation to decelerating flows instead, or modify the upwinding with additional terms.

Consider a modified upwind scheme that imposes a momentum constraint on the pressure terms in Equation 6.29, to satisfy Equation 6.24. A pressure influence at the integration point is introduced to prevent the problem in the previously mentioned example with an accelerating flow. For steady flow, Bernoulli’s equation can be writ-ten in the following form (Fox and McDonald, 1992):

Pu

Pu

lossesii

ii+ = + ++ /+ /ρ ρ

2

1 21 2

2

2 2 (6.30)

where losses ≥ 0 represent frictional losses. The velocity terms in Equation 6.30 can be factored in the following form:

P P u u u lossesi i i i- + - =+ / + /( )1 2 1 2ρ (6.31)

where u u ui i= + /+ /( )1 2 2. It is evident from Equation 6.30 and Equation 6.31 that setting a2 = a3 in the physical influence scheme (PINS) and a4 0≈ (negligible down-stream influences for high Pe number case) allows the losses to be isolated. Then Equation 6.27 may be rewritten as follows:

222

10( ) ( )

( )S a c u

x Plossesx

v

i

, , = -∆

≥qf γ ρ

(6.32)

Using this formulation, the Second Law requirement in Equation 6.24 is then satisfied.

Consider another example for more general duct flows with friction. A detailed comparison between PINS (subscript pins) and other conventional schemes (sub-script o), such as CDS, UDS, SUDS, and EDS, will be examined and interpretated in the context of general duct flows with friction. The wall shear stress is defined as τ ρw f u= /2 2, for incompressible flows where f refers to the friction factor (Fox and McDonald, 1992). Defining G = (S,q f,x) and Ec = u2/(cp ∆ T) (Eckert number), the difference between upwind schemes can be computed using the ideal gas law with the following result:

G Ga c u

x P

a

apins oo v

i

pins

o

- =-

∆- -, ,

,2

11

22 2

2

( ) (γ ρ γγγ-

1 1)

( )D

L fEclosses

(6.33)

where D and L refer to diameter and length, respectively.The critical points, Ec and D L/ , where this difference changes signs, occur when

Ecf

D

L

a

a aano

pins o

= - -

,

, ,

2 1 2

2 2

( )γγ

ddD

L

a

afEcpins

o

= -

-,

,

2

2

12 1γγ( )

(6.34)

7262_C006.indd 155 1/19/08 3:43:45 PM


Thus, conventional upwind schemes will exhibit entropy-stable behavior for low Eckert numbers below the critical points. These trends are illustrated in Figure 6.6, where the following sample constants have been selected: g = 1.4 (air), a2,pins = a3,pins = 1, and P c u losses a a xi v i pins o i≡ - - / ∆, ,( ) ( )( )γ ρ1 2 2

. Below the critical points, G Gpins o< but Gpins ≥ 0 for all Eckert numbers, so Go ≥ 0 is guaranteed also in this region. However, above the critical points, PINS satisfies the entropy inequality, Equation 6.24, but the other schemes may violate it. Both Figure 6.6 and Figure 6.7 show that the difference, G Gpins o- , increases with the friction factor at a specific Ec or D/L (diameter per length) ratio. This indicates that higher wall friction pro-duces more entropy for a pressure-weighted scheme than a scheme (such as UDS)

0 –8

–6

–4

–2

0

2

0.05 0.1 0.15 0.2Eckert Number

D/L=0.01: f=0.010 D/L=0.01: f=0.013 D/L=0.01: f=0.016 D/L=0.02: f=0.010D/L=0.02: f=0.013D/L=0.02: f=0.016

G(p

ins)

-G(o

) [J/m

ˆ3Ks

] Only PINS Ensures

Entropy Stability

All Schemes SatisfyEntropy Constraint

f Ig u r e6.6 Regions that satisfy the entropy constraint (in terms of the Eckert number).

0–4

–3

–2

–1

0

1

2

0.02 0.04 0.06 0.08 0.1D/L

G(p

ins)

-G(o

) [J/m

ˆ3Ks

] G(pins)>0 but-c<G(o)<c

Ec=0.2; f=0.010Ec=0.2; f=0.013Ec=0.2; f=0.016Ec=0.4; f=0.010Ec=0.4; f=0.013Ec=0.4; f=0.016

G(pins)>0 andG(o)>0

f Ig u r e6.7 Regions that satisfy the entropy constraint (in terms of D/L).

7262_C006.indd 156 1/19/08 3:43:51 PM


without this dependence. Friction, pressure loss, and entropy production are thus closely related to the integration point approximations. Because the current analysis only considers the momentum transport parts of Equation 6.24, the overall accuracy and entropy stability of an entire finite volume method has not been formally con-firmed. Each integration point discretization is generally constructed independently of the other transport equations. In closing, this case study has shown that pressure-weighted upwinding with PINS leads to entropy stability in the convective formula-tion of the momentum equation.

6.3 Ph ysIca l Pl au sIbIl It y o f nu mer Ica l resu l t s

6.3.1 en t r o pyCo r r eCt io n o Fnu mer iCa l diFFu Sio n

In the previous section, compliance with the Second Law was outlined for individual components of the numerical formulation (specifically the convective upwind scheme). In the absence of preventative measures to ensure physically plausible results, an alter-native is a corrective measure that uses errors based on computed negative entropy production to recalculate results, therein striving to satisfy the Second Law. This sec-tion describes a corrective procedure that first detects anomalous flow patterns in the flow field (like numerical oscillations) using the local entropy production rates, then performs a corrective procedure by applying a required diffusion coefficient to ensure positive entropy production. The analysis will be performed with a control volume-based finite element method (Naterer and Schneider, 1994).

Let Vj denote the volume associated with node j, so the integral form of the Sec-ond Law can be written as

VS q

tF q dsj

s j

∂∂+ ⋅ ≥∫( )

( )

0

(6.35)

To perform the integration, a typical four-noded, quadrilateral finite element was illustrated in Figure 6.1b. The element comprises four subcontrol volumes, each of which is associated with the node located at its outermost corner. The subcontrol-volume boundaries are defined by the element external surfaces and lines corre-sponding to local coordinate values of s = 0 and t = 0 (origin at the center of the element). Considering the shaded subcontrol-volume of Figure 6.1b, integration of Equation 6.35 over a subcontrol volume results in

Sj

ipj ipjj

F q ds F q s∫ ∑⋅ = ⋅ ∆=

( ) ( )

1

4

(6.36)

where in each term, ∆s is an outward facing normal at the midpoint of the appropri-ate edge.

Two alternative temporal discretizations will be considered in the formulation of the Second Law. The first approach is an explicit backward difference for the transient term. This backward difference may potentially lead to a violation of the Second Law inequality if the temporal discretization is inadequate. The second

7262_C006.indd 157 1/19/08 3:43:53 PM


approach is a “semidiscrete” method, wherein the entropy time derivative is rewrit-ten in terms of spatial derivatives through the chain rule, and no discrete time step is introduced (Naterer, 1989). Both methods were described previously in Section 2.7.3. Although the semidiscrete formulation is exact in one dimension (Merriam, 1988), it too may violate the Second Law through its midpoint approximations of entropy and conserved variable fluxes in multidimensions. It is desirable to use a discretiza-tion that minimizes the numerical entropy production rate such that entropy-violat-ing solutions are not concealed by artificial entropy production through temporal differencing.

As shown in Section 2.7.3, the difference between entropy production rates obtained from the semidiscrete and fully discrete formulations is a linear function of the entries of the Hessian of S q( ).

This remainder term (denoted by ( )Ps j < 0 ) can

be written in the following summation form (Naterer, 1989):

( )s jR j

l m

lm l mPV

th =

∆= =∑∑2

1

4

1

4

α α

(6.37)

where hlm denotes the entries of the Hessian matrix, H S q= ∂ ∂2 2/ , and α l l jn

l jnq q= -,

+,( )1 .

Because H is convex (negative definite), the quadratic form given by the double sum in Equation 6.37 must be negative for all ( )α α α α1 2 3 4, , , , and it follows that

( )s j

RP ≤ 0 (6.38)

This result shows that the fully discrete entropy production rate is less than or equal to the semidiscrete entropy production rate, independent of the time step or control volume size. At steady state, ( )j

njnq q+ -1

vanishes, and the entropy production rate is entirely determined by the spatial discretization. In this case, the equality in Equa-tion 6.38 holds. In a similar manner, it can be shown that the volumetric entropy production rate formed by implicit time advance is less than or equal to the semi-discrete entropy production rate. These results are equally valid for both Euler and Navier–Stokes equations, except that the entropy function, S q( ),

and entropy flux

F q( ) will be different.The Euler equations do not provide any natural dissipation mechanism (such

as viscosity in the Navier–Stokes equations) to diffuse numerical oscillations resulting from inadequate mesh refinement in regions of large gradients like shock waves. Instead, smoothing algorithms such as methods of Lohner et al. (1984) and MacCormack (1975) have been used to add artificial dissipation terms. Early pioneer-ing studies of Von Neumann and Richtmyer (1950) developed various techniques like the user-specified constants to control numerical oscillations and stability. Upwind differencing introduces an implicit artificial viscosity into a scheme. The effect of artificial viscosity reduces the spatial flow gradients. It arises from even derivative terms in the truncation error (called numerical dissipation). When odd derivative terms appear in the truncation error, the properties of various waves are distorted. This quasiphysical effect is called dispersion. A key benefit of an entropy-based error analysis is that the Second Law is sensitive to both of these errors, because the resulting effects are both nonphysical.

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From previous chapters, the following entropy transport and entropy production equations have been derived:

ρ µDs

Dt

q

T T= - ∇ ⋅ +

′′ Φ (6.39)

sP

k T T

T T = ∇ ⋅∇ +

2

µΦ (6.40)

where Φ refers to the velocity gradient portion of the dissipation function. Because ∇ T ∙ ∇ T, Φ, and the fluid properties in Equation 6.40 are all greater than or equal to zero, then Ps ≥ 0. However, discretization errors and nonphysical solution behavior in the numerical solution may lead to local discrete violations of the Second Law, thus potentially ( )Ps j < 0 in some control volume j. If the Second Law is violated locally, then a quantitative indication of the artificial viscosity required to correct the solution may be expressed in terms of sP from Equation 6.40. Using the Prandtl number (Pr = v/a), an “artificial viscosity” can be factored out from the previous entropy production equation to give

µ =∇ ⋅∇ +

s

p

Pc T T Pr T T

/( ) /2 Φ

(6.41)

In Equation 6.40, the local entropy production rate will be greater than or equal to zero, both analytically and numerically, because it is a sum of squared terms. However, temporal and spatial differencing of the entropy transport equation may lead to nonphysical numerical results and negative entropy production rates within a discrete control volume. If sP is computed as a negative value, then the numeri-cal solution behavior is not physically correct because the Second Law is violated, and Equation 6.41 would imply a negative viscosity, which would steepen gradients rather than smooth them. If μ and k are computed in Equation 6.40 using the magni-tude of sP from the entropy transport equation, then the “entropy-based” diffusion could prevent potentially nonphysical solution behavior, such as rarefaction shocks and numerical oscillations, because additional diffusion is a “smoothing” process.

To implement this approach, the corrective procedure should only be applied in regions containing the nonphysical solution behavior because there is no physical justification for modifying the solution elsewhere. Also, a mechanism is needed to determine how much is a sufficient amount of diffusion. The Second Law can be used as the required corrective mechanism. It is sensitive to the nonphysical numeri-cal results. Also from Equation 6.41, it can be used to provide a quantitative measure of the amount of diffusion required in the numerical procedure to correct any non-physical results.

Following each time step, the entropy production rate is computed based on the entropy transport equation. If a nodal value of sP is negative, then the local solution is not physically correct. Therefore, instead of proceeding to the next time step, a corrective iteration of the Navier–Stokes equations is performed to find the entropy production needed to prevent the computed entropy destruction. The entropy

7262_C006.indd 159 1/19/08 3:44:06 PM


production rate for a control volume is given by the left-hand side of Equation 6.35. For a desired production rate, given a prescribed temperature and dissipation distri-bution, the viscosity needed to supply this production rate can be computed from the discrete representation of Equation 6.41. Given that Φ and (∇ T ∙ ∇ T) can be repre-sented in terms of nodal velocities and temperatures,

Φ j

k

l

j ksu

k

k

l

j ksv

kC U C V= +=

,=

,∑ ∑1 1

(6.42)

( )∇ ⋅∇ ==

,∑T T C Tj

k

l

j kTT

k

1 (6.43)

where the coefficients are determined through appropriate discretization of velocity and temperature gradients. The required amount of viscosity can then be determined by

( )( )

µ js j

j

P

Den=-

(6.44)

where

Denc

PrC T T Cj

p

k

l

j kTT

k j

k

l

j ke=

/ +

=,

=,∑ ∑

1

2

1

ssuk

k

l

j kesv

k jU C V T+

/

=,∑

1 (6.45)

where l = 4, 2, and 1 for interior, boundary, and corner control volumes, respectively. This procedure attempts to overcome entropy destruction, but does guarantee that the Second Law is satisfied in a single iteration of the corrective procedure.

The previous method has not rigorously proven that the entropy-corrected vis-cosity will ensure compliance with the Second Law. Multiple iterations might be needed to ensure sufficient numerical diffusion. As a result, the numerical viscosity, μe, with a single iteration is not always sufficient to completely remove nonphysical solution behavior, so

µ µe

mec← (6.46)

can be used as an alternative to accelerate iterations, where cm is a correction factor. It typically has the range 1 < cm < 10. Because the numerical solution with cm = 1 does not guarantee local satisfaction of the Second Law, cm > 1 may be required. Past stud-ies have indicated that cm ≈ 1 for subsonic and transonic flows, 1 < cm < 5 for supersonic flows, and 1 < cm < 10 for hypersonic flows (Naterer and Schneider, 1994).

Equation 6.44 provides a distribution of viscosity, as well as the conductivity through the Prandtl number, for the correction iteration. However, because these distributions could be digital in nature, due to the elimination of all positive values of sP , it is possible that the distribution could lead to unexpected results. Entropy production alone, caused by local irreversibilities in the flow field or spatial

7262_C006.indd 160 1/19/08 3:44:11 PM


differencing or both, may occur in regions of nonphysical solution behavior. But in this case, there is no conclusive evidence that the solution is not possible. Also, in the presence of rapid motion, such as fast shock propagation, it is also possible that the detection-to-application delay may cause the addition of entropy produc-tion to “miss its target.” For these reasons, the formulation can further “diffuse” the viscosity distribution calculated from Equation 6.44 through Jacobi iterations using a diffusion operator. The maximum value of the calculated viscosity (after diffu-sion) is returned to its prediffusion value. In this way, the magnitude of the required viscosity is retained, while the distribution becomes “smoother.” Once the viscosity distribution has been smoothed, the control surface diffusive flows are reevaluated using the shape functions. The discrete Navier–Stokes equations are then resolved with these entropy diffusion terms to correct the nonphysical solution behavior.

6.3.2 Ca SeSt u d yo FSh o CkCa pt u r in g in a Sh o Cktu be

The method of entropy-based correction with a numerical viscosity (developed in the previous section) will be applied to a shock tube problem in this case study. Con-sider shock tube flow with initial conditions illustrated in Figure 6.8. The problem involves a 1-m-long duct containing air (assumed perfect gas) that is initially at rest and divided by a diaphragm into a high pressure region (1032 kPa) and a low pressure region (101.3 kPa). The diaphragm is located at x = 0.5 m. A finite element method is used with grid Courant numbers (a ∆ t/∆ x), in the low pressure region varying between 0.24 and 0.07. The boundary conditions are given by zero normal velocity and zero tangential stress at the walls and ends of the shock tube. The conservation of mass and energy equations are completed at the boundaries by using nodal repre-sentations of the required boundary surface flows.

In Figure 6.9, the predicted results with a control volume-based finite element method (Naterer and Schneider, 1994) indicate that numerical oscillations develop at the initial interface and shock front. The prediction of the rarefaction waves, shock speed, and positioning, as well as the shock resolution, is shown. The dip in the pres-sure solution, which occurs at the original pressure interface, can be removed through an artificial viscosity stability term of the form

q v= |∇ |α (6.47)

where a = -r(cLL)c and c is the local speed of sound. Also, L is a local characteris-tic mesh length scale. The above term was added to the momentum equations, and it represents an artificial diffusion term. The results of this addition are shown in

Low-Pressure Gas101.3 kPa

High-Pressure Gas 4 1032 kPa

1 Air

Diaphragm

1 m

f Ig u r e6.8 Schematic of shock tube problem.

7262_C006.indd 161 1/19/08 3:44:14 PM


Figure 6.9 for three different values of cL: 0.0, 1.0, and 3.0. This stability damping yields good accuracy at the original high-low pressure interface. However, the numer-ical oscillations at the shock front are not diminished by this stability damping.

To remove the over- and undershoots at the shock wave, the method of entropy-correction of the numerical viscosity from the previous section was used. The vis-cosity distribution predicted by the Second Law formulation is shown in Figure 6.10. It provides a highly localized viscosity. Six Jacobi iterations were used in the viscosity smoothing operations. The results indicate that the computed entropy production

f Ig u r e6.9 Pressure profiles at varying cl coefficients.

f Ig u r e6.10 Artificial viscosity distribution.

0.0 0.0

2.0

4.0

6.0

8.0

10.0

0.2 0.4 0.6 0.8 1.0 Distance (m)

Pres

sure

(Pa)

×105 Legendc1 – 0.0c1 – 1.0c1 – 3.0

0.0–1.0

0.0

1.0

2.0

3.0

4.0

5.0

0.2 0.4 0.6 0.8 1.0Distance (m)

Legendcm – 2.0cm – 5.0

Art

ifici

al V

iscos

ity (k

g/m

s)

7262_C006.indd 162 1/19/08 3:44:16 PM


provides a useful error indicator, which can be used in a corrective manner to improve shock capturing in compressible flows. The results of the applied viscosity field are shown in Figure 6.11 for values of cm = 0.0, 2.0, and 5.0. For cm = 3.0, the over- and undershoots are significantly diminished. The distribution is determined entirely from Second Law considerations, and there is little or no smearing of the shock front.

6.4 en t r o Py dIf f er en c eIn resId u a l er r o r In d Ica t o r s

6.4.1 Fo r mu l at io n o FaVer ag een t r o pydiFFer en Ce

The previous section has shown that negative entropy production can provide a use-ful error indicator for fluid flow simulations. More generally, fluid entropy differ-ences (averaged over the computational domain) can provide a general measure of residual “error” in fluid flow simulations. In many ways, this measure or metric has key advantages over other conventional methods. When solving the equations of fluid flow with a numerical technique, one would like to know when the solution has reached a steady state, presumably, the correct solution.

A numerical solution of a steady-state problem is converged when further calcu-lations will have little or no effect on the flow-field results. Whether the converged solution is correct is another issue. For any given flow variable, like mass density, momentum, or energy, versus time or iteration number, the solution convergence is indicated by a flat line after some iterations. In principle, for any flow variable x, the limit

limn n→∞

∂∂=ξ 0 (6.48)

0.0 –100.0

0.0

100.0

200.0

300.0

400.0

500.0

0.2 0.4 0.6 0.8 1.0 Distance (m)

Legend

cm – 2.0cm – 0.0

Velo

city

(m/s

)

cm – 5.0

f Ig u r e6.11 Predicted velocity profiles at varying cm coefficients.

7262_C006.indd 163 1/19/08 3:44:19 PM


indicates convergence where n refers to the iteration number. This limit indicates a change in the flow variable with each iteration or time step. In numerical simulations, the algorithm is limited to a finite number of iterations. This requires a “residual” that effectively represents the finite version of a tangent slope given by Equation 6.48. A residual is any nonnegative indicator of changes in the solution with time (or iteration). A finite difference representation of Equation 6.48 is

ξ ξ ξ ξn n

n n

n n

++-

+ -= -

11

1

(6.49)

A residual may be defined by Equation 6.49.Any flow or solution variable can be “representative” of the overall solution.

Consider the following definitions that are indicative of particular features of the flow field.

massdensity:

ρ ρρ

ρρ

* ln= -o o

(6.50)

The density ro at a reference state is given at some standard conditions (for air, stan-dard sea-level conditions can be used). A corresponding reference temperature and pressure To, Po are also used.

Kineticenergy:

kRTo o

ε ρρε* = -

(6.51)

In this equation, ε = ∑12

2uk , where uk are the velocity components and R is the gas

constant.

Internalenergy:

ieTTo o

* ln= ρρ

(6.52)

specificentropy:

s s CP P

v- =( )

0

0

0

lnρ ρ

γ

(6.53)

This equation holds for an ideal gas (Sonntag and van Wylen, 1982). Using the ideal gas law, P = rRT, together with the definition of the ratio of specific heats, g = Cp /Cv, the following nondimensional formula for the entropy is obtained:

7262_C006.indd 164 1/19/08 3:44:24 PM


n ondimensionalentropy:

Ss s

R

T

Tij

o

o

o o

ij

o

≡ -

= -

-

ρρ

ρρ γ

ρρ

11

ln ln

.

(6.54)

Mathematically, residuals are abstract measures of a “distance” between ele-ments in an abstract space. The abstract space represented in fluid flow simulations involves the flow variables, which are typically mass, momentum, and energy. The residual is a measure of the distance between the flow variables, at some point in a calculation, to their steady-state values. In computational fluid dynamics, it is com-mon to either take the maximum value of this distance function, or take its average over the computational domain. For a two-dimensional domain with grid cells of variable dimensions, the average can be expressed symbolically by the following operation:

⟨ ⟩ =∑ ∑ξ ξ∆ ∆ ∆ ∆x y x yi j ij i j (6.55)

Average Mass Density Difference:

RES n n

ρ ρ ρ≡ ⟨ - ⟩+| |1 (6.56)

The ⟨ ⟩ operator refers to averaging over the computational domain.

r msmassdensitydifference:

RES n nρ ρ ρ2

1 2≡ ⟨ - ⟩+( ) (6.57)

r msPressuredifference:

RES p ppn n≡ ⟨ - ⟩+( )1 2

(6.58)

averageentropydifference:

RES S SSn n

∆ ≡ ⟨ - ⟩+| |1

(6.59)

Other variations are also possible, but Equation 6.56 through Equation 6.58 represent the most common examples (Anderson, 1984). In the next section, sample results of the entropy-based residual error will be investigated for a specific case study.

6.4.2 Ca SeSt u d yo Fer r o r in d iCat o r Sin Su per So n iCFl o w

This section presents a case study that uses an entropy-based residual as an error indicator for compressible flow simulations. The explicit numerical scheme solves the Euler equations with a technique described by Camberos (1995). For a nozzle

7262_C006.indd 165 1/19/08 3:44:29 PM


flow problem, an implicit scheme is used with a Gauss–Seidel line relaxation tech-nique for the thin-layer Navier–Stokes equations (MacCormack, 1985). This section focuses on how flow variables and residuals change with each iteration.

The first example represents supersonic flow of an ideal gas over a two-dimen-sional wedge. The configurations and contours of constant density are shown in Figure 6.12. Because the method is explicit and first-order accurate, the oblique shock wave is quite thick, but oriented at the correct location, as predicted by theo-retical gas dynamics. In Figure 6.13, the iteration history for the representative flow

10°

x

Pi

y

M∞ = 2.0

f Ig u r e6.12 Flow-field contours of constant density for two-dimensional supersonic wedge flow.

250–0.3

–0.2

–0.1

0.0

0.1

0.2

500 750 1000

Internal Energy

Entropy Difference

Mass Density

Kinetic Energy

Entropy-Based Norm

Repr

esen

tativ

e Flo

w V

aria

bles

n

f Ig u r e6.13 Iteration history for representative flow variables: two-dimensional super-sonic wedge flow.

7262_C006.indd 166 1/19/08 3:44:32 PM


variables is shown. Note that a flat line is evident after about 300 time iterations (or time steps). According to this indicator, the solution is essentially converged. In addition, visible changes in the flow-field solution (not shown) represented in Fig-ure 6.12 are no longer evident. From Figure 6.14, it appears that after 300 iterations, the residual metric based on an average mass density difference has dropped about two orders of magnitude. This is true for other metrics as well, although they are about one order of magnitude less (down to 10-4, compared with 10-3 for density). The sudden drop in the residual that appears at around n = 350 is a spurious but benign result after the oblique shock wave reaches a stable location. The initial con-ditions were uniform incoming flow at a 10-degree flow angle toward a solid surface, so the oblique shock appears at the leading edge. It gradually propagates through the grid to its final location. Figure 6.14 shows that nearly machine zero is reached after about 750 iterations. A large number of iterations is typical of explicit numerical solutions to steady-state fluid flow problems.

Contours of constant Mach number are shown in Figure 6.15 for supersonic flow over a convex corner, which leads to a centered Prandtl-Meyer expansion fan. The numerical method is explicit and first-order accurate, so the expansion fan is quite thick, but oriented at the correct location, as predicted by theoretical gas dynamics. Figure 6.16 shows the iteration history for the representative flow variables. A flat line for all of the variables is evident after about 250 iterations. The solution (from this indicator) is essentially converged. Visible changes in the flow field solution (as rep-resented in Figure 6.15) are no longer evident. From Figure 6.16, not all of the flow variables have reached steady state. In particular, the line representing the kinetic

f Ig u r e6.14 Iteration history for residual metrics: two-dimensional supersonic wedge flow.

10–1

10–3

10–5

10–7

10–9

10–11

10–13

10–15

Resid

ual

250 500 750 1000n

(S)n+1_(S)n

(∆norm)(|∆p|)(∆p2)(∆p2)

7262_C006.indd 167 1/19/08 3:44:35 PM


energy appears to flatten out at around 100 iterations. This indicates that one should choose a representative flow variable that takes into account all of the state variables to guarantee that convergence has been reached. From Figure 6.17 at 250 iterations, the residual based on the average mass density difference has dropped about two orders of magnitude. This also holds for other residuals as well, although their values are about one order of magnitude less (down to 10-4 compared with 10-3 for the density). Figure 6.17 also shows that machine zero is reached after about 400 iterations, which is again typical of explicit numerical solutions for steady-state fluid flow problems.

Supersonic flow over blunt bodies involves the formation of bow shock waves. It provides the challenge of “shock capturing” when the equations of gas dynamics

M = 3.0

M = 3.55

f Ig u r e6.15 Flow-field contours of constant Mach number: two-dimensional Prandtl–Meyer expansion.

250 500 750 1000

Kinetic Energy

Mass Density

Entropy-Based Norm

Internal Energy

Entropy Difference

n

–0.3

–0.2

–0.1

0.0

0.1

0.2

0.3

Repr

esen

tativ

e Flo

w V

aria

bles

f Ig u r e6.16 Iteration history for representative flow variable for two-dimensional Prandtl–Meyer expansion.

7262_C006.indd 168 1/19/08 3:44:37 PM


are solved in conservation law form. Because the method used in these exam-ples is first-order accurate and explicit, the bow shock wave is thick, but located approximately at the correct location as expected from theoretical gas dynamics. The pressure jump obtained by the numerical solution, from the leading edge of the shock wave to the stagnation point at the square block, is within 5% of the pre-dicted value from normal shock wave theory. Contours of constant Mach number are shown in Figure 6.18, where the blunt body is the square block shown in black.

f Ig u r e6.17 Residual error metrics for two-dimensional Prandtl–Meyer expansion.

M = 3

f Ig u r e6.18 Mach contours for two-dimensional supersonic flow.

10–1

10–3

10–5

10–7

10–9

10–11

10–13

10–15

250 500 750 1000n

(S)n+1_(S)n

K0||qn_qave||2

|∆|MAX

(∆p2)(∆p2)

Resid

ual

7262_C006.indd 169 1/19/08 3:44:42 PM


Figure 6.19 shows the iteration history for the representative flow variables of mass density, kinetic energy, internal energy, and entropy difference. Note that a flat line is evident after about 600 iterations. The solution (based on this indicator) is essen-tially converged. In addition, visible changes in the flow-field solution (represented in Figure 6.18) are no longer evident. From Figure 6.20, at 600 iterations, the residual

5000–0.8

–0.6

–0.4

–0.2

0

Repr

esen

tativ

e Flo

w V

aria

bles

0.2

0.4

0.6

0.8

0.10

1000n

1500

Kinetic EnergyEntropy-Based Norm

Mass Density

Internal Energy

Entropy Difference

f Ig u r e6.19 Iteration history of flow variables.

10–1

10–3

10–5

10–7

10–9

10–11

10–13

Resid

ual

10–151000 2000

n3000 4000

(s)n+1–(s)n

K0||qn–qave||2

|(Δp)|

(Δp2)(Δp2)

f Ig u r e6.20 Iteration history of flow variables.

7262_C006.indd 170 1/19/08 3:44:47 PM


based on the average mass density difference has dropped about three or four orders of magnitude. Similar results are obtained for other residuals. Figure 6.20 shows that machine zero is essentially reached after about 3400 iterations.

For the blunt body problem, a grid size of 100 × 100 cells was used, with the blunt body occupying 10 × 10 grid cells. By reducing the size of the body to a single grid cell, and thereby moving the outflow boundaries farther from the simulated solid surface, the number of iterations required to reach machine zero is reduced to 700 (Figure 6.21). Compared with 3400 shown in Figure 6.20, this is a significant decrease.

In Figure 6.22, two-dimensional flow through a converging-diverging nozzle is examined. The thin-layer Navier–Stokes equations are solved with Gauss–Seidel line relaxation and an implicit technique described by MacCormack (1985). In Figure 6.22, the lower portion of the nozzle contour is shown, with the centerline indicated at the upper portion of the figure. The inlet conditions are subsonic. The velocity flow field is shown in Figure 6.22. For this problem, the solution is approxi-mately second-order accurate, and the line relaxation technique is implicit.

In Figure 6.23, only the residual metric based on the average entropy difference is shown. For this case, there are two methods for imposing the wall-boundary con-dition. One method is specifying the normal flux term equal to the pressure at the wall. The other method imposes a flux-splitting technique, with a layer of cells adja-cent to the wall to create a layer of ghost cells with the normal velocity component reflected. As shown in Figure 6.23, the two techniques have different convergence histories. Machine zero is reached after about 500 iterations, but the solution appears to converge much earlier, at about 50 to 70 iterations. No major difference between the techniques for imposing the wall-boundary condition was observed.

f Ig u r e6.21 Residual metrics for 1/10 size Blunt body.

10–2

10–4

10–6

10–8

10–10

10–12

10–14

10–16250 500 750 1000

n

(S)n+1_(S)n

|(∆p)|(∆p2)(∆p2)

K0||qn_qave||2

Resid

ual

7262_C006.indd 171 1/19/08 3:44:49 PM


0.00 C.L.

–0.02

–0.04

–0.06

–0.08

–0.10

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

f Ig u r e6.22 Converging–diverging nozzle flow implicit of thin-layer, Navier–Stokes equa-tions (TL-NSE) solution.

10–2

10–4

10–6

10–8

10–10

Entr

opy R

esid

ual

10–12

10–14

10–160 100 200 300

n400 500

Flux = Pressure at Wall

Flux-Splitting at Wall

f Ig u r e6.23 Residual error metrics for two-dimensional thin-layer Navier–Stokes implicit solution.

7262_C006.indd 172 1/19/08 3:44:56 PM


It is beneficial to reduce the number of iterations without compromising solution quality. A possible method would monitor the residual by calculating the difference in the tangent slope for entropy, as it changes with each iteration. This can be accom-plished once the residual has dropped two or three orders of magnitude from its initial value, to ensure that the iteration curve for the representative variable has flattened out sufficiently. An automatic procedure could be developed to reduce user input and monitoring of a fluid flow calculation that requires many iterations.

The results indicate that entropy can provide several advantages over other con-ventional methods for characterizing solution residuals. Fluid entropy, S = S(Q), is functionally dependent on all of the fluid state variables, Q = (q1, q2, q3, q4, q5), namely, mass density, momentum density, and total energy density. Also, entropy has a physical significance embodied by the Second Law of Thermodynamics, that requires nonnegative entropy production.

r ef er en c es

Anderson, D.A., Tannehill, J.C., and R.H. Pletcher. 1984. Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, New York.

Camberos, J.A. 1995. A Probabilistic Approach to the Computational Simulation of Gas Dynamic Processes. Ph.D. thesis. Stanford University, Stanford, CA.

Camberos, J.A. 1998. Calculation of residual error in explicit and implicit fluid flow simu-lations based on generalized entropy concept. Proceedings of the Joint American Society of Mechanical Engineers/Japan Society of Mechanical Engineers Meeting (PVP377-2). San Diego, CA, 279–286.

Fox, R.W. and A.T. McDonald. 1992. Introduction to Fluid Mechanics. 4th ed. John Wiley & Sons, New York, 123–124.

Harten, A. 1983. On the symmetric form of systems of conservation laws with entropy. J. Computational Phys., 49: 151–164.

Huebner, K. and E. Thornton. 1991. The Finite Element Method for Engineers. 2nd ed. John Wiley & Sons, Toronto, Canada.

Hughes, T.J.R., Franca, L.P., and M. Mallet. 1986. A new finite element formulation for com-putational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier–Stokes equations and the Second Law of Thermodynamics. Computer Methods Appl. Mechanics Eng., 54: 223–234.

Karki, K.C. and S.V. Patankar. 1989. Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations. AIAA J., 27(9): 1167–1174.

Lax, P.D. 1971. Shock Waves and Entropy. Contributions to Non-Linear Functional Analy-sis. Academic Press, New York, 603–634.

Leonard, B.P. 1979. A stable and accurate convective modeling procedure based on quadratic upstream interpolation. Computer Methods Appl. Mechanics Eng., 19: 59–98.

Lohner, R., Morgan, K., and O.C. Zienkiewicz. 1984. The solution of the non-linear hyper-bolic equation systems by the finite element method. Int. J. Numerical Methods Flu-ids, 4: 1043–1063.

MacCormack, R.W. 1985. Current Status of Numerical Solutions of the Navier-Stokes Equa-tions. AIAA Paper 85-0032. 23rd AIAA Aerospace Sciences Meeting. Reno, NV.

MacCormack, R.W. and B.S. Baldwin. 1975. A Numerical Method for Solving the Navier–Stokes Equations with Application to Shock–Boundary-Layer Interactions. AIAA Paper 75-1, 13th AIAA Aerospace Sciences Meeting. Pasadena, CA.

Majda, A. and S. Osher. 1979. Numerical viscosity and the entropy condition. Commun. Pure Appl. Math., 32: 797–838.

7262_C006.indd 173 1/19/08 3:44:57 PM


Merriam, M.L. 1987. Smoothing and the second law. Computer Methods Appl. Mechanics Eng., 64: 177–193.

Merriam, M.L. 1988. An Entropy Based Approach to Nonlinear Stability. Ph.D. thesis. Stanford University, Stanford, CA.

Minkowycz, W. J., Sparrow, E. M., Schneider, G. E., and R. H. Pletcher. 1988. Handbook of Numerical Heat Transfer. John Wiley & Sons, New York, 252–253.

Naterer, G.F. 1999. Constructing an entropy-stable upwind scheme for compressible fluid flow computations. AIAA J., 37(3): 303–312.

Naterer, G.F. and G.E. Schneider. 1994. Use of the second law for artificial dissipation in compressible flow discrete analysis. AIAA J. Thermophysics Heat Transfer, 8(3): 500–506.

Nellis, G.F. and J.L. Smith. 1997. Entropy-based correction of finite difference predictions. Numerical Heat Transfer B, 31(2): 177–194.

Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow. Hemisphere, New York, 30–31.

