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PROBLEM 4.42
KNOWN:
Onedimensional fin of uniform cross section insulated at one end with prescribed base
emperature, convection process on surface, and thermal conductivity.
t
FIND:
Finitedifference equation for these nodes:
(a) Interior node, m and (b) Node at end of fin, n,
here x = L.
w
SCHEMATIC:
A
SSUMPTIONS:
(1) Steadystate conditions, (2) Onedimensional conduction.
ANALYSIS:
(a) The control volume about node m is shown in the schematic; the node spacing and
control volume length in the x direction are both
Δ
x.
The uniform crosssectional area and fin
perimeter are A
c
and P, respectively.
The heat transfer process on the control surfaces, q
1
and q
2
,
represent conduction while q
c
is the convection heat transfer rate between the fin and ambient fluid.
erforming an energy balance, find
P
()
in
out
1
2
c
m

1m
m
+
cc
E
E
0
q
q
q
0
TT
T
T
kA
kA
hP x T
T
0.
xx
∞
−=
+
+
=
−−
++
Δ
−
ΔΔ
±±
m
=
M
ultiply the expression by
Δ
x/kA
c
and regroup to obtain
22
m1
m+1
m
hP
hP
T
T
x T
2
x
T
0
1<m<n
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This note was uploaded on 12/07/2010 for the course MAE Heat Trans taught by Professor Lee,j.s. during the Spring '10 term at Seoul National.
 Spring '10
 LEE,J.S.
 Heat Transfer

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