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PROBLEM 4.70
K
NOWN:
Straight fin of uniform cross section with insulated end.
FIND:
(a) Temperature distribution using finitedifference method and validity of assuming one
dimensional heat transfer, (b) Fin heat transfer rate and comparison with analytical solution, Eq. 3.76, (c)
ffect of convection coefficient on fin temperature distribution and heat rate.
E
SCHEMATIC:
ASSUMPTIONS:
(1) Steadystate conditions, (2) Onedimensional conduction in fin, (3) Constant
roperties, (4) Uniform film coefficient.
p
ANALYSIS:
(a) From the analysis of Problem 4.43, the finitedifference equations for the nodal
arrangement can be directly written.
For the nodal spacing
Δ
x = 4 mm, there will be 12 nodes.
With
A
>> w representing the distance normal to the page,
( )
2
22
2
3
3
c
hP
h 2
h 2
500W m
K 2
x
x
x
4 10
mm
0.0533
kA
k
w
kw
50W m K 6 10
m
−
−
⋅⋅
⋅
×
⋅Δ
≈
Δ
=
Δ
=
×
=
⋅××
A
A
Node 1:
or
2.053T
()
2
100 T
0.0533 30
2 0.0533 T
0
++
×−
+
=
1
1
+ T
2
= 101.6
Node n:
or
n1
n
T
T
1.60 2.0533T
0
+−
−
=
n
T
2.053T
T
1.60
−
−
−
+=
−
Node 12:
11
12
T
0.0533 2 30
0.0533 2 1 T
0
+
=
or
11
12
T
1.0267T
0.800
−
=−
Using matrix notation, Eq. 4.48, where [A] [T] = [C], the Amatrix is tridiagonal and only the nonzero
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 Spring '10
 LEE,J.S.
 Heat Transfer

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