�
�
�
�
�
�
⎧
⎪
∂T
⎪ ⎪ h(T − T ∞ ) = −k
⎪ ⎪
∂y
⎪ ⎨
∂(T − T 1 )
y = b, 0 < x < a : h[(T − T 1 ) − (T ∞ − T 1 )] = −k
⎪
∂y
⎪ ⎪ ⎪
∂θ
⎪ ⎪ ⎩ h(θ − θ ∞ ) = −k
∂y
The solution is subject to the three homogeneous BCs with θ replacing T,
∞
nπx
nπy
θ(x, y) =
C n sin
sinh
a
a
n=1
and at y = b, applying nonhomogeneous BCs to solve for C n :
∞
∂θ
nπx nπ
nπb
h
nπx
nπb
=
C n sin
cosh
= −
C n sin
sinh
− θ ∞
∂y
a a
a
k
a
a
n=1
n
nπx nπ
nπb h
nπb
h
C n sin
cosh
+ sinh
= θ ∞
a
a
a
k
a
k
n=1
a
(h/k)θ ∞ 0 sin(nπx/a) dx
C n =
a
((nπ/a) cosh(nπb/a) + (h/k) sinh(nπb/a)) sin
2 (nπx/a) dx
0
(h/k)θ ∞ (a/nπ) [1 − (−1) n ]
=
((nπ/a) cosh(nπb/a) + (h/k) sinh(nπb/a)) [(a/2)]
θ ∞ (2/nπ) [1 − (−1) n ]
=
(sinh(nπb/a) + (nπ/a)(h/k) cosh(nπb/a))
Therefore, we obtain
T − T 1
∞ (2/nπ) [1 − (−1) n ] sin(nπx/a) sinh(nπy/a)
=
(3.12)
T ∞ − T 1
sinh(nπb/a) + (nπ/a)(h/k) cosh(nπb/a)
n=1
52
Analytical Heat Transfer
3.3 Principle of Superposition for Nonhomogeneous BCs
Superposition
In some applications, we may have all three kinds of surface BCs applied
to a given problem. For example, Figure 3.4 shows a 2-D heat conduction
problem with given surface temperature, heat flux, and convection BCs,