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Fouriers Law

and the

Heat Equation

Chapter Two

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Fouriers Law

A rate equationthat allows determination of the conduction heat flux

from knowledge of the temperature distriutionin a medium

Fouriers Law

!ts most general "#ector$ form for multidimensional conduction is%

q k T =

!mplications%

& Heat transfer is in the direction of decreasing temperature

"asis for minus sign$'

& (irection of heat transfer is perpendicular to lines of constant

temperature "isotherms$'

& Heat flux #ector ma) e resol#ed into orthogonal components'

& Fouriers Law ser#es to define the thermal conducti#it)of the

medium

*k q T

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Heat Flux Components

"+',-$T T Tq k i k j k k r r z

=

rq q zq

C)lindrical Coordinates% ( ). .T r z

sin

T T Tq k i k j k k

r r r

=

"+'+,$

rq q q

/pherical Coordinates% ( ). .T r

Cartesian Coordinates% ( ). .T x y z

T T T

q k i k j k k x y z

= xq yq zq

"+'0$

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Heat Flux Components "cont'$

!n angular coordinates . the temperature gradient is still

ased on temperature change o#er a length scale and hence has

units of C*m and not C*deg'

( )or .

Heat ratefor one1dimensional. radial conductionin a c)linder or sphere%

& C)linder

+r r r r q A q rLq = =

or.

+r r r r q A q rq = =

& /phere

+2r r r r q A q r q = =

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Heat Equation

The Heat Equation A differential equation whose solution pro#ides the temperature distriution in a

stationar) medium'

3ased on appl)ing conser#ation of energ) to a differential control #olume

through which energ) transfer is exclusi#el) ) conduction'

Cartesian Coordinates%

4et transferof thermalenerg) into the

control #olume "inflow1outflow$

p

T T T T k k k q c

x x y y z z t

+ + + =

"+',0$

Thermal energ)

generationChange in thermal

energ) storage

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Heat Equation "5adial /)stems$

+

, ,

p

T T T T

kr k k q cr r r z z t r

+ + + =

"+'+6$

/pherical Coordinates%

C)lindr

ical Coordinates%

+

+ + + +

, , ,sin

sin sin p

T T T T kr k k q c

r r tr r r

+ + + =

"+'00$

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Heat Equation "/pecial Case$

7ne1(imensional Conductionin a 8lanar 9ediumwith Constant 8roperties

and4o :eneration

+

+

,T T

tx

=

thermal diffu osi#it f the med) iump

k

c

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3oundar) Conditions

3oundar) and !nitial Conditions For transient conduction. heat equation is first order in time. requiring

specification of an initial temperature distriution% ( ) ( )6

. .6t

T x t T x= =

/ince heat equation is second order in space. twooundar) conditions

must e specified' /ome common cases%

Constant /urface Temperature%

( )6. sT t T=

Constant Heat Flux%

6;x s

T

k qx =

=

Applied Flux Insulated Surface

6; 6x

T

x =

=

Con#ection

( )6; 6.xT

k h T T t x

=

=

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8roperties

Thermoph)sical 8roperties

Thermal Conducti#it)% A measure of a materials ailit) to transfer thermal

energ) ) conduction'

Thermal (iffusi#it)% A measure of a materials ailit) to respond to changes

in its thermal en#ironment'

8ropert) Tales%

/olids% Tales A', & A'0

:ases% Tale A'2

Liquids% Tales A'< & A'=

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Conduction Anal)sis

9ethodolog) of a Conduction Anal)sis

/ol#e appropriate form of heat equation to otain the temperature

distriution'

>nowing the temperature distriution. appl) Fouriers Law to otain the

heat flux at an) time. location and direction of interest'

Applications%

Chapter 0% 7ne1(imensional. /tead)1/tate Conduction

Chapter 2% Two1(imensional. /tead)1/tate Conduction

Chapter

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8rolem % Thermal 5esponse of 8lane ?all

8rolem +'2@ Thermal response of a plane wall to con#ection heat transfer'

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8rolem% Thermal 5esponse "Cont'$

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in con# s6E q A dt

=

d$ The total energ) transferred to the wall ma) e expressed as

( )( )in s 6E hA T T L.t dt

=

(i#iding oth sides ) AsL. the energ) transferred per unit #olume is

( ) 0in6

E hT T L.t dt *m

B L

=

8rolem% Thermal 5esponse "Cont'$

8 l 4 if : i d

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8rolem% 4on1niform :eneration due

to 5adiation Asorption

8rolem +'+- /urface heat fluxes. heat generation and total rate of radiation

asorption in an irradiated semi1transparent material with a

prescried temperature distriution'

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8rolem% 4on1niform :eneration "Cont'$

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8rolem% 4on1niform :eneration "Cont'$