�
�
�
At t = 0, θ = 1, U = r ,
∫
r o
0 r sin(λ n (r /r o )) dr

C n = ∫
r
0
o sin 2 (λ n (r /r o )) dr

2(sin(λ n ) − λ n cos(λ n ))

C n = λ n − sin(λ n )cos(λ n )
∞ 2 [sin(λ n ) − λ n cos(λ n )] sin(λ n (r /r o )) −λ 2 F 0
θ =
·
· e n
λ n − sin(λ n )cos(λ n )
λ n (r /r o )
n=1
4.3. Solve transient temperature profiles in a semiinfinite solid body using the
Laplace transform for the following BCs of constant surface heat flux.
If at time t = 0 the surface is suddenly exposed to a constant heat flux
""
q —for example, by radiation from a high-temperature source—the resulting
s
temperature response is
""
� 1/2
q
4αt
x
s
−x
T − T i =
e
2 /4αt − x erfc
k
π
(4αt ) 1/2
SOLUTION
1-D transient:

∂ 2 T
1 ∂T

=
∂x 2
α ∂t
Initial condition:

t = 0; T = T i .

Boundary conditions:
""
i. x = 0; −k ∂T = q s
∂x
ii. x → ∞; T = T i
Let θ = T − T i ,

∂ 2 θ
1 ∂θ

=
∂x 2
α ∂t
Initial condition:

t = 0; θ = 0.

Boundary conditions:
""
i. x = 0; −k ∂θ = q
∂x
s
ii. x → ∞; θ = 0
96
Analytical Heat Transfer