02A -Chapter 2 Black Text

8/6/2019 02A -Chapter 2 Black Text

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Fouriers Law

and theHeat Equation

Chapter Two


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Fouriers Law

A rate equation that allows determination of the conduction heat fluxfrom knowledge of the temperature distribution in a medium

Fouriers Law

Its most general (vector) form for multidimensional conduction is:

q k Tp p

dd!

Implications: Heat transfer is in the direction of decreasing temperature

(basis for minus sign).

Direction of heat transfer is perpendicular to lines of constant

temperature (isotherms).

Heat flux vector may be resolved into orthogonal components.

Fouriers Law serves to define the thermal conductivity of the

medium /k q Tp p

dd|


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Heat Flux Components

(2.18)T T Tq k i k j k k r r zJ

p p p px x xdd! x x x

rqdd qJdd zqdd

Cylindrical Coordinates: , ,T r zJ

sin

T T Tq k i k j k k

r r rU U J

p p p px x xdd! x x x

(2.21)

rqdd qUdd qJdd

Spherical Coordinates: , ,T r J U

Cartesian Coordinates: , ,T x y z

T T Tq k i k j k k x y z

p p p p

x x xdd! x x x

xqdd yqdd zqdd

(2.3)


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Heat Flux Components (cont.)

In angular coordinates , the temperature gradient is still

based on temperature change over a length scale and hence has

units ofrC/m and not rC/deg.

or ,J J U

Heat rate forone-dimensional, radial conduction in a cylinder or sphere:

Cylinder

2r r r r q A q rLqTdd dd! !

or,

2r r r r q q rqTd ddd dd! !

Sphere

24r r r r q q r qTdd dd! !


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Heat Equation

The Heat Equation A differential equation whose solution provides the temperature distribution in a

stationary medium. Based on applying conservation of energy to a differential control volume

through which energy transfer is exclusively by conduction.

Cartesian Coordinates:

Net transferofthermal energy into the

control volume (inflow-outflow)

p

T T T T k k k q c

x x y y z z tV

x x x x x x x ! x x x x x x x

(2.13)

Thermal energy

generationChange in thermal

energy storage


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Heat Equation (Radial Systems)

2

1 1p

T T T T kr k k q c

r r r z z t r VJ J


(2.20)

Spherical Coordinates:

Cylindri

cal Coordinates:

2

2 2 2 2

1 1 1sin

sin sinp

T T T T kr k k q c

r r tr r rU V

J J U UU U


(2.33)


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Heat Equation (Special Case)

One-Dimensional Conduction in a Planar Medium with Constant Properties

andNo Generation

2

2

1T T

tx E

x x!

xx

thermal diffu osivit f the medy ium

p

k

cE

V| p


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Boundary Conditions

Boundary and Initial Conditions For transient conduction, heat equation is first order in time, requiring

specification of an initial temperature distribution: 0, ,0tT x t T x! ! Since heat equation is second order in space, twoboundary conditions

must be specified. Some common cases:

Constant Surface Temperature:

0,s

T t T!

Constant Heat Flux:

0|x sT

k qx !

xdd !x

Applied Flux Insulated Surface

0 0x

T

x !

x

!x

Convection

0 0,xT

k h T T tx

! gx

! x


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Properties

Thermophysical Properties

Thermal Conductivity: A measure of a materials ability to transfer thermal

energy by conduction.

Thermal Diffusivity: A measure of a materials ability to respond to changes

in its thermal environment.

Property Tables:

Solids: Tables A.1 A.3

Gases: Table A.4

Liquids: Tables A.5 A.7


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Conduction Analysis

Methodology of a Conduction Analysis

Solve appropriate form of heat equation to obtain the temperaturedistribution.

Knowing the temperature distribution, apply Fouriers Law to obtain the

heat flux at any time, location and direction of interest.

Applications:

Chapter 3: One-Dimensional, Steady-State Conduction

Chapter 4: Two-Dimensional, Steady-State Conduction

Chapter 5: Transient Conduction


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Problem : Thermal Response of Plane Wall

Problem 2.46 Thermal response of a plane wall to convection heat transfer.

KNOWN: Plane wall initiall at a niform temperat re is s enl e pose to convective

heatin .

FIND: a ifferential e ation an initial an bo n ar con itions which ma be se to fin

the temperat re istrib tion T t ; b etch T t for the followin con itions: initial t

0), steady-state (t pg), and two intermediate times; (c) Sketch heat fluxes as a function oftime at the two surfaces; (d) Expression for total energy transferred to wall per unit volume

(J/m3).

SCHEMATIC:


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c) The heat flux, ddq x, txb g, asafunctionoftime, isshownonthesketchforthesurfacesx = 0 andx =L.

in conv s0

E q dtg

dd!

d) The total energy transferred to the wall may be expressed as

in s 0E hA T T L,t dtg

g!

Dividing both sides by AsL, the energy transferred per unit volume is

3in0

E hT T L,t dt J/m

V L

gg

!

Problem: Thermal Response (Cont.)


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Problem: Non-Uniform Generation due

to Radiation Absorption

Problem 2.28 Surface heat fluxes, heat generation and total rate of radiation

absorption in an irradiated semi-transparent material with a

prescribed temperature distribution.

KNOWN: emperature distribution in a semi-transparent medium subjected to radiati e flux.

FIND: (a) Expressions for the heat flux at the front and rear surfaces, (b) The heat generation rate

,q xbgand (c) Expression for absorbed radiation per unit surface area.

SCHEMATIC:


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Ona nitarea basis

Alternativel ,eval ate ddEg b integration over the volume o the medium,

.dd! !

z zE q d Ae d -

A

ae

A

ae

L -aL -aL

-aL

bg e j01

.dd! dd dd ! dd dd ! E E E q q LA

a

eg i t x x-aLbg bg

e

Problem: Non-Uniform Generation (Cont.)

02A -Chapter 2 Black Text

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