A SHORT COURSEREZA TOOSSI, PH.D, P.E.

CALIFORNIA STATE UNIVERSITY, LONG BEACH

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Heat Transfer Modeling and Applications

Outline2

Scope and Types Energy Equation

Formulation Conversion Mechanisms Dimensionless Parameters in Heat Transfer

Modes of Heat Transfer Conduction, Convection, and Radiation Correlations Combined Modes

Heat Transfer in Multiple Phases Conjugate Heat Transfer Composites Phase Change

Scales3

T 0 K – 1.4x1032 K L 1.6x10-35 m – 1.6x1026 m (15 bly)t 5.4x10-44 s – 2.7x1017 sm 10-69 kg - 1054 kg

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Mechanical Systems6

Mechanically determined problems Can be solved only by Newton’s Law of Motion and

Conservation of MassExamples:

Free fall of a body; F = ma Dynamics of rigid bodies in absence of friction Flow of ideal fluids between two parallel plates

Mechanically undetermined problems Additional laws are needed

Examples: Dynamics of rigid bodies with friction Dynamics of deformable body

Thermal Systems7

Thermodynamically determined problems Can be solved by the general laws of mechanics and

the first and second laws of thermodynamics.Example: Flow of steady 1-D isentropic and subsonic fluid through a

nozzle. mass d(AV) = 0 Momentum dp+ AdV = 0 Energy du +pd(1/) = 0

Thermodynamically undetermined problems Heat Transfer (modes of heat transfer) Gas Dynamics (equation of state)

Knudsen Number8

Continuum regime (Kn < 0.01)Slip flow regime (0.01 < Kn < 0.1)Transition regime (0.1 < Kn < 3)Free molecular flow regime (Kn > 3)

Energy Equation9

Formulations10

Differential

Integral

Integral-Differential

Thermal Nodes

Differential-Volume11

Solution:

Infinite flat plateFire Walking

Interface Temperature (no contact resistance)

Integral Volume (lumped)12

Heat losses under the condition of hypothermia

Take cvV = 5x105 J/K, Sv = 400 W, and QA= Qr = 800 W; T = 10oCGet t =3.47 hr

Surface Coating

Droplet Impingement on a Hot Surface Temperature is uniform within the droplet

(particle) Particle temperature varies during its

flight Sensible heating until particle reaches its

melting temperature. Melting occurs at T = Tsl

Time to reach melting temperature

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Integral-Differential14

Thermal Nodes15

- Slab

- Cylinder

- Sphere

Energy Conversion Mechanisms16

Heat Source

Phase ChangeChemical ReactionNuclear Fission and FusionSurface Friction HeatingViscous HeatingUltrasound HeatingMicrowave HeatingJoule HeatingThermolectric Heating (and Cooling)

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Phase Change18

Chemical Reaction19

ar is pre-exponential factor = func (f)

Nuclear Fission and Fusion20

Surface Friction Heating

Interfacial Energy Conversion of Mechanical Energy

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μF is the friction coefficientpc is the contact or joint pressureΔui is the interface relative velocity

Viscous Heating

Volumetric Conversion of Mechanical Energy Due to Fluid Viscosity

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Example: Viscous heating in a ball bearing. The bearing is 0.2 mm in diameter, and ui = 1 m/s. Engine oil at STP has a viscosity of f = 0.366 Pa/s

Ultrasound Heating23

Volumetric Conversion of Longitudinal Acoustic Waves to Thermal Energy

ac acoustic absorption coefficient

= 1.4 for blood

= 14 for muscle= 31 for skin= 161 for bone

as speed of sound

= 1,519 m/s in tissues

= 3,445 m/s in bones

f frequency

μ dynamic viscosity

γ specific heat ratio g = cp/cv

Pr Prandtl number

Microwave (Dielectric) Heating

Volumetric Conversion of Electromagnetic to Molecular Vibration (Heat)

ee (V/m) electric field intensity

ec dielectric loss factor

(relative permittivity)0 permittivity of free space

f (Hz) oscillation frequency

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Joule heating

Conversion of Electrical Energy to Heat

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Seebeck Thermoelectric26

Peltier Thermoelectric Cooling and Heating

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Thermoelectric Properties for Some Materials28

