h, T ∞
T (x,t)
T (x,t)
t
0
�
�
� �
� �
�
�
�
�
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4.3.3 Approximate Integral Method for Semiinfinite Solid Material
Let θ = T − T s ,
θ i = T i − T s , t = 0
∂θ(δ, t)
θ(0, t) = 0,
= 0, t > 0
∂x
∂θ
∂ 2 θ
= α
(4.30)
∂t
∂x 2
δ
δ
∂ 2 θ
∂θ dx =
∂t
dx
α
∂x 2
0
0
∂θ
∂θ
= α
− α
∂x
∂x
x=δ
x=0
where
δ
δ
δ
dδ
dδ
d
dδ
θ dx −
θ dx −
d
∂θ dx ≡
· θ i + 0
θ x=δ +
θ x=0 =
dt
dt
dt
dt
dt
∂t
0
0
0
δ
� �
� �
dδ
θ dx−
d
∂θ
∂θ
∴
· θ i = α
− α
dt
∂x
dt
∂x
x=δ
x=0
0
⎡
⎤
δ
� �
d
∂θ
⎦
θ dx − θ i δ = −α
⇒
⎣
dt
∂x x=0
0
Assume a temperature profile is a second-order polynomial,
θ(x, t) = a + bx + cx
2
85
Transient Heat Conduction
FIGURE 4.11
Solution of 1-D transient heat conduction convective surface boundary condition.