1
2
Δx
T 1
T 4
T 0
T 3
h, T ∞
Given surface temperature
Δx
Δy
Insulated
wall
Convection
T 1
x
y
h, T ∞
q x '
Δy
Uniform heat flux
1 Δy
2
Δx
T 2
T 1
T 3
T 0
q x '
T 4
T 3
T 2
T 1
Δy
Δy
Δy
Δx
Δx
Δx
i + 1, j
i – 1, j
i , j – 1
i , j + 1
m, n + 1
m + 1, n
m, n – 1
m – 1, n
T 0
107
Numerical Analysis in Heat Conduction
FIGURE 5.2
Finite difference method to solve 2-D heat conduction problem.
Uniform heat flux at the surface nodes:
Δy
T 1 − T 0
T 3 − T 0
Δy
T 2 − T 0
k
· 1 ·
+ kΔx · 1 ·
+ k
· 1 ·
2
Δx
Δy
2
Δx
Δy
""
+ q s Δx · 1 + q ˙
· Δx = 0
(5.5)
2
""
If insulation BCs applies to the surface nodes, then q = 0.
s
In general, T 0 , T 1 , T 2 , T 3 , and T 4 can be replaced by T m,n , T m−1,n ,
T m+1,n , T m,n−1 , and T m,n+1 or by T i,j , T i−1,j , T i+1,j , T i,j−1 , and T i,j+1 ,
where m = 1, 2, 3, . . ., n = 1, 2, 3, . . . or i = 1, 2, 3, . . ., j = 1, 2, 3, . . ., to obtain
T 1,1 , T 1,2 , T 1,3 , . . . , T 2,1 , T 2,2 , T 2,3 , . . ., and T 3,1 , T 3,2 , T 3,3 , . . . .
2
Δx
T 1
T 4
T 0
T 3
h, T ∞
Given surface temperature
Δx
Δy
Insulated
wall
Convection
T 1
x
y
h, T ∞
q x '
Δy
Uniform heat flux
1 Δy
2
Δx
T 2
T 1
T 3
T 0
q x '
T 4
T 3
T 2
T 1
Δy
Δy
Δy
Δx
Δx
Δx
i + 1, j
i – 1, j
i , j – 1
i , j + 1
m, n + 1
m + 1, n
m, n – 1
m – 1, n
T 0
107
Numerical Analysis in Heat Conduction
FIGURE 5.2
Finite difference method to solve 2-D heat conduction problem.
Uniform heat flux at the surface nodes:
Δy
T 1 − T 0
T 3 − T 0
Δy
T 2 − T 0
k
· 1 ·
+ kΔx · 1 ·
+ k
· 1 ·
2
Δx
Δy
2
Δx
Δy
""
+ q s Δx · 1 + q ˙
· Δx = 0
(5.5)
2
""
If insulation BCs applies to the surface nodes, then q = 0.
s
In general, T 0 , T 1 , T 2 , T 3 , and T 4 can be replaced by T m,n , T m−1,n ,
T m+1,n , T m,n−1 , and T m,n+1 or by T i,j , T i−1,j , T i+1,j , T i,j−1 , and T i,j+1 ,
where m = 1, 2, 3, . . ., n = 1, 2, 3, . . . or i = 1, 2, 3, . . ., j = 1, 2, 3, . . ., to obtain
T 1,1 , T 1,2 , T 1,3 , . . . , T 2,1 , T 2,2 , T 2,3 , . . ., and T 3,1 , T 3,2 , T 3,3 , . . . .