Page 102 - Design and Operation of Heat Exchangers and their Networks

90 Design and operation of heat exchangers and their networks

and
1
e t c ¼ e t h (3.108)
s +1
The inverse transformation of Eqs. (3.107), (3.108) is easy to be per-
formed, which yields the solutions in the space domain with the Bessel func-
tion of zero order:
Solutions with Bessel function of zero order (Nusselt, 1911; Anzelius, 1926)

x
ð

p ffiffiffiffiffiffi
y
0
x
t h x, yÞ ¼ 1 e e I 0 2 x y dx 0
0
ð
0
y
ð
p ffiffiffiffiffiffi
x y p ffiffiffiffiffi x y 0 0
¼ e I 0 2 xy + e e I 0 2 xy dy (3.109)
0
0
y
ð
p ffiffiffiffiffiffi
0
t c x, yÞ ¼ e x e y 0 I 0 2 xy dy 0 (3.110)
ð
0
The solution of the governing equation system (3.102)–(3.104) has sev-
eral different forms. Eqs. (3.109), (3.110) are only one pair of them. Other
forms of the solution are summarized as follows:
Solutions with derivative of Bessel function of zero order (Schumann, 1929):
∞ ∞
X X
n
n
t h x, yÞ ¼ 1 e x y x M n xyðÞ ¼ e x y y M n xyðÞ (3.111)
ð
n¼1 n¼0
∞ ∞
X X
n
n
t c x, yÞ ¼ 1 e x y x M n xyðÞ ¼ e x y y M n xyðÞ (3.112)
ð
n¼0 n¼1
n p ffiffi
d I 0 2 zÞ
ð
where M n zðÞ ¼ .
dz n
Solution with Bessel function of higher order (Goldstein, 1953):
∞ n=2
X y p ffiffiffiffiffi
t h x, yÞ ¼ e x y I n 2 xy (3.113)
ð
x
n¼0
Solutions with double power series (Binnie and Poole, 1937; Smith, 1934):
∞ n ∞ k
X ð 1Þ X ð 1Þ kn + k 1Þ!
ð
t h x, yÞ ¼ 1+ 2 y n 2 x k (3.114)
ð
n!
k!
n¼0 ðÞ k¼1 ðÞ