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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
x x x x x x x ! x x x x x x x
(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
x x x x x x x ! x x x x x x x
(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.)