(a)
h, T ∞
T i
t
–L
0
L
(b)
T i
t
T s
–L
0
L
Constant surface
Convective boundary condition
temperature boundary condition
74
Analytical Heat Transfer
FIGURE 4.4
1-D transient heat conduction.
Initial condition:
θ(x, 0) = θ i = T i − T ∞
Boundary conditions:
⎧ ∂θ(0, t)
⎪
{
⎨
= 0
∂x
c 1 = 0
⇒
⎪ ∂θ(L, t)
−kc 2 (− sin λL)λ = hc 2 cos λL
⎩ −k
= hθ(L, t)
∂x
k sin(λL) · λ = h cos(λL)
h
λ n = cot(λ n L),
k
hL
λ n L =
cot(λ n L) = Bi cot(λ n L).
k
λ n is determined by the convection BC.
−λ 2 αt
−λ 2 αt
θ = c 2 cos λ n x · c 3 e n = c n cos(λ n x) e n ,
where c n = c 2 c 3 .
Applying the initial condition
0
θ i = c n cos(λ n x) e ,
2
θ i cos(λ n x) = c n cos (λ n x),
� L
θ i 0 cos(λ n x) dx
θ i 2 sin λ n
c n =
=
,
�
0
L cos 2 (λ n x) dx
λ n + sin λ n cos λ n
where λ n L = λ n