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PROBLEM 4.45
K
NOWN:
Steadystate temperatures (K) at three nodes of a long rectangular bar.
FIND:
(a) Temperatures at remaining nodes and (b) heat transfer per unit length from the bar using
nodal temperatures; compare with result calculated using knowledge of q.
±
SCHEMATIC:
A
SSUMPTIONS:
(1) Steadystate, 2D conduction, (2) Constant properties.
ANALYSIS:
(a) The finitedifference equations for the nodes (1,2,3,A,B,C) can be written by
inspection using Eq. 4.35 and recognizing that the adiabatic boundary can be represented by a
symmetry plane.
()
2
73
2
2
neighbors
i
5 10 W/m
0.005m
qx
T
4T
q x / k
0
and
62.5K.
k2
0
W
/
m
K
×
Δ
−+
Δ
=
=
=
⋅
∑
±
±
Node A (to find
T
2
):
2
2BA
2T
2T
4T
q x / k
0
+
Δ
=
±
2
1
T
2 374.6
4 398.0 62.5 K
390.2K
2
=
−
×+
×−
=
<
Node 3 (to find
T
3
):
2
c2B
3
TTT
3
0
0
K
4
Tq
x
/
k0
+
++
−
+
Δ
=
±
3
1
T
348.5 390.2 374.6 300 62.5 K
369.0K
4
=
+++
+
=
<
Node 1 (to find
T
1
):
2
C2
1
300
2T
T
4T
q x / k
0
+
+−
+
Δ
=
±
1
1
T
300
2 348.5 390.2 62.5
362.4K
4
=+
×
+
+
=
<
(b) The heat rate out of the bar is determined by calculating the heat rate out of each control volume
around the 300 K nodes.
Consider the node in the upper lefthand corner; from an energy balance
in
out
g
a
a,in
g
g
E
E
E
0
or
q
q
E
where
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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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