�
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Energy balance at the interior nodes:
4
q i→0 + q ˙(Δx · Δy · 1) = 0
(5.1)
i=1
T 1 − T 0
T 2 − T 0
T 3 − T 0
kΔy · 1 ·
+ kΔy · 1 ·
+ kΔx · 1 ·
Δx
Δx
Δy
T 4 − T 0
+ kΔx · 1 ·
+ q ˙(Δx · Δy) = 0
(5.2)
Δy
If Δx = Δy,
1
q ˙ΔxΔx
T 0 =
T 1 + T 2 + T 3 + T 4 +
(5.3)
4
k
Energy balance at boundary nodes (not needed if the surface temperature
is given):
Convection boundary on the surface nodes:
T 1 − T 0
Δx
T 4 − T 0
Δx
T 3 − T 0
kΔy · 1 ·
+ k
· 1 ·
+ k
· 1 ·
Δx
2
Δy
2
Δy
Δx
+ hΔy(T ∞ − T 0 ) + q ˙
· Δy = 0
(5.4)
2
106
Analytical Heat Transfer
T 0
y
T (x, y)
T 1
Grids cells
T 0
T 0
x
FIGURE 5.1
Finite difference method to solve 2-D heat conduction problem.
�
Energy balance at the interior nodes:
4
q i→0 + q ˙(Δx · Δy · 1) = 0
(5.1)
i=1
T 1 − T 0
T 2 − T 0
T 3 − T 0
kΔy · 1 ·
+ kΔy · 1 ·
+ kΔx · 1 ·
Δx
Δx
Δy
T 4 − T 0
+ kΔx · 1 ·
+ q ˙(Δx · Δy) = 0
(5.2)
Δy
If Δx = Δy,
1
q ˙ΔxΔx
T 0 =
T 1 + T 2 + T 3 + T 4 +
(5.3)
4
k
Energy balance at boundary nodes (not needed if the surface temperature
is given):
Convection boundary on the surface nodes:
T 1 − T 0
Δx
T 4 − T 0
Δx
T 3 − T 0
kΔy · 1 ·
+ k
· 1 ·
+ k
· 1 ·
Δx
2
Δy
2
Δy
Δx
+ hΔy(T ∞ − T 0 ) + q ˙
· Δy = 0
(5.4)
2
106
Analytical Heat Transfer
T 0
y
T (x, y)
T 1
Grids cells
T 0
T 0
x
FIGURE 5.1
Finite difference method to solve 2-D heat conduction problem.