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PROBLEM 4.60
K
NOWN:
Rectangular plate subjected to uniform temperature boundaries.
F
IND:
Temperature at the midpoint using a finitedifference method with space increment of 0.25m
SCHEMATIC:
ASSUMPTIONS:
(1) Steadystate conditions, (2) Twodimensional conduction, (3) Constant
roperties.
p
ANALYSIS:
For the nodal network above, 12 finitedifference equations must be written.
It follows
that node 8 represents the midpoint of the rectangle.
Since all nodes are interior nodes, Eq. 4.29 is
appropriate and is written in the form
m
neighbors
4T
T
0.
−=
∑
For nodes on the symmetry adiabat, the neighboring nodes include two symmetrical nodes.
Hence, for
Node 4, the neighbors are T
b
, T
8
and 2T
3
.
Because of the simplicity of the finitedifference
equations, we may proceed directly to the matrices [A] and [C] – see Eq. 4.48 –
and matrix inversion
can be used to find the nodal temperatures T
m
.
4
1
0
0
1
0
0
0
0
0
0
0
1
4
1
0
0
1
0
0
0
0
0
0
0
1
4
1
0
0
1
0
0
0
0
0
0
0
2
4
0
0
0
1
0
0
0
0
1
0
0
0
4
1
0
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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