ChE 120B
2  1
Shell Balances — Spherical Geometry
S.S. Heat conduction with source term:
Try spherical geometry using a shell balance:
Input — Output + Source = 0
2


4

0
r
r
r
r
r
r
r
Q
Q
S
r
r
π
+Δ
−
+
Δ
=
Remember, we can substitute Fourier's Laws anytime to get:
r
dT
Q
k A
dr
= −
A = 4
π
r
2
(
)
2
2
2
4

4

4

0
r
r
r
r
r
dT
dT
k
r
k
r
S
r
r
dr
dr
π
π
π
+Δ
⎛
⎞
⎛
⎞
−
+
+
Δ
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
, rearranging we get:
2
2
2
0


lim
r
r
r
r
r
dT
dT
r
r
S
dr
dr
r
r
k
+Δ
Δ →
⎛
⎞
⎛
⎞
+
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
=
Δ
or
2
2
r
S
d
T
r
r
dr
r
k
∂
⎛
⎞
= −
⎜
⎟
∂
⎝
⎠
Integrating once we get
2
3
1
3
r
S
dT
r
r
C
dr
k
= −
+
or
1
2
3
r
rS
C
dT
dr
k
r
= −
+
We can use the obvious boundary condition that nothing real goes to
∞
:
as
0
dT
r
dr
→
→ ∞
unless
1
0
C
=
3
r
rS
dT
dr
k
= −
Alternative:
We can also say that the center of a sphere is a point of symmetry, so
0
0

r
T
r
=
∂
=
∂
; we get the same result.
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