Pulliam, T.H. and J.L. Steger. 1980. Implicit finite difference simulations of three dimen-sional compressible flow. AIAA J., 18(2): 159–167.

Rosen, M.A. and I. Dincer. 1999. Exergy analysis of waste emissions. Int. J. Energy Res., 23(13): 1153–1163.

Schneider, G.E. and M.J. Raw. 1987. Control-volume finite-element method for heat transfer and fluid flow using co-located variables. Part 1. Computational procedure. Numeri-cal Heat Transfer, 11(4): 363–390.

Sonntag, R.E. and G. van Wylen. 1982. Introduction to Thermodynamics. John Wiley & Sons, New York.

Von Neumann, J. and R.D. Richtmyer. 1950. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys., 21: 232–237.

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175

7 Numerical Stability and the Second Law


The classic paper by Courant et al. (1928) marked a milestone for numerical analysis. The archived literature often refers to the “CFL” (Courant, Friedrichs, Lewy) condition as a criterion that restricts the time step for linear differential equations, to achieve numerical stability. However, the CFL condition originally was not related to numerical stability, because that term was not phrased until the 1940s by a group associated with John von Neumann. Nevertheless, the terminol-ogy remained, and today we understand the CFL condition as a necessary, and in some cases sufficient, condition for both numerical stability and convergence of nonlinear equations.

The basic question of numerical stability deals with discretization error and round-off error. Discretization errors are analogous to systematic errors that arise in experimental measurements, whereas round-off errors are analogous to the unpredictable and unavoidable errors that occur in a measurement process itself. Minimizing discretization errors requires very accurate approximations of the dif-ferential equations. Round-off errors have a significant impact on the stability of a numerical method. Numerical stability deals with the growth of an overall round-off error. The growth of a single round-off error is a question most frequently studied because it can be answered more easily and practically than the overall error (Anderson et al., 1984). In the modern use of computational fluid dynamics (CFD) codes, heuristic arguments and rules of thumb are often used to establish a restriction on the time step for explicit methods and time-accurate solutions. Unfortunately, ad hoc trial and error are often needed to determine a method’s stability bounds.

This chapter examines how the logic intrinsic to the Second Law of Thermody-namics can be used to establish stability of a numerical algorithm. Pioneering numer-ical analysis of the Second Law provided mathematical constraints that determine physically relevant solutions to the differential equations. However, these equations may exhibit nonunique and discontinuous solutions (Oleinik, 1957, 1959). Expanding the essence of the Second Law will show that an entropy-based alternative to lin-ear stability analysis exists. Due to the universality of concepts associated with the Second Law, an entropy-based method can be applied to any of the governing differ-ential equations of thermal and fluid dynamics. This chapter will address the ques-tion of strong numerical stability, by extending the “modified equation” technique of Warming and Hyett (1974) and others. A modified equation for the balance of entropy will provide a powerful method for gauging a numerical method’s stability

7262_C007.indd 175 1/19/08 11:52:25 AM


properties, because numerical stability is directly related to the overall generation of entropy (Camberos, 1995). In addition, the Second Law provides a way to gauge the local quality of the solution. For example, spurious oscillations in the numerical solution often arise under conditions of entropy destruction (Merriam, 1989).

Transient numerical solutions of the equations of fluid dynamics often require a time-step constraint. The constraint can be reduced to an inequality relating the time step, grid spacing, and some reference wave velocity. Historically, the techni-cal literature in numerical analysis refers to this parametric cluster as the “Courant number” and the condition for the linear case as the “CFL condition.” Classically, numerical analysis relies on linearization and von Neumann’s use of a Fourier series to derive the CFL condition. In contrast, this chapter will use the Second Law to impose a restriction on the time step, for linear and nonlinear equations, as well as systems of equations like the equations of gas dynamics. By transforming the trun-cation error for the governing equation into an equation representing the balance of entropy, one can obtain an inequality that restricts the time step to satisfy the Second Law in a weak sense. In this chapter, the Second Law will be applied to the linear advection equation, then a nonlinear equation, and finally a system of equa-tions representing the one-dimensional equations of gas dynamics. It will be shown that the results agree with the classical approach for linear equations, but they differ for others, thereby showing that the Second Law has valuable utility in numerical analysis beyond its role in thermodynamics. This chapter will develop entropy crite-ria for explicit numerical algorithms with truncation errors. Generalized results will be established for implicit and higher-order methods, due to the universality of the Second Law and the concept of entropy.

7.2 St a bIl It y no r mS

Nonlinear stability can be analyzed in the context of the Second Law, whereby the difference equation must satisfy a global form of the entropy balance equation to guarantee numerical stability. Stability requires that the solution remain bounded in some norm, meaning that this norm either decreases or remains constant for the duration of the calculation. Suppose that the initial value problem for a set of con-servation laws has a unique equilibrium solution, q, defined as the average value of the state variables over the computational domain. For a closed domain, this average state remains constant, and it can be computed from the initial conditions. Statistical thermodynamic analysis can be used to establish that the average state can be repre-sented by an invariant probability distribution for the state variables. In equilibrium, this probability density is constant. Define the following averaging operator as the volume integral over the computational domain, Ω. The average state of an arbitrary variable, x, that establishes equilibrium is

ξ ξ ξ= ≡ /∫ ∫Ω ΩdV dV (7.1)

Concavity of entropy implies a balance of entropy containing a term indicative of nonnegative entropy generation. For an inviscid adiabatic fluid flow, the Second

7262_C007.indd 176 1/19/08 11:52:27 AM

Numerical Stability and the Second Law 177

Law can be used to prescribe local restrictions on the time step for explicit numerical simulations that give physically relevant solutions. Globally, the Second Law leads to the existence of some probability measure intrinsic to the distribution of state vari-ables in their proper solution space. In this regard, the Second Law can be expressed in terms of the vector of conserved quantities, q, as follows,

- = - - , ⋅ - ≥ln ( ) ( ) ( ) ( )g q S q S q S q qq 0 (7.2)

This principle may be interpreted as an expression for the existence of a ther-modynamic equilibrium state, where entropy is a maximum. Statistically, this is also a statement about a probability distribution, for which the average state is asso-ciated with a maximum probability. Equation 7.2 indicates the existence of some equilibrium distribution of the state variables, which maximizes a probability dis-tribution for those variables. The concavity property of entropy guarantees that the g(q) functional is a probability distribution with a maximum probability that cor-responds to the equilibrium state. This is the essence of the Second Law, in a form known as Gauss’s principle or Gauss’s law of error (Lavenda, 1991). The third term in Equation 7.2 vanishes on integration, due to the definition of the average state q and the conservation of state variables. As a result,

S q S q( ) ( )- ≥ 0 (7.3)

This expresses the principle of nondecreasing entropy, for an isolated thermodynamic system. Because the state of equilibrium corresponds to the state of maximum entropy,

S q S q S q

t t t t( ) ( ) ( )≥ ≥= =2 1

(7.4)

for t2 > t1. This result provides an upper bound to a state metric or norm, which is a direct measure of the mean-square variation in the state variables over the domain.

An approach to the construction of a suitable norm is to begin with a series expansion of the local entropy about the equilibrium state, when variations in the state variables are very small, that is, | - |<<ξ ξ 1. A Taylor series expansion, up to the second order, is given by

S q S q S q q q q S q qq

Tqq( ) ( ) ( ) ( ) ( )≈ + , ⋅ - + - ⋅ , ⋅ -1

2 (7.5)

where the row vector of first derivatives S,q and the Hessian S,qq are both evaluated at the average state. The determinant for the Hessian is

det [ ]

( )S

pqq, = - -γ 13 (7.6)

where p is the pressure.The Hessian constitutes a quadratic form that is negative definite and nonsingu-

lar when g > 0 and P > 0.

7262_C007.indd 177 1/19/08 11:52:32 AM


In the small fluctuation limit, the probability measure for a distribution of state variables in the global domain is approximately normal. It can be shown that the functional form, combining previous equations, becomes

g qS

q q S q qqq Tq q( )

( )exp ( ) ( )≈

- | , |- ⋅ , ⋅ - 2

123π

where | S,qq | indicates the determinant. The negative inverse of the Hessian S,qq equals the covariance matrix for the state variables q. The covariance matrix determines the statistical dependence or independence of two or more variables. In particular, it is a useful measure of mean-square variations in a data set, such as the calculated values of q from a numerical solution.

Mathematically, this leads to the following definition of a norm for q in terms of a domain integral as follows:

q q S qT

q q2 ≡ - ⋅ , ⋅ (7.7)

The mean-square variations in the solution will be expressed relative to equilibrium conditions. Given the norm defined by the previous equation, write a distance func-tional as follows:

q q q q S q qT

qq- ≡ - - ⋅ , ⋅ -2( ) ( ) (7.8)

Expanding the right side and simplifying,

q q q S q q S q q qT

qqT

qq- = - ⋅ , ⋅ + ⋅ , ⋅ = -2 2 2

(7.9)

An alternative formulation of the Second Law that explicitly relates the state of a thermodynamic system to its state at equilibrium introduces the concept of exergy or availability (Camberos, 2000a,b,c). This thermodynamic function represents an abstract functional that quantifies the thermodynamic distance from the state of equilibrium. Exergy, for the present analysis, is defined as

X S q S q S q qq≡ - - , ⋅ -( ) ( ) ( ) (7.10)

The concavity of entropy then translates into a convexity condition on exergy, such that

X q X q X q qq( ) ( ) ( )- - , ⋅ - ≤ 0 (7.11)

After averaging,

X q X q( ) ( )- ≤ 0 (7.12)

7262_C007.indd 178 1/19/08 11:52:37 AM


Subsequently for an isolated system, the principle of monotonic exergy decrease becomes

X q X q X q

t t t t( ) ( ) ( )≤ ≤= =2 1

(7.13)

for t2 > t1. This is consistent with the Second Law as a statement that minimum exergy determines the state of equilibrium.

From the definition of the distance functional and the near-equilibrium conditions,

2

2X q q q( ) ≈ - (7.14)

which can be expressed as

2

2 2X q q q( ) + ≈ (7.15)

For a bounded solution,

q q

t t t t= =≤2 1 , (7.16)

which is a statement of stability, given t2 > t1. The inequality may be enforced by postulating the existence of a scaling constant K0, determined from initial condi-tions, such that

2 20

2X q q K q( ) + ⟨ ⟩ ≥ ⟨ ⟩

(7.17)

The scaling constant may be calculated as

KX q q

q00

2

02

2= ⟨ ⟩ + ⟨ ⟩⟨ ⟩

( )

(7.18)

where q0 = q (t = t0). Because exergy is a maximum at the initial state and nonin-creasing thereafter, it can be concluded that

2 0

20

2X q K q+ ≥ (7.19)

implying a uniform bound on the mean-square variation in the data for all time, t > t0. In general, mean-square variations will fluctuate, but they cannot fluctuate beyond the upper bound in the previous equation, when a numerical scheme satis-fies the Second Law. As a result, satisfaction of the Second Law of thermodynamics provides a sufficient condition for nonlinear numerical stability.

7262_C007.indd 179 1/19/08 11:52:42 AM


7.3 En t r o py St a bIl It y o f fIn It EdIf f Er En c ESc h EmES

7.3.1 Lin ea r Sc a La r ad v ec t io n

The previous section used concavity properties of entropy to establish general sta-bility criteria, without referring to discrete parameters obtained from a numerical solution of the governing equations. In this section, specific stability criteria will be derived in terms of the discretization parameters (time step and grid spacing). Con-sider the following linear equation for one-dimensional scalar advection:

∂∂+ ∂∂=η

t

f

x0 (7.20)

where f(h) = ch represents the advection of a conserved scalar quantity, h, at a con-stant wave speed, c. Mathematically, one can readily obtain the solution to this equa-tion, given initial and boundary conditions.

For this problem, there exists an “entropy” S(h) and an “entropy flux” F(h) such that

genS

S

t

F

x= ∂∂+ ∂∂

(7.21)

which represents a balance of entropy. The “entropy” and the “entropy flux” are constructed for the dependent variable h by first postulating that there exists a func-tional S(h) such that S" < 0, which is a sufficient condition for concavity. Integra-tion by parts yields the following necessary condition, Gauss’s principle (Lavenda, 1991):

S S S( ) ( ) ( )( )η η η η η- - ′ - ≥ 0 (7.22)

The “maximum entropy” formalism of statistical optimization (Jaynes, 1991; Kapur and Kesavan, 1992), combined with the property of concavity given in Equation 7.22, results in an “entropy” that is a logarithmic function of the dependent variable:

S( ) ln( )η η ηη= / (7.23)

Other possibilities exist, like using -η2 or η . The important functional relation is given by a sufficient condition for concavity. Using the compatibility condition for the “entropy flux” and applying the chain rule gives

F cS( ) ( )η η= (7.24)

Setting η = 1 incurs no loss in generality.With the entropy pair defined, let us proceed to analyze discrete formulae for

approximating solutions to the differential equation. A control volume formulation equates the value of the dependent variable fluxes with the change of the integral

7262_C007.indd 180 1/19/08 11:52:48 AM


average over the cell the volume. Mathematically, the “weak” or integral solutions may admit discontinuities that can develop for nonlinear equations, even from smooth initial data. Weak solutions may lose their uniqueness; therefore some addi-tional criteria must pick out the solution that best reflects what would be considered physically relevant or observable. It is known that the “entropy condition” (compli-ance with the Second Law) picks out the correct (physically relevant) weak solution (Lax, 1954, 1973; Oleinik, 1959).

Using a standard, one-dimensional explicit discretization of the scalar advection equation,

η η η ηj

njn

jn

jnC C C+ +

--

+= + +11

01 (7.25)

This represents an integral approximation of the hyperbolic conservation law in Equation 7.20, based on a three-point numerical stencil. As written in this format, it contains a set of coefficients C+, C0, C- that must satisfy certain conditions. First, for the numerical solution to approximate the conservation law, the sum of the transi-tion coefficients must equal unity:

C C C+ -+ + =0 1 (7.26)

This must hold true for a single nonlinear equation and systems of equations, also. Second, the numerical solution must approximate the mathematical behavior of the original differential equation, called consistency (Anderson et al., 1984). This leads to the following second requirement:

∆∆

- =+ -x

tC C f( ) ( )η η (7.27)

A third “monotonic” condition (Crandall and Majda, 1980), although not necessary, may suffice for numerical stability:

1 00≥ , , ≥+ - C C C (7.28)

The transition coefficients C± may or may not satisfy monotonic criteria. But the coef-ficient C0 must satisfy it; otherwise the numerical solution will systematically vio-late the conservation principle. For linear equations, the CFL condition also implies satisfaction of the monotonic requirement in some cases. However, for nonlinear equations, monotonicity can lead to over- or underpermissive time-step limitations. Solutions obtained with a time step close to the limit imposed by the monotonic requirement may be stable, but also erroneous.

To expand the error incurred by the discrete numerical formula, assume a uni-form, equally spaced mesh with an exact solution available at time t, so that

η η η ηj

n C t x x C t x C t x x+ + -= , - ∆ + , + , + ∆1 0( ) ( ) ( ) (7.29)

7262_C007.indd 181 1/19/08 11:52:51 AM


The required Taylor series expansions are

η η η η η( ) ( )t x x t x x

x

x

x

x, ± ∆ = , ± ∆ ∂∂+ ∆ ∂

∂± ∆ ∂

∂

2 2

2

3 3

2 6 xx3+ (7.30)

for the spatial terms and

η η η η η( ) ( )t t x t x t

t

t

t

t+ ∆ , = , + ∆ ∂∂+ ∆ ∂

∂+ ∆ ∂

∂

2 2

2

3 3

2 6 tt3+ (7.31)

for the temporal terms. Collecting terms gives

Lt

C C Ct

x

tC C

xx

∆+ - + -=

∆- - - + ∂

∂+ ∆∆

- ∂∂

- ∆

η η η( ) ( )1 0

2

22 2 6

2

2

2

2

3 3

∆+ ∂

∂+ ∆ ∂

∂+ ∆∆

- ∂+ - + -

tC C

x

t

t

x

tC C( ) ( )

η η ηη η∂+ ∆ ∂

∂+

x

t

t3

2 3

36

(7.32)

Substituting Equation 7.26 and Equation 7.27, the conservation and consistency con-ditions, reduce the expression to

Lt

f

x

x

tC C

x

t

t∆+ -= ∂

∂+ ∂∂- ∆∆

+ ∂∂+ ∆ ∂

∂

+

η η η2 2

2

2

22 2( )

∆∆∆

- ∂∂+ ∆ ∂

∂++ -x

tC C

x

t

t

3 3

3

2 3

36 6( )

η η

(7.33)

Eliminating the higher-order time derivatives using the original conservation equa-tion, Equation 7.20, gives

Lt

f

x

x

tC C

c t

x∆+ -( )=∂

∂+ ∂∂- ∆∆

+ - ∆∆

∂η 2 2 2

2

2

2ηη η∂+ ∆∆

∆∆- ∆∆

∂∂+

x

x

t

c t

x

c t

x x2

3 3 3

3

3

36

(7.34)

Truncating up to the first-order leading error by setting L∆ ≈ 0 and rearranging,

∂∂+ ∂∂≈ ∆∆

+ - ∆∆

∂∂

+ -( )η ηt

f

x

x

tC C

c t

x x

2 2 2

2

2

2 22 (7.35)

This yields a “modified equation” for the numerical method, which prescribes the transition coefficients in Equation 7.25. This technique originated with Warming and Hyett (1974). The procedure retains the transition coefficient format, so that

7262_C007.indd 182 1/19/08 11:52:56 AM


any numerical method is already represented by Equation 7.34, without having to rederive the modified equation each time. Transition coefficients that give a first-order approximation will have a smoothing effect, due to the second-order derivative on the right side of Equation 7.35. From this formula, we will define a parameter (essentially a numerical viscosity) that has a critical role in formulating the cor-responding “modified equation” for the balance of entropy. The following analysis extends Warming and Hyett’s “heuristic stability analysis” by giving it a solid foun-dation in the Second Law.

Multiplying both sides of the “modified equation” by S' (derivative of S with respect to q) and using the chain rule of differentiation, plus the compatibility condi-tion, yields the following balance of entropy equation:

∂∂+ ∂∂≈ ′ ∂

∂S

t

F

xS

xε η2

2 (7.36)

where

ε = ∆∆

+ - ∆∆

+ -( )x

tC C

c t

x

2 2

2

(7.37)

defines the numerical viscosity parameter. Using the chain rule, the right-hand side of Equation 7.36 expands to

′ ∂∂=∂∂

′ ∂∂

- ′′ ∂

∂

Sx x

Sx

Sx

2

2

2η η η (7.38)

Integrating both sides of the entropy balance equation, using Equation 7.38 on the right side, over a small interval in space and time yields

genS dx dt

S

xdt S

xdx d

x

x

≈ ∂∂

- ′′ ∂∂∫∫ ∫ ε ε η

1

2 2

tt∫∫ (7.39)

After taking the limit [ ] [ ]x x t t1 2 1 2 0, × , → , the first term on the right side van-ishes, but the second term will not vanish if the limits of integration contain a dis-continuity. The balance of entropy for the one-dimensional advection equation will satisfy the Second Law only for nonnegative values of the parameter obtained from the leading error terms in the numerical approximation. Examining the transition coefficients for a given numerical method, one can establish the corresponding rela-tion for the time step, grid size, and wave velocity.

Table 7.1 contains the transition coefficients for several well-known finite differ-encing methods for Equation 7.25, which approximates the solution to the advection equation, Equation 7.20. Table 7.2 presents the results of the numerical viscosity parameter, Equation 7.37, given the transition coefficients in Table 7.1. The constraint

7262_C007.indd 183 1/19/08 11:53:00 AM


matches the classical CFL condition where | | ∆ / ∆c t x equals the Courant (or CFL) number. The “downwind” and central methods, from a Second Law perspective, destroy entropy intrinsically for any finite time step. The upwind and Lax–Friedrichs methods both approximate linear advection to first order, but the Lax–Friedrichs for-mula dissipates the solution more than the upwind method. Comparing the numerical viscosity parameter for these two methods explains why (for a given CFL number, like 0.9) the Lax–Friedrichs method gives 2 ε ∆ t/∆ x = 0.19, whereas the same CFL number gives a value of 0.09 for the upwind method. The Lax–Friedrichs method generates close to twice the amount of entropy as the upwind method (per time step). The last method (Lax–Wendroff) in the table contains transition coefficients that give a second-order approximation in both space and time, so the first-order numeri-cal viscosity parameter equals zero. For these methods, the monotonic requirement on the C0 transition coefficient establishes the time-step constraint shown in the table. Inspecting the next-order leading error terms (third order) justifies the choice.

t a bl E7.1t ransitioncoefficientsforVariousnumericalSchemesforlinearScalara dvection

t ransitioncoefficients

method C+ C 0 C-

Downwind - | | -12 ( )ν ν 1+ | |ν - | | +1

2 ( )ν νCentral 1

2 ν 1 - 12 ν

Upwind 12 ( )| | +ν ν 1- | |ν 1

2 ( )| | -ν ν

Lax–Friedrichs 12 1( )+ ν 0 1

2 1( )- ν

Lax–Wendroff 12 1( )+ ν ν 1 2- ν - -1

2 1( )ν ν

Note: ν ≡ ∆ /∆c t x .

t a bl E7.2t ime-StepconstraintsImposedbytheSecondlawforone–d imensionalScalara dvection

method 2 2εε∆∆ ∆∆t/ x c onstraint

Downwind - - ( )| |∆∆

∆∆

c tx

c tx

2 N/A

Central - ∆ / ∆( )2c t x N/A

Upwind | |∆∆

∆∆- ( )c t

xc t

x

2 | |∆∆ ≤c t

x 1

Lax–Friedrichs 1

2- ( )∆∆c t

x

| |∆∆ ≤c t

x 1

Lax–Wendroff 0 | |∆∆ ≤c t

x 1

7262_C007.indd 184 1/19/08 11:53:15 AM


By extending the Taylor series expansion to include second- and third-order terms, it can be shown that

∂∂+ ∂∂≈ ∂∂+ ∂∂+ ∂∂+η ε η ε η ε η

t

f

x x x x

2

2 2

3

3 3

4

4 (7.40)

where ν = ∆ / ∆c t x,

ε ν ν2

33

6= - ∆

∆-x

t( )

(7.41)

and

ε ν3

44

24= ∆∆

+ -+ -x

tC C[( ) ]

(7.42)

For the nth-order derivative, multiplying by S' and using the chain rule gives

′ ∂∂=∂∂

′ ∂∂

- ′′ ∂

∂∂-

-

-S

x xS

xS

x

n

n

n

n

nη η η η1

1

1

∂∂.

-xn 1 (7.43)

The general expression for the balance of entropy becomes

gen S

xS

x x x=∂∂

′ ∂∂+ ∂∂+ ∂∂+

ε η ε η ε η

2

2

2 3

3

3

- ′′ ∂∂

∂∂+ ∂∂+ ∂∂+

Sx x x x

η ε η ε η ε η2

2

2 3

3

3.

(7.44)

The first nonzero ek indicates the order of accuracy for a given method. A higher-order numerical method may still satisfy the entropy-generation inequality in the limit, but not necessarily for finite space and time increments. Entropy generation will depend on the sign of the wave speed and the local distribution of data.

According to the Warming–Hyett technique for obtaining the modified equation from the numerical formula, one should not use the original differential equation when replacing temporal derivatives with spatial ones, because a solution to the par-tial differential equation does not necessarily satisfy the difference equation. Using the Warming–Hyett (1974) approach to obtain the modified equation leaves the first-order parameter e unchanged, but gives

ε ν2

32

61 3 2= - ∆

∆- + ++ -x

tC C[ ( ) ] (7.45)

ε ν ν3

42 4

244 1 3 6= ∆

∆+ - - + -+ - + -x

tC C C C( ) [ ( )]

These are second- and third-order parameters that are substituted into the leading-error terms in Equation 7.40. These parameters differ from the results obtained by

7262_C007.indd 185 1/19/08 11:53:21 AM


replacing the higher-order temporal derivatives using the original conservation law. In any case, formulae for ek do not change the conclusions reached by a Second Law analysis. The second- and third-order terms associated with the third- and fourth-order derivatives may yet satisfy the inequality in Equation 7.39. Standard Fourier stability analysis and Warming–Hyett’s technique relate these terms with dispersion and dissipation of the solution, respectively. Contrary to this approach, the analogy to the Second Law indicates that errors associated with these two terms may both destroy or generate entropy. These effects are usually attributed only to numerical dispersion or dissipation. By examining the sign of local derivatives in the second term of the right side of Equation 7.44, one may pinpoint regions where a numerical method might fail to satisfy the Second Law, thus providing a means for predicting entropy destruction and the subsequent adverse effects on solution error and numeri-cal stability.

Figure 7.1 shows sample results of the numerical approximation of linear advec-tion with the upwind method. This solution is obtained with a CFL number of 0.5 using the updated expression in Equation 7.25 with the transition coefficients pre-sented in Table 7.1. The figure shows the exact (initial) data advected to the right (solid line) and the numerical approximation (circles connected by lines) after 10 time steps. Numerical dissipation reduces the magnitude of the peaks and spreads the data in the spatial domain. Compare this solution with results presented for the

f Ig u r E7.1 Exact (–) and numerical (– –) solution of linear advection equation with upwind method at n = 10.

2.2

2

1.8

1.6

1.4

1.2

1

0.8

η

0 0.25x

0.5 0.75 1

7262_C007.indd 186 1/19/08 11:53:23 AM


Lax–Wendroff method in Figure 7.2. This method is accurate to second order in both space and time, but the solution quality suffers from spurious oscillations that appear near sharp discontinuities, which is often observed with higher-order tech-niques. Both numerical solutions satisfy the CFL condition, and they are numeri-cally stable.

An explanation for the spurious oscillations in the Lax–Wendroff method is shown in Figure 7.3, illustrates entropy destruction at the first time step, as calcu-lated from the discrete approximation to Equation 7.21. For the linear advection equation, any method that destroys entropy locally suffers from the same defect. In Figure 7.3 and Figure 7.4, peaks in entropy generation and destruction occur at the first time step. A net destruction of entropy in the total domain leads to in numerical instabilities that cause the solution to diverge. The downwind and central differenc-ing methods, as represented by the transition coefficients in Table 7.1, exhibit this behavior. The Lax–Wendroff method does not suffer this defect because sufficient entropy generation occurs to offset the effects of entropy destruction.

In conclusion for linear advection: (i) a net destruction of entropy results in numerical instability; (ii) local violations of the Second Law, depending on severity, introduce spurious oscillations in the solution; and (iii) the Second Law approach reproduces the CFL condition established from conventional linear analysis.

f Ig u r E7.2 Exact (–) and numerical (– –) solution of linear advection equation with Lax–Wendroff method at n = 10.

2.2

2

1.8

1.6

1.4

1.2

1

0.8 0 0.25 0.5 0.75 1

η

x

7262_C007.indd 187 1/19/08 11:53:27 AM


12

8

4

0

–4

–8

–120 0.25 0.5 0.75 1x

(Sge

n)n j

Figure 7.3 Entropy generation for linear advection equation with Lax–Wendroff method at the time step n = 1.

12

8

4

0

–4

–8

–120 0.25 0.5 0.75 1

x

(Sge

n)n j

Figure 7.4 Entropy generation for linear advection equation with upwind method at the time step n = 1.

7262_C007.indd 188 1/19/08 11:53:30 AM


The leading-error numerical viscosity parameter exhibits a critical role that deter-mines whether a numerical method satisfies the Second Law. Satisfying the Second Law by prescribing nonnegative entropy generation is a robust way to establish time-step restrictions for the numerical scheme.

7.3.2 n o n Lin ea r Sc a La r ad v ec t io n

This section extends the previous analysis from linear to nonlinear problems involv-ing the scalar advection equation. Consider the variable u(t, x) (note: general scalar variable, not velocity field) with the following governing equation:

∂∂+ ∂∂=u

t

f

x0 (7.46)

where u t x( ), ∈ -∞,+∞( ), and the flux function is defined as

f u u( ) = 1

22 (7.47)

Using the same construction strategy as the linear case in the previous section, a pair of functions will represent the “entropy” and “entropy flux” for this case. The entropy functional will be defined as follows:

S u u( ) = - 2 (7.48)

Applying the compatibility condition ′ ′ = ′S u f u F u( ) ( ) ( ) and integrating give the following entropy flux function:

F u uS u( ) ( )= 2

3 (7.49)

The balance of entropy equation for this case is

genS

S

t

F

x= ∂∂+ ∂∂

(7.50)

Using an explicit time advance, the discretized transport equation can be expressed as

u C u C u C uj

njn

jn

jn+

-+

+-= + +1

10

1( ) ( ) ( ) (7.51)

where the solution-dependent transition coefficients must satisfy consistency and conservation as follows:

∆∆

- =+ -x

tC C u f u( ) ( ) (7.52)

C C C+ -+ + =0 1 (7.53)

7262_C007.indd 189 1/19/08 11:53:38 AM


For this nonlinear case, the numerical approximation expands in a Taylor series with the transition coefficients included in the expansion. Some methods prescribe com-plex formulae for the transition coefficients. For first-order methods that utilize a three-point stencil whereby the transition coefficients depend only on upstream val-ues of u, the technique for obtaining the modified equation is the same as the linear case, except that the transition coefficients must be included in the Taylor series expansion. The first three methods in Table 7.3 are shown of this kind. The second method listed in the table as “Baganoff” refers to an artificial dissipation technique derived by a statistical approach (Baganoff, 1983). The amount of dissipation is controlled by the parameters ( )r,ε . For consistency with the hyperbolic equation, Equation 7.46, the time step is calculated by the formula ∆ = ∆t x4ε for Baganoff’s method. In addition, other criteria apply to satisfy the monotonic condition and sta-bility. The condition obtained by the present technique agrees with Baganoff’s sta-tistical analysis (Baganoff, 1983). The modified equation for Lax and Wendroff’s technique requires a separate derivation (presented below).

For the first-order methods, spatial derivatives for the terms on the right side of the discretized transport equation lead to

C u t x x C u t x x

C u

x

x C u

x

( ) ( ), ± ∆ = , ± ∆ ∂∂+ ∆ ∂

∂±

2 2

22∆∆ ∂

∂+x C u

x

3 3

36

(7.54)

Using these expressions and the series expansion for u t t x( )+ ∆ , , collecting terms and simplifying the result by using the conservation and consistency conditions give the following discretized advection equation (up to second order):

Lu

t

f

x

x

t x

d

duC C u

u

x∆+ -≈

∂∂+∂∂-∆∆∂∂

+∂∂

2

2[( ) ]

+ ∆ ∂

∂+∆ ∂∂

∂∂

t u

t

x

x

d f

du

u

x2 6

2

2

2 2

2+ ∆ ∂

∂t u

t

2 3

36

(7.55)

t a bl E7.3t ransitioncoefficientsinVariousnumericalmethodsforanonlineara dvectionEquation


method C+ C 0 C-

Lax–Friedrichs 12 21+( )∆

∆tx u 0 1

2 21-( )∆∆tx u

Baganoff r u+ ε 1 2- r r u- εUpwind flux ∆

∆ | | +tx u u4 ( ) 1 2- | |∆

∆tx u ∆

∆ | | -tx u u4 ( )

Lax–Wendroff ∆∆

∆∆ -+( )t

xtx ju u4 1 1

2ˆ 1 1

4

212

12

- ( ) +( )∆∆ + -

tx j ju u uˆ ˆ - -( )∆

∆∆∆ +

tx

tx ju u4 1 1

2ˆ

7262_C007.indd 190 1/19/08 11:53:49 AM


The chain rule was used to modify the second- and third-order derivative terms. Similar to the linear case (previous section), with the Warming–Hyett technique, direct substitution of the original governing equation to eliminate second-order time derivatives lead to the same result for the leading first-order parameter. Using the chain rule and Equation 7.46 twice,

∂∂=∂∂

∂∂

2

2

u

t x

df

du

df

du

u

x (7.56)

Truncating Equation 7.55 to the leading order and moving terms to the right side lead to the following modified equation:

∂∂+ ∂∂≈∂∂

∂∂

u

t

f

x x

u

xε (7.57)

where (by definition)

ε( ) [( ) ]ux

t

d

duC C u

df

du

t

x≡ ∆∆

+ - ∆∆

+ -

2 2

2

(7.58)

This result reflects the nonlinearity of the governing equation.The corresponding modified equation for the balance of entropy is obtained by mul-

tiplying both sides of Equation 7.57 by S (u) and using the chain rule on the left side:

∂∂+ ∂∂≈ ′∂∂

∂∂

S

t

F

xS

x

u

xε (7.59)

Applying the chain rule to the right side of Equation 7.59 gives

′∂∂

∂∂

=∂∂

′ ∂∂

- ′′ ∂

∂S

x

u

x xS

u

xS

u

xε ε ε

2 (7.60)

The numerical entropy generation becomes

genS

xS

u

xS

u

x≈∂∂

′ ∂∂

- ′′ ∂

∂

ε ε2

(7.61)

Integrating this expression over a discrete space and time interval, the first term on the right side vanishes in the limit. The second term will not vanish if the interval contains a discontinuity. Therefore, entropy generation remains nonnegative, if and only if the leading-error parameter remains nonnegative. Similar to the linear case (previous section), selected methods can be used to see what restrictions are imposed on the time step or other parameters. Before deriving those results, the modified equation for the Lax–Wendroff method must be obtained.

7262_C007.indd 191 1/19/08 11:53:53 AM


The second-order modified equation for the Lax–Wendroff method, is

∂∂+ ∂∂≈∂∂

∂∂

u

t

f

x x

u

x

2

2 2ε (7.62)

where

ε2

3 3

6( )u

x

t

u t

x

u t

x≡ - ∆

∆∆∆- ∆∆

(7.63)

This result can be derived by directly replacing the third-order time derivative with the original governing equation and using the chain rule. The corresponding balance of entropy equation reduces to

genS

xS

x

u

xS

u

x≈∂∂

′∂∂

∂∂

- ′′ ∂

∂ε ε2 2

2∂∂∂+ ∂∂

u

x

u

xε2

2

2

with the derivative

d

du

x u t

x

ε22 2

61 3= - ∆ - ∆

∆

(7.64)

The sign for genS in the second-order Lax–Wendroff method therefore depends on

the sign of the local wave speed and the sign of the first and second derivatives. The method does not guarantee satisfaction of the Second Law as stipulated by gen

S ≥ 0. It appears evident from Equation 7.63 to set | | ∆ / ∆ ≤u t x 1, which is the same CFL condition given by linear analysis. Fortunately for this method, local violations of the Second Law are offset by sufficient entropy generation in adjacent regions, resulting in a net production of entropy so the method remains numerically stable. However, the upcoming results will show that solution quality is compromised.

Besides providing a means for nonlinear numerical analysis, an entropy-based approach also provides a way of developing numerical methods consistent with the Second Law. The dependence of the local first-order numerical viscosity on the solu-tion in Equation 7.58 can be used to construct a second-order method comparable to the Lax–Wendroff technique. Reordering the terms in Equation 7.58 to derive the following differential equation for the transition coefficients:

22

2 2

2

∆∆

+ ∆∆

= ++ -t

x

u t

x

d

duC C uε [( ) ] (7.65)

Integrating and solving for C C+ -+ gives

C C

t

x

u t

x+ -+ = ∆

∆+ ∆∆

232

2 2

2ε (7.66)

7262_C007.indd 192 1/19/08 11:54:00 AM


Combining the result with the consistency condition in Equation 7.52 gives two equations with two unknowns. The solution is obtained as

Cu t

x

u t

x

t

x+ = ∆

∆+

∆∆+ ∆∆

23

14 2

ε (7.67)

Cu t

x

u t

x

t

x- = ∆

∆-

∆∆+ ∆∆

23

14 2

ε (7.68)

Substituting the resulting transition coefficients into the discretized transport equa-tion yields a second-order accurate method, if e = 0. By controlling the parameter e based on local gradients, the numerical dissipation can be minimized in a way that satisfies the Second Law.

