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Chapter 2
Heat Conduction Equation
229
We consider a thin ring shaped volume element of width
Δ
z
and thickness
Δ
r
in a cylinder. The
density of the cylinder is
ρ
and the specific heat is
C.
In general, an
energy balance
on this ring element
during a small time interval
Δ
t
can be expressed as
t
E
Q
Q
Q
Q
z
z
z
r
r
r
Δ
Δ
=
−
+
−
Δ
+
Δ
+
element
)
(
)
(
&
&
&
&
But the change in the energy content of the element can be expressed as
Δ
Δ
Δ
ΔΔ
Δ
E
m
C
T
T
C
r
r
z
T
T
t
tt t
=
EE
tt
element
=−
−
=
−
++
+
()
(
)
ρ
π
2
Substituting,
(
&&
)(
)( )
QQ
Cr
rz
TT
t
rr
r
zz
z
−+
−=
−
+
Δ
Δ
ρπ
2
Δ
z
r+
Δ
r
r
Dividing the equation above by
2
rr z
Δ
Δ
gives
−
−
−
−
=
−
1
2
1
2
ππ
+
r z
r
r r
z
C
t
r
z
Δ
Δ
r
2
Noting that the heat transfer surface areas of the element for heat conduction in the
r
and
z
directions are
Ar
z
==
2
Δ
and
,
Δ
respectively, and taking the limit as
Δ
Δ
Δ
t
,
and
→
0 yields
t
T
C
z
T
k
z
T
k
r
r
T
kr
r
r
∂
∂
ρ
=
⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
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 Fall '10
 Dr.DanielArenas
 Thermodynamics, Energy, Mass, Heat

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