Thermoelectric Power Generation Unit

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Direct Electrical Power Generation by Heat Absorption at a Hot Junction Rejecting the Peltier Heat at the Cold

Junction Bismuth-telluride cold p-n junction S,p=230x10-6 V/k, S,n= -210x10-6 V/k, Je=10 A The Peltier heat absorbed at the cold

junction Tc = 120oC Q = -1.73 W

Tc = 20oC Q = -1.29 W

Tc = - 80oC Q = -0.85 W

Volumetric and Surface Radiation Absorption

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Classification 31

Solar Irradiation32

Flame Irradiation33

Laser Irradiation34

Summary of Heat Source Terms35

Modes of Heat Transfer36

Diffusion (transfer of heat within one medium or from one medium to another medium) Conduction (diffusion of heat in moving or

stationary rigid bodies) Convection (diffusion of heat in moving

deformable bodies)

Radiation (transfer of heat by electromagnetic waves)

Conduction37

Convection38

Heat Transfer Coefficient39

Convective heat transfer is affected by the geometry, surface condition, and fluid properties.

To find h, the detailed flow field must be known. Mass (1) Momentum (3) Energy (1) Equation of State (1)

Solve for u, v, w, T, P, and ρ Find qw = h A (TW -T∞) {and also w = du/dyw}

Π-Buckingham Theorem (f = n – m) h = h (k, , cp, , L, , T, gT, hst, t) Nu = Nu (Gr, Pr, Ste, Fo, /L)

Dimensionless Parameters in HT40

Dimensionless Parameters in HT41

Boundary Conditions

1. Prescribed Temperature2. Prescribed Heat Flux (Fourier’s Law)

3. Insulation4. Convective (Newton’s Cooling Law)

5. Radiative (Stefan-Boltzmann Law)

6. Prescribed Heat Flux Acting at a Distance7. Interface (Continuity, Conservation Law)

8. Moving Boundary

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How to insulate a boundary?

q 0 Efficient electric heater

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Guard Heater

When is the assumption of isothermal or insulated wall justified?

h 0; h ∞ Flow through a thin heated tube

Insulated wall [ ] Ti Tm (Boiling)

k 0; k ∞ Lumped vs. distributed (Biot number)

Insulated wall k large (lumped) k small (distributed)

Bare tube, h0<< hi (qo0)

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Critical thickness of Insulation45

Moving Boundary

Ice layer forming on surface of a pond on a clear night with calm wind

Tsky = 0 K; T∞ = 3oC; Tℓ = 4oC; Tice = 0oC)

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Forced Convection (External Flow)47

Forced Convection (Internal Flow)48

Free Convection: Flat Plates

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Free Convection: Cylinders and Spheres

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Conjugate Heat Transfer51

Both flow field in the fluid and temperature field in a solid must be known Requires two physical models for fluid and the solid

materials Point-to-point connectivity at the boundary zone Energy must be conserved at the interface.

This requires interfacial matching of appropriate boundary conditions at the interface

In general energy and momentum equations are coupled

Interfaces52

Solid-Solid Pressure Contact (glue, contact resistance) Relative motion(grinding wheel, gears)

Solid-Liquid Phase Change at the interface (ablation, melting of

ice)

Solid-Gas No phase change (fins) Phase Change at the interface (condensation)

Liquid-Gas Phase Change at the interface (pool nucleate

boiling)

Examples of CHT Problems

Heat transfer through composites and porous media.

Laminar boundary-layer flow past a plate of finite thickness with internal heat generation.

Steady and unsteady duct-flow in which axial wall conduction is important.