The numerical methods selected for comparison represent a sampling of a vari-ety of techniques available. The few methods chosen clearly indicate the advan-tages of the Second Law analogy for establishing a nonlinear restriction on the time step. Numerical analysis based on local linearization yields the standard CFL condition. The results in Table 7.4 confirm this condition as consistent with the Second Law analogy. It guarantees nonnegative entropy generation, due to the concavity property of entropy over the solution domain. In the artificial dis-sipation technique, the constant coefficient r lies between zero and two. It yields the same condition as other methods. The Lax–Wendroff technique gives e = 0, so the time step is set by inspection of e2 and the monotonic restriction on C0. An interesting result for the nonlinear case shows that the monotonic requirement may yield an overpermissive time step condition. Examining the transition coefficients and the monotonic requirement leads to a CFL-like condition with ν ≤ 2 for the Lax–Friedrichs, Baganoff, and upwind methods. For the Lax–Wendroff methods, the monotonic requirement gives ν ≤ 2 . The results in Table 7.4 indicate that a CFL condition based on monotonicity would destroy entropy for these methods. Numerical simulations confirm that the quality of the solution degrades with entropy destruction, even if the method remains numerically stable.

t a bl E7.4t ime-StepconstraintsImposedbytheSecondlawforone-d imensionalnonlinearScalara dvection

method 2 2εε∆∆ ∆∆t/ x c onstraint

Lax–Friedrichs 12

- ( )∆∆u tx

| | ∆∆ ≤u t

xmax 1

Baganoff 22

r u tx- ( )∆∆ | | ∆

∆ ≤u tx rmax 2

Upwind | |∆∆

∆∆- ( )u t

xu t

x

2 | | ∆∆ ≤u t

xmax 1

Lax–Wendroff 0 | | ∆∆ ≤u t

xmax 1

7262_C007.indd 193 1/19/08 11:54:08 AM


Sample numerical results for the nonlinear scalar equation were obtained for a domain x ∈ ,[ ]0 1 with the following initial conditions:

u x

x

x x

x

( ) ( )0

1 0 0 0 1

10 2 3 0 1 0 4

0 0 4 1

, =. ≤ ≤ .

+ / . < < .. ≤ ≤ ..

0

For the finite volume formulation, 100 cells were used with uniform grid spacing of ∆ = .x 0 01 (Camberos, 1995). The time increment was evaluated at each calculation by searching data for the largest local wave speed, then obtaining a numerical value for the maximum time step allowed, according to the condition in Table 7.4. The theoretical solution for this case predicts that the characteristic waves will coalesce into a shock wave at tn = 0.3 (nondimensional time), which subsequently moves at a constant speed of cs.

The scalar form of the Rankine–Hugoniot jump condition is f(uL) - f(uR) = cs(uL - uR), where the subscripts L and R represent the downstream and upstream values, respectively. Solving for the given initial conditions yields a shock propaga-tion speed of cs = 1/2. The weak form of the governing equation admits nonunique solutions. As mentioned previously, a separate “entropy condition” is required to mathematically enforce the desired solution. The desired solution corresponds to a compression shock wave. For convex flux-functions, f(u), one version of the entropy condition in the literature is ′ > > ′f u c f uL s R( ) ( ) (LeVeque, 1992). The theoretical shock solution satisfying this condition requires that the shock propagates in the direction indicated from umax to umin (left to right in Figure 7.5). Another form of the entropy condition for scalar equations corresponds to the weak form of Equa-tion 7.49, and the inequality gen

S ≥ 0 is enforced. Numerical methods satisfying this inequality will admit only discontinuities consistent with the desired solution.

Numerical results for two methods are presented in Figure 7.5 through Figure 7.8. Various numerical simulations revealed that the upwind method consistently gave the best results, unmatched even by second-order methods like the Lax–Wendroff method. Figure 7.5 shows the exact and numerical solution (with the upwind method) at the point of shock formation. Characteristic lines are shown in Figure 7.6, clearly indicating coalescence at the predicted time. The slope also gives dt/dx = 2, consis-tent with the theoretical value for shock propagation. The numerical solution for the upwind method is again shown in Figure 7.7, superimposed with the numerically predicted entropy generation. The peak in the entropy generation corresponds to the cell where the shock wave is “captured.” No regions of negative entropy genera-tion are evident, and the solution quality is reasonably good. Figure 7.8 shows the numerical solution for the Lax–Wendroff method, superimposed with the numerical entropy generation. The peak in the entropy generation is higher than results from the first-order upwind method, but it also results in a region of entropy destruction (lagging the discontinuity). The cause of the spike in the solution is the entropy destruction occurring in the corresponding cell. Both methods converge to the physi-cally relevant solution in the limit (by virtue of satisfying the weak form of the Second Law). However, only the upwind method matched the theoretical solution

7262_C007.indd 194 1/19/08 11:54:12 AM


InitialDistribution

Numerical

Exact, n = 29

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0.2–

unj

0 0.25 0.5 0.75 1x

f Ig u r E7.5 Numerical solution of the nonlinear scalar equation with the upwind method at tn = 0.3.

Shock Motionat Steady Speed

Shock Formation

tn

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0 0.2 0.4 0.6x

0.8 1

f Ig u r E7.6 Space-time contours of the numerical solution of the nonlinear scalar equation with upwind method.

7262_C007.indd 195 1/19/08 11:54:18 AM


for finite time and space increments. Thus, for the nonlinear case, the nonnegative “entropy generation” criterion establishes a restriction on the time step for numerical calculations, which differs from monotonic criteria, but matches quasilinear analysis corresponding to the CFL condition.

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

–0.20.0 0.2 0.4 0.6 0.8 1.8

22

Shock Motion

(Sgen)nj 20

18

16

14

12

10

8

6

4

2

0

–2

unj

x

f Ig u r E7.7 Numerical solution and entropy generation for nonlinear scalar equation with upwind method.

1

0

1.6

1.4

1.2

0.8

0.6

0.4

0.2

–0.2

–0.4 0.0 0.2 0.4 0.6 0.8 1.0

–1.0

–5

0

5

10

15

20

25

30

35

Shock Motionun

j

x

(Sgen)nj

f Ig u r E7.8 Numerical solution and entropy generation for nonlinear scalar equation with Lax–Wendroff method.

7262_C007.indd 196 1/19/08 11:54:22 AM


7.3.3 c o u pLed no n Lin ea r eq u at io n S

This third case will focus on a system of coupled equations that represent the conser-vation of mass, momentum, and energy for an ideal gas. Consider the following form of the Euler equations of gas dynamics in conservation law form,

∂∂+ ∂∂=q

t

f

x0 (7.69)

where

q u

e

f

u

u P

u e P

=

, = +

+ /

ρρρ

ρρρ ρ

2

( )

represent the algebraic state and flux vectors, respectively. Define the flux Jacobian function as follows:

A q

f

q( ) = ∂

∂ (7.70)

It has certain properties that arise from the first-order homogeneous nature of the flux vector as a function of the state variables. Note that A q f⋅ = and A can be decom-posed as A Y Y= ⋅ ⋅ -Λ 1 where Y contains the eigenvectors and L has the eigenvalues of A. For an ideal gas, the thermodynamic entropy functional is known. Assuming an ideal gas with constant specific heats, the nondimensional formula for the specific entropy is

s T=-

-11γ

ρln ln (7.71)

where T and ρ are the normalized temperature and mass density, respectively. The balance of entropy equation is

genS

S

t

F

x= ∂∂+ ∂∂

(7.72)

where the functional relation for entropy is S q s( ) = ρ , and F q us( ) = ρ for the entropy flux. The flux function F satisfies the following compatibility condition required by the Second Law:

F S fq q q, - , ⋅ , = 0 (7.73)

The subscripts represent derivatives with respect to the algebraic vector of state vari-ables. The ideal gas formula for the entropy leads to a concave function of the state

7262_C007.indd 197 1/19/08 11:54:31 AM


variables. Writing the formula in terms of q and taking the Hessian (a matrix of second derivatives) demonstrate its negative definite property.

Using a standard one-dimensional explicit discretization of the system of equations,

q C q C q C qjn

jn

jn

jn+

-+ -

+= ⋅ + ⋅ + ⋅11

01( ) ( ) ( )

(7.74)

This equation reflects a three-point numerical stencil in a transition coefficient form. In this case, matrix transition coefficients arise instead of scalars. The transition coefficients must still satisfy the conservation and consistency requirements. From conservation requirements, the transition coefficient matrices must add up to the identity matrix of the same size,

C C C I+ -+ + =0 (7.75)

For consistency purposes, we require that

∆∆

- ⋅ =+ -x

tC C q f q( ) ( ) (7.76)

As described previously in the scalar case, the transition coefficient C± may or may not satisfy monotonicity, but for conservation it is necessary that I C≥ ≥0 0. Expand-ing the numerical discretization in Equation 7.74 to the known solution at time level, t, in a Taylor series leads to

C q t x x C q x

C q

x

x C q

x

x

( ), ± ∆ = ± ∆∂ ⋅∂

+ ∆∂ ⋅∂

± ∆2 2

2

3

2 66

3

3

∂ ⋅∂

+C q

x

(7.77)

and similarly for q t t x( )+ ∆ , . Substituting this result into the previous equation, col-lecting terms, and simplifying by using the conservation and consistency conditions gives (up to second order)

Lq

t

f

x

x

t x xC C q∆+ -≈

∂∂+∂∂-∆∆∂∂∂∂

+ ⋅

+

2

2[( ) ]

∆∆ ∂∂+∆ ∂∂

⋅∂∂

+∆ ∂t q

t

x

xA

q

x

t

2 6 6

2

2

2 2

2

2 3qqt∂ 3

(7.78)

In contrast to the nonlinear scalar equation, the first-order error term with spatial derivatives can be evaluated as follows:

∂∂

+ ⋅ =∂∂

+ ⋅ + ++ - + -

+ -

xC C q

xC C q C C( ) ( ) ( ) ⋅⋅ ∂

∂q

x

7262_C007.indd 198 1/19/08 11:54:37 AM


The transition coefficients are constructed by the flux vector, f, which is homoge-neous of degree one. Thus,

∂∂⋅ =A

xq 0

(7.79)

The sum of the transition coefficients, C C+ -+ , then obeys the following equation:

∂∂

+ ⋅ =+ -

x

C C q( ) 0 (7.80)

Using this result and replacing the second-order time derivatives with the original differential equation,

∂∂=∂∂

∂∂

2

22q

t xA

q

x (7.81)

gives

∂∂+ ∂∂≈∂∂

⋅ ∂∂

q

t

f

x x

q

x[ ]ε

(7.82)

where the numerical viscosity parameter is

[ ] ( )ε = ∆∆

+ - ∆∆ + -x

tC C A

t

x

22

2

22 (7.83)

The modified entropy balance equation is obtained after multiplying Equation 7.82 by S,q and using the chain rule. The result becomes

∂∂+ ∂∂≈ , ⋅

∂∂

⋅ ∂∂

S

t

F

xS

x

q

xq [ ]ε (7.84)

which represents the modified equation for the balance of entropy. Expressing the result in terms of the entropy generation rate gives

genS

xS

q

xq

xSq

T

qq≈∂∂

, ⋅ ⋅ ∂∂

+ ∂∂⋅ - , ⋅[ ] ( [ε ε ]]) ⋅ ∂

∂q

x (7.85)

For the entropy variable, S,q equals an algebraic row vector and S,qq represents a second-order tensor, which is the Hessian of S with respect to q. The Hessian of the entropy for an ideal gas is symmetric and negative definite. The following result was

7262_C007.indd 199 1/19/08 11:54:43 AM


derived by Merriam (1989):

S Y YqqT, = - ⋅- -( )1 1 (7.86)

where the superscript T represents the matrix transpose operation. The condition stipulated by the Second Law from Equation 7.85 requires that

x S xT

qq⋅ - , ⋅ ⋅ ≥( [ ])ε 0 (7.87)

If - , ⋅S qq [ ]ε is positive definite, then equality holds if and only if x = 0. This effec-tively imposes a limit on the time step, given a particular grid size and initial distribu-tion. After substituting Equation 7.86 for the Hessian, the matrix core of Equation 7.87 becomes

- , ⋅ = ⋅ ⋅ ⋅ ⋅- - -S Y Y Y Yqq

T[ ] ( ) [ ]ε ε1 1 1 (7.88)

Defining a new vector z = Y ⋅ x reduces Equation 7.86 to

z Y Y zT ⋅ ⋅ ⋅ ⋅ ≥-( [ ] )1 0ε (7.89)

The matrix Y -1 ⋅ [ε] = Y is mathematically similar to [ ]ε , and hence they have the same eigenvalues. Furthermore, the construction of transition coefficients satisfying the consistency requirement guarantees that Y Y- ⋅ ⋅1 [ ]ε is diagonal and therefore also symmetric. This implies that

δ ε δmax minz z z Y Y z z zT T T⋅ ≥ ⋅ ⋅ ⋅ ⋅ ≥ ⋅-( [ ] )1

(7.90)

where δ δmax min≥ are the largest and smallest eigenvalues of the symmetric matrix. From Equation 7.83, for a given set of transition coefficients, the minimum eigen-value δmin coincides with the maximum eigenvalue of the flux Jacobian A. Thus, δmin ≥ 0 will guarantee the condition given by Equation 7.89, and hence also satisfy the Second Law as required by gen

S ≥ 0 in Equation 7.85.The transition coefficients for various methods are presented in Table 7.5.

The upwind method includes the classical Steger–Warming flux vector split-ting method (FVS) (Steger and Warming, 1981), Roe’s “flux-difference splitting” (FDS) (Roe, 1981), and other related methods. To obtain the correct formula for the error parameter, an equivalent differentiable term replaces the dissipation non-differentiable error term for Steger–Warming FVS. The time step limit observed in numerical simulations to keep the Steger–Warming method stable places an upper value of 0.7 for the Courant number. This differs from results of either lin-ear analysis or the monotonic condition, both of which limit the Courant number to an upper value of one. Steger–Warming’s method exhibits numerical instability when the Courant number lies too close to its maximum (linearly obtained) value. The method labeled “arithmetic averaging” is a variant of FVS and flux-difference splitting. The transition coefficients contain flux Jacobians evaluated at intermediate

7262_C007.indd 200 1/19/08 11:54:52 AM


values of the state variables: q q qj j j+ / += + / .1 2 1 2( ) The fifth method listed in Table 7.5 represents the “kinetic split flux” technique. Construction of the split fluxes (or, equivalently, the transition coefficients) relies on generating one-sided moments of the Boltzmann equation from kinetic theory (Camberos, 1997a,b). The technique utilizes a given probability distribution function that reflects local values of the den-sity, velocity, and temperature. For the Euler equations, the probability distribution reflects the local equilibrium assumption and hence equals a Maxwell–Boltzmann probability distribution in the molecular velocity variable, with the mass density, macroscopic velocity, and temperature as constraints.

The following abbreviations are used in Table 7.6:

I Iσσπ

≡ 2 2

(7.91)

t a bl E7.5t ransitioncoefficientsfortheone-d imensionalEulerEquationswithVariousnumericalSchemes


method C+ C 0 C-

Lax–Friedrichs 12 ( )I At

x+ ∆∆ 0 12 ( )I At

x- ∆∆Steger–Warming ∆

∆ | | +tx A A2 ( ) I At

x- | |∆∆

∆∆ | | -t

x A A2 ( )

Arithmetic averaging ∆∆ -| | +t

x jA A2 1

2( ) I C C- ++ -( ) ∆

∆ +| | -( )t

xj

A A2 12

Roe FDS ∆∆ -| | +( )t

x jA A2 1

2

ˆ I A Atx j j

- | | + | |∆∆ - +

2 1

212

ˆ ˆ ∆∆ -| | +( )t

x jA A2 1

2

ˆ

Kinetic split flux ∆∆ +( )t

x A A2 σ I Atx- ∆∆ σ

∆∆ -t

x A A2 ( )σ

Lax–Wendroff ∆∆

∆∆ - +( ) ⋅t

xtx j

A I A2 12

I A A Atx j j

- + ⋅∆∆ - +

2

2 12

12

2 ∆

∆∆∆ + -( ) ⋅t

xtx j

A I A2 12

t a bl E7.6t ime-StepconstraintsImposedbytheSecondlawfortheone-d imensionalEulerEquationsmethod Y t

t- ⋅ ⋅1 2

2[ ]εε ∆∆∆∆Y c onstraint

Lax–Friedrichs I t

x- ( )∆∆Λ22 | | ≤∆∆λ t

x 1

Steger–Warming I tx

txσ

∆∆

∆∆- ( )Λ2

2 λ σ π2 2∆∆ ≤t

x

Roe FDS & AA Λ Λ∆∆

∆∆- ( )t

xtx

22 | | ≤∆∆λ t

x 1

K–Split flux Λ Λσ∆∆

∆∆- ( )t

xtx

22 λ λαβ2 ∆

∆ ≤tx

Lax–Wendroff 0 | | ≤∆∆λ tx 1

7262_C007.indd 201 1/19/08 11:55:12 AM


where I equals the identity matrix. Also

Λ Λσ α βσ γγ

= + --

2 0 0

021

0

0 0 2

(7.92)

with

α

σβ

π σ≡

, ≡ - erf

u u2

22

2

2exp

(7.93)

and

λ α λ βσαβ = | | +max 2 (7.94)

The parameter σ defines a characteristic thermal velocity as follows:

σ γρ

2

12 1 3 2

212

2( )qq

q q qP= - - = .( ) (7.95)

Table 7.6 presents results of the Second Law analysis for the equations of gas dynamics. From the table, the Lax–Friedrichs method requires a time-step con-straint, essentially equivalent to a CFL condition based on linear theory. For the Steger–Warming method, the constraint gives a CFL number of 0.674 for uniform initial conditions of zero velocity. This number is surprisingly close to the limit observed in numerical simulations, thereby suggesting a possible theoretical expla-nation. For uniform initial conditions of zero velocity, a = 0 and β π= /2 , the Second Law constraint for the kinetic split flux method reduces to

λ σ

πmax2 2

2∆∆≤t

x (7.96)

which equals twice the value of the CFL number imposed on the Steger–Warming method by ad-hoc arguments. This result suggests that a connection between the two methods may exist.

7.4 St a bIl It y o f Sh o c k ca pt u r In g mEt h o d S

Differences in the numerical solution of the Euler equations become evident by test-ing methods against various cases. Consider a shock-structure problem (Liepmann et al., 1962) governed by the viscous Navier–Stokes equations. Although the Euler equations of gas dynamics contain no mathematical terms representing viscous dis-sipation or heat conduction, their numerical solution does contain diffusive effects. These effects of numerical dissipation and heat conduction will be made evident by solving the shock-structure problem.

7262_C007.indd 202 1/19/08 11:55:17 AM


Gas dynamics theory provides a solution to the “jump equations” across a planar shock wave (Liepmann and Roshko, 1957). They are given by the following Rankine– Hugoniot relations for an ideal gas:

ρρ

γγ

2

1

1

2

12

12

11 2

= = +- +

u

u

M

M

( )( ) (7.97)

P

PM2

1121

21

1= ++

-( )γγ (7.98)

T

T

P

P2

1

2

1

1

2

= ρρ (7.99)

s sP

P2 12

1

2

1

11

- =-

γρρ

γ

ln (7.100)

where M1 and M2 refer to the upstream and downstream Mach number, respectively.The initial and boundary conditions are set according to these relations with the

jump centered in the computational domain, assuming a linear connection across 10 grid cells. Camberos (1995) solved these equations numerically with a finite-volume algorithm and piecewise constant data using the expression of Equation 7.74 and the various transition coefficients in Table 7.5. An idealized solution is given by the exact jump relations with a shock width of zero thickness. Theoretically, indepen-dent of the dissipative mechanisms, the entropy change across the shock wave has a maximum value (Pike, 1985). The maximum value of the specific entropy for the shock wave depends only on the upstream Mach number and the ratio of spe-cific heats, just like other variables in the Rankine–Hugoniot relations. This maxi-mum can be found from the conservation equations and setting the velocity equal to

u u1 2 , where u1 and u2 represent the upstream and downstream velocities, respec-tively. The formula for the entropy peak is given by

S MMmax

mx=-

-+++

+

ργ

γγ γ

γ

111

211

2

12

12

ln( )

(7.101)

where

ρ ρ γγmx =+- +1

12

12

11 2

( )( )

M

M

(7.102)

equals the mass density at the same conditions which yield the entropy maximum. Note that Equation 7.101 is a maximum for all Mach numbers, only for the specific entropy.

Consider a stationary shock wave, as described previously for supersonic flow at a Mach number of 1.5. Gas dynamics theory predicts a positive jump in the pressure, temperature, and mass density, whereas a negative jump is predicted for the velocity

7262_C007.indd 203 1/19/08 11:55:23 AM


(the flow slows down to subsonic conditions). The entropy also undergoes a positive jump because a shock wave represents an intrinsically irreversible (entropy-generat-ing) process. Thus, a finite amount of entropy generation occurs, and the entropy balance equation can be used to predict its magnitude. The numerical value of the entropy maximum for M1 = 1.5 equals 0.2214, which is 38% greater than the entropy jump of 0.1356 given by Equation 7.100.

In the following examples, Equation 7.69 is solved numerically with the approxi-mation of Equation 7.74 together with the transition coefficients for various methods listed in Table 7.5. The spatial domain was ξ ∈ ,[ ]0 1 , with initial and boundary con-ditions given by fixed upstream and downstream values of the mass density, pres-sure, and velocity according to Equation 7.97 to Equation 7.99, along with a linear variation spanning 10 cells for ξ ∈ . , .[ ]0 4 0 6 . The grid consists of 100 cells that give ∆ = .ξ 0 01 uniformly. Time increments are set according to the constraints prescribed in Table 7.6. Numerical solutions converge in about 1000 time increments, giving a steady-state solution that satisfies the Rankine–Hugoniot jump conditions for a stationary, planar shock wave.

As shown in Figure 7.9 and Figure 7.10, the Lax–Friedrichs method solves the shock structure problem with a smooth transition from upstream (ahead of the shock) to downstream values. The result is close to the solution of the Navier–Stokes equa-tions for a viscous, heat-conducting gas. For the gas dynamic equations, the shock structure is entirely due to numerical effects, indicating the numerical dissipation present in this method. In Figure 7.9, no entropy peak is present, and Equation 7.98 is not satisfied. Entropy generation has a very low peak, spread over 24 to 26 cells. For this example, the number of cells with nonzero entropy generation can be used as a measure of the shock width. To represent an entropy peak, a minimum of 3 points

Pres

sure

, Tem

pera

ture

, Vel

ocity

Entr

opy

2.6

2.4

2.2

2

1.8

1.6

1.4

1.2

1

0 0.25 0.5 0.75 1

0.20

0.14

0.08

0.02

T/T1

P/P1

u/u1S – S1

f Ig u r E7.9 Numerical “shock-structure” solution of Euler equations with Lax–Friedrichs method.

7262_C007.indd 204 1/19/08 11:55:30 AM


is required, so an ideal method would have a shock width of 2 cells. The Lax–Friedrichs method (with this metric) gives a very wide shock.

A solution to the stationary shock problem given by the Steger–Warming method is shown in Figure 7.11 and Figure 7.12. The transition from upstream to downstream

14

12

10

8

6

4

2

0

0 0.25 0.5 0.75 1

0

0.05

0.1

0.15S – S1

0.2

0.25

(Sge

n)n j

f Ig u r E7.10 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution of Euler equations with Lax–Friedrichs method.

2.6

2.4

2.2

2

1.8

1.6

1.4

1.2

1

1 0.25 0.5 0.75 1

0.02

0.08

0.14

0.20

Pres

sure

, Tem

pera

ture

, Vel

ocity

Entr

opy

T/T1

P/P1

u /u1S – S1

f Ig u r E7.11 Numerical shock structure solution of Euler equations with Steger–Warming flux-vector splitting.

7262_C007.indd 205 1/19/08 11:55:35 AM


values appears to have a sharp discontinuity (Figure 7.11) followed by a smooth tran-sition to downstream conditions. Nonzero entropy generation occurs in the shock interior, as displayed in Figure 7.12, indicating a shock thickness of about 8 or 9 cells. An entropy peak is evident in the figure, but it is much smaller than the value pre-dicted by Equation 7.101. Also, the peak in entropy generation appears to have a value of 4 (nondimensional units), whereas theory predicts a value of 6.5 for a minimum- width shock wave of 2 cells required to capture the entropy peak.

Figure 7.13 and Figure 7.14 present the results of Roe’s FDS method. Although not shown, the arithmetic averaging and Lax–Wendroff methods give converged results identical to those shown in Figure 7.13 and Figure 7.14 (Lax–Wendroff entropy generation peaked at about 12). The discontinuity is well resolved by these methods, changing from upstream to downstream conditions within one cell spacing. Unfortunately, none of these methods captures the peak in the entropy predicted by Equation 7.101. Judging from the nonzero entropy generation shown in Figure 7.14, the shock wave is actually 2 cells wide. The peak entropy, as it changes from upstream to downstream conditions across the shock wave, is predicted by ideal gas dynamics. This is consistent with past studies by Pike (1985), where “any scheme for obtaining steady solutions of the Euler equations, which conserves mass and energy and obeys the equation of state, will be correct in exhibiting an entropy maximum at the shock wave.” But it differs from a standard view that successful shock capturing should produce monotonic results by means of a minimum of intermediate values. Changes in the entropy across a shock wave are not monotonic, but these are not made evident in the test cases typically presented.

Only the kinetic split flux technique appears to approximately satisfy Equation 7.101. Results with this method are presented in Figure 7.15 and Figure 7.16.

14

12

10

8

6

4

2

0

0 0.25 0.5 0.75 1

0

0.05

0.01

0.15

0.2

0.25

S – S1

(Sge

n)n j

f Ig u r E7.12 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution of Euler equations with Steger–Warming flux-vector splitting.

7262_C007.indd 206 1/19/08 11:55:38 AM


2.6

2.4

2.2

1.8

1.6

1.4

1.2

1

0 0.25 0.5 0.75 1

0.02

0.08

0.14

0.20

2

Pres

sure

, Tem

pera

ture

, Vel

ocity

Entr

opy

T/T1

P/P1

u/u1S – S1

f Ig u r E7.13 Numerical shock-structure solution of Euler equations with Roe’s flux- difference splitting.

14

12

10

8

6

4

2

0

0 0.25 0.5 0.75 1

0

0.05

0.1

0.15

0.2

0.25

S – S1

(Sge

n)n j

f Ig u r E7.14 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution of Euler equations with Roe’s flux-difference splitting.

7262_C007.indd 207 1/19/08 11:55:43 AM


2.6

2.4

2.2

2

1.8

1.6

1.4

1.2

1

1 0.25 0.5 0.75

0.02

0.10

0.18

0.26

1

Pres

sure

, Tem

pera

ture

, Vel

ocity

Entr

opy

T/T1

P/P1

u/u1S – S1

f Ig u r E7.15 Numerical shock-structure solution of Euler equations with kinetic split-flux method.

14

12

10

8

6

4

2

0

0 0.25 0.5 0.75 1

0

0.05

0.1

0.15

0.2

0.25

S – S1

(Sge

n)n j

f Ig u r E7.16 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution of Euler equations with kinetic split-flux method.

7262_C007.indd 208 1/19/08 11:55:45 AM


The transition from upstream to downstream conditions is smooth, but not as smeared as the Lax–Friedrichs method. In addition, the entropy clearly peaks inside the shock wave, although the value obtained is about 28% higher than results pre-dicted by Equation 7.101. The peak entropy generation appears to be centered in the shock interior, and it has a value of about 4.5, compared with 6.5, predicted by the-ory. The shock wave is about 12 cells wide, as noted by nonzero entropy generation in Figure 7.16. The positive results obtained with this method suggest that combining the kinetic split flux technique with some type of averaging of the flux Jacobians in the transition coefficients (like Roe’s FDS) may produce an optimal method that captures both the entropy peak and the shock wave with minimal spreading.

Transient results are displayed in Figure 7.17 and Figure 7.18. Figure 7.17 shows the transient entropy and entropy production after 100 time increments for the Steger–Warming FVS method. The initial and boundary conditions led to a spike in the state variables. Figure 7.18 shows the same result for the Lax–Wendroff method, which clearly exhibits the oscillations of this technique. As reported previously for the scalar cases, regions of spurious oscillations in the solution coincide with entropy destruction. This highlights the difference between entropy generation and entropy change, which are often confused in the literature. Spurious, nonphysical oscilla-tions in the solution of the gas dynamics equations occur from local violations of the Second Law, which stipulates that entropy generation must be nonnegative. Peaks in the entropy change across a control volume are not violations of the Second Law. The example in this section has shown that numerical methods exhibiting a maxi-mum in the entropy at the shock wave are not entirely spurious.

S – S1

14

12

10

8

6

4

2

0

0 0.25 0.5 0.75 1

0

0.05

0.1

0.15

0.2

0.25

(Sge

n)n j

f Ig u r E7.17 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution with Steger–Warming FVS at 100 time-step iterations.

7262_C007.indd 209 1/19/08 11:55:46 AM


In closing, this chapter has demonstrated the utility of the Second Law as a reli-able method for establishing time-step constraints and numerical stability. It also serves as a time-step indicator for implicit calculations in transient simulations. One could well interpret the results of a Second Law approach as a logical component of numerical or mathematical analysis that establishes numerical stability, conver-gence, existence, and uniqueness of weak solutions to a given problem. By the same argument, one could extend the conjecture to apply the same conclusions to systems of equations, regardless of equation type (parabolic, elliptic, or hyperbolic). This would follow early pioneering efforts to fully understand the implications of entropy and the Second Law in numerical analysis. It would also firmly establish the utility of the Second Law beyond its role in thermodynamics, or an analogy to that role. Instead, it would become an essential element in numerical modeling itself, not lim-ited to its physical and historical origins.

r Ef Er En c ES

Anderson, D.A., Tannehill, J.C., and R.H. Pletcher. 1984. Computational Fluid Mechanics and Heat Transfer. Hemisphere, New York, 70–71.

Baganoff, D. 1983. Stochastic Processes in Aeronautics. Department of Aeronautics and Astronautics. ©1983 by D. Baganoff, Stanford University, Stanford, CA (unpublished), 37–39.

Camberos, J.A. 1995. Probabilistic Approach to the Computational Simulation of Gasdy-namic Processes. Doctoral dissertation, Department of Aeronautics and Astronautics (SUDAAR No. 668), Stanford University, Stanford, CA, 102–105.

S – S1

14

12

10

8

6

4

2

0

–2

–4

0 0.25 0.5 0.75 1

–0.05

0

0.05

0.1

0.15

0.2

(Sge

n)n j

f Ig u r E7.18 Entropy (– –) and entropy generation (– –) for stationary shock numerical solution with Lax and Wendroff’s method at 100 time-step iterations.

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Camberos, J.A. 1997a. Comparison of Split Fluxes Generated with Selected Probability Dis-tributions Functions. Presented at the 13th Computational Fluid Dynamics Confer-ence, AIAA Paper No. 97–2095. Snowmass Village, CO (unpublished).

Camberos, J.A. 1997b. Comparison of Selected Probability Distribution Functions for Gasdynamic Simulations Inspired by Kinetic Theory. Presented at the 35th Aero-space Sciences Meeting AIAA Paper No. 97-0340. Reno, NV (unpublished).

Camberos, J.A. 2000a. On the Construction of Entropy Balance Equations for Arbitrary Thermophysical Processes. Paper draft in preparation for submission to the 39th AIAA Aerospace Sciences Meeting in January 2001 (unpublished).

Camberos, J.A. 2000b. Nonlinear time-step constraints based on the Second Law of Thermo-dynamics. AIAA J. Thermophysics Heat Transfer, 14(3): 231–244.

Camberos, J.A. 2000c. An alternative interpretation of work potential in thermophysical pro-cesses. AIAA J. Thermophysics Heat Transfer, 14(2): 177–185.

Courant, R., Friedrichs, K.O., and H. Lewy. 1928. On the partial difference equations of mathematical physics. IBM J. Res. Dev. (1967), 11: 215-234; originally published in Mathematische Annalen, 100: 32–74.

Crandall, M.G. and A. Majda. 1980. Monotone difference approximation for scalar conserva-tion laws. Math. Computation, 34(149): 1–21.

Jaynes, E.T. 1991. Probability theory as logic. In Maximum-Entropy and Bayesian Methods, P.F. Fougére, Ed., Kluwer, Dordrecht, 1–16.

Kapur, J.N. and H.K. Kesavan. 1992. Entropy Optimization Principles with Applications. Academic Press–Harcourt Brace Jovanovich, San Diego, CA, 66–67.

Lavenda, B.H. 1991. Statistical Physics: A Probabilistic Approach. John Wiley & Sons, New York.

Lax, P.D. 1954. Weak solutions of nonlinear hyperbolic equations and their numerical com-putation. Commun. Pure Appl. Math, 7: 159–193.

Lax, P.D. 1973. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conference Series in Applied Mathematics.

LeVeque, R.J. 1992. Numerical Methods for Conservation Laws. Birkhäuser-Verlag, Berlin, 36–40.

Liepmann, H.W. and A. Roshko. 1957. Elements of Gasdynamics. John Wiley & Sons, New York, 59–60.

Liepmann, H.W., Narasimha, R., and M.T. Chahine. 1962. Structure of a plane shock layer. Phys. Fluids, 5(11): 1313–1324.

Merriam, M.L. 1989. An Entropy-Based Approach to Nonlinear Stability. Ph.D. thesis. Stanford University, Stanford, CA.

Oleinik, O.A. 1957. Discontinuous solutions of non-linear differential equations. AMS Trans-lation Series 2, 26: 95–172 (1963). Russian original: Uspehi Mat. Nauk (N.S.), 12(2): 3–73.

Oleinik, O.A. 1959. Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. AMS Translation Series 2, 33: 285–290 (1963). Russian original: Uspehi Mat. Nauk (N.S.), 14(2): 165–170.

Pike, J. 1985. Notes on the structure of viscous and numerically-captured shocks. Aeronaut. J. R. Aeronaut. Soc., November: 335–338.

Steger, J.L. and R.F. Warming. 1981. Flux vector splitting of the inviscid gasdynamic equa-tions with application to finite-difference methods. J. Computational Phys., 40: 263–293.

Warming, R.F. and B.J. Hyett. 1974. The modified equation approach to the stability and accu-racy analysis of finite-difference methods. J. Computational Phys., 14: 159–179.

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213

8 Entropy Transport with Phase Change Heat Transfer


This chapter examines the role of entropy and the Second Law of Thermodynamics in selected problems involving phase change heat transfer, particularly with solidifi-cation, melting, and film condensation. It is not intended to cover all aspects of multi-phase problems, but rather a sample of selected topics in specific areas where entropy and the Second Law have particular importance. Typical applications arise in heat exchangers, multiphase processing in chemical equipment, and so forth. Rosen and coworkers (1999, 2004) have developed methods of exergy analysis for applications to systems like industrial steam process heaters and thermal energy storage systems. This chapter examines the role of entropy and exergy as design tools for developing improvements to such engineering systems, particularly involving phase change and multiphase flows.

