Post on 01-Jun-2018
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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'$