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PROBLEM 4.3
K
NOWN:
Temperature distribution in the twodimensional rectangular plate of Problem 4.2.
FIND:
Expression for the heat rate per unit thickness from the lower surface (0
≤
x
≤
2, 0) and result
ased on first five nonzero terms of the infinite series.
b
SCHEMATIC:
A
SSUMPTIONS:
(1) Twodimensional, steadystate conduction, (2) Constant properties.
A
NALYSIS:
The heat rate per unit thickness
from
the plate
along the lower surface is
()
x2
out
y
2
1
y0
x0
T
qd
q
x
,
0
k
d
x
k
T
T
yy
∂∂
==
=
=
=
=
′′
=−
−
=
−
∫∫
∫
d
x
θ
(1)
w
here from the solution to Problem 4.2,
n1
1
21
11
s
i
n
h
n
y
L
TT 2
nx
sin
TT
n
Ls
i
n
h
n
W
L
π
ππ
+
∞
=
−+
−
⎛⎞
≡=
⎜⎟
−
⎝⎠
∑
.
(2)
E
valuate the gradient of
θ
from Eq. (2) and substitute into Eq. (1) to obtain
out
2
1
n
L
c
o
s
h
n
y
L
2n
x
qk
T
T
s
i
n
d
nL
s
i
n
h
n
W
L
x
+
=
∞
=
=
=
′
∑
∫
2
out
2
1
n
T
T
c
o
s
ns
i
n
h
n
W
L
L
+
∞
=
=
x
⎡
⎤
⎢
⎥
′
−
⎢
⎥
⎣
⎦
∑
out
2
1
T
T
1
c
o
s
n
i
n
h
n
L
+
∞
=
′
⎡
⎤
−
⎣
⎦
∑
<
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 Spring '08
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