Solidification and melting arise in many engineering applications, including materials processing, ice accretion on structures, and thermal energy storage in electronic assemblies. The design and prediction of these phase change processes typically involve solutions of the conservation equations (mass, momentum, energy, and species equations). A variety of numerical procedures, such as finite differences (Salcudean and Guthrie, 1979), finite elements (Pardo and Weckman, 1990), finite volumes (Bennon, Incropera, 1988), and combined finite volume-element methods (Naterer and Schneider, 1996), have been developed for these problems. Numerical models provide effective tools for better understanding of transport processes during solidification and melting. This includes solute segregation, thermosolutal convec-tion, and interdendritic and shrinkage flows. Flood and Davidson (1994) observed the formation of centerline macrosegregation in aluminum cast ingots including the sensitivity to ingot thickness and casting speed. Rady and coworkers (1997) used a finite volume method to predict thermal and solutal buoyancy during solidification of hypereutectic and hypoeutectic binary alloys. Additional phenomena involving interdendritic flows, including solute redistribution in the mushy zone, were exam-ined by Maples and Poirier (1984). Modeling developments in these phase change problems were summarized and discussed in a comprehensive review by Salcudean and Abdullah (1988).

In solid–liquid phase change problems, entropy can serve as an effective parame-ter for understanding and describing various physical processes. For example, interface properties like interface “roughness” are determined from the entropy change during phase transition. At microscopic scales, a rough or “nonfaceted” surface exhibits a low

7262_C008.indd 213 1/19/08 11:57:28 AM


entropy of fusion. During solidification, dendritic arms grow in a direction cor-responding to the maximum thermal irreversibility, because it is aligned with the heat flow direction (i.e., direction of the local temperature gradient). As a result, entropy is an important characteristic of the two-phase permeability, when describ-ing how pockets or channels of liquid are formed within the solid matrix. Another complex process is thermal recalescence, which involves a transient temperature rise during cooling of a freezing crystal it occurs from a latent heat release that exceeds the other modes of cooling. In this case, a positive rate of entropy change can indicate the duration and magnitude of the local reheating. These processes have been observed by many researchers, but less attention has been given to the role of entropy during the processes. Bejan (1996) applied minimization of entropy generation to various multiphase systems, including refrigeration, energy stor-age systems, and power generation. Charach and Rubinstein (1989) investigated entropy generation during phase change heat conduction. Much additional oppor-tunity exists for incorporating entropy and the Second Law into phase change analysis.

As discussed in previous chapters, past computational fluid dynamics (CFD) studies have shown that the Second Law can improve solution accuracy (Lax, 1971) and upwinding accuracy (Naterer, 1999) in fluid flow simulations. In the context of phase change heat transfer, these results can be extended to establish the unique-ness of interface resolution, subject to different convergence tolerances imposed on a numerical model. Entropy production can also establish an optimal and convergent phase distribution during numerical iterations, without randomly cycling through phases, because only entropy-producing solutions are physically possible. Arbitrary convergence tolerances can be reduced or eliminated with the Second Law. Numer-ous techniques have been developed for convergence acceleration in nonlinear prob-lems, including relaxation factors and multigrid methods (Minkowycz et al., 1988). Interface tracking by sequential steps was proposed by Schneider and Raw (1984), whereby two phase rules were used to coordinate the orderly progression of phase transition between adjacent control volumes. In this chapter, it will be shown that these iterative procedures are closely linked to the Second Law.

This chapter will derive an alternative entropy-based framework (or heat-entropy analogies) for various transport processes during phase change. This includes inter-phase momentum exchange and recalescence phenomena. The intrinsic generality of entropy as an abstract concept provides opportunities for deeper insight into com-plicated phenomena. Previous analogies have established connections between heat transfer and friction coefficients (i.e., Reynolds analogy between heat and momen-tum), and similar opportunities can be realized with entropy. It will be shown that transport phenomena involving one variable (temperature) may be inferred through consideration of the other analogy variable (entropy). This type of similarity can be particularly useful if prediction of a certain variable is difficult or time-consuming, whereas analysis involving the other analogy variable may be more readily imple-mented. Additional benefits arising from the Second Law in computational models, such as numerical stability, may be realized.

In addition to past numerical studies, experimental data provides vital insight for detailed understanding of phase change processes. Previous experimental studies

7262_C008.indd 214 1/19/08 11:57:29 AM

Entropy Transport with Phase Change Heat Transfer 215

of solidification and melting have often used aqueous mixtures, such as sodium chloride and water (Hayashi and Komori, 1979) or ammonium chloride and water (Szekely and Jassal, 1978), due to their translucency, low melting temperatures, and similar behavior to dendritic solidification in metals. Important transport processes have been observed in these studies (Burton et al., 1995; Clyne and Kurz, 1981; Voller and Brent, 1989), including dendritic formations (Yoo and Viskanta, 1992), planar interface movement, microgravity formations, and solid matrix permeability (Naterer, 2000). This chapter will investigate the importance of entropy and the Second Law in these processes. Applications ranging from materials processing to energy storage will be considered. An example is phase change materials (PCMs) for thermal management of electronic assemblies (Vesligaj and Amon, 1999). The temperature difference between an electronic component and the PCM, ∆T, at a fixed heat transfer rate, Q, is reduced when the entropy production rate, Q∆T 2/T 2, is mini-mized. In this example and others, the unique insight provided by the Second Law will be examined.

8.2 En t r o py tr a n spo r t Eq u at Io n s f o r so l Id If Ica t Io n a n d MEl t In g

The governing equations for solid–liquid phase transition are the conservation equations (mass, momentum, and energy), in conjunction with an appropriate phase diagram, equations of state, and supplementary equations relating micro-scopic and macroscopic quantities. Continuum equations can be written for the conserved quantities, xk, where xk refers to a vector of conserved quantities, includ-ing mass and energy, and the subscript k refers to phase k, that is, k = 1 (solid) and k = 2 (liquid). The mixture equations are obtained by summing the individual continuum equations over both phases within a control volume, including solid and liquid phases, and rewriting the variables in terms of mixture variables. A mixture quantity is defined as the mass fraction-weighted sum of individual phase components. For example,

v v v= +f fl l s s (8.1)

k f k f kl l s s= + (8.2)

refer to the mixture velocity and thermal conductivity, respectively. If a conserved quan-tity is written without a subscript involving a phase, then it refers to a mixture quantity.

After performing the summation of conservation equations over both phases, the mixture equations for mass and momentum, respectively, can be expressed in the following manner:

∂∂+ ∇ ⋅ =ρ ρ

t( )v 0 (8.3)

∂∂+ ∇ ⋅ = -∇ + ∇ ⋅ ∇

+ +( )

( )ρ ρ ρ

ρµv

vv v F Ft

p ll b p

(8.4)

7262_C008.indd 215 1/19/08 11:57:32 AM


where Fb and Fp refer to body forces and phase interaction forces, respectively. In the present analysis, the body forces, Fb, are given by

F g gb T CT T C C= - + -ρ β ρ β( ) ( )0 0 (8.5)

where g, bT, and bC represent the gravity vector and thermal and solutal expansion coefficients, respectively. The phase interaction forces, Fp, will be determined from appropriate supplementary relations.

The conservation equations for species and energy, respectively, are given by

∂∂+ ∇ ⋅ = ∇ ⋅ + + ∇ ⋅( )

( ) ( ) (ρ ρ ρ ρ ρC

tC f D C f D C Cl l l l s s s sv --Cl )v (8.6)

( )( ) ( ) ( )

∂∂+ ∇ ⋅ = ∇ ⋅ ∇ + ∇ ⋅ -ρ ρ ρh

th k T h hlv v

(8.7)

where h refers to enthalpy. In phase k, this enthalpy is written as

h C T c d h C TkT

T

r k r k( ) ( ) ( ), = + ,∫ , ,0

ζ ζ (8.8)

In Equation 8.8, cr,k (T) refers to the reference specific heat of phase k. The final terms in Equation 8.6 and Equation 8.7 are written in a way that simplifies their eval-uation as source terms, Sc and Se, respectively, in a conventional numerical model. The previous governing equations must be solved in conjunction with a phase equi-librium diagram (see Figure 8.1). The following assumptions will be used in the

CS CLComposition

Te

Tm

Tem

pera

ture

(Eutectic)

SolidusSolid

Mushy

Liquid

Liquidus

f Ig u r E8.1 Binary alloy phase diagram.

7262_C008.indd 216 1/19/08 11:57:36 AM


upcoming analysis: continuous liquid–solid mixture without internal gas voids; two-dimensional, incompressible, laminar, Newtonian flow; and a stationary solid phase during phase transition.

In addition to the conservation equations, interfacial constraints will be required for balances of conserved quantities and entropy across the moving phase interface. In numerical models, interface tracking typically requires iterative solutions, because the interface position is generally unknown and its motion has a nonlinear behavior. The interfacial constraints will be utilized with entropy as a basis for effective inter-face tracking. The binary phase diagram will be used to determine the equilibrium temperature and concentration at the solid–liquid interface. In Figure 8.2, a typical schematic of the solid–liquid interface is illustrated (note: n, Vi, dA, and dfsdV refer to the normal direction, interface velocity, area, and solid fraction increment multi-plied by a change in volume, respectively).

The heat transfer from the liquid phase into the phase interface, HTl, consists of conduction (Fourier’s law) and advection components,

HT k dAdT

dndt V e dAdtl ll

l l l= - + ρ (8.9)

A similar heat transfer expression can be written in the solid phase. Consider a con-trol volume at the phase interface with a thickness dn. Then the change of energy that accompanies the advance of the interface arises from the energy difference of an entirely liquid volume, dAdn, and a final solid volume, so

dE HT HT e dAdn e dAdnl s l l s s≡ - = -ρ ρ (8.10)

Based on the results in Equation 8.9 and Equation 8.10, it can be shown that the fol-lowing interfacial energy constraint is obtained

- + = - D + DkdT

dnk

dT

dnV e e Vl

ls

ss s f s f iρ ρ (8.11)

f Ig u r E8.2 Schematic of phase interface.

dn

dA

Vs,i

Solid

n

dfsdV

Heat Flow

Entropy Production Due toShear Action

Liquid

7262_C008.indd 217 1/19/08 11:57:40 AM


where Def = el - es is the latent heat of fusion and Vi refers to the interface velocity. In the present analysis, the solid phase will be assumed stationary, so the first term on the right side of Equation 8.11 is neglected.

Similarly, the interfacial constraints for the other conserved variables, such as mass, solute concentration, and momentum, can be obtained from appropriate bal-ances across the phase interface. For example, the interfacial relation for the concen-tration of component c in a multicomponent mixture, Cc, is obtained as follows:

- + = - +, ,ρ ρ ρ ρl ll

cs s

s

cs s c ls s c ls iD

dC

dnD

dC

dnV C C V (8.12)

where the difference between phase concentrations, C C Cc ls c l c s, , ,= - , is obtained from the binary phase equilibrium diagram (see Figure 8.1).

In the case of entropy transport, the following transport equation in phase k is obtained:

D f s

Dt

f k T

T

fk k k k k

c

Nk c k c( )ρ ζ

= ∇ ⋅ ∇

+ ∇ ⋅

=

,∑1

j ,,,

+k

s kTP

(8.13)

where s kP , refers to the entropy production rate and ζc k k c ke C, ,= ∂ /∂ is the chemical potential of constituent c in phase k. It can be interpreted as an increase of work potential in the fluid, if dCc,k of constituent c is added to the mixture. On the right side of Equation 8.13, the terms represent the entropy flow associated with the heat flux and species (mass) flux, jc,k, and the entropy production rate, respectively.

Entropy is not measured directly, so an additional relationship involving entropy and the conserved quantities, such as energy (or temperature), solute concentration, etc., is needed. The following Gibbs equation for a multicomponent mixture in terms of phase k will be used:

dsde

T

pdv

T

dC

Tkk k

c

Nc k c k= + -

=

, ,∑1

ζ

(8.14)

where vk represents specific volume of phase k. A latent heat term is not included in Equation 8.14 because the equation is written within a single phase k.

Assuming an incompressible substance in each phase, rewriting Equation 8.14 in terms of a substantial derivative and rearranging terms,

TD f s

Dt

D f e

Dtg

D f Ck k k k k k

c

N

c kk k c( ) ( ) (ρ ρ ρ

= -=

,,∑

1

kk

Dt

) (8.15)

Substituting terms from Equation 8.6 and Equation 8.7 into this equation,

D f s

Dt Tk T P

Tk k k

k k e k

c

N

c

( ):

ρ τ ζ= ∇ + ∇ + -,=∑1 12

1

v ,, , ,-∇ ⋅ +k c k c kP j (8.16)

7262_C008.indd 218 1/19/08 11:57:48 AM


Expanding the divergence terms in this equation, using the product rule and com-paring the result to Equation 8.13, we obtain the following result for the entropy production in phase k:

s kk k e k

c

N

c kPf k T

T TPT T,,

=,= ∇ + + - ⋅∇∑

( )2

21

1µ ζΦj cc k

c

N

c k c k

c

Nc k c k

TT P

T,=

, ,=

, ,+ ⋅∇ -∑ ∑12

1 1

ζζ

j

(8.17)

The production terms on the right side of Equation 8.17 vanish after summation over the phases because production or destruction of energy or species in one phase is accompanied by destruction or production in the other phase. However, this does not apply to entropy because processes such as heat transfer, viscous dissipation, and fluid mixing are irreversible and thus produce entropy within an individual phase.

The entropy interfacial constraint can also be written in terms of the local entropy production rate at the phase interface. The entropy transfer from the liquid phase into the interface, ETl, may be written as

ET k

dA

T

dT

dndt V s dAdtl l l l l= - + ρ (8.18)

which consists of entropy flow arising from heat conduction, as well as advection, because liquid motion carries entropy into the interface. A similar expression, ETs, can be obtained for the solid phase. Similarly, as the energy analysis, the entropy dif-ference between a liquid volume, dAdn, at the interface and a subsequent solidified volume can be written as

dS s dAdn s dAdn ET ET P dAdnl l s s l s l s i= - = - + ,ρ ρ ρ (8.19)

where the interfacial entropy production, Ps,i, accounts for the entropy produced per unit mass due to heat transfer and shear action along the dendrite arms, when the dendrite moves a distance dn during the time interval dt.

Combining the previous equations and rearranging terms,

( )ρ ρ ρl l s ss

l l

s

s sl l ls s

dn

dt

k

T

dT

dn

k

T

dT

dnV s- = - + + -- + ,ρ ρs s s l s iV s P

dn

dt (8.20)

Using the continuity equation, the following result for the entropy production at the phase interface is obtained:

P se

T

k

V

dT

dn T Ts is

lf

f l

l i l l s, = D -

D

+ -

ρρ ρ

1 1

(8.21)

7262_C008.indd 219 1/19/08 11:57:53 AM


where Dsf = sl - ss refers to the entropy of fusion. The entropy of fusion is approxi-mately constant for most metals and alloys, Dsf ≈ 8.4 [J/molK]. The lowercase symbol for entropy, s, refers to the intensive (specific) variable, whereas the upper-case nota-tion refers to the extensive variable. Richard’s rule states that the entropy of fusion is approximately equal to the heat of fusion divided by the phase change temperature.

8.3 HEat a n d En t r o py an a l o g IEs In pHa sEc Ha n g Epr o c EssEs

The Reynolds analogy between momentum and heat flow relates the Nusselt number to the friction coefficient. This section develops similar analogies between entropy and heat flow. Transport phenomena involving one variable (temperature) will be inferred through behavior of the other analogy variable (entropy). If predicting a certain variable is difficult or time-consuming, but analysis with the other analogy variable is more readily implemented, then the analogy can be particularly benefi-cial. Two specific examples of heat and entropy analogies will be given, involving processes of interdendritic permeability and thermal recalescence.

8.3.1 Ir r ev er sIbIl It yo f In t er d en d r It Ic Per mea bIl It y

Modeling of the two-phase momentum equations requires a supplementary rela-tion for the momentum phase interactions, Fp, for closure of the solid–liquid phase change equations. For fluid flow through a porous medium, Darcy’s law (Bird et al., 1960), FpK = vfl (vl - vs), may be used for the phase interaction force dependence on the porous medium permeability, K, and liquid fraction, fl. In Darcy’s law, a fixed dendritic section (porous medium) is required. The assumption of a stationary solid material (i.e., vs = 0) in the governing equations is needed for consistency with Darcy’s law.

Previous models of solid–liquid phase change have often used the following Blake-Kozeny equation (Bird et al., 1960) for the solid permeability:

K Kf

fl

l

=-

0

3

21( )

(8.22)

where K0 is an empirical coefficient. This model was developed from a physical analogy between interdendritic flow and Hagen-Poiseuille viscous flow (Bird et al., 1960), through a noncircular tube with an equivalent hydraulic radius based on the local liquid fraction (see Figure 8.3). At high values of fl (i.e., fl > 0.5), a crossflow perpendicular to the dendrite arms may produce a higher pressure difference than the results of the Blake-Kozeny prediction, because of wake interactions and inter-dendritic viscous effects. The following alternative model can be used to account for the crossflow effects (Naterer, 2000):

K Kf

fl

l

=-

0 1

(8.23)

7262_C008.indd 220 1/19/08 11:57:54 AM


It can be shown that both Equation 8.22 and Equation 8.23 approach the proper limits as fl Æ 0 (solid) and fl Æ 1 (liquid).

Consider a flow alignment factor, c, to represent a weighting factor between the axial relation, Equation 8.22, and the crossflow relation, Equation 8.23,

K Kf

fK

f

fl

l

l

l

=-

+ -

-

0

3

2 011

1χ χ

( )( )

(8.24)

where the first and second terms represent axial and crossflow permeabilities, respec-tively. For example, c = 0 corresponds to a crossflow, and c = 1 represents an axial flow, with the appropriate permeability factors used in each limiting case.

Entropy and the Second Law may provide key insight about how c can be best cal-culated to ensure physically plausible trends of interdendritic flow. Because the direc-tion of dendritic growth is related to the local thermal irreversibility, entropy can be used to interpret axial and crossflow contributions to the interdendritic permeability. Consider a weighting between v and the dendritic growth direction, based on ther-mal irreversibility and entropy. In dendritic solidification, the primary dendrite arms grow in the direction of the local temperature gradient (Flemings, 1974). This gradi-ent gives the direction of steepest ascent (or descent for a negative gradient vector). Because the primary dendrite arm growth is aligned with the local temperature gra-dient, this growth occurs toward the steepest temperature slope. This direction cor-responds to the maximum thermal irreversibility. This thermal irreversibility, s tP , , can be subdivided into x- and y-direction components, s txP , and s tyP , , respectively, in the following manner:

s t s txPk

T

T

x

k

T

T

yP, ,= ∂

∂+ ∂

∂= +

2

2

2

2

ss tyP ,

(8.25)

Although entropy production is a scalar (not a vector), we may define an equivalent entropy vector, s t,P , consisting of the above components of entropy production in the x- and y-directions, given by Ps tx, i and Ps tx, j (note: unit vectors i , j), respectively.

fL

Solid

Liquid

f Ig u r E8.3 Permeability analogy for interdendritic axial conditions.

7262_C008.indd 221 1/19/08 11:58:10 AM


For example, a large component s txP , (in comparison to s tyP , ) suggests that entropy production arises mainly from heat flow in the x-direction. Based on this concept, a flow alignment weighting factor is defined as follows:

χ =| ⋅ |

| || |; ≡ +,

,, , ,

v P

v PP i

s t

s ts t s tx s tyP P

ˆ

ˆˆ ˆ ˆj

(8.26)

where c can be interpreted in terms of thermal irreversibility (or temperature gradient) relative to the direction of local interdendritic flow. The square root in Equation 8.26 is used for a more direct analogy between entropy production in Equation 8.25 and temperature gradient (or heat flow).

The equivalent vector s t,P is similar to the heat flux vector, with two exceptions: (i) minus sign for Fourier heat flux, and (ii) magnitude of k T T| ∇ | /2 2, rather than k T| ∇ |. A higher crossflow weighting is given when the interdendritic flow is normal to the direction of maximum thermal irreversibility (direction of dendritic growth). Conversely, the axial permeability is adopted when the velocity is aligned with the direction of the maximum thermal irreversibility. Using a physical interpretation based on entropy and the Second Law, the alignment weighting factor will be shown (in an upcoming case study) to provide a robust method for accurate interface track-ing of phase change processes.

8.3.2 t he r ma l rec a l esc en c ea n d dImens Io n l essen t r o Pyrat Io

This second example demonstrates that another multiphase process (thermal recales-cence) can be interpreted or modeled through analogies and insight provided by the Second Law, which would not otherwise be gained through the conservation equa-tions alone. During dendritic solidification, latent heat is released to the surrounding liquid–solid mixture. If it exceeds the rate of external cooling, then a local temperature rise, or recalescence, may be observed. Together, with solute diffusion, this reheating may initiate melting of smaller dendrite arms at the expense of growing primary arms (i.e., dendritic coarsening). Coarsening during crystal formation and sedimentation in solidification processes may contribute to recalescence and latent heat release. In both cases, thermodynamic irreversibilities with entropy production arise during the heat transfer processes. Because reheating and coarsening may affect the final mate-rial properties, such as material strength, these processes have significance during solidification processing of materials. In the following analysis, it will be shown that entropy serves as an important variable in these processes.

Consider a simplified heat balance for a crystal (or dendrite) during solidifica-tion. The transient temperature change arises from the net heat exchange with the surrounding solid–liquid mixture (described by an overall heat transfer coefficient, h) and release of latent heat from the freezing crystal, that is,

ρ ρVc

T

thA T T V e

f

tp f fl∂

∂= - - D ∂

∂( )

(8.27)

where V, A, and Tf refer to a characteristic crystal (or dendrite) volume, correspond-ing surface area, and surrounding mixture temperature, respectively. Expanding the

7262_C008.indd 222 1/19/08 11:58:15 AM


last derivative in Equation 8.27 through the chain rule, we obtain

∂∂=

-+ ∂ /∂ D /

T

t

hA

Vc

T T

f T e cp

f

l f pρ 1 ( ) (8.28)

A characteristic length scale, Lc, may be represented by the ratio V/A (reciprocal appears in the above equation).

Equation 8.28 can be written in a dimensionless form by selecting appropriate ref-erence scales for temperature and time (i.e., t Lref c= /2 α ). The dimensionless entropy, temperature, time, and Stefan number (c T ep fD /D ) will be denoted by s*, θ, t*, and Ste, respectively. Writing Equation 8.28 in dimensionless form, it can be shown that

θθ= ′

≡

h L

k

k

kNu

k

kc

l

l

s

l

s (8.29)

where

′ = -+ / ∂ /∂

h h

Ste fl

11

1 ( )θ (8.30)

refers to a modified heat transfer coefficient that accounts for simultaneous external cooling and release of latent heat from the solidified crystal to the interdendritic fluid. Also, the overdot in Equation 8.29 represents differentiation with respect to time.

Comparing Equation 8.29 with the rate of entropy change arising from the sensible heat portion of the Gibbs equation (i.e., Maxwell-type relation where cdT = Tds*),

∂∂= =

∗

∗

s

tNu

k

kl

s

θθ (8.31)

The change of liquid fraction with temperature is calculated based on a supple-mentary relation, such as an approximated linear variation of fl with q through the two-phase region. In a similar way as analogies between heat and momentum transport (i.e., Reynolds analogy), this result suggests a type of analogy between heat and entropy.

The process of dendritic coarsening is closely related to recalescence during solid–liquid phase change. During coarsening, small or secondary dendrite arms (or crystals) shrink or melt at the expense of heat transferred from the large and growing main dendrites. The previous results suggest that entropy may serve as an effective variable in characterizing the coarsening and recalescence. Because entropy can-not be measured directly, it requires measurements indirectly through other means. The magnitude of recalescence is observed by a measured temperature rise. The corresponding dimensionless temperature gradient at the dendrite arm (character-ized by the Nusselt number) can be interpreted in terms of the local rate of entropy change. This entropy change (measured indirectly) incorporates heat flow from the interdendritic fluid, as well as latent heat released from the crystal or dendrite.

7262_C008.indd 223 1/19/08 11:58:21 AM


Both mechanisms are important aspects for understanding recalescence and coars-ening, during remelting of small, secondary dendrite arms.

Further information regarding the effective heat transfer coefficient, ′h (and Nu), could be gained by performing many phase change experiments with different initial temperatures (leading to different Grashof numbers) and then seeking a suit-able dimensionless correlation for the Nusselt number, Nu, in terms of the Grashof number, Gr, and other parameters. This is similar to the approach when single phase convection correlations are constructed for external or internal flows. For example, the initial and wall temperatures can be measured (with resulting Grashof number), then the entropy changes and other parameters can be found by a specific phase change experiment. The effective heat transfer coefficient can be estimated from Equation 8.31. These experiments can then be repeated for other fluids and presented in a final correlated form.

In the previous analysis, a uniform heating or cooling rate was assumed. Entropy can serve as an effective variable for recalescence analysis, because the local heating or cooling rate can be related to the correlation involving heat transfer coefficient and entropy in Equation 8.31. The heat-entropy analogy can provide an improved estimate of the relevant heating or cooling rate, based on the result of how the Nusselt number is correlated with the local rate of entropy change. In particular, the aver-age heat transfer coefficient during this period can be estimated by integration over the range where a positive entropy derivative is observed in the data (i.e., conditions corresponding to recalescence or coarsening). In this way, the time period of reca-lescence can be expressed in terms of a heat-entropy analogy.

Consider the freezing of an ammonium chloride and water mixture in a rect-angular enclosure with an initial temperature and solute (ammonium chloride) concentration of 318 K and 0.72, respectively (Naterer, 2001). Thermal and solutal buoyancy in the liquid region generates two counterrotating convection cells on the left and right sides of the cavity, respectively (see Figure 8.4). Upward transport of crystals by these convection cells and sedimentation of crystals along the vertical midplane create a growing mushy layer (NH4Cl dendrites and liquid) along the lower boundary of the domain. The crystals along the vertical midplane descend to create an inverted v-shaped sedimentation layer. Due to different sizes and struc-tures of crystals, the descent of various crystals is initiated at different positions.

Recalescence is a thermal phenomenon that occurs when the rate of latent heat release from a freezing crystal (or dendrite) exceeds the rate of heat transfer from the surrounding solid–liquid mixture. This creates a local heating effect that has impor-tant impact on the properties of solidified materials, such as mechanical strength. For example, it can lead to material defects such as dendrite tearing through repeti-tive freezing-melting cycles. A freezing crystal (or detached dendrite arm) may lead to reheating and melting of the surrounding solid back to liquid. Combined with solute transfer, the local melting of smaller dendrite arms may occur at the expense of growing primary arms (called dendritic coarsening). From temperature fluctua-tions alone, the occurrence of these processes may not be evident. The heat-entropy analogy aims to provide new insight in these regards. Thermal irreversibilities occur by heat transfer and phase change in these processes. The previous analysis gave a relationship between the rate of entropy change and the ratio between the

7262_C008.indd 224 1/19/08 11:58:22 AM


transient temperature derivative and temperature itself. Comparing Figure 8.5a and Figure 8.5b at location 3, it can be observed that the temperature rise after about 0.26 h (i.e., period of recalescence; Figure 8.5a) is closely coincident with the cross-over to a positive value of entropy change in Figure 8.5b.

This analogy between temperature and entropy can shed new light on the associ-ated thermodynamic processes. The magnitude of the highest entropy change indi-cates the maximum rate of reheating and coarsening, whereas the duration of the positive entropy change indicates the length of time over which the recalescence occurs. The magnitude of the entropy change includes the following thermal irre-versibilities: (i) release of latent heat by the freezing material, and (ii) coarsening by heat absorption or melting of the smaller arms, combined with heat release from the

**

*

*

** *

*

*

**

*

ConvectionCell

(a)

f Ig u r E8.4 Schematic of (a) convection cell and (b) thermocouple positions.

Liquid

Solid

Thermocouple WireTest Midplane

Cold Boundaries(Connected to

Heat Exchanger)

Thermocouple Grid

1

2

3

4

5

6

7

8

12

11

10

9

16

15

14

13

(b)

7262_C008.indd 225 1/19/08 11:58:24 AM


f Ig u r E8.5 (a) Measured temperatures, and (b) rate of entropy change for solidification with free convection in a cavity.

–15.0

0.0

0 2300.0

(17)

(1) (2) (3) (4)

(20)

Legend Liquidus

4600.0 t(s) (a)

6900.0 9200.0

15.0

30.0

Tem

pera

ture

(°C

) 45.0

60.0

75.0

T0 = 343 K, C0 = 0.68

ds/d

t (W

/kgK

)

0.40

0.20

0.00

–0.20

–0.40

0.27 0.46 0.66Time (hr)

(b)

0.86 1.05

Location (3)

Location (4)

Location (7)

Location (8)

–0.60

7262_C008.indd 226 1/19/08 11:58:28 AM


growing larger arms. The magnitude of these combined irreversibilities decreases after the point of maximum ds/dt in Figure 8.5b.

At higher initial concentrations of water, the level of initial undercooling of the liquid is lower (i.e., liquidus temperature minus the wall temperature is lower). As a result, slower solid growth occurs with less pronounced temperature fluctuations from thermal recalescence. A period of recalescence (D >t 0) occurs when the rate of latent heat release within a freezing crystal (or dendrite) exceeds the rate of heat transfer from the solid–liquid mixture around the crystal, thereby creating a local heating effect.

Thermal irreversibilities arise during recalescence when heat is released by freezing material and transferred to the surrounding liquid or solid. Similarly, irre-versibilities occur during coarsening by heat absorption or melting of smaller arms, as well as heat release by growing larger arms. These effects can be related to the magnitude and slope of the entropy change. The magnitude of the maximum entropy change indicates the highest rate of reheating and coarsening, whereas the duration of positive entropy change indicates the approximate time period of recalescence. As the coarsening time increases, an increasing number of smaller arms disappear while the main dendrite arms grow larger and their spacing increases.

8.4 n u MEr Ica l st a bIl It y o f pHa sEcHa n g Eco Mpu t at Io n s

8.4.1 mo d el In g o f tw o -Ph a seen t r o PyPr o d u c t Io n

In addition to heat-entropy analogies based on entropy in the previous section, the Second Law can provide unique insight for improving the performance of a numeri-cal formulation. In this section, a Second Law formulation is presented with predic-tive and corrective capabilities for the improvement of phase change predictions. The formulation can serve as an effective complement to the discretized conserva-tion equations in the overall numerical procedure. This section will focus on numer-ical modeling and implications of entropy and the Second Law in the numerical formulation.

The Second Law for a multiphase mixture can be written in the following form:

s k

kkP

S

t, ≡∂∂+ ∇ ⋅ ≥ F 0

(8.32)

In Equation 8.32, s kP , , S(fk), and F(fk) refer to the entropy production rate in phase k, entropy (as a function of the vector of conserved quantities, fk), and the entropy flux in phase k, respectively. Expanding the entropy flux in terms of advective and diffusive components, Equation 8.32 can be written as

D s

Dt

k T

T

gk k k k k

c

Nk c k c( )λ ρ λ λ

= ∇ ⋅ ∇

+ ∇ ⋅

=

,∑1

j ,,,

+k

s kTP

(8.33)

where lk, sk, and g e Cc k k c k, ,= ∂ ∂ refer to the mass fraction, specific entropy and chem-ical potential of constituent c, in phase k, respectively. The summation includes each chemical potential from c = 1 to c = N constituents, i.e., N = 2 for a binary alloy. The species flux, jc,k, for constituent c in phase k is determined by Fick’s law.

7262_C008.indd 227 1/19/08 11:58:32 AM


On the left side of Equation 8.33, the substantial derivative includes both transient and advection terms. The entropy flux (Fk) in Equation 8.32 includes an advective component from the substantial derivative in the left side of Equation 8.33, as well as diffusive (energy and species) components in the first two terms on the right side of Equation 8.33.

A positive–definite expression is needed for the evaluation of the entropy pro-duction rate. From Section 8.2, this expression can be determined from the Gibbs equation and the entropy transport equation. The following mixture expression can be derived after summing over both solid and liquid phases:

s

k

k k

c

N

c k cPk T T

T T Tg = ∇ ⋅∇ + - ⋅∇

= =, ,∑ ∑

1

2

21

1λ µ( ) Φj kk

c

N

c k c kTg T+ ⋅∇

=, ,∑1

21

j

(8.34)

where Φ refers to the viscous dissipation function. From Section 8.2, the interfacial entropy production is,

P S

E

T

k

V

dT

dn T Ts is

lf

f l

l i l l s, = D -

D

+ -

ρρ ρ

1 1

≥ 0 (8.35)

where D = -S s sf l s refers to the entropy of fusion (approximately equal to the heat of fusion divided by the phase change temperature). The entropy production in Equation 8.35 includes effects of viscous dissipation due to shear action along a den-drite arm, as it moves a particular distance over a time interval.

The discretized form of the Second Law can be written as

s

in

in

ip

ip ipPS S

tV S ≡ -

D

D + D ≥

+

∑1

0F ( ) (8.36)

where the summation over “ip” integration points refers to surface flux calculations in a finite volume method.

A reconstruction step is required in Equation 8.36 because the distribution of conservation variables φ( )x, t must be approximated from integration point and nodal values. It will be assumed that f is piecewise constant within a control volume. This quasiequilibrium assumption can complete the reconstruction step, without violat-ing the Second Law. The Gibbs equation will be used to write the transient entropy derivative in Equation 8.36, in terms of variables obtained from solutions of the conservation equations, such as temperature and liquid fraction, as follows:

S S

t

c

T

T T

tSi

nin

p

in

in

in

fl

+ +,-

D= -

D

+ D

1 1ρρ

λ iin

l in

t

+,-

D

1 λ (8.37)

In this approach, entropy computations can be distinguished between sensible and latent heat contributions.

7262_C008.indd 228 1/19/08 11:58:37 AM


An entropy equation of state is required for implementation of the Second Law. For solid–liquid phase transition, the entropy varies with temperature and concen-tration. Based on the form of the Gibbs equation for an incompressible substance, consider a piecewise logarithmic equation of state in the following form:

s T C s c logT

Tr k r kr k

( ), = +

, ,

, (8.38)

where the subscripts r and k denote reference and phase, respectively. In the solid phase (k = 1), the following reference values will be used:

sr , =1 0

c cr s, =1

T Tr e, =1 (eutectic)

In the mushy (two-phase) region (k = 2), the reference specific heat must include both sensible and latent entropy contributions. Also, the reference entropy is determined at the solidus line. The following reference values are obtained (Naterer, 2000):

s c log T Tr s sol e, = /2 ( )

cr

c T c TT T

c c Slog

s liq l sol

liq sol

l s f,

--

- +D= ( ) +2 (TT Tliq sol/ )

T Tr sol, =2

Finally, the Gibbs equation and the binary phase diagram (Figure 8.6) can be uti-lized to derive the following reference values in the liquid phase (k = 3):

s log T Tr

c T c TT T liq sol

s liq l sol

liq sol,--= ( ) /3 ( ) ++ / + - + Dc log T T c c Ss sol e l s f( )

c c

T T

r l

r liq

,

,

=

=

3

3

Examples of typical entropy equations of state are illustrated in Figure 8.6a and Figure 8.6b (note logarithmic scale). Thermophysical properties of the material in Figure 8.6a and Figure 8.6b include cs = 167[J/kg K] and cl = 167[J/kg K]. The melt temperature, eutectic temperature, and liquidus-eutectic intersection (see Figure 8.1) are T = 343[K], T = 341[K], and CL = 0.4, respectively. In Figure 8.6a, L = 32.6[KJ/kg] and CS = 0.2, whereas in Figure 8.6b, L = 3.26[KJ/kg] and CS = 0.38 (see Figure 8.1).

