where
∂ 2 θ 1
q ˙
+ = 0
∂x 2
k
∂ 2 θ 2x
1 ∂θ 2x
=
∂x 2
α ∂t
∂ 2 θ 2y
1 ∂θ 2y
=
∂y 2
α ∂t
∂ 2 θ 2z
1 ∂θ 2z
=
∂z 2
α ∂t
A general form of the solution of 1-D transient heat conduction with the
convection BC applicable to a slab, a cylinder, and a sphere, as shown in
Figure 4.1, can be written [2] as
∞
θ
−λ 2 Fo
n
θ i
=
C n f (λ n η) e
(4.16)
n=1
∞
Q
−λ 2 Fo
n
= 1 −
C n F(λ n ) e
(4.17)
Q o
n=1
77
Transient Heat Conduction
–L
L
x
y
q
.
z
FIGURE 4.6
Multidimensional transient heat conduction with heat generation.
Geometry
θ(x, t)
C n
f (λ n η)
F(λ n )
Slab
(∂ 2 θ/∂x 2 )
= (1/α)(∂θ/∂t)
(2 sin λ n /(λ n
+ sin λ n cos λ n ))
cos(λ n (x/L))
sin λ n /λ n
Cylinder (1/r)(∂/∂r)(r(∂θ/∂r))
= (1/r)(∂θ/∂t)
(2J 1 (λ n )/(λ n [J 2
o (λ n )
+J 2
1 (λ n )]))
J o (λ n (r/r o ))
(2J 1 (λ n )/λ n )
Sphere
(1/r 2 )(∂/∂r)(r 2 (∂θ/∂r))
= (1/r)(∂θ/∂t)
(2[sin λ n − λ n cos λ n ])
�
(λ n − sin λ n cos λ n )
(sin(λ n (r/r o ))
�
(λ n (r/r o ))
3(sin λ n
−λ n cos λ n )/λ 3
n