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Chapter 2
Heat Conduction Equation
221
We consider a thin element of thickness
Δ
x
in a large plane wall (see Fig. 213 in the text). The
density of the wall is
ρ
, the specific heat is
C,
and the area of the wall normal to the direction of heat
transfer is
A
. In the absence of any heat generation, an
energy balance
on this thin element of thickness
Δ
x
during a small time interval
Δ
t
can be expressed as
&&
QQ
E
t
xx
x
−=
+Δ
Δ
Δ
element
where
Δ
Δ
ΔΔ
EE
E
m
C
T
T
C
A
x
T
tt
t
tt t
element
=−
=
Δ
T
−
=
−
++
+
()
ρ
Substituting,
C
A
x
TT
t
x
−
+
+
Δ
Δ
Δ
Δ
Dividing by
A
Δ
x
gives
−
−
=
−
1
A
x
C
t
x
Taking the limit as
and
yields
Δ
x
→
0
Δ
t
→
0
t
T
C
x
T
kA
x
A
∂
=
⎟
⎠
⎞
⎜
⎝
⎛
1
since, from the definition of the derivative and Fourier’s law of heat conduction,
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Mass, Heat

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