© Fluent Inc. 8/10/2015G1 Fluids Review TRN-1998-004 Heat Transfer

© Fluent Inc. 04/19/23G1

Fluids ReviewTRN-1998-004

Heat Transfer



Outline

Introduction Modes of heat transfer Typical design problems Coupling of fluid flow and heat transfer Conduction Convection Radiation



Introduction

Heat transfer is the study of thermal energy (heat) flows Heat always flows from “hot” to “cold” Examples are ubiquitous:

heat flows in the body home heating/cooling systems refrigerators, ovens, other appliances automobiles, power plants, the sun, etc.



Modes of Heat Transfer

Conduction - diffusion of heat due to temperature gradient Convection - when heat is carried away by moving fluid Radiation - emission of energy by electromagnetic waves

qconvection

qconduction

qradiation



Typical Design Problems

To determine: overall heat transfer coefficient - e.g., for a car radiator highest (or lowest) temperature in a system - e.g., in a gas turbine temperature distribution (related to thermal stress) - e.g., in the walls of a

spacecraft

temperature response in time dependent heating/cooling problems - e.g., how long does it take to cool down a case of soda?



Heat Transfer and Fluid Flow

As a fluid moves, it carries heat with it -- this is called convection Thus, heat transfer can be tightly coupled to the fluid flow solution Additionally:

The rate of heat transfer is a strong function of fluid velocity Fluid properties may be strong functions of temperature (e.g., air)



Conduction Heat Transfer

Conduction is the transfer of heat by molecular interaction In a gas, molecular velocity depends on temperature

hot, energetic molecules collide with neighbors, increasing their speed In solids, the molecules and the lattice structure vibrate



Fourier’s Law

“heat flux is proportional to temperature gradient”

where k = thermal conductivity in general, k = k(x,y,z,T,…)

y

T

x

TkTkq

A

Q

hot wall cold wall

dx

dT

1

temperature profile

x

heat conduction in a slab

units for q are W/m2



Generalized Heat Diffusion Equation

If we perform a heat balance on a small volume of material…

… we get:

qTkt

Tc

2

c

k

thermal diffusivity

Theat conductionin

heat conductionoutq

heat generation

rate of changeof temperature

heat cond.in/out

heatgeneration



Boundary Conditions

Heat transfer boundary conditions generally come in three types:

T = 300Kspecified temperature

Dirichlet condition

q = 20 W/m2

specified heat fluxNeumann condition

q = h(Tamb-Tbody)external heat transfercoefficientRobin condition

Tbody



Conduction Example

Compute the heat transfer through the wall of a home:

shinglesk=0.15 W/m2-K

sheathingk=0.15 W/m2-K

fiberglas insulationk=0.004 W/m2-K

2x6 studk=0.15 W/m2-K

sheetrockk=0.4 W/m2-K

Tout = 20° F Tout = 68° F

Although slight, you can see the “thermal

bridging” effect through the studs



Convection Heat Transfer

Convection is movement of heat with a fluid E.g., when cold air sweeps past a warm body, it draws away warm air

near the body and replaces it with cold air

often, we want to know the heat transfer coefficient, h (next page)

flow over a heated block



Newton’s Law of Cooling

Tbody

T

ThTThq body )(

average heat transfer coefficient (W/m2-K)h

q



Heat Transfer Coefficient

h is not a constant, but h = h(T) Three types of convection:

Natural convection fluid moves due to buoyancy

Forced convection flow is induced by external means

Boiling convection body is hot enough to boil liquid

3

1

4

1

, ThTh

consth

2Th

Typical values of h:

4 - 4,000 W/m2-K

80 - 75,000

300 - 900,000

Thot Tcold

Thot

Tcold

Tcold

Thot



Looking in more detail...

Just as there is a viscous boundary layer in the velocity distribution, there is also a thermal boundary layer

t

wT

UT ,

y

)( yT

velocity boundarylayer edge

thermal boundarylayer edge



Nusselt Number

Equate the heat conducted from the wall to the same heat transfer in convective terms:

Define dimensionless quantities:

Then rearrange to get:

)(

TThy

Tk wf

L

yy

TT

TTT

w

w

Nu

f

w

w

k

hL

Ly

TTTT

Nusselt number

“dimensionless heat

transfer coefficient”

conductivityof the fluid



Energy Equation

Generalize the heat conduction equation to include effect of fluid motion:

Assumes incompressible fluid, no shear heating, constant properties, negligible changes in kinetic and potential energy

Can now solve for temperature distribution in boundary layer Then calculate h using Fourier’s law:

qTkTt

Tc

2u

0

yww y

T

TT

k

TT

qh

From calculatedtemperaturedistribution



Correlations for Heat Transfer Coefficient

As an alternative, can use correlations to obtain h E.g., heat transfer from a flat plate in laminar flow:

where the Prandtl number is defined as:

Typical values are: Pr = 0.01 for liquid metals Pr = 0.7 for most gases Pr = 6 for water at room temperature

333.05.0 PrRe332.0Nu xx

k

cPr

ydiffusivit thermal

ydiffusivit momentum



Convection Examples

Developing flow in a pipe (constant wall temperature)

T wT T wT T wT

TwT

x

bulk fluid temperature

heat flux from wall

TwT



Convection Examples

Natural convection (from a heated vertical plate)

u

TTw

gravity

As the fluid is warmed by the plate, its density decreases and a buoyant force arises which induces flow in the vertical direction. The force is equal to:

,T

)(T

g)(

The dimensionless group that governs natural convection is the Rayleigh number:

3

RaTLg



Radiation Heat Transfer

Thermal radiation is emission of energy as electromagnetic waves Intensity depends on body temperature and surface characteristics Important mode of heat transfer at high temperatures Can also be important in natural convection problems Examples:

toaster, grill, broiler fireplace sunshine



Surface Characteristics

1

q W/m2 (incident energy flux) q (reflected)

q (transmitted)

q (absorbed)

absorptance

reflectance

transmittance

translucent slab



Black Body Radiation

A “black body”: is a model of a perfect radiator absorbs all energy that reaches it; reflects nothing therefore = 1, = = 0

The energy emitted by a black body is the theoretical maximum:

This is Stefan-Boltzmann law; is the Stefan-Boltzmann constant (5.6697e-8 W/m2-K4)

4Tq



“Real” Bodies

Real bodies will emit less radiation than a black body:

Example: radiation from a small body to its surroundings both the body and its surroundings emit thermal radiation the net heat transfer will be from the hotter to the colder

4Tq emissivity (between 0 and 1)

)( 44 TTAQ wnet

T

q

wTA

wqnetQ



When is radiation important?

Radiation exchange is significant in high temperature problems: e.g., combustion

Radiation properties can be strong functions of chemical composition, especially CO2, H2O

Radiation heat exchange is difficult solve (except for simple configurations) — we must rely on computational methods



Heat Transfer — Summary

Heat transfer is the study of thermal energy (heat) flows: conduction convection radiation

The fluid flow and heat transfer problems can be tightly coupled through the convection term in the energy equation when properties (, ) are dependent on temperature

While analytical solutions exist for some simple problems, we must rely on computational methods to solve most industrially relevant applications Can I go back to

sleep now?

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© Fluent Inc. 8/10/2015G1 Fluids Review TRN-1998-004 Heat Transfer