Page 110 - Design and Operation of Heat Exchangers and their Networks

98 Design and operation of heat exchangers and their networks

∂G n x, yÞ
ð
¼ F n G n (3.162)
∂x
y n
F n 0, yð Þ ¼ ð n 0Þ (3.163)
n!
G n x,0ð Þ ¼ 0 (3.164)
Obviously, for n¼0, the previous equation system becomes the govern-
ing equation system for the crossflow with both fluids unmixed
(Eqs. 3.102–3.104). Therefore, we can express F 0 and G 0 as follows:
∞
m
X m X k
y
x
F 0 x, yð Þ ¼ e x y ð 3:118Þ, (3.165)
m! k!
m¼0 k¼0
∞
m 1
X m X k
x
y
G 0 x, yð Þ ¼ e x y ð 3:117Þ, (3.166)
m! k!
m¼1 k¼0
Using the recurrence relations given by Romie (1987),
ð
F n x, yð Þ ¼ G n x, yÞ + G n 1 x, yÞ (3.167)
ð
∞ m +1 m
X x y
F 1 x, yð Þ ¼ e x y (3.168)
ð m +1Þ!m!
m¼0
∞ m
xy
X ðÞ
x y
G 1 x, yð Þ ¼ e 2 (3.169)
m!
ðÞ
m¼0
∂G 1 x, yÞ
ð
G 2 x, yð Þ ¼ ¼ F 1 x, yð Þ G 1 x, yÞ
ð
∂y
∞ m
xy
X x ðÞ
¼ e x y 1 (3.170)
m +1 ðÞ 2
m!
m¼0
1
G n x, yð Þ ¼ ½ ð y x 2n +1ÞG n 1 x, yð Þ +2y n +1ÞG n 2 x, yÞ
ð
ð
n
+ yG n 3 ð x, yފ n 1Þ (3.171)
ð
we can obtain F n (x, y) for n¼1, 2, …, ∞. We can also use the series
expansions
∞
X k p
x y k=2 ffiffiffiffiffi
F n x, yð Þ ¼ e ð y=xÞ I k 2 xy n 0ð Þ (3.172)
n
k¼n