�
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given by
∂ 2 T
1 ∂T
=
∂x 2
α ∂t

The surface BC is

T(0, t) = T s

And the interior BC is prescribed by

T(∞, t) = T i
The initial condition is
T(x, 0) = T i
By applying the similarity method, we may transform the PDE, which
involves two independent variables (x and t), to an ODE expressed in terms
of a single similarity variable (η).
√
√
As x ∼ αt ( αt is the diffusion length), we define the similarity variable
√
η = (x/ 4αt); therefore T(x, t) = T(η),
√
x = η 4αt
(4.18)
Let θ = ((T − T i )/(T s − T i )).
The 1-D transient heat equation can be expressed as
∂ 2 θ
1 ∂θ
=
∂x 2
α ∂t
Applying the similarity variable into θ, that is, θ(x, t) = θ(η)
Therefore,
∂θ
dθ ∂η
dθ
−x
=
=
√
∂t
dη ∂t
dη 2t 4αt
∂θ
dθ ∂η
dθ
1
=
=
√
∂x
dη ∂x
dη
4αt
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∂ 2 θ
d ∂θ ∂η
d ∂θ
1
1
1 ∂ 2 θ
=
=
√
√
=
∂x 2
dη ∂x ∂x
dη ∂η
4αt
4αt
4αt ∂η 2

Inserting the above terms into the heat equation, we obtain

1 d 2 θ
1
−x
dθ

=
4αt dη 2
α 2t(4αt) 1/2 dη
79
Transient Heat Conduction