Liquid jet spray into a quiescent gas (with and without impingement)

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Effective Conductivity (LTE)54

Local thermal equilibrium (<T>g= <T>s <k>eff)

• Low porosity, low conductivity beads

Composites (Layered)

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1-D Treatment Layered Non-layered

Composites (Non-layered)56

Solid-gas composites or porous solids (fiberglass insulation) Solid-solid composites (fiberglass filled with resin or thermo-

plastic polymers) Liquid-liquid composites (liquid emulsion)

Heat transfer through porous media57

High porosity, open-celled metal foams Assumption of the local thermal equilibrium is not reasonable (ks

>> kg)

Betchen, L. , and Straatman A., “Interface conditions for a non-equilibrium heat transfer model for conjugate fluid /porous/solid domains,” Numerical Heat Transfer, Part A, 49, pp. 543–565.

Coating Substrate

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Lim, J. S., Bejan, A., and Kim, J. H., “The optimal thickness of a wall with convection on one side,“ Int. J. Heat mass Transfer, Vol. 35, 1992, pp..1673-1679.

Uncoated Plate (Similarity Solution)

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Coated Plate60

Natural Convection along a Vertical Wall

1. Isothermal wall, Isothermal pool of fluid (simplest case)

2. Wall with internal heat generation, Isothermal pool of fluid (Cooling of nuclear fuel rod by liquid metals)

3. Isothermal wall, Thermally stratified fluid (heated wall bounded by the ceiling)

4. Conjugate boundary layer (single-pane window)

5. Channel flow (chimney)

6. Consideration of free convection (mixed convection)

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Thermobuoyant Flow along a Vertical Wall

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Mixed Convection63

Lloyd, J. R., and Sparrow, E. M., “Combined forced and free convection flow on vertical surfaces, Int. J. Heat Mass Transfer, Vol. 13, 1970, pp. 434-438.

Cooling of a Nuclear Fuel Element by Upward Stream of Liquid Metals

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S. Jahangeer, M.K. Ramis, G. Jilani, « Conjugate heat transfer analysis of a heat generating vertical plate,” Int J Heat and Mass Transfer, 50 (2007) 85–93

Cooling of a Nuclear Fuel Element by Liquid Metals (Continued)

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Cooling of a Nuclear Fuel Element by Liquid Metals (Continued)

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Cooling by Free Convection (Re 0)

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M. D. Kelleher and K. YANG,” A Steady Conjugate Heat Transfer analysis of a heat generating vertical plate,” Int. J. Heat Mass Transfer, 50 (2007) 85-93.

Heat conduction equation for the slab

Analytical Solution

Conduction in Slab Poisson Equation Use Fourier sin transform

Free Convection Similarity Solution Define stream function

Interfacial Condition

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Results69

Single-Pane Windows

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R. Anderson and A. Bejan, “Heat transfer through single and double vertical walls in natural convection: theory and experiment,” Int. J. Heat Mass Transfer, Vol. 24, 1981, pp. 1611-1620

Impermeable Wall Imbedded in a Porous Medium

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Bejan, A., and Anderson, R., “Heat transfer across a vertical impermeable partition imbedded in a porous medium,” Int. J. Heat Mass Transfer, Vol. 24, 1981, pp. 1237-1245.

Vertical Channel Flow(Chimney)

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Anderson, R., and Bejan, A., “Heat transfer through single and double vertical walls in natural convection: theory and experiment”, Int. J. Heat Mass Transfer, Vol. 24, 1981, pp. 1160-1245.

Boussinesq Approximation

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What is the effect of buoyant force due to temperature variation within the system?

Solution: Vertical Channel Flow

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Moving Boundary Problems75

Solid-Liquid Transition76

Analytical Solutions in Phase Change Problems

Contact Melting (melting of a solid under its own weight)

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Solidification (One-Region Problem)

A solid PCM at Ti = Tm is suddenly exposed to a constant heat flux q”0. Find

the transient location of the solid-liquid interface.

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Solidification (Two-Region Problem)

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Convective Effects80

Numerical Simulation in Phase Change Problems

Analytical 1D and some 2D conduction-controlled

Numerical Strong (Classical ) numerical solution

Velocity u and pressure p satisfy Navier Stokes equations pointwise in space-time.

