(
)�
(
)�
(
)�
(
)�
Finite-difference explicit form (lower bond):
T
p
− T
p
T
p
− T
p
Δx +
Δx
m+1,n
m,n
m−1,n
m,n
Δx
T m
p
,n+1 − T m
p
,n Δy + T m
p
,n−1 − T m
p
,n Δy q ˙ 1 T
p+1 − T
p
m,n
m,n
+
+ =
Δy
k α
Δt
(5.22)
Let Δx = Δy, q ˙ = 0, Fo = αΔt/Δx 2 , and temperature at (P + 1) step is
determined by the preceding time temperature at P step as
(
)
T
p+1
T
p
m,n = Fo m+1,n + T m
p
−1,n + T m
p
,n+1 + T m
p
,n−1 + (1 − 4Fo)T m
p
,n
(5.23)
For the stability criterion, (1 − 4Fo) ≥ 0, that is, Fo ≤ 1/4.

For the 1-D transient heat conduction problem, one obtains

T
p+1
m
= Fo (T m
p
+1 + T m
p
−1 ) + (1 − 2 Fo)T m
p
For stability, (1 − 2Fo) ≥ 0, that is, Fo ≤ 1/2

Finite-difference implicit form (upper bond):

(
)
(
)
T
p+1 − T
p+1
T
p+1 − T
p+1
/Δx +
/Δx
m+1,n
m,n
m−1,n
m,n
Δx
(
)
(
)
T
p+1
T
p+1
m,n+1 − T m
p+
,n
1 /Δy + m,n−1 − T m
p+
,n
1 /Δy q ˙ 1 T
p+1 − T m
p
,n
m,n
+
+ =
Δy
k α
Δt
(5.24)
Let Δx = Δy, q ˙ = 0, Fo = αΔt/Δx 2 , and temperature at (P + 1) step is
determined by
T
p+1 − T
p+1 + T
p+1 − T
p+1 + T
p+1

m+1,n
m,n
m−1,n
m,n
m,n+1

− T
p+1 + T
p+1
1 (
T
p+1 − T
p
)
m,n
m,n−1 − T m
p+
,n
1 =
m,n
m,n
Fo
(
)
T
p+1 + T
p+1 + T
p+1
(1 + 4 Fo)T
p+1 − Fo
m,n+1 + T
p+1
= T
p
(5.25)
m,n
m+1,n
m−1,n
m,n+1
m,n
For a 1-D transient heat conduction problem, one obtains
(1 + 2 Fo)T m
p+1 − Fo(T m
p
+1 + T m
p
−1 ) = T m
p
It can be solved by a computer matrix, and there is no instability issue.
118
Analytical Heat Transfer