�
�
�
�
�
�
�
�
�
�
�
�
1 (
)
T 8 =
T 5 + T 7 + T 9 + T b
4
1 (
)
T 9 =
T 6 + T 8 + T 8 + T b
4
For the boundary, nodal temperatures are
k Δy
k Δx
k Δy
k Δx
k Δx
−
+
+ hΔy T 1 +
T 2 +
T 4 = −
T a − hΔyT ∞
Δx
Δy
Δx
2Δy
2Δy
k Δx
k Δy
k Δx
k Δy
k Δx
T 1 −
+
+ hΔy T 4 +
T 5 +
T 7 = −hΔyT ∞
2Δy
Δx
Δy
Δx
2Δy
k Δx
k Δy
k Δx
k Δy
k Δx
T 4 −
+
+ hΔy T 7 +
T 8 = −
T b − hΔyT ∞
2Δy
Δx
Δy
Δx
2Δy
In matrix form [A] [T ] = [C ],
⎡
k Δy
k Δx
k Δy
k Δx
−
+
+ hΔy
⎢
Δx
Δy
Δx
2Δy
⎢
⎢
⎢
1
−4
1
⎢
⎢
2
−4
⎢
⎢
⎢
k Δx
k Δy
k Δx
⎢
−
+
+ hΔy
⎢
2Δy
Δx
Δy
⎢
[A] = ⎢
1
1
⎢
⎢
⎢
1
⎢
⎢
⎢
k Δx
⎢
⎢
2Δy
⎢
⎢
⎣
⎤
⎥
1
⎥
⎥
⎥
1
⎥
⎥
⎥
k Δy
k Δx
⎥
⎥
Δx
2Δy
⎥
⎥
−4
1
1
⎥
⎥
⎥
2
−4
1 ⎥
⎥
⎥
k Δy
k Δx
k Δy
⎥
−
+
+ hΔy
⎥
Δx
Δy
Δx
⎥
⎥
1
1
−4
1 ⎥
⎦
1
2
−4
120
Analytical Heat Transfer