7262_C008.indd 229 1/19/08 11:58:44 AM


8.4.2 It er at Iv ePh a seru l esa n d t he sec o n d law

Because the phase change process is nonlinear, numerical iterations are needed within a time step to achieve solution convergence. Determining the phases in each control volume requires a solution of the energy equation, but the phase distribution is needed before a solution can be obtained. Thus, a tentative phase distribution is required prior to the solution of the energy equation. If the computed solution leads to a different phase distribution, then further iterations are required until conver-gence between tentative and computed phases is achieved. The phase iterations will

f Ig u r E8.6 Entropy equation of state with (a) L = 32.6 kJ/kg, CS = 0.2; (b) L = 3.26 kJ/kg, CS = 0.38.

100

10

1 0.1

0.01 0.35

0.3 0.25

0.2 0.15

0.1 0.05

0 (a)

68 68.5

69 69.5

70 75

0.01 0.1 1 10 100

100

10 1

0.1 0.01

0.35 0.3

0.25 0.2

0.15 0.1

0.05 0

(b)

68 68.5

69 69.5

70 75

0.01 0.1 1 10 100

Concentration Temperature

Entropy

Concentration Temperature

Entropy

7262_C008.indd 230 1/19/08 11:58:49 AM


be established based on discrete analogies of the Second Law (the following three rules):

1. A control volume must pass through a melt region, during phase transition between solid and li quid phases.

2. Phase transition in a control volume cannot occur without a phase transi-tion to the same phase in an adjacent volume first.

3. The tentative phase within a control volume must give a positive entropy production rate for the discrete time step. If the Second Law is violated in a control volume, then an entropy-based correction is required in the numerical formulation.

It has been shown that the first two rules can be interpreted as discrete analogies of the Second Law (Naterer, 2001). For example, rule (1) remains consistent with non-negative entropy production in the interfacial entropy constraint, Equation 8.35. It is not surprising that the Second Law can be interpreted in various different ways. For example, Carnot’s statement of the Second Law requires that heat cannot be converted completely and continuously into work. On the other hand, Kelvin stated that it is impossible by means of an inanimate material to derive a mechanical effect from any portion of the material, by cooling it below the temperature of the coldest of the surrounding objects. In a similar way, the previous phase rules express alter-native consequences of the Second Law in the case of a solid–liquid phase change. The previous rules (1) and (2) represent effective procedural guidelines for phase-temperature iterations, as well as discrete analogies of the Second Law.

Under certain circumstances, a volume may ultimately exist in a phase different from its neighboring volumes (i.e., supercooled dendrite pocket ahead of a liquidus interface) without violating the second rule during its progression between phases. To clarify this situation, consider a sequence of solution iterations illustrated by steps (0) to (5) for a binary alloy in Figure 8.7 with solution convergence at step (5). In this example, different initial mean solute compositions, C1, C2, and C3 in volumes 1, 2, and 3, respectively, create the liquidus temperature, TL, and solidus temperature, Ts, step functions (see Figure 8.7). After solution (1), rule 2 enforces volumes 3 to 7 to return to a liquid (L) phase because none of their neighbors existed in a melt (M) phase at step (0). A similar result for volumes 4 to 7 occurs after solution (2). After solution (3), rule 1 permits a solid (S) volume 3 due to the nature of the liquidus and solidus temperatures between two melt (M) volumes. Following a subsequent rule 2 correction after solution 4, the final state illustrates a converged solution with a solid volume 3 between two volumes, 2 and 4, with melt (M) phases. The solution converges because the tentative and modified phases agree with one another after the application of the phase rules. This example demonstrates how physical reasoning embodied by phase rules and the Second Law provides a sound basis for numerical iterations of a phase change problem.

Although rules 1 and 2 do not ensure positive computational entropy production in rule 3, they permit a closer compliance with the Second Law, in comparison to other iterative techniques based on ad-hoc convergence tolerances or purely numeri-cal manipulations such as underrelaxation factors. Local satisfaction of the Second

7262_C008.indd 231 1/19/08 11:58:49 AM


Law provides several potential benefits, such as convergence enhancement, and a physically based understanding of discrete error analysis. The third rule aims to ensure compliance with the Second Law. In the next section, a predictive technique, as well as additional entropy-based diffusivity corrections, will be applied in the computations.

8.4.3 en t r o Pyco r r ec t Io n o f nu mer Ic a l co n d u c t Iv It y

In this section, an entropy-based approach will apply both corrective steps for accu-racy improvements (entropy-based diffusivity), as well as predictive steps (nonlinear time constraint) for stable computations. The magnitude of the computed negative entropy production rate can provide a quantitative measure of the degree of discreti-zation error in the control volume. The third phase rule in the previous section repre-sents a quantitative rule, whereas the first two rules provide qualitative or procedural guidelines. If the Second Law is violated within a discrete control volume, then a quantitative indication of the diffusivity required to correct the solution may be

Solution L M M M S M S Rule 1 L M M M M M S Rule 2 L L L L L M S

Solution L M M M S M S Rule 1 L M M M M M S Rule 2 L L L M S M S

Solution L M M M S M S Rule 1 L M M M M M S Rule 2 L L L L M M S

Solution L M M M S M S Rule 1 L M M M M M S Rule 2 L M M M S M S

Solution L M M M S M S Rule 1 L M M M M M S Rule 2 L L M M S M S

Phase L L L L L L S

CS = 0.23CL = 0.54C1 = 0.15C2 = 0.30C3 = 0.10x = 0

T = 1x = 1T = 0

Liquid Solid

Interface

Problem Domain Phase Diagram (Ag-Sn)

x

x

C3 C1 CS C2 CL

TM

TL

TS

TL

TS

TL

TS

TL

TS

TL

TS

x

xx

x

x

x

0

5 4 3 2 1

1

2 3

54 SolutionConvergence

f Ig u r E8.7 Binary alloy solidification from side boundary.

7262_C008.indd 232 1/19/08 11:58:51 AM


expressed in terms of the negative entropy production rate (absolute value) from Equation 8.34 and Equation 8.36. This effective diffusivity will be substituted into the discrete equations, and a corrected solution will be obtained. It is anticipated that this corrected solution can achieve closer compliance with the Second Law.

Summing over the solid and liquid phases, and rewriting Equation 8.34 in terms of an entropy-corrected conductivity, ks,

k P

T

T T

T c

Prs sp≈

∇ ⋅∇+

2

Φ (8.39)

where cross-diffusion (Soret) effects have been neglected. In the numerical proce-dure, the solution is obtained by first solving the conservation two-phase equations independently of the Second Law. Following each time step, the entropy production rate is then computed based on the Second Law in Equation 8.36. If a nodal value of sP is negative, then discretization errors occur or the local solution exhibits non-physical behavior. A computed negative entropy production at the phase interface may indicate a discretization error arising from temporal or spatial differencing or both. Thus, instead of proceeding to the next time step, a corrective diffusion step is taken in the next solution of the discrete equations, to prevent the computed entropy destruction. The required quantitative amount of corrective diffusion is inferred by the magnitude of the discretization error outlined by Equation 8.39.

In the implementation of Equation 8.39, temperature gradients and the viscous dissipation function are evaluated from the numerical solution and the interpolation functions. Then, an entropy-based conductivity, ks, is combined with the molecular conductivity in the following manner:

k k Max ks→ + ,| |ε (8.40)

where e represents an upper limit on the corrective step. Because this entropy-based approach is decoupled from the implicit solution of the conservation equations, the term e may be interpreted as a relaxation factor in the corrective procedure.

Numerical dispersion is a quasiphysical effect that may distort phase relations between various thermal or fluid waves, through odd derivative terms appearing in the truncation error. A comprehensive error analysis, based on a Taylor series truncation for multidimensional problems, is unavailable for nonlinear phase change problems with fluid flow. The current entropy-based approach provides an alternative basis for a unified approach to error analysis. Differences between computed and physical entropy, particularly those differences leading to computational entropy destruction, may allow us to quantify the amplitude and frequency of the discretization error.

Entropy-based corrections of the thermal diffusivity were introduced when nega-tive entropy production rates were computed at the current time level. If the entropy-based diffusivity coefficients alternated between zero and large nonzero values, then a “smooth” diffusivity distribution could be obtained in an appropriate manner. A concern would be poor results arising from the application of a highly irregu-lar (digital) diffusivity field. A smoothing algorithm could be applied to overcome this potential problem. Positive diffusivity coefficients are computed and located at

7262_C008.indd 233 1/19/08 11:58:54 AM


global nodes whenever the Second Law is violated in the corresponding finite con-trol volume. If the resulting diffusivity variations resemble a “digital” distribution (i.e., diffusivity alternates between zero and nonzero values), then diffusion can be applied over the entire mesh to smooth the distribution. For example, a Jacobi itera-tive procedure can be used to solve the diffusion (of conductivity) equation.

8.4.4 en t r o Pyco n d It Io n f o r temPo r a l st a bIl It y

In addition to corrective steps described in the previous section to improve accu-racy and convergence of numerical computations, entropy and the Second law can provide key insight for predictive measures to ensure stability of phase change computations. This section presents a stability analysis based on the Second Law, which can be applied to time-step selection in phase change problems. Consider the following one-dimensional transport equation for a general scalar, f:

L u cat x xx r( ) ( )φ ρφ ρ φ φ φ φ= + - - - =, , ,G 0

(8.41)

where the subscript comma notation refers to differentiation and G refers to a refer-ence value. In Equation 8.41, G and c refer to diffusion and source term coefficients, respectively. We will use ξ to refer to the entire source term, i.e., last term on the right side of Equation 8.41. Integrating Equation 8.41 over a discrete volume (one dimensional) and time step,

Lt

x Md i

n

i

n

i( ) (

φ ρ

φ φφ=

-D

D + -

-

+ /

1

1 2GG G, - /,- - - D

x ixM x φ φ φ ξ) ( )

1 2

(8.42)

where M u= ρ refers to the mass flow rate per unit width and depth. The subscripts i - /12 and i + /12 refer to west and east integration points, respectively. Also, L and the superscripts a and d refer to operator, analytic, and discrete, respectively. In the discrete formulation, Ld ( )φ gives a nonzero residual, because the approximate solu-tion, φ, in general, will be different from the exact solution. In the following analysis, the overbar tilde notation will be subsequently dropped, and it will be understood that the discrete operator acts on the approximate solution, φ .

Using standard finite difference approximations for the convection and diffusion terms,

Lt

Mx

d in

in

i i( )φ ρ φ φ α φ φ= -D

+

-D

--

11 ++ - -

D

- - +D

+

+ +

Mx

x

i i

i i i

( )1

2

1

1 1

2

α φ φ

φ φ φG - Dξ x

(8.43)

Here α = 1 represents upwinding, and α = /1 2 represents central differencing. This factor may include a weighting bias on the local Peclet number to include the

7262_C008.indd 234 1/19/08 11:59:02 AM


proper balance between upstream and downstream influences. Using Taylor series expansions,

φ φ φ φin

in

t tttt= + D + D +-

, ,1

2

2

(8.44)

φ φ φ φi i x xxxx

+ , ,= + D + D +1

2

2

(8.45)

A similar expansion may be written for the scalar value at the upstream location about point i. Substituting these expansions into the discrete operator,

L L u xx

d a

ir xx x( ) ( )φ φ α ρ φ φ= + -

D -

D +, ,

1

12

G

-, ,

-

D -

D ir xx xu x

x 12α ρ φ φG

,

+ Di

ttt12ρ φ

(8.46)

where ur represents a reference, or characteristic (i.e., lagged), linearization velocity. Also, higher-order terms have been neglected.

Also, taking derivatives in Equation 8.41,

ρφ ρ φ φ ρ φ, , , ,= - + -tt r xt xxt xu cG (8.47)

ρφ ρ φ φ ρ φ, , , ,= - + -xt r xx xxx tu cG (8.48)

In the present analysis, only transient, convection, and source-type terms will be subsequently analyzed because these terms will predominantly affect the numerical stability. Then, substituting Equation 8.47 and Equation 8.48 into Equation 8.46,

L L u u t x tc ud a

r r xx r( ) ( ) ( ) (φ φ ρ φ ρ φ= + D - D + D, ,12

12

2 xx rc c+ -φ φ) (8.49)

Thus the discrete operator depends on the second-order spatial derivative of the sca-lar, such as temperature, and higher-order terms (neglected).

In addition, the Second Law of Thermodynamics may be written as

s t xP S F = +, , (8.50)

where S(f) and F(f) represent the entropy and entropy flux, respectively. As discussed in previous chapters, these entropy functions must satisfy two important properties: (i) downward concavity, and (ii) compatibility. Let us now premultiply the discreti-zation error, Ld(f), by S,f in Equation 8.49 to give the following expression for the

7262_C008.indd 235 1/19/08 11:59:07 AM


entropy-weighted discretization error:

S L S L S u u t xd ar r xx, , , ,= + D - D +φ φ φφ φ ρ φ ρ( ) ( ) ( )

12

12DD + -

,tc u c cr x r( )2 φ φ φ (8.51)

Also, from the chain rule of multivariable calculus,

S S Sxx x x x, , , , , , ,= -φ φ φφφ φ φ( ) ( )( )2 (8.52)

where the last term is positive definite as a result of downward concavity of the entropy function. Expressing f in dimensionless terms with a Taylor series expan-sion about the phase interface, then seeking an analogous result as Equation 8.52 from the chain rule for the first-order derivative of f, it can be shown that

S S x Sx x x, , , , , ,= - Dφ φ φφφ φ φ( ) ( )( )2 (8.53)

If we integrate Equation 8.52 over a discrete volume, then the first term on the right side becomes the difference between the entropy gradient at the two edges of the control volume, whereas the remaining terms represent average values of entropy derivatives and entropy production. The difference between entropy gradient terms diminishes, in comparison to the remaining terms, when the grid spacing is refined.

Approximating the right side (entropy-weighted discretization error) in Equation 8.51 as zero, and using the chain rule to write the first term as the entropy production rate,

s r r xxP S u x u t HOT S = D - D +

+ D, , ,φ φρ φ ρ1

212

( ) ttc u c x HOTr x( )- - D +

,2 φ (8.54)

where HOT refers to higher-order terms. Thus, substituting Equation 8.52 and Equation 8.53 into Equation 8.54, it can be shown that

s r r x rP u x u t S tc u c = - D - D - D - -, ,

12

12

22ρ φ ρφφ( ) ( ) ( DD D ≥, ,x xS x) ( )φφ φ 2 0 (8.55)

As a result of the downward concavity property of entropy, and the requirement of positive entropy production in the Second Law, the following result is obtained:

D ≤ D+ D

tu x

u c xr

r( )2 (8.56)

Thus, the Second Law identifies an appropriate time step for the numerical compu-tations. Results from the previous stability constraints are illustrated in Figure 8.8a

7262_C008.indd 236 1/19/08 11:59:12 AM


and Figure 8.8b. It can be observed that the time step must be reduced when the grid spacing increases, or the phase change coefficient, c, increases (i.e., interface velocity and source terms increase), to maintain stable computations. In a similar manner as the CFL Courant condition for single phase problems, the time step is reduced when ∆x approaches zero because a disturbance would not propagate beyond the extent of the control volume boundaries for stable time advance.

8.4.5 c a sest u d yo f mel t In g In a n en c l o su r e

This section presents a case study with several example problems that apply the methods developed in the previous section to demonstrate the valuable utility of the Second Law in phase change analysis. The application problems in this sec-tion will illustrate both physical and computational aspects of entropy production, as well as their roles in predictive capabilities of the overall numerical formulation.

0.4

Unstable

Unstable

Stable

Stable

0.3

0.2

0.1

0

6

5

4

3

2

1

00 0.1 0.2 0.3

c0.4 0.5

0 0.01 0.02∆x

∆t

∆t

(a)

(b)

0.03 0.04

f Ig u r E8.8 Stability curves for (a) a fixed interface velocity and (b) fixed grid spacing.

7262_C008.indd 237 1/19/08 11:59:13 AM


A control volume-based finite element method (Naterer and Schneider, 1996; Naterer, 2000) will be used in the numerical analysis. Although the present results will consider heat transfer with melting and solidification, the entropy principles can be extended to other heat transfer problems, such as phase change with boiling or condensation. The following thermophysical properties for materials in the applica-tion problems will be given in the liquid (subscript l) and solid (subscript s) phases. For the Ag (silver)–Sn (lead) alloy, cp,s = 250[J/kgK], cp,l = 285[J/kgK], ml = 7.1 × 10-8[kg/ms], ks = kl = 315[W/mK], r = 105000[kg/m3], L = 120[kJ/kg], Te = 494[K], Tm = 1234[K], ce = 0.54, and kp = 0.43. For the Lipowitz (Cerrobend) material, cp,s = cp,l = 167[J/kgK], vl = 3.31 × 10-7[m2/s], ks = 19[W/mK], Kl = 5.5[w/mK], L = 32.6[kJ/kg], and Tm = 374[K].

Consider an application problem with melting from an upper boundary (see Figure 8.9) (Gau and Viskanta, 1984). In this example, an initially solid material at Tc = 68[ ]o C is suddenly exposed to a hot upper boundary (Th = 92[ ]o C ). The melting temperature is Tm = 70[ ]o C , and both vertical boundaries are well insulated. Melt-ing of the solid begins near the upper boundary and proceeds downward as time advances.

Interface movement results are illustrated in Figure 8.10a. Sharp changes in cur-vature of the temperature profiles characterize the interface movement over time. Each sharp change of curvature occurs at the melting front. As time advances, the temperature gradient and heat flux decrease in the bulk liquid region, because the interface moves farther inward. In Figure 8.10a, close agreement between the com-puted results and experimental data (Gau and Viskanta, 1984) is achieved when the grid spacing is reduced.

Liquid

Solid

Tc

Th

X

Y

f Ig u r E8.9 Schematic of melting problem.

7262_C008.indd 238 1/19/08 11:59:17 AM


Entropy-corrected computations of interface movement and entropy production results are illustrated in Figure 8.10a and Figure 8.10b, respectively. Corrective steps in the computations are based on the entropy formulation. Without the entropy- corrected model, the direct computations underpredict the interface position at

f Ig u r E8.10 Computed (a) interface position and (b) entropy production.

1

1000

800

600

400

200

–200

0

0 0.2 0.4 0.6 0.8 1

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8

Fo (a)

(b)

1 1.2 1.4 1.6 1.8

Experiment Direct Computations (10 × 07) Corrected Computations

With Entropy- Corrected Model

y/Y

y/Y

P s (W

/m3 K)

Without Entropy- Corrected Model

Entropy-Stable Result(Nonnegative Entropy Production)

Direct Computations (10 × 07) Corrected Computations

7262_C008.indd 239 1/19/08 11:59:19 AM


large Fourier (Fo) (nondimensional time) values. The entropy-correction technique improves the predictions (see Figure 8.10a). At Fo = 0.32, the entropy production is illustrated in Figure 8.10b. The large positive entropy production occurs near y/Y = 0.3 because the phase interface closely coincides with this position at Fo = 0.32 (see Figure 8.10a). At the interface, an entropy of phase change is absorbed by the melt-ing solid and then transferred outward by heat conduction.

In the direct computations, a positive entropy production is observed at the inter-face, but behind the advancing interface, some computational entropy destruction occurs. This result coincides with discretization errors, due to coarse spatial differenc-ing that leads to overpredicted interface locations in Figure 8.9 with 10 × 7 elements. Improved predictions can be achieved with the entropy-corrected computations (see Figure 8.10b). In the corrected computations, the diffusivity required to correct the direct calculations is based on the magnitude of negative computed entropy production (absolute value) whenever the Second Law is violated in a discrete control volume.

When both physical and computational parts of entropy production are com-bined, then interpretation of the entropy prediction becomes more difficult because nonphysical data associated a nonobservable event may have been combined with positive physical entropy production. In other words, a physically plausible result may occur, even though the computations destroyed entropy, thereby creating a non-physical situation. Unfortunately, this situation cannot be readily identified when the numerical and physical parts of the computations are not separated. Conventional analysis of errors, such as “diffusive” or “dissipative” errors, would not apply to nonobservable processes. For example, a nonobservable process would be negative kinetic energy. The abstract meaning of entropy computations requires a special interpretation in such cases.

The Second Law may reveal nonphysical interactions between thermal or fluid waves, which might otherwise be too complex to assess in terms of physical plausibil-ity. For example, consider a soliton, which represents a wave that behaves like a par-ticle. It can be shown that when high, fast waves are sent behind low, slow waves, then each series of waves may preserve its identity through the interactions, even though these interactions (soliton waves) are nonlinear. The outcome of such complex wave interactions may not be easily understood. The Second Law is a physical principle available to determine the correct behavior. In the current example, the propagation of thermal waves is disturbed through its interaction with the phase interface and its sub-sequent release or absorption of latent heat. The Second Law can be used to identify nonphysical processes through the manifestation of negative entropy production.

8.4.6 c a sest u d yo f fr eeco n v ec t Io n a n d so l Id If Ic at Io n

In this final example, solid–liquid phase transition with natural convection is pre-dicted in a two-dimensional domain (see Figure 8.11a) (Naterer, 2000). This problem considers solidification of an initially liquid metal at T = 70[ ]o C , subject to Dirichlet boundary conditions along the vertical boundaries and adiabatic boundary condi-tions along the upper and lower boundaries. Zero velocity conditions are specified along all boundaries. In Figure 8.11a, Y = .8 89[ ]cm , X = .8 89[ ]cm , Tc = 68[ ]o C , and Th = 92[ ]o C .

7262_C008.indd 240 1/19/08 11:59:22 AM


Solidification begins at the cold right boundary, and the phase interface propa-gates inward as time advances. Thermal buoyancy creates an upward flow near the left boundary and downward flow near the phase interface and mushy regions in the right section of the domain. This resulting clockwise recirculation cell has a key role in the phase interface advancement and heat transfer characteristics. Recall that the following entropy-based time-step constraint, which was derived for numerical stability,

D ≤ D+ D

tu x

u c xr

r( )2 (8.57)

98.0

95.0

92.0

89.0

86.0

83.0

80.0

77.0

74.0

71.0

68.0

Width (m)0 0.012 0.025 0.038 0.05

0 = Y/Yo.2

.4.6

.81

(b)

f Ig u r E8.11 (a) Schematic of solidification problem and (b) onset on numerical oscillation.

Free ConvectionRecirculation cell

Liquid

Solid

TcTh

Y

X

7262_C008.indd 241 1/19/08 11:59:29 AM


where ur and cDx in Equation 8.57 refer to a reference velocity (or characteristic, lagged, linearization velocity) and a characteristic velocity associated with phase change, such as the interface velocity, respectively. In this two-dimensional phase change example, the time-step constraint becomes Dt ≤ 5[s]. This stability condition provides a general order of magnitude for recommended time steps.

Temperature results at t = 40[s] from two different time advances (Dt = 20[s] and Dt = 5[s], respectively) were investigated. The results represent two different cases: (i) time step exceeds nonlinear stability constraint, and (ii) time step complies with the stability constraint. Convergence criteria (interequation residual tolerance and maximum iteration cycles) are identical in both of the above cases; only the time step is modified between different simulations. Instead of an ad-hoc time step selection, the Second Law provided guidance in the time-step selection for subsequent numeri-cal stability in the computations.

The time step adopted in the latter result (Dt = 5[s]) comes within a close prox-imity of the above entropy-based time-step guideline, and it yielded stable computa-tions. However, the former case (Dt = 20[s]), which exceeds the time-step constraint, leads to an onset of instability, or oscillation, arising near the phase interface (see Figure 8.11b). The entropy-based time-step constraint provides an approximation of the general order of magnitude required for stable computations with phase change. In these computations, interequation iterations are performed between the momen-tum and energy equation (i.e., phase-temperature iterations based on discrete analo-gies of the Second Law). Additional predictive and corrective steps based on the Second Law are performed to ensure numerical stability and convergence.

8.5 t HEr Ma l co n t r o l o f pHa sEcHa n g Ew It HIn v Er sEMEt Ho d s

8.5.1 f o r mu l at Io n o f a n In v er semet h o d

The previous sections have examined the importance of entropy and the Second Law in direct problems, where boundary conditions are known and internal temperatures, velocities, and other dependent variables are sought. Inverse problems represent another class of important problems that specify desired behavior of the dependent variables within the domain, then require boundary conditions to be determined. Inverse problems often suffer from greater instability problems than direct solutions because any perturbations in boundary conditions are magnified into the domain to generate numerical oscillations. This section will consider heat and entropy transport in inverse modeling of phase change processes with solidification. Inverse methods provide an effective way to control phase change processes by changing boundary temperatures to achieve a desired progression of isotherms within the domain.

A fixed domain method will be used, and the phase interface moves uniformly at a desired progression rate. An entropy-based correction is applied to improve the numerical stability of the inverse computations. The formulation considers local vio-lations of the Second Law (arising from spatial or temporal discretization errors) as a criterion for a corrective strategy in the computations. The magnitude of nega-tive entropy production is used for a quantitative correction of the apparent thermal

7262_C008.indd 242 1/19/08 11:59:29 AM


conductivity. It will be shown that this approach provides an effective alternative to previous inverse stabilizing techniques, such as future time stepping.

As an example of a typical inverse problem, consider solidification in a closed region (see Figure 8.12) that is initially occupied by a pure liquid of temperature Tin(x), where Tm denotes the melting temperature. The governing equation consists of one-dimensional heat conduction with solid–liquid change, as developed previously in this chapter. Because the top, bottom, and right boundaries of the cavity are insulated, no temperature gradients (and no thermal buoyancy) arise in the liquid region. The posi-tion of the interface will be controlled by the temperature of the left boundary. The conductivity, k, density, r, and specific heat, c, are independent of temperature. Also, the melting temperature, Tm, is given. In this example, it is assumed that the interface moves in the x-direction, and the shape of the interface remains a vertical straight line (moving into the domain). Also, the velocity of the phase interface can be given as a desired quantity. The temperature at the left boundary is uniform, and it only varies with time, i.e., T0 = T0 (t). Although the domain is illustrated in a two-dimensional geometry (Figure 8.12), this example can be treated as a one-dimensional heat transfer problem.

In an inverse problem, with the exception of the boundary having the unknown controlling temperature, all boundary conditions can be treated as a direct problem,

f Ig u r E8.12 Schematic of (a) multiphase control volume and (b) inverse Stefan problem.

Interface

V

Liquid

Solid

Tm

(b)

T(t)

(a)

Liquid

Solid

7262_C008.indd 243 1/19/08 11:59:32 AM


with Dirichlet, Neumann, or Robin boundary conditions. At the controlling bound-ary, the temperature distribution remains unknown, and it represents the final solution of the inverse problem. This temperature is obtained through iterations, fol-lowing an initial estimate. At the beginning of each time step, a temperature estimate is initially taken. For example, it may be set as the temperature value at the previous time step (lagged estimate). Then, during each subsequent step in the simulation, the controlling boundary temperature (at x = 0 in our example) is updated at each itera-tion until the predicted interface movement agrees, within a given tolerance, with the specified (or desired) phase interface movement. In the current formulation, the following iterative update of the boundary temperature T0, is used:

T TT T

Rm m p

mpm

p0

10

11

1

1

( )+ ++

+

+= +

- (8.58)

where m, p + 1 (subscript), and R refer to the iterative counter, nodal point p + 1, and the sensitivity coefficient, respectively.

The sensitivity coefficient is defined as follows:

RT

Tpp

++=

∂∂1

1

0 (8.59)

This coefficient essentially measures the influence of changes in the boundary tem-perature, T0, on the temperature at nodal point p + 1. The range of the sensitivity coefficient is 0 ≤ R ≤ 1. The value of R may be interpreted as a temperature connection between nodal point p + 1 and the boundary value. For example, when the sensitivity coefficient, R, becomes larger, then the influence of the changes at the boundary tem-perature on the nodal value at point p + 1 becomes stronger. Nodal values closer to the boundary yield larger sensitivity coefficients. This coefficient will be used to update the boundary temperature in the current inverse model.

The finite volume equations for the sensitivity coefficient, R, can be derived after taking derivatives with respect to T0 on both sides of the discretized energy equation (Xu and Naterer, 2001). The inverse heat transfer problem is ill-posed, since arbitrarily small errors in the temperature measurements or interface position can be projected back to the boundary as magnified large errors. For inverse phase change problems, when the interface moves farther from the boundary, it becomes more difficult to control the interface by adjusting the boundary temperature. As a result, numerical oscillations may arise in the computations. Voller (1992) developed a future time stepping method to address this problem. In contrast, the following sec-tion examines how the Second Law can be used to stabilize computations in inverse problems.

8.5.2 en t r o Pyco r r ec t Io n f o r nu mer Ic a l st a bIl It y

For an inverse problem with solidification from the left boundary, the interface moves farther away from the left boundary over time, so the effect of the boundary temperature on the interface movement becomes weaker. The thermal influences

7262_C008.indd 244 1/19/08 11:59:34 AM


may not be effectively carried from the boundary to the interface due to the large distance between them. As a result, it becomes more difficult to control the interface movement solely by the temperature at the left boundary. Numerical oscillations in the inverse solution may start to occur.

When the interface moves farther away from the left boundary, the sensitivity coefficients become smaller. As the values of the sensitivity coefficient become small, the resulting roundoff error may lead to numerical instability. Meanwhile, from the iterative update equation for the boundary temperature, it can be shown that when the sensitivity coefficient becomes very small, the updated boundary temperature will exhibit large changes between one iteration and a subsequent iteration. The solu-tion may become unstable if the iterative values change drastically. Consequently, the entropy production rate may become negative whenever numerical instabilities arise. In this situation, the Second Law would be violated. Thus, the algorithm should be modified to stabilize the calculations. The entropy production can be used as a predictive tool in this regard. If the local value of entropy production becomes nega-tive, then a numerical instability may arise, and an entropy-based correction of the computations should be made. The corrective procedure would only be applied when nonphysical solution behavior occurs. In this way, we can reduce overall computa-tional time and avoid corrections of the solution at every time step.

The entropy production is computed after the solution of the energy equation is obtained. If a nodal value of sP is negative, then the local solution is not physically plausible. Instead of proceeding to the next time step, a correction is performed based on the magnitude of entropy production within the discrete volume. From the posi-tive-definite expression for entropy production, the conductivity can be expressed in terms of the entropy production, sP , as follows:

kT P

T Ts=

∇ ⋅∇

2 | |

(8.60)

In this way, the effective conductivity, k, is related to the local entropy production rate and temperature gradient. If the Second Law is violated locally, then the con-ductivity can be corrected based on the computed entropy production rate. Then this entropy-based conductivity would be used to calculate the sensitivity coefficients again, from which modified sensitivity coefficients are obtained. These new sen-sitivity coefficients can be used when updating the boundary temperature during each iteration. This method can prevent potentially nonphysical oscillations during solidification computations in inverse problems.

In the inverse problem, the governing equations are identical to the equations of the corresponding direct problem. The difference arises because the interface posi-tion is given, instead of unknown, and the temperatures at the controlling boundary are unknown, rather than known boundary conditions. The other boundary condi-tions are the same conditions arising from the direct problem. Initially, during each time step, a tentative (guessed) temperature at the controlling boundary is used, and the equations are solved as a direct problem. The temperatures at the controlling boundary are then updated continuously in an iterative manner, until the predicted

7262_C008.indd 245 1/19/08 11:59:36 AM


movement of the phase interface agrees within a given tolerance with the desired path. Then, the solution at the given time step is completed.

The numerical procedure for the inverse solution can be summarized as follows:

1. Specify the movement of the phase interface (interface velocity), and choose a constant time step based on this interface velocity.

2. Based on the velocity of the interface and the time step, a fixed numerical grid is specified, so at any time step the interface moves from one grid point to the next grid point.

3. Within each time step, an estimate of the unknown controlling boundary temperature is given, and the energy conservation equation is solved in a direct manner for the temperature, T.

4. The sensitivity coefficients are obtained, and the unknown (controlling) boundary temperature is updated. Then, the energy equation is solved again. An entropy-based correction of conductivity is made for the sensi-tivity coefficients whenever the Second Law is locally violated.

5. Repeat steps 3 and 4 for each iteration when solving the energy conserva-tion equation. The solution is terminated when the predicted movement of the interface agrees, within a given tolerance, with the specified (desired) interface movement.

Although the Second Law is decoupled from the implicit solution of the energy equa-tion, the current formulation provides an entropy-based corrective step, which strives to achieve Second Law compliance and stable computations in the inverse problem.

8.5.3 c a sest u d yw It h so l Id If Ic at Io n o f a Pu r emat er Ia l

Consider a specific case study depicted in Figure 8.12, where solidification of a pure material (initially at the phase change temperature, T0 = 0) begins at the left bound-ary and proceeds rightward into the domain (Xu and Naterer, 2001). Cooling from the boundary at x = 0 leads to solidification of the liquid metal. The inverse problem requires the estimation of the boundary temperature, T (0,t), at x = 0, which gener-ates a constant interface velocity, V, during advancement of the solid–liquid inter-face. The liquid portion within the mold remains at (or slightly above) the melting temperature. A slight undercooling is required for the onset of solidification. The numerical results in this section will be compared with an analytic solution (Charach and Rubenstein, 1989) to assess the accuracy and performance of the formulation.

The following problem parameters (dimensionless) are adopted in this example: a = 1, c = 1, L = 0.5, V = 2.0, where L, a, and c refer to latent heat, thermal diffusiv-ity, and specific heat, respectively. An analytical solution of this problem, involving solidification with a constant interface velocity, was reported by Carslaw and Jaeger (1967) as follows:

T x t TL

c

Vt

Vx( , ) = = - -

0

2

1 expα α

(8.61)

7262_C008.indd 246 1/19/08 11:59:38 AM


and

T x t T( , ) = 0 (8.62)

where x ≥ Vt. At x = 0, the surface temperature required to produce the specified interface motion is

T t T

L

c

Vt( , )0 10

2

= + -

expα

(8.63)

for t > 0. The analytical solution for this particular case study can be obtained from substitution of the previous parameters into the previous general solution to give

T t exp t( ) [ ( )]0

12

1 4, = - (8.64)

This result becomes unrealistic after long time periods due to the exponential behavior of the time-dependent boundary condition. It would be difficult to achieve in a large casting after long periods of time, because the boundary temperature is eventually required to reach an extremely low temperature at x = 0, to maintain a constant velocity of the solid–liquid interface. However, it represents practical conditions dur-ing early stages of solidification in problems of materials processing.

In Figure 8.13, the boundary temperature, T(0,t), for three specified interface velocities for pure gallium is illustrated. The phase interface position can be determined

f Ig u r E8.13 Boundary temperature at varying interface velocities.

50

–50

–100

–150

V = 0.04 mm/s

Boun

dary

Tem

pera

ture

(°C

)

V = 0.06 mm/s V = 0.08 mm/s

–200

–250

–300 0 500 1000

Time (s)1500 2000

0

7262_C008.indd 247 1/19/08 11:59:41 AM


0

–20

–40

–60

–80

–100

–120

–1400 0.5

Time(a)

1 1.5

Predicted Results

Boun

dary

Tem

pera

ture

Exact Solution

f Ig u r E8.14 Boundary temperature at dimensionless time steps of (a) 0.1, (b) 0.05, and (c) 0.005 s.

