T s
T s
T i
T s
y
z
x
T s
4.2.3 1-D Transient Heat Conduction in a Rectangle with Heat Generation
The governing equation for 1-D transient heat conduction with heat
generation is
∂ 2 θ q ˙
1 ∂θ
+ =
(4.12)
∂x 2
k
α ∂t
The temperature profile can be solved by separation of variables. For
example, to use the separation of variables method, we define
θ = θ 1 (x) + θ 2 (x, t)
(4.13)
d 2 θ 1
q ˙
+ = 0
dx 2
k
where θ 1 can be solved by 1-D heat conduction with internal heat generation
and θ 2 can be solved by the aforementioned method of separation of variables
with a given convection or surface temperature BCs.
For 3-D transient heat conduction with heat generation, as shown in
Figure 4.6, the following equations can be used to solve temperature profiles
T(x, y, z, t) or θ(x, y, z, t):
∂ 2 θ
∂ 2 θ ∂ 2 θ q ˙
1 ∂θ
+
+
+ =
(4.14)
∂x 2
∂y 2
∂z 2
k
α ∂t
θ = θ 1 (x) + θ 1 (y ) + θ 1 (z) + θ 2x (x, t) · θ 2y (y, t) · θ 2z (z, t)
(4.15)
76
Analytical Heat Transfer
FIGURE 4.5
Multidimensional transient heat conduction.
where θ x (x, t), θ y (x, t), and θ z (x, t) can be solved by the aforementioned
method of separation of variables with a given surface temperature or
convection BCs.