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PROBLEM 4.59
KNOWN:
Upper surface and grooves of a plate are maintained at a uniform temperature T
1
, while the
ower surface is maintained at T
l
2
or is exposed to a fluid at T
∞
.
FIND:
(a) Heat rate per width of groove spacing (w) for isothermal top and bottom surfaces using a
initedifference method with
Δ
x = 40 mm, (b) Effect of grid spacing and convection at bottom surface.
f
SCHEMATIC:
(b)
ASSUMPTIONS:
(1) Steadystate, twodimensional conduction, (2) Constant properties.
ANALYSIS:
(a) Using a space increment of
Δ
x = 40 mm, the symmetrical section shown in schematic
(b) corresponds to onehalf the groove spacing.
There exist only two interior nodes for which finite
difference equations must be written.
Node a:
()
a1
b
2
1
4T
T
T
T
T
0
−+
+
+=
or
ab
4T
200 T
20
200
0
+
4T
T
420
−
=
(1)
Node b:
b1
a
2
a
4T
T
T
T
T
0
+
+
=
or
(
ba
4T
200
2T
20
0
+
=
)
2T
4T
220
−
(2)
Multiply Eq. (2) by 2 and add to Eq. (1) to obtain
7T
b
= 860
or
T
b
= 122.9
°
C
From Eq. (1),
4T
a
 122.9 = 420
or
T
a
= (420 + 122.9)/4 = 135.7
°
C.
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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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