0

–0.1

–0.2

–0.3

–0.4

–0.5

–0.6

–0.70 0.05 0.1 0.15 0.2

Time(c)

Predicted ResultsBoun

dary

Tem

pera

ture

Exact Solution

0 –5

–10 –15 –20 –25 –30

–35 –40 –45

0 0.2 0.4 0.6 Time

(b)

Predicted Results Exact Solution Bo

unda

ry T

empe

ratu

re

0.8 1 1.2

7262_C008.indd 248 1/19/08 11:59:44 AM


based on the given interface velocity and the specified time level. As time advances, the phase interface moves farther away from the left boundary, and the solution becomes unstable. Oscillations appear and they lead to solution divergence (note: only stable regions are illustrated in Figure 8.14a through Figure 8.14c). It appears that oscillations occur earlier in the case of smaller time steps. The boundary tem-perature computations become unstable earlier with smaller time steps. In the numerical simulations, whenever the distance between the phase interface and left boundary becomes too large, it becomes difficult to control this interface movement solely through the left boundary temperature.

Numerical oscillations appear in Figure 8.15a with Dt = 0.05. Before t = 1.1, the computations proceed well. But after t = 1.1, oscillations arise, and their magnitude

40

20

0

–20

–40

–60

–80

–100

–1200 0.2 0.4 0.6 0.8 1 1.2 1.4

Time(a)

Predicted ResultsBoun

dary

Tem

pera

ture

Exact Solution

f Ig u r E8.15 Boundary temperature (∆t = 0.05) (a) without and (b) with entropy correction.

0

–50

–100

–150

–200

–2500 0.5 1 1.5

Time(b)

Predicted Results

Boun

dary

Tem

pera

ture

Exact Solution

7262_C008.indd 249 1/19/08 11:59:46 AM


increases with further time advancement. The computations from the Second Law give indications of these trends. An entropy-based approach is used to provide a criterion for corrective steps in the formulation. During each time step, the entropy production rate is computed, then the Second Law requires that it must remain posi-tive within a discrete volume. If oscillations appear in violation of the Second Law, the predictive entropy-based correction is applied in the computations. The effective thermal conductivity is corrected based on the local entropy production rate. This value is then adopted, and the sensitivity coefficient in the inverse model is modified accordingly.

The results in Figure 8.15b show that the entropy-based correction performs well and improves the numerical stability of results. Oscillations are reduced for sub-sequent time steps, in comparison to Figure 8.15a. The modifications permit stable computations of boundary temperature for longer time periods. It is worthwhile to compare this approach with another conventional technique for stabilizing inverse computations, namely, future time stepping. In future time stepping, the boundary temperature is assumed fixed for r future time steps, and the system of discrete equa-tions is solved over this time range. Then, the boundary temperature is updated, and iterations continue until a sum of squares difference (involving the interface tem-perature at a future time level and the phase change temperature) is minimized. This section has indicated that the Second Law provides a physically based alternative to future time stepping. It underlies a physical mechanism that can stabilize results that exhibit nonphysical behavior, such as numerical oscillations, for problems involving solidification and melting.

8.6 En t r o py pr o d u c t Io n w It HfIl Mco n d En sat Io n

8.6.1 f o r mu l at Io n o f heat tr a nsf er a n d Ir r ev er sIbIl It ydIst r Ibu t Io n

The previous sections have analyzed entropy and the Second Law in solidification and melting problems. Before closing this chapter, another multiphase system will be examined, namely, heat transfer with condensation. Previous sections have focused on numerical analysis with the Second Law, whereas this section will use analytical methods and entropy generation minimization. Pioneering work on laminar film condensation along an isothermal surface was conducted by Nus-selt (1916). This analysis neglected the advection terms to obtain an approximate solution, in terms of forces and heat balances within the condensate film. Spar-row and Gregg (1959) showed that inertial effects only have a significant influ-ence on the heat transfer rate when the Prandtl number (Pr) is less than 10. Koh and coworkers (1961) obtained an exact boundary layer solution, when the shear forces at the liquid/vapor interface were taken into account. The effects of gravi-tational forces and interfacial stress were found to be negligible for high Prandtl numbers (Pr > 10).

An improved correlation of film thickness was developed by Rohsenow (1956), when thermal advection effects become significant. A modified latent heat of vaporization was introduced, which depends on the Prandtl number (Sadasivan and Lienhard, 1987). Dier and Lienhard (1974) showed that results for a vertical plate

7262_C008.indd 250 1/19/08 11:59:47 AM


could be extended to inclined plates, provided that g is replaced by g · cos(q), where q is the angle between the vertical direction and the wall. Although analytical solu-tions can be derived for laminar film condensation, correlations involving turbulence typically require approximate or empirical methods.

Film condensation is accompanied by thermodynamic irreversibilities of fluid friction and heat transfer over finite temperature ranges. There exists a direct rela-tionship between entropy production of a process and the amount of power consumed or lost by the process (Bejan, 1996). A traditional approach to the study of laminar film condensation involves a solution of the governing conservation equations. The Second Law may serve as an important additional tool for optimization in the design of thermal systems involving laminar film condensation, such as finned surfaces in a heat exchanger. The method of entropy generation minimization (EGM) has been applied previously to single-phase convective heat transfer. Bejan (1979) analyzed the entropy production of various configurations and flow regimes. It was shown how certain flow and geometric parameters can be selected to minimize the irrevers-ibilities of the thermofluid processes. In this section, an optimization correlation is presented for film condensation on a flat plate, including results of entropy and the Second Law.

Consider laminar film condensation of a pure saturated vapor at Tsat on a vertical isothermal plate held at a temperature of Tw. Uniform thermophysical properties will be assumed. Also, it is assumed that the condensate film flows in the x-direction along the plate (see Figure 8.16) due to either gravitational or shear-driven flow effects. The product of the friction coefficient (based on the interfacial shear stress at the liquid/vapor interface), cf, and the Froude number, Fr, identifies the relative magni-tudes of these effects. For cf · Fr << 1, gravitational effects are dominant, whereas shear-driven effects are dominant when cf · Fr >> 1.

Vapor Region

y

x

g

τ ρv

ρ

δ

Liquid Condensate Film

Case 1

WallTw

Tsat

Case 2CV

Temperature Profile

Г + (dГ/dx)dx

m”v hfgq”x

q”xdx

Velocity Profiles

Г

f Ig u r E8.16 Schematic of film condensation.

7262_C008.indd 251 1/19/08 11:59:49 AM


The first case will examine conditions of cf · Fr << 1 (without effects of inter-facial shear stress). The reduced equations for mass, x-momentum, and energy are given by

∂∂+ ∂∂= ∂

∂= - - ∂

∂=u

x

v

y

u

yg k

T

yx v0 02

2

2

2µ ρ ρ( );

(8.65)

The boundary conditions are

y = 0, v = 0, u = 0, T = T

y = , T = T ,

w

sat

,

δ kT∂∂∂=•

ym hv fg

(8.66)

where u, v, m, gx, r, k, Tsat, Tw, mv, hfg, and d refer to x-velocity, y-velocity, dynamic viscosity, gravity (x-component; see Figure 8.16), density, thermal conductivity, satu-ration temperature, wall temperature, vapor mass flow rate, enthalpy of vaporization, and film thickness, respectively.

The rate of entropy production, Ps, within the liquid (excluding the interface) is

Pk

T

T

y T

u

ys =∂∂+ ∂

∂2

2 2µ

(8.67)

with streamwise velocity and temperature gradients assumed to be much smaller than y-direction gradients. Assuming a constant wall heat flux, q´ , and a zero interfacial velocity gradient,

T T Tq

kysat- = = ′′ -D ( )δ

(8.68)

ug y yx v= - -

( )ρ ρ δµ δ δ

2 212

(8.69)

Combining Equation 8.67 through Equation 8.69 and assuming that T - Tsat is much smaller than Tw or Tsat,

Pk

q

T

g y

Tssat

x v

sat

= ′′

+ - -1

22( ( )( ))δ ρ ρ δ

µ

(8.70)

On the right side of Equation 8.70, the first term represents the entropy generated by heat transfer, and the second term represents the fluid friction irreversibility. The entropy production is maximum at the wall, as viscous effects are highest there. Substituting k · Pr/cp for m in Equation 8.70, it can be shown that the second term

7262_C008.indd 252 1/19/08 11:59:55 AM


in Equation 8.67, which represents the friction irreversibility due to viscous drag, becomes less significant as the Prandtl number increases. In that case, entropy pro-duction is mainly due to heat transfer. This result agrees with earlier findings of Sparrow and Gregg (1959), whereby the inertial effects become negligible for Prandtl numbers of 10 and higher. The velocity boundary layer is comparatively larger than the thermal boundary layer for Pr >> 1, so that inertial effects on the Nusselt number in the condensation analysis become relatively small.

Define an irreversibility distribution ratio, f, which represents the ratio of fric-tion irreversibility, Ps V, , to thermal irreversibility, Ps T, , in Equation 8.70. At the wall, this ratio is

φρ ρ δ

= = ⋅-′′

P

P

g c T

qs V

s T

x p v sat,

,

( )1 3

2Pr (8.71)

As mentioned previously for Pr >> 1, inertial effects become negligible as the hydro-dynamic boundary layer thickness becomes much larger than the thermal bound-ary layer thickness. In that case, it is expected that the friction irreversibilities, Ps V, , are much lower than the thermal irreversibilities, Ps T, . The inertial term becomes negligible, as the velocity gradients at the wall are reduced when the boundary layer thickness increases. Thus, f << 1 when Pr >> 1, which is confirmed by the functional form of Equation 8.71. Previous researchers (Sparrow and Gregg, 1959) have shown the inertial terms to be significant when Pr << 1. This trend is confirmed by Equa-tion 8.71. If friction irreversibilities are comparatively larger when inertial terms are important, then f >> 1, which requires that Pr << 1 for given problem parameters.

The second case will examine conditions with cf · Fr >> 1 (predominant effects of interfacial shear stress). In Figure 8.16, the shear stress, t, acts at the phase inter-face. The same governing, boundary, and entropy equations, Equation 8.65 through Equation 8.67, are solved except that a condition of ∂ ∂ =2 2 0u y/ is applied at the edge of the condensate film, thereby yielding

ug y y

yx v= - -

+( )ρ ρ δ

µ δ δτµ

2 212

(8.72)

Pk

q

T

g y

Tssat

x v

sa

= ′′

+ - - +1

22( ( )( ) )ρ ρ δ τ

µ tt

(8.73)

It should be noted that Equation 8.73 does not include the entropy change of phase transformation at the interface, because it only applies within the liquid film (not at the phase interface). The last portion of Equation 8.73 arises from irreversibilities due to the interfacial shear stress.

Applying mass and energy balances for the control volume, CV, in the liquid (see Figure 8.16),

k

T Th

d

dxsat

fg

-

=δ

G

(8.74)

7262_C008.indd 253 1/19/08 12:00:00 PM


For this case of cf · Fr >> 1, the interfacial stress term is dominant, so Equation 8.72 becomes

u y= τµ

(8.75)

Then, the condensate mass flow rate per unit width, G, may be determined as follows:

G = =∫ ρ ρτ

µδ

δudy

0

2

2 (8.76)

Using Equation 8.76 in Equation 8.74 and solving for the film thickness,

δ µρτ

= -

31 3

k T T x

hsat

fg

( )/

(8.77)

For a given condensate thickness, d, the average liquid velocity will be lower when the interfacial stress is applied. This leads to a smaller mass flow rate of condensate. Because the heat transfer rate, q´ , is proportional to the mass flow rate multiplied by the latent heat (mv´ ⋅ hfg ), the rate of heat transfer will decrease. This heat transfer rate becomes

′′ = - -

-

q k T Tk T T

hxsat

sat

fg

( )( )

/3

1 3µ

ρτ--1 3/

(8.78)

so the Nusselt number is

Nuq T x

k Jaxx= ′′ = ⋅

( / ) Re Pr /D3

1 3

(8.79)

where the Reynolds number is based on the shear velocity, τ ρ/ .Consider entropy generation minimization of the geometrical and flow param-

eters when shear effects are dominant (cf · Fr >> 1; case 2). Based on the previous assumptions and Equation 8.73, the entropy production per unit width becomes

P P dyk

q

T Ts ssat sat

' = = ′′

+∫0

221δ

δ τµ

δ (8.80)

Substituting d and combining Equation 8.79 and Equation 8.80, it can be shown that

Pk

q

T T

Jas

sat sat

'

Pr= ′′

+

1 3

22τ

µ

-1 3 2 3

1 3/ /

/ρτµ

x

(8.81)

7262_C008.indd 254 1/19/08 12:00:06 PM


Integrating over a plate length of L, expressing the heat transfer per unit length as q´ = q´ L, and simplifying in terms of the Reynolds number, ReL, it follows that

Pk

q

T

Jas

sat

= ′

34

1 32 1

Pr

/332 3

1 343

43

RePr

ReLsat

LT

Ja- +

//µτ

ρ//3

(8.82)

From this result, the surface length is an example of a parameter that can be changed to minimize the entropy production. Setting ∂ ∂ =Ps L/ Re 0, we find the following optimum that minimizes Ps :

ReL opt B, .= 0 707 (8.83)

where the duty parameter, B, for a plate width of W is defined by

Bq W

k Tsat

=( )

//ντ 1 2

(8.84)

Equation 8.83 is a criterion to deliver more effective heat exchange in problems involving film condensation. It can be rewritten in terms of the optimal plate length, based on the definition of the Reynolds number. As an example, for a fixed rate of heat transfer within a finned heat exchanger, it gives the optimal length to mini-mize the thermal irreversibility and required temperature difference to achieve that amount of heat transfer. Alternatively, these results can be expressed in terms of minimized destruction of energy availability (or exergy). The rate of exergy destruc-tion is T Psat s

, so an equivalent result of Equation 8.83 is obtained for the minimized exergy destruction.

If Equation 8.83 is substituted into Equation 8.81, an expression is obtained for the optimal (minimized) entropy production. The ratio of the actual entropy produc-tion to the minimized entropy production represents the entropy generation number, Ns, which is determined to be

N

P

Pss

s opt

L

L opt

= =

+

-

, ,

/

. .0 666 0 3

2 3Re

Re333

4 3Re

ReL

L opt,

/

(8.85)

For a fixed rate of heat transfer, q , within a finned heat exchanger, a large sur-face length would reduce the temperature difference (between steam and wall) required to achieve q , thereby reducing the refrigeration power needed to main-tain Tw. But the interfacial shear and vapor friction increase for a larger surface length, thereby increasing the input power required to maintain a certain vapor flow rate at steady state. On the other hand, a smaller surface length reduces the total vapor friction and input power, but at the expense of a higher thermal irreversibility and refrigeration power, due to the higher temperature difference needed to maintain q . The best compromise is reached when the entropy genera-tion number is minimized, thereby minimizing the net power input from both vapor and refrigeration sides.

7262_C008.indd 255 1/19/08 12:00:12 PM


8.6.2 c a sest u d yo f fl at Pl at eco n d ens at Io n

Results will be presented for an example involving condensation of steam at 1 atm (Adeyinka and Naterer, 2004). Comparisons will be made with previous studies involving other optimized flow configurations, as described by Bejan (1996) and Fowler and Bejan (1994). In Figure 8.17, the variation of entropy generation with length (or diameter, D, for the case of a tube) shows that a minimum entropy genera-tion number, Ns, occurs when ReL reaches the optimized value. For the case of film condensation and others shown in Figure 8.18 at low values of L (or D), the friction irreversibility is relatively small, due to the small surface area of fluid friction. But the thermal irreversibility is high because a high temperature difference occurs over a thin region to transfer a given heat flow, q . On the other hand, the friction

1

10

100

1,000

L/Lopt, D/Dopt

NS

Plate, Condensation

Plate, Single Phase

Internal Flow Tube

1E–03 1E–02 1E–01 1E+00 1E+01 1E+02 1E+03

f Ig u r E8.17 Entropy generation number.

11 10 100 1,000

10

100

1,000

10,000

B

Re L,

opt

Plate, Laminar, Condensation

Plate, Turbulent,Single Phase

Plate, Laminar, Single Phase

Air

Air

Water

Water

f Ig u r E8.18 Optimized Reynolds number.

7262_C008.indd 256 1/19/08 12:00:16 PM


irreversibility is higher at large values of L (due to the large surface area involving fluid friction), whereas the thermal irreversibility is lower because a smaller tempera-ture difference between the wall and surrounding fluid is needed to transfer the given q´ for a larger surface area. As a result, a minimized Ns exists for each case shown in Figure 8.17.

The entropy generation number for flat plate condensation within the film is lower than the case involving tube flow. The surface curvature implies that the area avail-able to heat and momentum transfer decreases in the radial direction when the fluid is heated by the wall. As a result, the near-wall temperature and velocity gradients become higher for the duct flow at a specified Reynolds number, so the entropy pro-duction increases. The entropy production for the single phase boundary layer appears smaller than the condensation result. Linear velocity and temperature profiles, with constant shear stress and heat flux values within the condensate film, were derived for case 2, whereas nonlinear profiles with decreasing stresses and heat fluxes (perpen-dicular to the wall) arise in single phase boundary layers (Fowler and Bejan, 1994).

The entropy generation number in Figure 8.17 for single phase laminar flow is symmetric about the minimum, as both friction and thermal irreversibilities appear to have equal contributions to the net entropy production (note: on a relative basis after nondimensionalization). However, a higher contribution of friction irrevers-ibility at high values of ReL, as compared to the thermal irreversibility, leads to asymmetry about the minimum for film condensation. When a given heat flow is transferred over a surface area, this effect is distributed spatially to reduce the required temperature difference and thermal irreversibility. However, the fluid fric-tion is a cumulative effect when the surface area increases, thereby showing different characteristics with variations of the surface length. Similar interpretations can be made for the asymmetry observed in the duct flow with convective heat transfer in Figure 8.17.

In Figure 8.18, the minimized ReL (or plate length) for film condensation increases at higher values of the duty parameter, B. The duty parameter for film condensation involves the total heat transfer, q, and interfacial shear stress, t, in Equation 8.84. Its definition for other flow configurations is documented by Bejan (1996). As B increases, then q also increases, so a larger plate is needed to minimize the entropy production (i.e., positive slope in Figure 8.18) by reducing the thermal irreversibility associated with a larger heat flow. For the single phase flow cases in Figure 8.18, the turbulent friction irreversibility rises faster, (steeper velocity gradient), and the wall or fluid temperature difference, ∆T, falls faster than the laminar case. This occurs when turbulent mixing entails a lower ∆T needed to transfer a given q. Thus, ReL for the turbulent case is lower than the laminar case at low values of q and B. But at high values of q and B, the thermal irreversibility component becomes more significant. Then the laminar friction irreversibility rises faster, whereas the thermal irrevers-ibility falls faster. As a result, ReL is lower for the laminar case at low values of B, but crosses and exceeds the turbulent profile at sufficiently high values of B.

In Figure 8.18, the slopes of the curves for film condensation and turbulent single phase flow are nearly equal. The relative rates at which the friction irreversibility rises and the thermal irreversibility falls with increasing B (or q) appear closely coin-cident. However, the condensation slope is lower than the laminar single phase case,

7262_C008.indd 257 1/19/08 12:00:16 PM


suggesting that the friction irreversibility increases at a faster rate or the thermal irreversibility falls faster with B (or q) in the former case. Similar observations are made when the slopes of laminar and turbulent flows are compared. Although the film condensation results for ReL were derived independently of Pr, earlier results for single phase flow included a Prandtl number dependence (Fowler and Bejan, 1994). As shown in Figure 8.18, water has a higher viscosity and thermal conductivity than air at 300 K, so the friction irreversibility rises faster and the thermal irreversibility falls faster (i.e., lower DT to meet a given q) for water. Thus, the minimum Ps occurs at a lower ReL for water at a fixed value of B, as shown in Figure 8.18.

These results have practical implications for the design of two-phase heat exchangers. An example of a case with cf · Fr >> 1 (case 2) is spacecraft thermal systems in microgravity, that is, heat pipes and capillary pumped loops. In variable- conductance capillary pumped loops, the surface length in contact with the condens-ing vapor is lowered to suppress the temperature rise during high thermal loads, or operation in a hot environment. Similarly, this length increases to impede the temper-ature drop during low thermal loads or operation in a cold environment (Furukawa, 1999). An entropy-based analysis provides useful insight to improve performance of systems involving condensation heat transfer.

r Ef Er En c Es

Adeyinka, O.B. and G.F. Naterer. 2004. Optimization correlation for entropy production and energy availability in film condensation. Int. Commun. Heat Mass Transfer, 31(4): 513–524.

Bejan, A. 1979. Study of entropy generation in fundamental convective heat transfer. ASME J. Heat Transfer, 101: 718.

Bejan, A. 1996. Entropy Generation Minimization: The Method of Thermodynamic Optimi-zation of Finite-Time Systems and Finite-Time Processes. CRC Press, Boca Raton, FL, Chap. 8.

Bennon, W.D. and F.P. Incropera. 1988. Numerical analysis of binary solid–liquid phase change using a continuum model. Numerical Heat Transfer, 13: 277–296.

Bird, R., Stewart, W., and E. Lightfoot. 1960. Transport Phenomena. John Wiley & Sons, New York.

Burton, R., Yang, G., Dong, Z.F., and M.A. Ebadian. 1995. An experimental investigation of the solidification process in a V-shaped sump. J. Heat Mass Transfer, 38(13): 2383–2393.

Carslaw, H.S. and Jaeger, J.C. 1967. Conduction of Heat in Solids, Oxford University Press, New York.

Charach, C. and I.L. Rubinstein. 1989. On entropy generation in phase-change heat conduc-tion. J. Appl. Phys., 66(9): 4053–4061.

Clyne, T.W. and W. Kurz. 1981. Solute redistribution during solidification with rapid solid stated. Metallurg. Trans., 12A: 965.

Dhir, V.K. and J.H. Lienhard. 1974. ASME Journal of Heat Transfer, 1971: 93, 97.Flemings, M.C. 1974. Solidification Processing. McGraw-Hill, New York, Chap. 5.Flood, S.C. and P.A. Davidson. 1994. Natural convection in aluminum direct chill cast ingot.

Mater. Sci. Technol., 10: 741–751.Fowler, A. and A. Bejan. 1994. Correlation of optimal sizes of bodies with external forced

convection heat transfer. Int. Commun. Heat Mass Transfer, 21: 17.Furukawa, M. 1999. AIAA 33rd Thermophysics Conference. Paper 3445. Norfolk, VA.

7262_C008.indd 258 1/19/08 12:00:18 PM


Gau, C. and R. Viskanta. 1984. Melting and solidification of a metal system in a rectangular cavity. Int. J. Heat Mass Transfer, 27(1):113–123.

Hayashi, Y. and T. Komori. 1979. Investigation of freezing of salt solutions in cells. J. Heat Transfer, 101: 459–464.

Koh, J.C., Sparrow, E.M., and J.P. Hartnett. 1961. Int. J. Heat Mass Transfer, 2: 69.Lax, P.D. 1971. Shock waves and entropy, in Contributions to Non-Linear Functional Analy-

sis. Academic Press, New York, 603–634.Maples, A.L. and D.R. Poirier. 1984. Convection in the two-phase zone of solidifying alloys.

Metallurg. Trans., 15B: 163–172.Minkowycz, W.J., Sparrow, E.M., Schneider, G.E., and R.H. Pletcher. 1988. Handbook of

Numerical Heat Transfer. John Wiley & Sons, New York, Chap. 8.Naterer, G.F. 1999. Constructing an entropy-stable upwind scheme for compressible fluid

flow computations. AIAA J., 37(3): 303–312.Naterer, G.F. 2000. Predictive entropy based correction of phase change computations

with fluid flow. Part 1. Second Law formulation. Numer. Heat Transfer B, 37(4): 393–414.

Naterer, G.F. 2001. Applying heat-entropy analogies with experimental study of inter-face tracking in phase change heat transfer. Int. J. Heat Mass Transfer, 44(15): 2917–2932.

Naterer, G.F. and G.E. Schneider. 1996. PHASES model of binary constituent solid– liquid phase transition. Part 1. Numerical method. Numerical Heat Transfer B, 28(2): 111–126.

Nusselt, W. 1916. Die Oberflachen Kondensatin des Wasser dampfes. Zeitschrift des Vereines Deutscher Ingenieure, 60: 541.

Pardo, E. and D.C. Weckman. 1990. A fixed grid finite element technique for modelling phase change in steady state conduction — Advection problems. Int. Journal Numer-ical Methods Eng., 29: 969–984.

Rady, M.A., Satyamurty, V.V., and A.K. Mohanty. 1997. Thermosolutal convection and mac-rosegregation during solidification of hypereutectic and hypoeutectic binary alloys in statically cast trapezoidal ingots. Metallurg. Mater. Trans. B, 28: 943–952.

Rohsenow, W.M. 1956. Heat transfer and temperature distribution in laminar film condensa-tion. Trans. ASME, 78: 1645.

Rosen, M.A. and I. Dincer. 2004. A study of industrial steam process heating through exergy analysis. Int. J. Energy Res., 28(10): 917–930.

Rosen, M.A., Pedinelli, N., and I. Dincer. 1999. Energy and exergy analyses of cold thermal storage systems. Int. J. Energy Res., 23(12): 1029–1038.

Sadasivan, P. and J.H. Lienhard. 1987. Sensible heat correction in laminar film boiling and condensation. ASME J. Heat Transfer, 109: 545.

Salcudean, M. and Z. Abdullah. 1988. On the numerical modelling of heat transfer during solidification processes. Int. J. Numerical Methods Eng., 25: 445–473.

Salcudean, M. and R.I.L. Guthrie. 1979. A three dimensional representation of fluid flow induced in ladles or holding vessels by the action of liquid metal jets. Metallurg. Trans. B, 10: 423–428.

Schneider, G.E. and M.J. Raw. 1984. An implicit solution procedure for finite difference modelling of the Stefan problem. AIAA J., 22: 1685–1690.

Sparrow, E.M. and J.L. Gregg. 1959. A boundary layer treatment of laminar film condensa-tion. ASME J. Heat Transfer, 81: 13.

Szekely, J. and A. Jassal. 1978. An experimental and analytical study of the solidification of a binary dendritic solution, Metallurg. Trans., 9b: 389–398.

Vesligaj, M.J. and C.H. Amon. 1999. Transient thermal management of temperature fluctua-tions during time varying workloads on portable electronics. IEEE Trans. Compo-nents Packaging Technol., 22(4): 541–550.

7262_C008.indd 259 1/19/08 12:00:20 PM


Voller, V.R. 1992. Enthalpy method for inverse Stefan problem. Numerical Heat Transfer, B, 21: 41–55.

Voller, V.R. and D. Brent. 1989. The modelling of heat, mass and solute transport in solidifi-cation systems. Int. J. Heat Mass Transfer, 32(9): 1719–1731.

Xu, R. and G.F. Naterer. 2001. Inverse method with heat and entropy transport in solidifica-tion processing of materials. J. Mater. Processing Technol., 112(1): 98–108.

Yoo, H. and R. Viskanta. 1992. Effect of anisotropic permeability on the transport process during solidification of a binary mixture. Int. J. Heat Mass Transfer, 35(10): 2335–2346.

7262_C008.indd 260 1/19/08 12:00:22 PM

261

9 Entropy Production in Turbulent Flows


This chapter concludes the book by examining entropy and the Second Law for turbulent flows. It presents an overview of modeling and experimental methods for determining entropy production in turbulent flows. The turbulent entropy equation will be derived from the Reynolds averaged Clausius–Duhem equality (Hauke, 1995), which expresses entropy in terms of mean and fluctuating components in the Reynods averaging. A small thermal turbulence assumption (STTAss) will be used in the turbulence analysis (Kramer-Bevan, 1992). Under the STTAss, the fluctuating component of temperature is assumed small compared with the mean temperature, which allows the mean turbulent entropy production to be expressed in terms of viscous mean and turbulent fluctuating parts.

Experimental measurement of the turbulent dissipation rate can be obtained with different methods, such as a turbulent kinetic energy balance (Hussein and Martinuzzi, 1995), direct measurement of strain rate tensors (Andreopoulos and Honkan, 1996), turbulent energy spectrum, Taylor’s frozen turbulence hypothesis, dimensional analysis (Kresta and Wood, 1993), or a more recent large eddy par-ticle image velocimetry (PIV) method (Adeyinka and Naterer, 2007). A detailed review of past advances regarding the measurement of turbulence dissipation has been presented by Sheng et al. (2000). A major limitation of pointwise methods is the laborious measurement needed to acquire whole-field data. The whole-field method of PIV offers certain advantages over standard methods of anemometry for entropy-related experimental analysis. For pointwise methods, a direct evalu-ation of the dissipation rate from its definition would require resolution of the fluctuating strain rate tensor, which is possible only with multiple hot-wire probes in the flow field. In contrast, the PIV method provides a whole-field measurement technique, while allowing nonintrusive and time-varying measurements of instanta-neous velocity and temperature fields. Because the PIV technique provides whole-field data for velocity and temperature fields, it can lead to spatial measurements of turbulent entropy production throughout a flow field. Measured velocities by PIV are estimated over finite grids, so the turbulence statistics are influenced by the type of low-pass filter (FlowMap, 1998). For this reason, conventional dissipation rate approximations are limited when analyzing PIV data. The large eddy PIV method does not preclude the possibility of obtaining high resolution velocity measurements, where the detailed turbulent structures are captured (Liu et al., 1991). It can obtain the turbulent dissipation rate in whole-field regions, where the dynamic range of velocity measurements captured by PIV is limited by spatial resolution. This chapter will investigate both modeling and experimental methods for measuring turbulent

7262_C009.indd 261 1/19/08 12:02:35 PM


entropy production rates throughout an incompressible flow field. Results will be particularly examined for channel flow problems.

9.2 r eyn o l d s Av er Ag ed en t r o py tr An spo r t eq u At Io n s

Using the entropy transport equation from Section 3.2 and subdividing entropy into mean and fluctuating components, the following result can be obtained for the Reynolds averaged Clausius–Duhem equality (Jansen, 1993):

∂∂

+ ∂∂

+ − ∂∂

=

∂′ ′

ts

xu s u s

k

T

T

x

k

Tii i

i

( )ρ ρ ρ2

2TT

x T

u

xi

ij i

j∂+ ∂

∂t

(9.1)

where the overbar (i.e., s ) and prime (i.e., s ) notations refer to mean and fluctuating components associated with the Reynolds averaging, respectively. Because T and ui (and consequently the viscous dissipation term) have mean and fluctuating compo-nents in the denominator and numerator, it becomes difficult to explicitly express the mean entropy production in terms of other mean flow variables alone. Two main methods for evaluating the mean entropy production will be briefly addressed below.

In the first approach, the two sides of Equation 9.1 can be rearranged as follows:

si

i ii

Pt

sx

u s u s kx

ln TT

T = ∂∂

+ ∂∂

+ + ∂∂

+′ ′′

( ) ( )ρ ρ ρ 1

≥ 0 (9.2)

The first term can be simplified by substituting ∂ / ∂( )lnT xi for ( )∂ / ∂ /T x Ti before the time averaging. The time-averaged positive definite entropy equation becomes

si i i i

P kx

lnTx

lnT kx

lnTx

lnT = ∂∂

∂∂

+ ∂∂

∂∂′( ) ( ) ( ) ( ))′ +

∂∂+∂∂

∂∂

µ 1

Tux

ux

ux

i

j

j

i

i

j

+ ∂∂+∂∂

∂′ ′

µ 1T

u

x

u

x

ui

j

j

i

ii

j

i

j

i

xux T

u′ ′

∂

+ ∂

∂∂

21µ

′′ ′ ′

′

∂+ ∂∂∂∂

+∂∂

xux T

u

x

ux T

j

i

j

j

i

j

i

µ

µ

1

1 ∂∂+ ∂∂+∂∂

′ ′ ′ ′u

x T

u

x

u

xi

j

i

j

j

i

µ 1 ∂∂∂

≥

′u

xi

j

0

(9.3)

A close examination of Equation 9.3 reveals the physical processes leading to turbulent entropy production. The first two terms on the right side are entropy pro-duction terms that arise from thermal fluctuations and transport. The terms in the

7262_C009.indd 262 1/19/08 12:02:40 PM

Entropy Production in Turbulent Flows 263

first squared brackets represent the entropy production due to mean viscous effects. The terms in the second squared brackets represent entropy produced due to the dissipation of turbulent kinetic energy. The terms in the last squared brackets rep-resent the mechanism of entropy produced by the interaction of fluctuating viscous effects and temperature fluctuations. The remaining terms represent the conversion of entropy production, due to mean viscous effects, to irreversibilities of fluctuating viscous-temperature effects and back.

By defining the mean viscous stress and the fluctuating viscous stress, respec-tively, as

iji

j

j

i

ux

ux

t µ= ∂∂+∂∂

(9.4)

t µiji

j

j

i

u

x

u

x′

′ ′

= ∂∂+∂∂

(9.5)

then Equation 9.3 becomes

si i i i

P kx

lnTx

lnT kx

lnTx

lnT = ∂∂

∂∂

+ ∂∂

∂∂′( ) ( ) ( ) ( ))′ ′

′+

∂∂+

∂∂

+

1 1T

ux T

u

xij

i

jij

i

j

i

t t

jji

j

i

jijT

u

xux T T

t t′ ′ ′

′′

∂∂+ ∂∂ +1 1 1

∂∂≥′

′t ij

i

j

u

x0

(9.6)

No models exist at the present time for complete correlations involving the (1/T ) terms. Any such correlations would be difficult to validate or measure with some degree of accuracy.

Modeling of the mean entropy generation can be simplified by the following approach, whereby the Clausius–Duhem equality is averaged. The left side is multi-plied by temperature to give

T Pk

T

T

x

u

xsi

iji

j

= ∂∂+ ∂∂

2

t (9.7)

Kramer-Bevan (1992) presented a derivation of the time-averaged form of Equation 9.7, with the following result:

T P T P kx

lnTT

xk

xlnT

Ts s

i i i

+ = ∂∂

∂∂+ ∂∂

∂∂

′ ′ ′′

( ) ( )xx

ux

u

xiij

i

jij

i

j

+ ∂∂+ ∂∂

′′

t t (9.8)

7262_C009.indd 263 1/19/08 12:02:44 PM


In Equation 9.8, the physical processes of conversion of entropy production, arising from mean viscous effects, to irreversibilities of fluctuating viscous and temperature effects, have been captured in the ′ ′T Ps

correlation. Other terms remain as previously described for Equation 9.3. This equation is more straightforward than Equation 9.2 and Equation 9.3, provided that suitable empirical models can be developed for ′ ′T Ps

and thermal gradient correlations. Kramer-Bevan (1992) proposed a closure approxi-mation for a subset of possible flow fields by using an STTAss. The following section will develop modeling for the ′ ′T Ps

correlation.To derive a general, combined equation for the mean entropy generation, time

averaging is performed to yield

T P T P Tt

sx

u s u s kxs s

ij

i

+ = ∂∂

+ ∂∂

+ + ∂∂

′ ′ ′ ′( ) ( )ρ ρ ρ lln TT

T

Tx

u si

i

1+

+ ∂∂

′

′ ′( )ρ ++ ∂∂

+ ∂∂

+ ∂∂

+′ ′ ′ ′ ′ ′Tx

u s Tx

u s kTx

ln Ti

ii

ii

( ) ( )ρ ρ 1TT

T

′

(9.9)

By comparing Equation 9.2 with Equation 9.9, it can be shown that

T P Tx

u s Tx

u s Txs

ii

ii

i

′ ′ ′ ′ ′ ′ ′= ∂∂

+ ∂∂

+ ∂∂

( ) ( ) (ρ ρ ρuu s kTx

ln TT

Tii

′ ′ ′′

+ ∂∂

+

) 1

(9.10)

Using the chain rule of calculus,

T P T us

xsT

u

xT s

uxs i

i

i

i

i

i

′ ′ ′ ′ ′′

′ ′= ∂∂+ ∂

∂+ ∂

∂+ ρ ρ ρ ρρ ρi

i ii

i

u Ts

xT

xu s

kTx

ln TT

T

′′

′ ′ ′

′′

∂∂+ ∂∂

+ ∂∂

+

( )

1

(9.11)

By assuming incompressibility, the mean and instantaneous velocity fields are sole-noidal, and Equation 9.11 reduces to

T P T us

xu T

s

xT

xu ss i

ii

i ii

′ ′ ′ ′ ′′

′ ′= ∂∂+ ∂

∂+ ∂∂

ρ ρ ρ( ′′ ′′

+ ∂∂

+

) kT

xln T

T

Ti

1 (9.12)

This equation provides the full expression for the ′ ′T Ps correlation. The following

section will consider modeling of individual terms in Equation 9.12.

