Lecture Objectives:Review - Heat transferConvection Conduction Radiation

Analysis of a practical problem

Example Problem radiant barrier in attic

Example Problem heat transfer in window construction

Convection

Convection coefficient h [W/m2K]ConductionConvection Natural convectionForced convectionL characteristic lengthh natural convectionk air conductionL- characteristic lengthorNusselt number:areaSpecific heat fluxHeat flux

Which surface in this classroom has the largest forced convectionA. Window B. CeilingC. WallsD. Floor

Which surface has the largest natural convection

How to calculate h ?

What are the parametrs that affect h ?

What is the boundary layer ?

Laminar and Turbulent Flowforced convection

Forced convection governing equations1) Continuity

2) Momentumu, v velocities n air viscosityNon-dimensionless momentum equation

Using

L = characteristic length and U0 = arbitrary reference velocityReLReynolds number

Forced convection governing equationsEnergy equation for boundary layerNon-dimensionless energy equationsT temperature, a thermal diffusivity a=k/rcp, k-conductivity, r - density, cp specific cap. Wall temperatureAir temperature outside of boundary layerInertial forceViscous forceMomentum diffusivity Thermal diffusivityReynolds number Prandtl number

Simplified Equation for Forced convection

For laminar flow: For turbulent flow:For air: Pr 0.7, n = viscosity is constant, k = conductivity is constant General equationSimplified equation:Or:

Natural convection

GOVERNING EQUATIONSNatural convection Continuity

Momentum which includes gravitational force

Energyu, v velocities , n air viscosity , g gravitation, b1/T - volumetric thermal expansion T temperature, air temperature out of boundary layer, a temperature conductivity

Characteristic Number for Natural ConvectionNon-dimensionless governing equations

Using

L = characteristic length and U0 = arbitrary reference velocity Tw- wall temperatureThe momentum equation becomeMultiplying by Re2 number Re=UL/n Gr

Grashof number Characteristic Number for Natural Convection The Grashof number has a similar significance for natural convection as the Reynolds number has for forced convection, i.e. it represents a ratio of buoyancy to viscous forces.Buoyancy forcesViscous forces General equation

Even more simpleNatural convection simplified equationsFor laminar flow: For turbulent flow:For air: Pr 0.7, n = constant, k= constant, b= constant, g=constant Simplified equation:Or:T - air temperature outside of boundary layer, Ts - surface temperature

Forced and/or natural convectionIn general,Nu = f(Re, Pr, Gr)natural and forced convectionforced convectionnatural convection

Combined forced and natural conventionChurchill and Usagi approach :This equation favors a dominant term (h1 or h2), and exponent coefficient n determines the value for hcombined when both terms have the same order of value

Example of general forced and natural convectionEquation for convection at cooled ceiling surfacesn

What kind of flow is the most common for indoor surfacesA. Laminar B. TurbulentC. TransitionalD. Laminar, transitional, and turbulent

What about outdoor surfaces?

Conduction

Conductive heat transferSteady-state

Unsteady-state

Boundary conditions

Dirichlet Tsurface = Tknown

Neumann

LTairk - conductivity of materialTS1TS2h

Boundary conditionsBiot number

conventionconduction

Importance of analytical solution

What will be the daily temperature distribution profile on internal surface for styrofoam wall? A.

B.External temperature profileTtime

What will be the daily temperature distribution profile on internal surface for tin glass? A.

B.External temperature profileTtime

Conduction equation describes accumulation

Important numbersInertial forceViscous forceReynolds number Momentum diffusivity Thermal diffusivityPrandtl number Buoyancy forcesViscous forces ConductionConvection Nusselt number thermal internal resistance surface film resistance Grashof number Biot number Reference book: Fundamentals of Heat and Mass Transfer, Incropera & DeWitt

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