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Chapter 2
Heat Conduction Equation
223
We consider a thin spherical shell element of thickness
Δ
r
in a sphere (see Fig. 217 in the text).
. The
density of the sphere is
ρ
, the specific heat is
C,
and the length is
L.
The area of the sphere normal to the
direction of heat transfer at any location
is
where
r
is the value of the radius at that location
.
Note that the heat transfer area
A
depends on
r
in this case
,
and thus it varies with location.
When there is
no heat generation, an
energy balance
on this thin spherical shell element of thickness
Δ
r
during a small
time interval
Δ
t
can be expressed as
Ar
=
4
2
π
&&
QQ
E
t
rr
r
−=
+Δ
Δ
Δ
element
where
Δ
Δ
ΔΔ
EE
E
m
C
T
T
C
A
r
T
tt
t
tt t
element
=−
=
Δ
T
−
=
−
++
+
()
ρ
Substituting,
&
g
A
r C
A
r
TT
t
r
−+=
−
+
+
Δ
Δ
Δ
where
.
Dividing the equation above by
A
Δ
r
gives
=
4
2
−
−
=
−
1
A
r
C
t
r
Taking the limit as
and
yields
Δ
r
→
0
Δ
t
→
0
t
T
C
r
T
kA
r
A
∂
=
⎟
⎠
⎞
⎜
⎝
⎛
1
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This note was uploaded on 01/14/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.
 Fall '10
 Dr.DanielArenas
 Thermodynamics, Mass, Heat

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