Weak (Fixed-Grid) solution Enthalpy Method (Shamsunder and Sparrow, 1975) The Equivalent Heat Capacity Method ( Bonacina et al .,

1973) The Temperature-Transforming Model ( Cao and Faghri,

1990)

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Enthalpy Method

Two-Region Melting of a Slab Assume densities of the liquid

and solid phase are equal.

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Discretization 83

Algorithm (explicit scheme)

1. Choose ∆t and ∆x to meet Neumann’s stability criterion

2. Determine the initial enthalpy at every node hjo (j = 1)

3. Calculate the enthalpy after the first time step at nodes (j = 2 ,..., N -1) by using equation (1).

4. Determine the temperature after the first time step at node (j = 1 ,..., N) by using equations (2) and (3).

5. Find a control volume in which the enthalpy falls between 0 and hsl , and determine the location of the solid-liquid interface by using equation (4).

6. Solve the phase-change problem at the next time step with the same procedure.

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Algorithm (implicit scheme)

Unconditionally stable but is more complex because two unknown variables enthalpy and temperature are involved. [See Alexiades , A ., and Solomon , A . D ., 1993 , Mathematical Modeling of Melting and Freezing Processes , Hemisphere , Washington , DC .]

Transform the energy equation into a nonlinear equation with a single variable h. [See Cao , Y ., and Faghri , A ., 1989 , " A Numerical Analysis of Stefan Problem of Generalized Multi-Dimensional Phase-Change Structures Using the Enthalpy Transforming Model ," International Journal of Heat and Mass Transfer , Vol . 32 , pp . 1289-1298.]

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Equivalent Heat Capacity Method

3-D Conduction controlled melting/solidification

Heat capacity during the phase change is infinite. Assume Cp and k change linearly from liquid to solid

Advantage: Simplicity Disadvantage: Unstable if right choices for ∆x, ∆t, and ∆T are not made.

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Temperature-Transforming Model

Combination of the two methods [Cao , Y ., and Faghri , A ., 1990a , " A Numerical Analysis of Phase Change Problem including Natural Convection ," ASME Journal of Heat Transfer, Vol . 112 , pp . 812-815.]

Use finite volume approach by Patankar to solve the diffusion equation.

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Enthalpy Method: Solidification in Square Section

Melting/solidification with natural convection Assumptions

Newtonian incompressible fluid with constant properties, except the density that is evaluated s linear function of temperature (Bousinessqe approximation)

Effective conductivity in the mushy zone Isotropic Heat transfer by conduction, convection and phase

change

Algorithm Finite volume method with body-fitted coordinates

with a SIMPLER scheme to join pressure and velocities

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CARLOS HERNÁN SALINAS LIRA1, SOLIDIFICATION IN SQUARE SECTION, Theoria, Vol. 10: 47-56, 2001.

Governing Equations89

Numerical Method

Finite volume method Staggered grid Poisson equation with the

stretching function SIMPLER algorithm (Patankar,

1980) Central differencing scheme Crank-Nicholson procedure for the convective

terms

Gauss-Seidel method with successive iterations

Convergence with local deviation of 0.006% for u, v, p, and T.

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Results91

Porous Media: Averaging Techniques for Multiphase Transport

Eulerian Averaging Averaged over space, time, or both within the domain of

integration Based on time-space description of physical phenomena Consistent with the c.v. analysis used to develop governing equations. Eulerian time-averaging

Eulerian volume-averaging Phase-averages:

Intrinsic phase average Extrinsic phase average

Lagrangian Averaging Follow a particle and average its properties during the flight

Molecular Statistical Averaging Boltzmann statistical distribution rather than individual particle

is the independent variable.

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Porous Media : One-Region Melting

Jany , P ., and Bejan , 1988 , " Scaling Theory of Melting with Natural Convection in an Enclosure ," International Journal of Heat and Mass Transfer , Vol . 31 , pp . 1221-1235.

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Governing Equations:

Solution: Porous Media : One-Region Melting

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Summary

FormulationSourcesModes

Conduction, Convection, and Radiation Mixed Modes Phase Change Conjugate Heat Applications

Thank you…

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