7262_C009.indd 264 1/19/08 12:02:49 PM


9.3 ed d y vIsc o sIt y Mo d el s o f MeAn en t r o py pr o d u c t Io n

A few simplified models, based on the solution of the Reynolds-averaged Navier–Stokes (RANS) equations and an eddy viscosity for mean entropy generation, have been documented in the literature (Adeyinka and Naterer, 2004; Drost and White, 1991; Moore and Moore, 1983). The linear eddy viscosity model assumes a Bouss-inesq relationship between the turbulent stresses (or second moments) and the mean strain rate tensor, through an isotropic eddy viscosity. These models attempt to reduce complexity, but it is difficult to ascertain whether the essence of relevant irreversibilities has been captured with sufficient accuracy, due to the lack of experi-mental data. Moore and Moore (1983) suggested that following correlations for mean entropy production, thermal diffusion, and viscous dissipation, respectively:

T Pk

T

T

x

T

xsi i

i = ∂∂+ ∂∂

+

′2 2

jji

jij

i

j

ux

u

xt t∂∂+ ∂∂

′′

(9.13)

kT

xk

T

xit

i

2 2∂∂

= ∂

∂

′

(9.14)

t µµtij

i

j

tij

i

j

u

xux

′′∂

∂= ∂

∂

(9.15)

In Equation 9.14 and Equation 9.15, kt and m t denote the turbulent molecular conduc-tivity and the turbulent molecular viscosity, respectively. This model misses most of the correlation in Equation 9.8, due to the assumption that the temperature fluctua-tions are small compared with the mean temperature.

Inconsistencies with this formulation occur close to the wall, so Kramer-Bevan (1992) proposed a different model for the viscous dissipation correlation,

t εiji

j

u

x′

′∂∂=

(9.16)

where ε is the “true” dissipation of turbulent kinetic energy. The definition of ε dif-fers from the dissipation of turbulent kinetic energy in the standard k - e model. The resulting model of entropy production becomes

T Pk k

T

T

xuxs

t

iij

i

j

= + ∂∂+ ∂

∂+

2

t ε

(9.17)

In contrast to Moore’s model, which uses the positive definite entropy equa-tion, the STTAss is based on time averaging of the entropy transport equation. It assumes that the fluctuating component of temperature is small compared with the mean temperature. When formulating this model, the fluctuating temperature is replaced by a Taylor series expansion of those functions. The expansions are

7262_C009.indd 265 1/19/08 12:02:54 PM


truncated after the linear terms, thereby yielding the following equations for mean entropy production and mean specific entropy:

s

ii

v t

t iP

ts

xu s

T

c

Prk

T

x = ∂∂

+ ∂∂

− +

∂∂

( )ρ ρ µ1

≥ 0

(9.18)

s s c ln TT

Rlnr vs

r

s

r

= + − ρρ

(9.19)

The turbulent Prandtl number, Prt, arises in Equation 9.18 because the entropy-velocity correlation has been modeled with a Reynolds analogy. Under the STTAss, extra terms arise in the entropy transport equation, with an increase of the diffusion term. This is equivalent to adding an effective diffusivity, cv m t / Prt, to the thermal diffusivity in the laminar model.

9.4 t u r bu l en c eMo d el In g w It h t h esec o n d lAw

The exact equation for the dissipation of turbulent kinetic energy (TKE) is useful to understand the meaning and importance of various terms, but usually it cannot be rigorously modeled in its full detailed form (Hanjalic and Jakirilic, 2002). Modeling of the exact equation is traditionally carried out by drastic simplification, and it usually involves a laborious empirical approximation of five or more closure coefficients. This section attempts to obtain the dissipation of TKE using the Second Law under the STTAss. In this approach, the local entropy production in convection-dominated flow can be found based on mean quantities (velocity and temperature) obtained from the solution of the RANS equations, using both the transport and positive defi-nite forms of the entropy equation. Because the dissipation of TKE appears in the positive definite mean entropy production equation, its local value can be computed throughout the flow domain by the Clausius–Duhem equation. A formulation for the proposed model will be presented for the eddy viscosity and second moment turbulent closure.

Combining Equation 9.2, Equation 9.8, and Equation 9.12, we obtain the following combined entropy equation for turbulent flow:

T P Tt

sx

u s u s kx

ln TT

si

ii

= ∂∂

+ ∂∂

+ + ∂∂

+′ ′′

( ) ( )ρ ρ ρ 1TT

kx

lnTT

xk

xi i

= ∂∂

∂∂+ ∂∂

( )ii i

iji

jij

i

j

i

lnTT

xux

u

x

T u

( )′′

′′

′ ′

∂∂+ ∂

∂+ ∂∂

−

t t

ρ ∂∂∂− ∂

∂− ∂∂

− ∂∂

′′

′ ′ ′ ′s

xu T

s

xT

xu s kT

xln

ii

i ii

i

ρ ρ( ) TTT

T1+

′

(9.20)

The fourth term on the right side of Equation 9.20 represents the dissipation of TKE. This term, called e, can be interpreted as a physical mechanism by which exergy (TPs

)

7262_C009.indd 266 1/19/08 12:02:57 PM


is destroyed in turbulent flow. This interpretation agrees with the traditional interpre-tation that associates e with the rate at which TKE is converted to internal energy in the flow. The terms after the second equality in Equation 9.20 reveal the physical pro-cesses leading to exergy destruction in turbulent flow. The total exergy destroyed in turbulent flow is the sum of exergy destroyed due to irreversible heat transfer (terms 1, 2, and 8), viscous dissipation (terms 3 and 4), turbulent enthalpy transfer (term 5), and the work done by fluctuating temperatures against turbulent entropy transfer by mass exchange (terms 6 and 7). All of these irreversible processes dissipate mechani-cal energy to internal energy.

Equation 9.20 indicates the importance of maintaining positivity of e in the numer-ical simulations. The time-averaged entropy equation does not shed much light with regard to modeling of e, except when simplified by the STTAss. Complete modeling of the Clausius–Duhem equation can only be achieved through experiments to cali-brate closure coefficients, when approximating the nonlinear fluctuating terms. Two following approaches (linear eddy viscosity and differential second moment closures [DSM]) will be described for modeling and simplification of Equation 9.20.

The terms in the time-averaged entropy equation, Equation 9.20, can be deter-mined from a linear eddy viscosity model as follows:

∂∂

+ ∂∂

− +

∂∂

t

sx

u sT

c

Prk

T

xii

v t

t i

( )ρ ρ µ1

= + ∂

∂

+ ∂1

2

2

Tk k

T

x Tti

ij i( )γ t uux

T

k

T

T

x

k

x T

j

i i

∂

+ + ∂∂

+ ∂∂∂∂

′ε2

2

21

xxT

c

T x

u T

Tu

xi

v

i

ii( )′

′ ′− ∂

∂

+ ∂∂

22ρ

ii

v

ii

T

T

c

T xu T

′

′ ′

+ ∂∂

2

22

2ρ ++ ∂

∂′

ii

ux

T 2

(9.21)

On the left side of Equation 9.21, the terms represent the transient change of mean entropy (first term) and the transport of entropy by mass and heat flow (sec-ond term in square brackets). On the right side of Equation 9.21, the terms refer to entropy production associated with thermal molecular and turbulent diffusion of the mean temperature field (first term in square brackets), viscous dissipation of the mean velocity field (second term), and irreversibilities through dissipation of TKE (third term). Within the braces, the terms represent entropy production correspond-ing to irreversible temperature fluctuations (first and second terms) and irreversible interactions between fluctuating velocity and temperature fields (remaining terms). The individual terms in braces can be obtained through the following correlation governing the dynamics of T′ 2 (Tennekes and Lumley, 1972),

i

i ii

i

ux

T

xu T

x

T∂∂

= ∂∂

− ∂∂

′

′ ′′2

22

212 2

α

−

∂∂− ∂

∂

′ ′′

u TT

x

T

xii i

α2

(9.22)

where a is the thermal diffusivity.

7262_C009.indd 267 1/19/08 12:02:59 PM


The differential second moment closure (DSMC) directly solves the transport equa-tions for the Reynolds stresses in the momentum equation. This approach is used to obtain the scalar fluxes in turbulent flow, involving the transport of passive scalars. The com-puted turbulent heat flux can then be used directly in Equation 9.20 to give

∂∂

+ ∂∂

+ − ∂∂

=′ ′

ts

xu s

c

Tu T

k

T

T

x

k

ii

vi

i

( )ρ ρ ρ2

TT

T

xT u

s

x Tux

T

k

ii

i

ij i

j

2∂∂+ ∂

∂+ ∂

∂

+ +

′ ′ρ t

ε22

2

2

21

T

T

x

k

x T xT

i i i

∂∂

+ ∂∂∂∂

′′

( )

′ ′ ′− ∂

∂

∂∂

ρc

T x

u T

Tu

x

T

Tv

i

ii

i

2 2

+ ∂∂

+ ∂∂

′ ′ ′

ρc

T xu T u

xTv

ii i

i2 22 2( )

(9.23)

This approach dispenses with the eddy viscosity to express the turbulent shear stress in terms of mean flow quantities.

Similarities in turbulent irreversibilities can be observed in Equation 9.21 and Equation 9.23. From left to right in Equation 9.23, the terms represent the tran-sient change of mean entropy (first term) and the transport of entropy by mass and heat flow (second term in square brackets). Unlike Equation 9.21, the heat flow is not modeled with a turbulent conductivity in this case. On the right side of Equation 9.23, the terms refer to entropy production corresponding to thermal molec-ular diffusion of the mean temperature field (first term), diffusive entropy transport in the mean flow due to velocity fluctuations (second term), viscous dissipation of the mean velocity field (third term), and dissipation of TKE (fourth term). In a similar way as previously described, the terms within braces represent entropy production corresponding to irreversible temperature fluctuations (first and second terms) and irreversible interactions between fluctuating velocity and temperature fields (remain-ing terms).

9.5 MeAsu r eMen t o f tu r bu l en t en t r o py pr o d u c t Io n

9.5.1 Fo r mu l at io n o FDissipat io n rat e

Unlike near-isothermal laminar flows (such as unheated pipe flows) where the only physical process that produces entropy is the mean viscous dissipation, the rate of dissipation of TKE is needed to compute entropy production in turbulent flows. A segment for extracting mean and turbulent quantities from velocity data is needed to measure the turbulent entropy production rates throughout a flow field. This section investigates modeling of the turbulent dissipation rate, for purposes of finding the turbulent entropy production rates. The effect of mean and fluctuating quantities on the total mechanical energy of a turbulent flow can be separated by the Reynolds averaging procedure. By subtracting the balance equation for the kinetic energy of

7262_C009.indd 268 1/19/08 12:03:01 PM


the mean motion, the following expression for the balance of kinetic energy of tur-bulence is obtained,

D

Dt

k

xu

p ku u

uxi

i i jj

i

2 2

2 2= − ∂∂

+

−

∂∂+ ∂′ ′ ′

ρν∂∂

∂∂+∂∂

−∂∂

∂∂

′′ ′ ′ ′

xu

u

x

u

x

u

x

u

xij

i

j

j

i

j

i

iνjj

j

i

u

x+∂∂

′

(9.24)

Equation 9.24 requires that the net convection of TKE (term 1) balances the flow work or work done by the total dynamic pressure (term 2), net work of turbulent stresses (term 3 minus term 4), minus the dissipation of TKE (last term). In the absence of periodic oscillation in the flow, the total dissipation in turbulent flows is a sum of mean (viscous shear stress) and random (dissipation of TKE) parts. The viscous shear stress performs deformation work, which increases the internal energy of the fluid at the expense of TKE. Because turbulence consists of a continuous spectrum of scales ranging from more energetic large scales to dissipative small scales, a continuous sup-ply of energy from the large scales or “eddies” is required to maintain turbulence. Otherwise, turbulence decays rapidly, and loss analysis of the fluid system reduces to an analysis involving only the mean viscous dissipation, as in laminar flows.

By expansion, the 12-term dissipation of TKE tensor, e, in Equation 9.24 can be expressed as

ε ν= ∂∂∂∂+∂∂∂∂

′ ′ ′ ′u

x

u

x

u

x

u

xi

j

j

i

j

i

j

i

(9.25)

Measurement of all terms in Equation 9.25 is difficult. A simplified form will be used based on the theory of homogenous turbulence and isotropy (Hinze, 1975). In homogeneous turbulence, the first term in Equation 9.25 vanishes due to incom-

pressibility, i.e., u u x xj i i j′ ′∂ / ∂ ∂ =2 0, resulting in the following 9-term tensor for e,

ε ν=∂∂∂∂

′ ′u

x

u

xj

i

j

i (9.26)

The essence of homogeneous turbulence is idealized, whereby that the mean properties of turbulence (including the mean velocity) are independent of transla-tions of the coordinate axis. However, it provides a reasonable basis for estimating experimental turbulence quantities (Batchelor, 1982). The assumption of homoge-neous turbulence also implies a relationship between the viscosity and mean square vorticity through Equation 9.26, so,

ε νω ω= k k (9.27)

where w k is the vorticity. Equation 9.27 is the entropy-based dissipation described by Tennekes and Lumley (1972).

7262_C009.indd 269 1/19/08 12:03:04 PM


Isotropic turbulence assumes that small turbulent scales are statistically inde-pendent of rotation and reflection of the coordinate axes at sufficiently high Reynolds numbers. A further simplification with the isotropic turbulence assumption can be obtained from Equation 9.26 in the following two-dimensional form:

ε ν= ∂∂

+ ∂∂

+ ∂∂∂′ ′ ′

62

1

1

2

1

2

1

2

u

x

u

x

u

x

u22

2

2

1

1

15′ ′

∂

= ∂

∂

x

u

xν

(9.28)

The Kolmogorov length scale represents the smallest length scale of turbulence, h = (v3/ e)1/4. Another length scale associated with the energy dissipated by turbulent eddies is the Taylor microscale, l, where,

λ 2 12

1 12

=∂ / ∂

′

′

u

u x( ) (9.29)

Rearranging Equation 9.28 in terms of the Taylor microscale leads to

ε ν

λ=

′15 1

2

2

u (9.30)

A similar dimensional analysis based on the integral length scale, l, and an assumption of mechanical equilibrium gives

ε =

′A

u

l13

(9.31)

where A is a proportionality constant of the order of unity. Equation 9.31 can be used to predict the dissipation rate, when only one integral length scale character-izes the flow region. It does not require the dissipation of TKE to be equal to the production of TKE, as its derivation is independent of the presence of turbulence production.

Another class of dissipation estimation methods (commonly used in laser dop-pler anemometry [LDA]) involves uses a time-series analysis and the turbulence energy spectrum. The following homogeneous turbulence relation applies:

ε ν= ,∫2

0

42k E k t dk( ) (9.32)

with a corresponding isotropic version given by

ε ν=

∞

∫150

12

1 1 1k E k dk( ) (9.33)

where E refers to the power spectrum, k is the wavenumber, and the subscripts “1” denote the one-dimensional values.

7262_C009.indd 270 1/19/08 12:03:09 PM


9.5.2 l a r g eeDDypa r t ic l eimag eVel o c imet r y

There are similarities between the correlation analysis of PIV and large eddy simu-lation (LES). LES computes the dynamics of the large energy-containing scales of motion, up to a certain cutoff wavelength, while modeling only the effect of the small, unresolved flow structures on the larger resolved scales. The underlying prin-ciple is that large-scale motions are affected by the geometry and not universal. The small-scale motions have a weaker influence on the Reynolds stress, and they have a universal character, which can be represented by simple subgrid scale (SGS) models. The approach in LES requires the solution of the Navier–Stokes equations for the fil-tered velocity field on a computational grid, with the objective of resolving the actual flow field with fewer discrete volumes. In the same way, the correlation techniques in PIV give velocities that are results of a spatial average over a discrete volume or interrogation area. In LES, the filter size is proportional to a cutoff wavelength in the inertial subrange of the turbulence energy spectrum. The size of the interroga-tion area determines the filter width, which averages the smaller scales of motion in PIV. Because the spatial filtering properties of PIV are similar to LES, the benefits of SGS modeling in LES will be helpful to determine the small-scale turbulence characteristics from PIV data.

With the filter in the inertial subrange, the turbulence dissipation rate in LES can be approximated by the following SGS dissipation rate:

ε ε t≈ = − , = ∂∂+ ∂∂

SGS ij ij ij

i

j

j

iS S

ux

ux

212

(9.34)

where ijS is the filtered rate of strain tensor and t ij is the SGS stress. Several SGS stress models have been used in previous LES studies at high Reynolds numbers. The first subgrid model to be widely used was reported by Smagorinsky (1963). Other models that were developed to improve the Smagorinsky model include the dynamic model of Germano et al. (1991; Lilly, 1992; Meneveau et al., 1996), Bardina scale similarity model (Bardina et al., 1980), Clark gradient model (Clark et al., 1979), structure function model of Métais et al. (1992), and the transport equation model (Mason, 1989; Sullivan et al., 1994). The next section will focus on the Smagorinsky and Gradient models, as well as compare the accuracy of different models.

The PIV technique permits the measurement of instantaneous velocity data in a whole-field manner, which allows direct calculation of the turbulence dissipation rate from spatial derivatives of velocity. However, the spatial range of PIV cannot usually be extended down to the required near-wall resolution for exact measure-ments, due to limitations imposed by the hardware, such as the size of recording media and the maximum allowable sampling speed (Adrian, 1997; Saarenrinne et al., 2001). Saarenrinne and Piirto (2000) proposed a restrictive requirement in PIV (depending on the flow), where the size of the PIV interrogation window and the laser light thickness do not exceed 30% of the lateral Taylor’s microscale and five times the Kolmogorov length scale, respectively.

7262_C009.indd 271 1/19/08 12:03:11 PM


Dimensional analysis based on equilibrium turbulence assumes local isotropy. It can be used to estimate the turbulence energy dissipation in stirred vessels (Kresta, 1998). The method uses Equation 9.31, and it yields accurate qualitative results despite implementation difficulties due to the variation of length scales in certain flows. In flows where the Taylor microscale can be estimated, dimensional analysis based on Equation 9.30 has been used successfully (Saareninne and Piirto, 2000). The dissipation rate in the turbulent kinetic equation can be determined from terms represented by the mean flow convection, diffusion and production of turbulent energy, and neglected terms of viscous diffusion. The applicability of this method is limited by an appropriate model for the pressure diffusion term, which is difficult to measure experimentally (Turan and Azad, 1989). Although other terms in the TKE equation involve large-scale quantities, the limitation imposed by spatial resolution has restricted the application of the method to simple geometries.

Another method for measuring turbulence dissipation uses Taylor’s frozen tur-bulence hypothesis, which allows Equation 9.28 to be rewritten in terms of a time series differential of the velocity fluctuation, that is,

ε ν ν= ∂ / ∂ = ∂ / ∂′ ′15 151 12

12 2( ) ( ) /u x u t u (9.35)

To obtain a reliable value of e, a calibration of the time derivative is necessary. It can be determined based on the energy spectrum function in Equation 9.32. Turan and Azad (1989) developed a “zero-wire-length dissipation method,” which defined the one-dimensional spectrum of the longitudinal velocity fluctuation by the following integral,

01 1 1

2

1

∞′∫ =E k dk u( ) (9.36)

But the sampling rate of a PIV system is often not high enough to allow this spectral analysis.

The large eddy PIV method is a promising method for the previous measure-ments. Sheng et al. (2000) established an appropriate resolution of time and length scales with this method. The authors developed a method to use full-field velocity data to estimate the dissipation rates. The large eddy PIV method is based on a dynamic equilibrium assumption, between the spatial scale that can be resolved by PIV and the subgrid length scales. When the interrogation or filter size is much smaller than the integral length scale of the flow, the turbulence dissipation rate can be approximated by Equation 9.34. In the following case study, the Clark Gradient model and the Sma-gorinsky model will be used for the SGS stress. For the Gradient model,

t ij

i

k

j

k

uu

uu

= D ∂∂∂∂

112

2 (9.37)

where D is the width of the interrogation area. The Smagorinsky model is given by

t ij s ijC S S= − D | |( )2 (9.38)

7262_C009.indd 272 1/19/08 12:03:14 PM


where | S | is the characteristic filtered rate of strain, 2 ij ijS S , and Cs is the Smagorin-sky coefficient (proportional to D), taken to be 0.07 (Adeyinka and Naterer, 2004).

The large eddy PIV method and other simplified models based on the isotropic assumption in this section do not preclude the possibility of obtaining high reso-lution measurements, where detailed turbulent structures are captured (1991). The method provides a useful estimate of turbulence dissipation in regions where the dynamic range of velocity measurements captured by PIV is limited by the spatial resolution.

9.5.3 c a sest u Dyo Ftu r bu l en t ch a n n el Fl o w

In this section, measured results in the previous formulations of mean turbulent entropy production will be compared against past DNS (direct numerical simula-tion) data. Large eddy PIV is used to determine the velocity, dissipation rate, and entropy production data (Adeyinka and Naterer, 2004). The DNS solution assumes negligible viscous dissipation in the energy equation. Therefore, attention is focused on the positive definite model involving the dissipation of TKE (right side of Equa-tion 9.23), because the entropy transport equation requires inclusion of the viscous dissipation in the energy equation for accurate modeling.

Turbulent flow between two parallel plates at four different Reynolds numbers, based on the friction velocity, will be examined. Computations of the friction fac-tor, f, at Ret = 180, 395, and 590, will be compared with DNS data of Moser et al. (1999). The data of Kuroda et al. (1989) were used to compute f at Ret = 100. The computed friction factors are compared in Table 9.1. The present results show excel-lent agreement with Darcy’s friction factor, computed from the Colebrook equation. The Colebrook equation is documented by White (1991). The results are illustrated at various Reynolds numbers, based on the bulk velocity in Figure 9.1. They confirm that the present turbulence modeling of entropy production (particularly in terms of e) has been accurately formulated.

A comparison with Moore’s model is presented (Figure 9.2), with regard to the spatial distribution of entropy production in the channel. The integral value of entropy production computed from Moore’s model in Equation 9.13 and Equation 9.15, based on the production of TKE, is within 1% of the currently formulated model. Figure 9.2 shows that Moore’s model underpredicts the entropy production close to the wall and overpredicts entropy production away from the wall, before it decreases to zero in the middle of the channel. The additional curve in Figure 9.2 shows that the viscous mean dissipation is the main component of entropy production

t Abl e9.1f rictionfactorsatdifferentRet

Ret

100 180 395 590

f (based on tw) 0.0383 0.0325 0.0260 0.0232

f (based on present modeling) 0.0388 0.0324 0.0252 0.0225

7262_C009.indd 273 1/19/08 12:03:15 PM


f Ig u r e9.1 Friction factor based on entropy production correlation.

f Ig u r e9.2 Local distribution of integrated entropy production in the channel.

0.30 Colebrook: Turbulent Smooth Ducts Present Model Colebrook: Laminar Flow Entropy-Based: Laminar 0.25

0.20

0.15

0.10

0.05

0.00 100 1000 10000 100000 1000000

ReD

Fric

tion

Fact

or (T

urbu

lent

)1.00

0.10

0.01

Fric

tion

Fact

or (L

amin

ar)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.00 0.10 0.20 0.30 0.40 y/h

Non

dim

ensio

nal E

ntro

py P

rodu

ctio

n

Moore Model Present Viscous Mean

7262_C009.indd 274 1/19/08 12:03:18 PM


near the wall, but other components become more significant at further distances away from the wall. In particular, the mean viscous dissipation accounts for more than 80% of the total entropy production at approximately y+ < 9, where y+ = yut /v. This percentage decreases to zero in the center of the channel.

In Figure 9.3, the mean velocity profile in the fully developed region is presented at Ret = 187, 295, and 395. The velocity profiles (shown in the inset) at different transverse locations collapse onto each another, due to fully developed conditions. The mean velocities are normalized by the centerline velocity in Figure 9.3, and the y-coordinate is normalized by the half-channel height. Figure 9.4 shows the distribution of mean velocity profiles in terms of wall variables. The wall shear stress is determined by the Clauser plot technique, which assumes a universal loga-rithmic profile in the overlap region. The experimental data confirm the extent of the logarithmic layer, as the Reynolds number increases. The mean profiles for Ret = 295 and 399 agree out to y+ ≈ 250. At Ret = 187, the standard constants (k = 0.4 and B = 5.0) give a logarithmic slope with a slight offset from a best fit (k = 0.4 and B = 5.5), in agreement with DNS data. These results are consistent with previous experimental measurements, which associate such flow behavior with low Reynolds number effects. The spatial resolution of PIV is limited by the size of the interrogation area, so measurements by Anteyinka and Naterer (2007) could not be made any closer to the wall than y+ = 8.18. The data compare well with DNS results, thereby providing useful validation of the formulation.

f Ig u r e9.3 Mean velocities.

0

0.4

0.8

1.2

0.0 0.4 0.8 1.2

U/U

c

Reτ = 295

Reτ = 395Reτ = 187

y/h

1.20

1.80

0.40

0.00 0 0.4 0.8

y/h 1.2

x/h = 160

x/h = 160.3

x/h = 160.8 U/U

c

7262_C009.indd 275 1/19/08 12:03:21 PM


The turbulent fluctuating velocities are normalized by the friction velocity and plotted in Figure 9.5 at three different cross sections. Figure 9.6 compares the dis-tributions of u+ and v+ obtained at Ret = 187 with the PIV results of Liu et al. (1991) and the DNS results of Kim et al. (1987). Good qualitative agreement is observed among the results. Compared with the DNS results, the peak value for the fluctuat-ing streamwise velocity is underpredicted for the present results at y+ = 13. The peak value shows close agreement with previous PIV results of Liu et al. (1991). The mea-sured data also shows higher values that the DNS results in the channel core.

The fluctuating velocities are plotted against y/h in Figure 9.7 at all Reynolds numbers. The u+ profiles collapse onto the Ret = 399 curve away from the wall, at approximately y/h > 0.36 for Ret = 187 and y/h > 0.2 for Ret = 295. All profiles vary nearly linearly in Figure 9.7 for u+ between 0.4 < y/h < 0.9 at the three Reynolds numbers tested and 0.2 < y/h < 0.9 for v+. The linear range for v+ at Ret = 187 is not immediately apparent. This observation is consistent with past studies of Moser et al. (1999) that suggested a collapse of the u+ profiles to a high Reynolds number outer-layer limit at y+ > 80. No such collapse is observed when the inner variables are used as the normalizing quantities. The qualitative trends of fluctuating velocities also compare well with the DNS data shown in the inset.

The Kolmogorov length scale, h, estimated from its definitions and DNS data, is between 6 and 18 mm at the highest Reynolds number and between 14 and 38 mm at Ret = 187. With a 32 × 32 PIV interrogation region, the spatial resolution of the PIV measurements is approximately 280 mm. Thus, the spatial resolution is about 16 times the Kolmogorov length scale at the channel core and 48 times close to the wall. The resolution of the velocity field is too small to accurately determine spatial

f Ig u r e9.4 Velocities normalized by inner variables.

0

5

10

15

20

25

30

1 10 100 1000 y+

u+

Reτ = 187 (PIV)

Reτ = 295 (PIV)

Reτ = 395 (PIV)

Reτ = 395 (DNS)

Log Law

u+ = y+

7262_C009.indd 276 1/19/08 12:03:23 PM


0.5

1

1.5

2

2.5

0.0 0.40 0.80 1.20y/h

u+

x/h = 160

x/h = 160.3

x/h = 160.8

f Ig u r e9.5 Turbulent velocities at Ret = 395.

f Ig u r e9.6 Turbulent velocities at Ret = 187.

Reτ = 187 (PIV)Reτ = 180 (DNS)Reτ = 183 (Lui et al.)

4

3.2

2.4

1.6

0.8

0

u+ , v+

0 0.4 0.8 1.2y/h

7262_C009.indd 277 1/19/08 12:03:27 PM


derivatives of the fluctuating velocity field and dissipation rate with the total dissipa-tion method. Nevertheless, simplified expressions for e and isotropic conditions can be used to estimate e. The dissipation rate is estimated by the dimensional analysis relation, Equation 9.31, and the large eddy PIV approach. The SGS stress is obtained from the Smagorinsky and Gradient models. The accuracy of the estimation meth-ods can be verified by comparisons with the DNS data of Moser et al. (1999).

The measured dissipation rates are compared with the corresponding DNS solu-tion at Ret = 187 in Figure 9.8. The dissipation rate in Figure 9.8 and all subsequent figures is normalized by ut ν4/ . The different methods show close agreement with DNS data, and they give correct distributions of the TKE in the channel. A high dis-sipation region is concentrated near the wall. The DNS data show an inflection point, not captured by PIV, closer to the wall at y+ = 12. The dissipation rate reaches a mini-mum in the center of the channel, and it becomes the only mechanism for energy loss in the channel centerline for turbulent flows. Greater deviations from the DNS data are noticed for all estimation methods closer to the wall, due to the anisotropic nature of the flow and smaller dissipation length scales in this region.

Dissipation rates computed from the DNS results of Kuroda et al. (1989) at Ret = 100 and Moser et al. (1999) at Ret = 180, 395, and 590 are plotted in Figure 9.9. In Figure 9.10, the dissipation rate has been estimated from the dimensional analysis relation at all Reynolds numbers investigated. The integral length scale, l, is defined as the distance from the wall to a point where the streamwise velocity is 99% of

f Ig u r e9.7 Turbulent velocities plotted in outer variables.

3.2

2.4

1.6

0.8

0 0 0.4 0.8 1.2 1.6

U+ , V

+

U+ , V

+

y/h

Reτ = 187Reτ = 295

Reτ = 399

Reτ = 180Reτ = 395

3.2

2.4

1.6

0.8

0 0 0.4 0.8 1.2

y/h

7262_C009.indd 278 1/19/08 12:03:31 PM


f Ig u r e9.8 Dissipation of turbulent kinetic energy at Ret = 187.

f Ig u r e9.9 Direct numerical simulation results.

0

0.05

0.1

0.15

0.0 0.4 0.8 1.2

ε+

Dimensional Analysis

Smagorinsky Model

Gradient Model

Reτ = 180 (DNS)

y/h

0

0.05

0.1

0.15

0.2

0.25

0.0 0.4 0.8 1.2 y/h

ε+

Reτ = 100Reτ = 180Reτ = 395Reτ = 590

7262_C009.indd 279 1/19/08 12:03:34 PM


the centerline velocity. The DNS results suggest lower values of e in the middle region, with higher values at the wall and steeper gradients as the Reynolds number increases. The dissipation rate shows similar trends in the wall layer in Figure 9.10.

The large eddy PIV dissipation estimates are shown in Figure 9.11 (Smagorinsky model) and Figure 9.12 (Gradient model). The filter size for the correlation analysis is ∆ = 280 mm, whereas the integral length scale is l ≈ 8 mm at Ret = 180. The Kolmogorov length scale is h = 18 mm at the channel centerline. Thus, ∆ << l, and the filter size is sufficiently larger than the Kolmogorov length scale to warrant the use of large eddy PIV. The dissipation rate closely agrees with the DNS result at low Reynolds numbers, but it is underpredicted at higher Reynolds numbers.

One would expect better performance with increasing Reynolds numbers because the flow shows higher tendencies toward local isotropy as the Reynolds number increases. This discrepancy is partly due to the overall accuracy of PIV measure-ments at high Reynolds numbers. At high Reynolds numbers, the PIV dynamic range required to accurately resolve the smaller Kolmogorov length scales becomes very high, and the spatial resolution of the PIV fails to capture certain aspects of the flow structure. Better performance of all estimation methods at Ret = 187 can be attributed to larger Kolmogorov length scales and a consequent higher flow resolution. Measure-ments too close to the wall suffer from reflections and poor accuracy of the velocity at high Reynolds numbers. The two SGS models show similar predictions, suggesting a weak dependence of the large eddy PIV method on the SGS stress model.

f Ig u r e9.10 Dimensional analysis-based e estimation.

Reτ = 187

Reτ = 295

Reτ = 399

0.25

0.2

0.15

0.1

0.05

0

ε+

0 0.5 1 1.5y/h

7262_C009.indd 280 1/19/08 12:03:37 PM


f Ig u r e9.11 Large eddy PIV-based e estimation with Smagorinsky SGS model.

Reτ = 187

Reτ = 295

Reτ = 399

0.25

0.2

0.15

0.1

0.05

0 0 0.5 1 1.5

y/h

ε+

Reτ = 187

Reτ = 295

Reτ = 399

0.25

0.2

0.15

0.1

0.05

0 0 0.5 1 1.5

y/h

ε+

f Ig u r e9.12 Large eddy PIV-based e estimation with Gradient SGS model.

7262_C009.indd 281 1/19/08 12:03:43 PM


Measured oscillations are effectively reduced by filtering of the velocity data. In Figure 9.13, a 3 × 3 average filter is used for smoothing of the velocity vectors, before calculating the viscous dissipation. A comparison of the experimental data with DNS results is presented in Figure 9.14, with regard to the spatial distribution of the mean viscous dissipation and the total entropy production at Ret = 187. All three estimation methods agree closely. The entropy production is scaled by ρ νtu T4 / . A region of high entropy production is evident close to the wall.

The distribution of TKE dissipation is plotted as a percentage of the total mechanical energy loss in Figure 9.15, to provide a Second Law insight into the energy conversion in turbulent flows. The measured results show close agreement with DNS results. The percentage of TKE decreases from a maximum at the chan-nel centerline to approximately 14% of the total mechanical energy loss, just outside the logarithmic region, toward the wall. The percentage of e+ is fairly constant in the outer region and logarithmic layers. This distribution implies that the viscous stress due to molecular viscosity dominates at the wall. Unlike laminar flow, where the viscous shear stress increases linearly across the fluid layer from the channel center to the wall and entropy production is distributed evenly over the entire channel, the viscous stress is concentrated in a region between the buffer layer and the wall in turbulent flows, thereby leading to a much higher mean shear stress and entropy production at the wall. This explains the need for higher pumping power in turbulent flows to move fluid through a duct at a given mass flow rate.

f Ig u r e9.13 Viscous dissipation at Ret = 187.

Reτ = 180 (DNS)Reτ = 187 (PIV)

0.8

0.4

0 0 0.5 1 1.5

y/h

Φ+

7262_C009.indd 282 1/19/08 12:03:47 PM


f Ig u r e9.14 Turbulent entropy production at Ret = 187.

Reτ = 178.12 (DNS)

Dimensional Analysis

Gradient Model

Smagorinsky Model

Viscous Mean (DNS)

1.50

1.00

0.50

0.00

Ps+

0 0.5y/h

1

f Ig u r e9.15 Percentage of e+ in total entropy production across the channel at Ret ≈ 187.

