PROBLEM 4.50
KNOWN:
Long rectangular bar having one boundary exposed to a convection process (T
∞
, h) while the
ther boundaries are maintained at a constant temperature (T
s
).
o
FIND:
(a) Using a grid spacing of 30 mm and the GaussSeidel method, determine the nodal
temperatures and the heat rate per unit length into the bar from the fluid, (b) Effect of grid spacing and
onvection coefficient on the temperature field.
c
SCHEMATIC:
ASSUMPTIONS:
(1) Steadystate, twodimensional conduction, (2) Constant properties.
ANALYSIS:
(a) With the grid spacing
Δ
x =
Δ
y = 30 mm, three nodes are created.
Using the finite
difference equations as shown in Table 4.2, but written in the form required of the GaussSeidel method
(see Section 4.5.2), and with Bi = h
Δ
x/k = 100 W/m
2
⋅
K
×
0.030 m/1 W/m
⋅
K = 3, we obtain:
Node 1:
()
(
12
s
2
2
11
1
T
T
T
BiT
T
50 3 100
T
350
Bi
2
5
5
∞
=+
+
=
+
+
×
=
+
+
)
(1)
Node 2:
(
2
1
s3
13
1
T
T
2T
T
T
T
2 50
T
T
100
44
4
=
+
+
=
++
×
=
)
(2)
Node 3:
(
32
s
2
2
1
T
T
3T
T
3 50
T
150
4
×
)
(3)
Denoting each nodal temperature with a superscript to indicate iteration step, e.g.
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 Spring '10
 LEE,J.S.
 Heat, Heat Transfer, 1 W, 0.01°C, 5 mm, 2 K, 0.02°C

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