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Applying the Laplace transform,
∂ 2 θ ˜
s
− θ ˜ = 0
(1)
∂x 2
α
Boundary conditions:
−k ∂θ ˜
q s
""
i. x = 0;
=
∂x
s
˜
ii. x → ∞; θ = 0.
Solving,
√
√
−
s/αx
θ ˜ = C 1 e s/αx + C 2 e
(2)
Applying BC (ii), C 2 = 0.
√
""
Applying BC (i), C 1 = (q /s)(1/k s/α).
s

Substituting this into above (2),

q "" 1
√
s
− s/αx
θ ˜ =
√
e
(3)
s k s/α
Rearranging,
√
""
˜
q s /k α −(x/
√
α)
√
s
θ =
e
s 3/2
Applying the Laplace inverse,
""
2
q √
t
x
x
x
s
θ =
α 2
exp −
− √ erfc
√
k
π
4αt
α
2 αt
""
� 1/2
2
q
4αt
x
x
s
T − T i =
exp −
− x erfc
k
π
4αt
(4αt ) 1/2
4.4. Solve transient temperature profiles in a semiinfinite solid body using the
Laplace transform for the following BCs of convective heat transfer to the
surface.
If at time t = 0 the surface is suddenly exposed to a fluid at temperature
T ∞ , with a convective heat transfer coefficient h, the resulting temperature
response is
T − T i
x
x
h
= erfc
− e hx/k +(h/k ) 2 αt erfc
+ (αt ) 1/2
T ∞ − T i
(4αt ) 1/2
(4αt ) 1/2
k
SOLUTION
1-D transient
∂ 2 T
1 ∂T
=
∂x 2
α ∂t
97
Transient Heat Conduction