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PROBLEM 4.33
K
NOWN:
Plane surface of twodimensional system.
FIND:
The finitedifference equation for nodal point on this boundary when (a) insulated; compare
esult with Eq. 4.42, and when (b) subjected to a constant heat flux.
r
SCHEMATIC:
ASSUMPTIONS:
(1) Twodimensional, steadystate conduction with no generation, (2) Constant
roperties, (3) Boundary is adiabatic.
p
ANALYSIS:
(a) Performing an energy balance on the control volume, (
Δ
x/2)
⋅Δ
y, and using the
onduction rate equation, it follows that
c
(1,2)
in
out
1
2
3
E
E
0
q
q
q
0
−=
+
+
=
±±
()
m1,n
m,n
m,n1
m,n
m,n+1
m,n
TT
xx
ky
1
k
1
k
1
0
x2y2y
−−
ΔΔ
⎡⎤
Δ⋅
+
⋅
+
⋅
=
⎢⎥
ΔΔΔ
⎣⎦
.
−
(3)
Note that there is no heat rate across the control volume surface at the insulated boundary.
ecognizing that
Δ
x =
Δ
y, the above expression reduces to the form
R
(4)
<
m1,n
m,n1
m,n+1
m,n
2T
T
T
4T
0.
++
The Eq. 4.42 of Table 4.2 considers the same configuration but with the boundary subjected to a
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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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