120

100

80

60

40

20

0

Perc

enta

ge ε+ /P

s+

0 0.5 1 1.5y/h

PIV

Reτ = 100 (DNS)

Reτ = 180 (DNS)

7262_C009.indd 283 1/19/08 12:03:51 PM


r ef er en c es

Adeyinka, O.B. and Naterer, G.F. 2007. Measured turbulent entropy production with large eddy particle image velocimetry. Exper. in Fluids, 42(6): 881–891.

Adeyinka, O.B. and Naterer, G.F. 2004. Modeling of entropy production in turbulent flows. ASME J. Fluids Eng., 126(6): 893–899.

Adrian, R.J. 1997. Dynamic ranges of velocity and spatial resolution of particle image velo-cimetry. Measurement Sci. Technol., 8: 1393–1398.

Andreopoulos, Y. and A. Honkan. 1996. Experimental techniques for highly resolved measure-ments of rotation, strain, and dissipation rate tensors in turbulent flows. Measurement Sci. Technol., 7: 1462–1476.

Bardina, J., Ferziger, J.H., and W.C. Reynolds. 1980. Improved Subgrid-Scale Models for Large-Eddy Simulation, AIAA Paper 80–1357, AIAA 13th Fluid and Plasma Dynamics Conference, Snowmass, CO.

Batchelor, G.K. 1982. The Theory of Homogeneous Turbulence. Cambridge University Press, London.

Clark, R.A., Ferziger, J.H., and W.C. Reynolds. 1979. Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid Mechanics, 91: 1–16.

Drost, M.K. and M.D. White. 1991. Numerical predictions of local entropy generation in an impinging jet. ASME J. Heat Transfer, 113: 823–829.

FlowMap Particle Image Velocimetry Instrumentation: Installation and User Guide. 1998. Dantec Dynamics, Demark.

Germano, M., Piomelli, U., Moin, P., and W.H. Cabot. 1991. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A, 3: 1760–1765.

Hanjalic, K. and S. Jakirilic. 2002. Second-moment turbulence closure modelling. In: Closure Strategies for Turbulent and Transitional Flows Part A. B. Launder and N. Sandham, Eds., Cambridge University Press, London.

Hauke, G.A. 1995. A Unified Approach to Compressible and Incompressible Flows and a New Entropy-Consistent Formulation for the K-Epsilon Model. Ph.D. thesis. Stan-ford University, Stanford, CA.

Hinze, J.O. 1975. Turbulence. 2nd ed. McGraw-Hill, New York.Hussein, H.J. and R.J. Martinuzzi. 1995. Energy balance for turbulence flow around a surface

mounted cube place in a channel. Phys. Fluids, 8: 764–780.Jansen, K.E. 1993. The Role of Entropy in Turbulence and Stabilized Finite Element Meth-

ods. Ph.D. thesis. Stanford University, Stanford, CA.Kim, J., Moin, P., and R. Moser. 1987. Turbulence statistics in fully developed channel flow

at low Reynolds number. J. Fluid Mechanics, 177: 133–166.Kramer-Bevan, J.S. 1992. A Tool for Analysing Fluid Flow Losses. M.Sc. thesis. University

of Waterloo, Waterloo, Ontario, Canada.Kresta, S. 1998. Turbulence in stirred tanks: anisotropic, approximate, and applied. Can. J.

Chem. Eng., 76(3): 563–575.Kresta, S.M. and P.E. Wood. 1993. The flow field produced by a pitched blade turbine: char-

acterization of the turbulence and estimation of the dissipation rate. Chem. Eng. Sci., 48: 1761–1774.

Kuroda, A., Kasagi, N., and M. Hirata. 1989 (August 28–29). A Direct Numerical Simulation of the Fully Developed Turbulent Channel Flow. International Symposium on Com-putational Fluid Dynamics, Nagoya, Japan, 1174–1179.

Lilly, D.K. 1992. A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids A, 4: 633–635.

Liu, Z.C., Landreth, C.C., Adrain, R.J., and T.J. Hanratty. 1991. High resolution measurement of turbulent structure in a channel with particle image velocimetry. Exp. Fluids, 10: 301–312.

7262_C009.indd 284 1/19/08 12:03:52 PM


Mason, P. 1989. Large-eddy simulation of the convective atmospheric boundary layer. J. Atmospheric Sci., 46: 1492–1516.

Meneveau, C., Lund, T., and W.H. Cabot. 1996. A Lagrangian dynamic subgrid-scale model of turbulence. J. Fluid Mech., 319: 353–385.

Métais, O. and M. Lesieur. 1992. Spectral large-eddy simulation of isotropic and stably strati-fied turbulence. J. Fluid Mech., 239: 157–194.

Moore, J. and J.G. Moore. 1983 (March 27–31). Entropy Production Rates from Viscous Flow Calculations Part I. A Turbulent Boundary Layer Flow. ASME Gas Turbine Confer-ence. ASME Paper 83-GT-70. Phoenix, AZ.

Moser, R.D., Kim, J., and N.N. Mansour. 1999. Direct numerical simulation of turbulent flow up to Ret = 590. Phys. Fluids, 11(4): 943–945.

Saarenrinne, P. and M. Piirto. 2000. Turbulent kinetic energy dissipation rate estimation from PIV velocity vector fields. Exp. Fluids, 29(7): S300–S307.

Saarenrinne, P., Piirto, M., and H. Eloranta. 2001. Experiences of turbulence measurement with PIV. Measurement Sci. Technol., 12: 1904–1910.

Sheng, J., Meng, H., and Fox, R.O. 2000. A large eddy PIV method for turbulence dissipation rate estimation. Chem. Eng. Sci., 55: 4423–4434.

Smagorinsky, J. 1963. General circulation experiments with the primitive equations. I. The basic equations. Monthly Weather Rev., 91: 99–164.

Sullivan, P.P., McWilliams, J., and C.-H. Moeng. 1994. A subgrid-scale model for large-eddy simulation of planetary boundary-layer flows. Boundary Layer Meteorol., 71: 247–276.

Tennekes, H. and J.L. Lumley. 1972. A First Course in Turbulence. MIT Press, Cambridge, MA.Turan, O. and R. Azad. 1989. Effect of hot-wire probe defects on a new method of evaluating

turbulence dissipation. J. Phys. E: Scientific Instruments, 22(4): 254–261.White, F.M. 1991. Fluid Mechanics, 5th ed. McGraw-Hill, New York.

7262_C009.indd 285 1/19/08 12:03:53 PM

287

Appendix

Ta bl e a .1Conversion of Units and Constants

Conversion Factors

Acceleration 1 m/s2 = 4.252 × 107 ft/hr2 = 3.2808 ft/s2

Area 1 m2 = 1550.0 in2 = 10.764 ft2

Density 1 kg/m3 = 0.06243 lbm/ft3

Dynamic viscosity 1 kg/m ⋅ s = 1N ⋅ s/m2 = 2419.1 lbm/ft ⋅ hrEnergy 1 kJ = 0.9478 Btu =737.56 ft ⋅ lbfForce 1 N = 1 kg ⋅ m/s2 = 0.22481 lbfHeat flux 1 W/m2 = 1 kg/s3 = 0.3171 Btu/hr ⋅ ft2

Heat transfer coefficient 1 W/m2 K = 0.1761 Btu/ft2 hr °FHeat transfer rate 1 W = 3.4123 Btu/hr = 1.341 × 10–3 hpKinematic viscosity 1 m2/s = 10.7636 ft2/s = 104 stokesLatent heat 1 kJ/kg = 0.4299 Btu/lbmLength 1 m = 39.37 in = 3.2808 ftMass 1 kg = 2.2046 lbm = 1.1023 × 10–3 US tonsMass diffusivity 1 m2/ s = 10.7636 ft2/s = 3.875 × 104 ft2/hrMass flow rate 1 kg/s = 7936.6 lbm/hrMass transfer coefficient 1 m/s = 1.181 × 104 ft/hrPressure, stress 1 Pa = 1 N/m2 = 1.4504 × 10–4 lbf/in2

Specific heat 1 kJ/kgK = 0.2388 Btu/lbm ⋅ °F = 0.2389 cal/g ⋅ °CTemperature K = °C + 273.15 = (5/9)(°F + 459.67)

R = °F + 459.67 = (9/5) (°K)Temperature difference 1 K = 1°C = (9/5) °FThermal conductivity 1 W/mK = 0.57782 Btu/hr ⋅ ft ⋅ °FThermal diffusivity 1 m2/s = 10.7636 ft2/s = 3.875 × 104 ft2/hrThermal resistance 1 K/W = 0.5275° F/hr ⋅ BtuVelocity 1 m/s = 3.2808 ft/s = 3.6 km/hrVolume 1 m3 = 264.17 gal (U.S.) = 1000 LVolume flow rate 1 m3/s = 1.585 × 104 gal/min = 2118.9 ft3/min

SI Unit Conversions

Prefix (Symbol) Multiplier

Tera (T) 1012

Giga (G) 109

Mega (M) 106

kilo (k) 103

milli (m) 10–3

micro (m) 10–6

nano (n) 10–9

pico (p) 10–12

femto (f) 10–15

7262_A001.indd 287 1/19/08 12:06:52 PM


ConstantsAtmospheric pressure (Patm) = 101,325 N/m2 = 14.69 lbf /in2

e = 2.7182818Gravitational acceleration (g) = 9.807 m/s2

1 mole = 6.022 × 1023 molecules = 10–3 kmol

p = 3.1415927Universal gas constant (R) = 8.315 kJ/kmol ⋅ K = 1.9872 Btu/lbmol ⋅ R

Ta bl e a .2Properties of Metals at STPa

Metal

Melting Point (°C)

boiling Point (°C)

Thermal Conductivity

(W/mK)

Specific Heat, cp

(kJ/kgK)

Coefficient of expansion (×106/K)

Density, r

(kg/m3)

Heat of Fusion (kJ/kg)

Aluminum 660 2441 237.0 0.900 25 2700 397.8Antimony 630 1440 18.5 0.209 9Beryllium 1285 2475 218 1.825 12Bismuth 271.4 1660 8.4 0.126 13Cadmium 321 767 93 0.230 30Chromium 1860 2670 91 0.460 6 7150 330.8Cobalt 1495 2925 69 0.419 12 8860 276.4Copper 1084 2575 398 0.385 16.6 8960 205.2Gold 1063 2800 315 0.130 14.2 19300 62.8Iridium 2450 4390 147 0.130 6Iron 1536 2870 80.3 0.452 12 7870 272.2Lead 327.5 1750 34.6 0.130 29 11300 23.0Magnesium 650 1090 159 1.017 25Manganese 1244 2060 7.8 0.477 22Mercury -38.86 356.55 8.39 0.138Molybdenum 2620 4651 140 0.251 5 10200 288.9Nickel 1453 2800 89.9 0.444 13 8900 297.3Niobium 2470 4740 52 0.268 7Osmium 3025 4225 61 0.130 5Platinum 1770 3825 73 0.134 9Plutonium 640 3230 8 0.134 54Potassium 63.3 760 99 0.753 83Rhodium 1965 3700 150 0.243 8Selenium 217 700 0.5 0.322 37Silicon 1411 3280 83.5 0.712 3Silver 961 2212 427 0.239 19 10500 111.0Sodium 97.83 884 134 1.226 70Tantalum 2980 5365 54 0.142 6.5Thorium 1750 4800 41 0.126 12Tin 232 2600 64 0.226 20 7280 59.0Titanium 1670 3290 20 0.523 8.5 4500 418.8Tungsten 3400 5550 178 0.134 4.5Uranium 1132 4140 25 0.117 13.4Vanadium 1900 3400 60 0.486 8Zinc 419.5 910 115 0.389 35

a Data reprinted with permission from Hewitt et al. (1997) and Weast (1970).

7262_A001.indd 288 1/19/08 12:06:53 PM

Appendix 289

Ta bl e a .3Properties of Nonmetalsa

Material Density,

r (kg/m3)

Thermal Conductivity,

k (W/mK)

Specific Heat, cp

(kJ/kgK)

Asbestos millboard 1400 0.14 0.837Asphalt 1100 1.67Brick, common 1750 0.71 0.920Brick, hard 2000 1.3 1.00Chalk 2000 0.84 0.900Charcoal, wood 400 0.088 1.00Coal, anthracite 1500 0.26 1.26Concrete, light 1400 0.42 0.962Concrete, stone 2200 1.7 0.753Corkboard 200 0.04 1.88Earth, dry 1400 1.5 1.26Fiber hardboard 1100 0.2 2.09Fiberboard, light 240 0.058 2.51Firebrick 2100 1.4 1.05Glass, window 2500 0.96 0.837Gypsum board 800 0.17 1.09Ice (0°C) 900 2.2 2.09Leather, dry 900 0.2 1.502Limestone 2500 1.9 0.908Marble 2600 2.6 0.879Mica 2700 0.71 0.502Mineral wool blanket 100 0.04 0.837Paper 900 0.1 1.38Paraffin wax 900 0.2 2.89Plaster, light 700 0.2 1.00Plaster, sand 1800 0.71 0.920Plastics, foamed 200 0.03 1.26Plastics, solid 1200 0.19 1.67Porcelain 2500 1.5 0.920Sandstone 2300 1.7 0.920Sawdust 150 0.08 0.879Silica aerogel 110 0.02 0.837Vermiculite 130 0.058 0.837Wood, balsa 160 0.050 2.93Wood, oak 700 0.17 2.09Wood, white pine 500 0.12 2.51

a Data reprinted with permission from Weast (1970).

7262_A001.indd 289 1/19/08 12:06:54 PM


Ta bl e a .4Properties of a ir at a tmospheric Pressurea

Temperature, T (K)

Density, r (kg/m3)

Specific Heat, cp

(kJ/kgK)Viscosity, r (kg/ms)


k (W/mK) Pr

150 2.367 1.010 10.28 × 10–6 0.014 0.758200 1.769 1.006 13.28 × 10–6 0.018 0.739250 1.413 1.005 15.99 × 10–6 0.022 0.722260 1.359 1.005 16.50 × 10–6 0.023 0.719270 1.308 1.006 17.00 × 10–6 0.024 0.716275 1.285 1.006 17.26 × 10–6 0.024 0.715280 1.261 1.006 17.50 × 10–6 0.025 0.713290 1.218 1.006 17.98 × 10–6 0.025 0.710300 1.177 1.006 18.46 × 10–6 0.026 0.708310 1.139 1.007 18.93 × 10–6 0.027 0.705320 1.103 1.007 19.39 × 10–6 0.028 0.703330 1.070 1.008 19.85 × 10–6 0.029 0.701340 1.038 1.008 20.30 × 10–6 0.029 0.699350 1.008 1.009 20.75 × 10–6 0.030 0.697400 0.882 1.014 22.86 × 10–6 0.034 0.689450 0.784 1.021 24.85 × 10–6 0.037 0.684500 0.706 1.030 26.70 × 10–6 0.040 0.680550 0.642 1.040 28.48 × 10–6 0.044 0.680600 0.588 1.051 30.17 × 10–6 0.047 0.680700 0.504 1.075 33.32 × 10–6 0.052 0.684800 0.441 1.099 36.24 × 10–6 0.058 0.689900 0.392 1.121 38.97 × 10–6 0.063 0.6961000 0.353 1.142 41.53 × 10–6 0.068 0.702

a Data reprinted with permission from Hewitt et al. (1997) and Weast (1970).

7262_A001.indd 290 1/19/08 12:06:55 PM

Appendix 291

Ta bl e a .5Properties of Other Gases (1 atm, 298 K)a

GasDensity,

r (kg/m3)

Specific Heat, cp

(kJ/kgK)

GasConstant (J/kgoC)


k (W/mK)

Dynamic Viscosity, m (kg/ms)

Acetylene, C2H2 1.075 1.674 319 0.024 1.0 × 10–5

Ammonia, NH3 0.699 2.175 488 0.026 1.0 × 10–5

Argon, Ar 1.608 0.523 208 0.0172 2.0 × 10–5

n-Butane, C4H10 2.469 1.675 143 0.017 0.7 × 10–5

Carbon Dioxide, CO2 1.818 0.876 189 0.017 1.4 × 10–5

Carbon Monoxide, CO 1.144 1.046 297 0.024 1.8 × 10–5

Chlorine, Cl2 2.907 0.477 117 0.0087 1.4 × 10–5

Ethane, C2H6 1.227 1.715 276 0.017 9.5 × 10–5

Ethylene, C2H4 0.072 1.548 296 0.017 1.0 × 10–5

Fluorine, F2 0.097 0.828 219 0.028 2.4 × 10–5

Helium, He 0.164 5.188 2077 0.149 2.0 × 10–5

Hydrogen, H2 0.083 14.310 4126 0.0182 0.9 × 10–5

Hydrogen Sulfide, H2S 10.753 0.962 244 0.014 1.3 × 10–5

Methane, CH4 0.662 2.260 518 0.035 1.1 × 10–5

Methyl Chloride, CH3Cl 2.165 0.837 165 0.010 1.1 × 10–5

Nitric Oxide, NO 1.229 0.983 277 0.026 1.9 × 10–5

Nitrogen, N2 1.147 1.040 297 0.026 1.8 × 10–5

Nitrous Oxide, N2O 1.802 0.879 189 0.017 1.5 × 10–5

Oxygen, O2 1.309 0.920 260 0.026 2.0 × 10–5

Ozone, O3 1.965 0.820 173 0.033 1.3 × 10–5

Propane, C3H8 1.812 1.630 188 0.017 8.0 × 10–5

Propylene, C3H6 1.724 1.506 197 0.017 8.5 × 10–5

Sulfur Dioxide, SO2 2.622 0.460 130 0.010 1.3 × 10–5

Xenon, Xe 5.375 0.481 63.5 0.0052 2.3 × 10–5

(Continued)

7262_A001.indd 291 1/19/08 12:06:56 PM


Ta bl e a .5 (CONTINUeD)Properties of Other Gases (1 atm, 298 K)a

Gas

boilingPoint(°C)

l atent Heat of evaporation

(kJ/kg)

Melting Point (°C)

l atent Heatof Fusion(kJ/kg)

Heat of Combustion

(kJ/kg)

Acetylene, C2 H2 -75 614.0 -82.2 53.5 50,200Ammonia, NH3 -33.3 1373.0 -77.7 332.3 —Argon, Ar -186 163.0 —

n-Butane, C4H10 -0.4 386.0 -138 44.7 49,700Carbon Dioxide, CO2 -78.5 572.0 —Carbon Monoxide, CO -191.5 216.0 -205 10,100Chlorine, Cl2 -34.0 288.0 -101 95.4 —Ethane, C2H6 -88.3 488.0 -172.2 95.3 51,800Ethylene, C2H4 -103.8 484.0 -169 120.0 47,800Fluorine, F2 -188.0 172.0 -220 25.6 —Helium, He 4.22 K 23.3 —Hydrogen, H2 20.4 K 447.0 -259.1 58.0 144,000Hydrogen Sulfide, H2S -60 544.0 -84 70.2 18,600Methane, CH4 510.0 -182.6 32.6 5327Methyl Chloride, CH3Cl -23.7 428.0 -97.8 130.0Nitric Oxide, NO -151.5 -161 76.5 —Nitrogen, N2 -195.8 199.0 -210 25.8 —Nitrous Oxide, N2O -88.5 376.0 -90.8 149.0 —Oxygen, O2 -182.97 213.0 -218.4 13.7 —Ozone, O3 -112.0 -193 226.0 —Propane, C3H8 -42.2 428.0 -189.9 44.4 50,340Propylene, C3H6 -48.3 438.0 -185 50,000Sulfur Dioxide, SO2 -10.0 362.0 -75.5 135.0 —Xenon, Xe -108.0 96.0 -140 23.3 —


7262_A001.indd 292 1/19/08 12:06:57 PM

Appendix 293

Ta bl e a .6Properties of Other Gases (effects of Temperature)a

GasTemperature

T (°C)Density,

r (kg/m3)

Specific Heat, cp

(kJ/kgK)


k (W/mK)


Ammonia, NH3 0 0.956 2.176 0.022 9.18 × 10–6

20 0.894 2.176 0.024 9.82 × 10–6

50 0.811 2.176 0.027 1.09 × 10–5

Argon, Ar -13 1.87 0.523 0.016 2.04 × 10–5

-3 1.81 0.523 0.016 2.11 × 10–5

7 1.74 0.523 0.017 2.17 × 10–5

27 1.62 0.523 0.018 2.30 × 10–5

77 1.39 0.519 0.020 2.59 × 10–5

227 0.974 0.519 0.026 3.37 × 10–5

727 0.487 0.519 0.043 5.42 × 10–5

1227 0.325 0.519 0.055 7.08 × 10–5

Butane, C4 H10 0 2.59 1.591 0.013 6.84 × 10–6

100 1.90 2.026 0.023 9.26 × 10–6

200 1.50 2.454 0.036 1.17 × 10–5

300 1.24 2.812 0.052 1.40 × 10–5

400 1.05 3.127 0.069 1.64 × 10–5

500 0.916 3.402 0.090 1.87 × 10–5

600 0.812 3.642 0.113 2.11 × 10–5

Carbon Dioxide, CO2 -13 2.08 0.813 0.014 1.31 × 10–5

-3 2.00 0.823 0.014 1.36 × 10–5

7 1.93 0.832 0.015 1.40 × 10–5

17 1.86 0.842 0.016 1.45 × 10–5

27 1.80 0.851 0.017 1.49 × 10–5

77 1.54 0.898 0.020 1.72 × 10–5

227 1.07 1.014 0.034 2.32 × 10–5

Carbon Monoxide, CO -13 1.31 1.041 0.022 1.59 × 10–5

-3 1.27 1.041 0.023 1.64 × 10–5

7 1.22 1.041 0.024 1.69 × 10–5

17 1.18 1.041 0.025 1.74 × 10–5

27 1.14 1.041 0.025 1.79 × 10–5

77 0.975 1.043 0.029 2.01 × 10–5

227 0.682 1.064 0.039 2.61 × 10–5

Ethane, C2 H6 0 1.342 1.646 0.019 8.60 × 10–6

100 0.983 2.066 0.032 1.14 × 10–5

200 0.776 2.488 0.047 1.41 × 10–5

300 0.640 2.868 0.065 1.68 × 10–5

400 0.545 3.212 0.085 1.93 × 10–5

500 0.474 3.517 0.108 2.20 × 10–5

600 0.420 3.784 0.132 2.45 × 10–5

Ethanol, C2H5OH 100 1.49 1.686 0.023 1.08 × 10–5

200 1.18 2.008 0.035 1.37 × 10–5

300 0.974 2.318 0.050 1.67 × 10–5

(Continued)

7262_A001.indd 293 1/19/08 12:06:58 PM


Ta bl e a .6 (CONTINUeD)Properties of Other Gases (effects of Temperature)a

GasTemperature

T (°C)Density,

r (kg/m3)

Specific Heat, cp

(kJ/kgK)


k (W/mK)


400 0.828 2.611 0.067 1.97 × 10–5

500 0.720 2.891 0.086 2.26 × 10 –5

Helium, He 0 0.368 5.146 0.142 1.86 × 10–5

20 0.167 5.188 0.149 1.94 × 10–5

40 0.156 5.188 0.155 2.03 × 10–5

Hydrogen, H2 -13 0.0944 14.133 0.162 8.14 × 10–6

-3 0.0910 14.175 0.167 8.35 × 10–6

7 0.0877 14.226 0.172 8.55 × 10–6

27 0.0847 14.267 0.177 8.76 × 10–6

77 0.0819 14.301 0.182 8.96 × 10–6

727 0.04912 14.506 0.272 1.26 × 10–5

Methane, CH4 0 0.716 2.164 0.031 1.04 × 10–5

100 0.525 2.447 0.046 1.32 × 10–5

200 0.414 2.805 0.064 1.59 × 10–5

300 0.342 3.173 0.082 1.83 × 10–5

400 0.291 3.527 0.102 2.07 × 10–5

500 0.253 3.853 0.122 2.29 × 10–5

600 0.224 4.150 0.144 2.52 × 10–5

Nitrogen, N2, 77 1.14 1.041 0.026 1.79 × 10–5

227 9.75 1.042 0.030 2.00 × 10–5

727 6.82 1.056 0.040 2.57 × 10–5

Oxygen, O2 -13 1.50 0.915 0.023 1.85 × 10–5

-3 1.45 0.916 0.024 1.90 × 10–5

7 1.39 0.918 0.025 1.96 × 10–5

27 1.35 0.918 0.026 2.01 × 10–5

77 1.30 0.920 0.027 2.06 × 10–5

227 1.11 0.929 0.031 2.32 × 10–5

727 7.80 0.972 0.042 2.99 × 10–5

Propane, C3 H8 0 1.97 1.548 0.015 7.50 × 10–6

100 1.44 2.015 0.026 1.00 × 10–5

200 1.14 2.456 0.040 1.25 × 10–5

300 0.939 2.833 0.056 1.40 × 10–5

400 0.799 3.159 0.074 1.72 × 10–5

500 0.694 3.446 0.095 1.94 × 10–5

600 0.616 3.695 0.118 2.18 × 10–5


7262_A001.indd 294 1/19/08 12:06:59 PM

Appendix 295

Ta bl e a .7Properties of l iquids (300 K, 1 atm)a

l iquid

l atent Heat of Fusion (kJ/kgK)

boiling Point (K)

l atent Heat of evaporation

(kJ/kgK)

Coefficient of expansion

(1/K)

Acetic acid 181 391 402 0.0011Acetone 98.3 329 518 0.0015Alcohol, ethyl 108 351.46 846 0.0011Alcohol, methyl 98.8 337.8 1100 0.0014Alcohol, propyl 86.5 371 779Benzene 126 353.3 390 0.0013Bromine 66.7 331.6 193 0.0012Carbon disulfide 57.6 319.40 351 0.0013Carbon tetrachloride 174 349.6 194 0.0013Chloroform 77.0 334.4 247 0.0013Decane 201 447.2 263Dodecane 216 489.4 256Ether 96.2 307.7 372 0.0016Ethylene glycol 181 470 800Fluorine, R-11 297.0 180.0Fluorine, R-12 34.4 243.4 165Fluorine, R-22 183 232.4 232Glycerine 200 563.4 974 0.00054Heptane 140 371.5 318Hexane 152 341.86 365Iodine 62.2 457.5 164Kerosene 251Linseed oil 560Mercury 11.6 630 295 0.00018Octane 181 398 298 0.00072Phenol 121 455 0.00090Propane 79.9 231.08 428Propylene 71.4 225.45 342Propylene glycol 460 914Toluene 71.8 383.6 363Turpentine 433 293 0.00099Water 333 373 2260 0.00020


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Ta bl e a .8Properties of Saturated Watera

T (°C)

P (kPa)

rf

(kg/m3)rv

(kg/m3)hfg

(kJ/kg)cp,f

(kJ/kgK)mf ⋅ 106

(kg/ms)kf

(W/mK) Prf

sf

(N/m)

0.01 0.612 999.8 0.005 2501 4229 1791 0.561 13.50 0.075710 1.228 999.7 0.009 2477 4188 1308 0.580 9.444 0.74220 2.339 998.2 0.017 2453 4182 1003 0.598 7.010 0.072730 4.246 995.6 0.030 2430 4182 798 0.615 5.423 0.071240 7.381 992.2 0.051 2406 4183 653 0.631 4.332 0.069650 12.34 988.0 0.083 2382 4181 547.1 0.644 3.555 0.068060 19.93 983.2 0.130 2358 4183 466.8 0.654 2.984 0.066270 31.18 977.8 0.198 2333 4187 404.5 0.663 2.554 0.064580 47.37 971.8 0.293 2308 4196 355.0 0.670 2.223 0.062790 70.12 965.3 0.423 2283 4205 315.1 0.675 1.962 0.0608100 101.3 958.4 0.597 2257 4217 282.3 0.679 1.753 0.0589110 143.2 951.0 0.826 2230 4233 255.1 0.682 1.584 0.0570120 198.5 943.2 1.121 2202 4249 232.2 0.683 1.444 0.0550130 270.0 934.9 1.495 2174 4267 212.8 0.684 1.328 0.0529140 361.2 926.2 1.965 2145 4288 196.3 0.683 1.232 0.0509150 475.7 917.1 2.545 2114 4314 182.0 0.682 1.151 0.0488160 617.7 907.5 3.256 2082 4338 169.6 0.680 1.082 0.0466170 791.5 897.5 4.118 2049 4368 158.9 0.677 1.025 0.0444180 1001.9 887.1 5.154 2015 4404 149.4 0.673 0.977 0.0422190 1254.2 876.2 6.390 1978 4444 141.0 0.669 0.937 0.0400200 1553.7 864.7 7.854 1940 4489 133.6 0.663 0.904 0.0377220 2317.8 840.3 11.61 1858 4602 121.0 0.650 0.857 0.0331240 3344.7 813.5 16.74 1766 4759 110.5 0.632 0.832 0.0284260 4689.5 783.8 23.70 1662 4971 101.5 0.609 0.828 0.0237280 6413.2 750.5 33.15 1543 5279 93.4 0.581 0.848 0.0190300 8583.8 712.4 46.15 1405 5751 85.8 0.548 0.901 0.0144320 11279 667.4 64.6 1239 6536 78.4 0.509 1.006 0.0099340 14594 610.8 92.7 1028 8241 70.3 0.469 1.236 0.0056360 18655 528.1 143.7 721 14686 60.2 0.428 2.068 0.0019373 21799 402.4 242.7 276 21828 46.7 0.545 18.69 0.0001

a Data reprinted with permission from Hewitt et al. (1997).

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Appendix 297

Ta bl e a .9a tomic Weights of elementsa

elementkg/

kmol elementkg/

kmol elementkg/

kmol

Hydrogen, H 1.01 Bromine, Br 79.90 Thulium, Tm 168.93Helium, He 4.00 Krypton, Kr 83.80 Ytterbium, Yb 173.04Lithium, Li 6.94 Rubidium, Rb 85.47 Lutetium, Lu 174.97Beryllium, Be 9.01 Strontium, Sr 87.62 Hafnium, Hf 178.49Boron, B 10.81 Yttrium, Y 88.91 Tantalum, Ta 180.95Carbon, C 12.01 Zirconium, Zr 91.22 Tungsten, W 183.84Nitrogen, N 14.01 Niobium, Nb 92.91 Rhenium, Re 186.21Oxygen, O 16.00 Molybdenum, Mo 95.94 Osmium, Os 190.23Fluorine, F 19.00 Technetium, Tc 97.91 Iridium, Ir 192.22Neon, Ne 20.18 Ruthenium, Ru 101.07 Platinum, Pt 195.08Sodium, Na 22.99 Rhodium, Rh 102.91 Gold, Au 196.97Magnesium, Mg 24.31 Palladium, Pd 106.42 Mercury, Hg 200.59Aluminum, Al 26.98 Silver, Ag 107.87 Thallium, Tl 204.38Silicon, Si 28.09 Cadmium, Cd 112.41 Lead, Pb 207.2Phosphorus, P 30.97 Indium, In 114.82 Bismuth, Bi 208.98Sulfur, S 32.07 Tin, Sn 118.71 Polonium, Po 208.98Chlorine, Cl 35.45 Antimony, Sb 121.76 Astatine, At 209.99Argon, Ar 39.95 Tellerium, Te 127.60 Radon, Rn 222.02Potassium, K 39.10 Iodine, I 126.90 Francium, Fr 223.02Calcium, Ca 40.08 Xenon, Xe 131.29 Radium, Ra 226.03Scandium, Sc 44.96 Caesium, Cs 132.91 Actinium, Ac 227.03Titanium, Ti 47.87 Barium, Ba 137.33 Thorium, Th 232.04Vanadium, V 50.94 Lanthanum, La 138.91 Protactinium, Pa 231.04Chromium, Cr 52.00 Cerium, Ce 140.12 Uranium, U 238.03Manganese, Mn 54.94 Praseodymium, Pr 140.91 Neptunium, Np 237.05Iron, Fe 55.85 Neodymium, Nd 144.24 Plutonium, Pu 239.05Cobalt, Co 58.93 Promethium, Pm 144.91 Americium, Am 243.06Nickel, Ni 58.69 Samarium, Sm 150.36 Curium, Cm 247.07Copper, Cu 63.55 Europium, Eu 151.97 Berkelium, Bk 249.07Zinc, Zn 65.39 Gadolinium, Gd 157.25 Californium, Cf 251.08Gallium, Ga 69.72 Terbium, Tb 158.93 Einsteinium, Es 252.08Germanium, Ge 72.61 Dysprosium, Dy 162.50 Fermium, Fm 257.10Arsenic, As 74.92 Holmium, Ho 164.93 Mendelevium, Md 258.10Selenium, Se 78.96 Erbium, Er 167.26 Nobelium, No 259.10

a Data reprinted with permission from Hewitt et al. (1997).

ReFeReNCeS

Hewitt, G.F., Shires, G.L., and Y.V. Polezhaev, Eds. 1997. International Encyclopedia of Heat and Mass Transfer. CRC Press, Boca Raton, FL.

Weast, R.C., Ed. 1970. CRC Handbook of Tables for Applied Engineering Science. CRC Press, Boca Raton, FL.

7262_A001.indd 297 1/19/08 12:07:02 PM

299

Nomenclature

B Bias and precision components of errorcv, cp Specific heatsc, C Wave speed; advection speed coefficientd, D Diffusion coefficiente, E Specific total energy (per unit mass); total energyEc Eckert numberf Flux function; mass fractionF Body force or source term, flux of entropy, or Faraday’s constantg Probability distribution function, gravitational vectorG Flux of exergyH EnthalpyHe Electric field strengthI Exergy flowj Heat flux compressible flowJ Jacobian of transformationJa Jacob numberk Thermal conductivity; turbulent kinetic energyKn Local Knudsen numberl, ℓ, L Thickness; reference length; Pixel dimensionM Molecular weightn Number densityN Thermomagnetic number, finite element shape functionNs Entropy generation parameterp, P PressurePe Peclet numberPr Prandtl numberq Heating, heat flow rateQ Generic function of molecular velocityR Gas constantRe Reynolds number; local Reynolds numbers, S Specific entropy; EntropySte Stefan numberu, v, w Cartesian velocity componentsU Total internal energyvk Molecular velocity

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W Workp1 Phase fractionX Exergy rate

Su bSc r ipt S a n d S u per Sc r ipt S

0 Initial; exchangea Advection quantity; anode; analyticact ActivationB Body interactionsc CathodeCD Central distributionCE Chapman–Enskogcoll Molecular collisionconc ConcentrationCV Control Volumed Diffusion quantity; discrete quantitydes Sestructioneff Effectivef FrictionF Faraday constantgen Generationi, j, k Coordinate direction indices

J′ Bessel function

m MembraneMB Maxwell–Boltzmannmin Minimum valuen Time step indexns No slip conditionp Phase interactions; parallelPDF Probability density functionref Referencerev Reversiblesupp Suppliedt Transient; time-dependent; turbulentT Matrix transposew Wall valuex, y, z Cartesian coordinate directions

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Nomenclature 301

Gr eek Symbo l S

a Charge transfer coefficient

b T, bC Thermal and solution expansion coefficients

g Ratio of specific heats; reaction order

G General diffusion coefficient

d Kronecker delta function; diffusion ratio

D Difference operator

e Total digital image error; turbulent dissipation rate

h Second-Law effectiveness; polarization

q Skewness coefficient

k Isothermal compressibility

l Interrogation length scale; Eigenvalue; Taylor microscale

m , me Dynamic viscosity coefficient; magnetic permeability

x Constitutive variable; slip coefficient

r Mass density

s Mechanical stress

t Shear stress

f Specific exergy

F Viscous dissipation function

c Flow alignment factor

Y Flow exergy

w k Vorticity

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Entropy-Based Design and Analysis of Fluids Engineering Systems

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