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Analytical Heat Transfer
5.2 Finite-Difference Energy Balance Method for 1-D
Transient Heat Conduction
The heat equation for 1-D transient heat conduction with heat generation is
∂ 2 T
q ˙
1 ∂T
+ =
.
∂x 2
k
α ∂t
This equation is a parabolic equation. As discussed in Chapter 4, this equation can be solved analytically by using the separation of variables method,
similarity method, Laplace transform method, or integral method. In this
chapter, we would like to solve 1-D and 2-D transient conduction problems with various BCs by using the finite-difference explicit method and the
finite-difference implicit method [1].
5.2.1 Finite-Difference Explicit Method
This is finite difference in space with an explicit form (lower bond). At a given
interior point, heat conductions from the neighborhood points are based on
the previous time-step temperatures in order to increase that point temperature during the incremental time-step change. This method is limited by the
instability problem, but easy to understand and calculate.
The finite-difference format of the above 1-D transient heat conduction
equation can be written as
(T 1
P − T 2
P )/Δx + (T 3
P − T 2
P )/Δx q ˙
1 T
P+1
− T P
+ =
2
2
(5.13)
Δx
k
α
Δt
Example 5.3
Energy balance at the interior nodes, for example, node 2, as shown in Figure 5.7:
q = energy storage
Lower bond:
T 1
P − T P
T 3
P − T P
ρC p Δx · y · 1 ·(T
P +1 − T 2
P )
2
2
2
k · y · 1
+ k · y ·1
+ q ˙ · Δx · y · 1 =
.
Δx
Δx
Δt
(5.14)
Let q ˙ = 0, α = (k /ρC p ), one obtains
T 1
P + T 3
P − 2T P =
1
(T
P +1 − T 2
P )
2
2
Fo
T
P +1 = Fo(T 1
P + T 3
P ) + (1 − 2Fo)T P
(5.15)
2
2
Analytical Heat Transfer
5.2 Finite-Difference Energy Balance Method for 1-D
Transient Heat Conduction
The heat equation for 1-D transient heat conduction with heat generation is
∂ 2 T
q ˙
1 ∂T
+ =
.
∂x 2
k
α ∂t
This equation is a parabolic equation. As discussed in Chapter 4, this equation can be solved analytically by using the separation of variables method,
similarity method, Laplace transform method, or integral method. In this
chapter, we would like to solve 1-D and 2-D transient conduction problems with various BCs by using the finite-difference explicit method and the
finite-difference implicit method [1].
5.2.1 Finite-Difference Explicit Method
This is finite difference in space with an explicit form (lower bond). At a given
interior point, heat conductions from the neighborhood points are based on
the previous time-step temperatures in order to increase that point temperature during the incremental time-step change. This method is limited by the
instability problem, but easy to understand and calculate.
The finite-difference format of the above 1-D transient heat conduction
equation can be written as
(T 1
P − T 2
P )/Δx + (T 3
P − T 2
P )/Δx q ˙
1 T
P+1
− T P
+ =
2
2
(5.13)
Δx
k
α
Δt
Example 5.3
Energy balance at the interior nodes, for example, node 2, as shown in Figure 5.7:
q = energy storage
Lower bond:
T 1
P − T P
T 3
P − T P
ρC p Δx · y · 1 ·(T
P +1 − T 2
P )
2
2
2
k · y · 1
+ k · y ·1
+ q ˙ · Δx · y · 1 =
.
Δx
Δx
Δt
(5.14)
Let q ˙ = 0, α = (k /ρC p ), one obtains
T 1
P + T 3
P − 2T P =
1
(T
P +1 − T 2
P )
2
2
Fo
T
P +1 = Fo(T 1
P + T 3
P ) + (1 − 2Fo)T P
(5.15)
2
2