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Chapter 5
Numerical Methods in Heat Conduction
where
k
= 0.84 W/m.
°
C,
,
T
αρ
==
×
−
kC
/.
039 10
6
m/
s
2
i
=
T
o
= 3°C
h
i
= 6 W/m
2
.
°
C,
h
o
= 20 W/m
2
.
°
C,
Δ
x
= 0.002 m, and
Δ
y
= 0.01 m.
The upper limit of the time step
Δ
t
is determined from the stability criteria that requires the
coefficient of
in the
T
expression (the primary coefficient) be greater than or equal to zero for all
nodes. The smallest primary coefficient in the 9 equations above is the coefficient of
T
in the
expression since it is exposed to most convection per unit volume (this can be verified). The equation for
node 6 can be rearranged as
T
m
i
m
i
+
1
i
9
T
i
6
1
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ
+
Δ
+
+
Δ
Δ
+
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ
+
Δ
+
Δ
Δ
−
=
+
2
5
2
9
3
0
6
2
2
1
6
2
1
1
2
1
x
T
y
T
T
T
x
k
h
t
T
x
y
x
k
h
t
T
i
i
i
o
i
o
i
α
Therefore, the stability criteria for this problem can be expressed as
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ
+
Δ
+
Δ
≤
Δ
→
≥
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
Δ
+
Δ
+
Δ
Δ
−
2
2
2
2
1
1
2
1
t
0
1
1
2
1
x
y
x
k
h
x
y
x
k
h
t
o
o
Substituting the given quantities, the maximum allowable value of the time step is determined to be
or,
s
8
.
4
m)
01
.
0
(
1
)
m
002
.
0
(
1
m)
002
.
0
)(
C
W/m
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This note was uploaded on 01/22/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.
 Fall '10
 Dr.DanielArenas
 Thermodynamics, Mass, Heat

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