Page 103 - Design and Operation of Heat Exchangers and their Networks

Steady-state characteristics of heat exchangers 91

∞ n ∞ k
ð
X ð 1Þ n X ð 1Þ n + k 1Þ! k
t c x, yÞ ¼ n 2 y 2 x (3.115)
ð
k!
n!
ðÞ ðÞ
n¼1 k¼0
Solutions with exponential function and double power series (Nusselt, 1930):
∞
n 1
X n X k
y
x
t h x, yÞ ¼ 1 e x y (3.116)
ð
n! k!
n¼1 k¼0
∞
n 1
X n X k
y
x
t c x, yÞ ¼ e x y (3.117)
ð
n! k!
n¼1 k¼0
Eq. (3.116) can be rewritten in a more compact form as (Luo, 1998):
∞
n
X n X k
y
x
t h x, yÞ ¼ e x y (3.118)
ð
n! k!
n¼0 k¼0
Eqs. (3.116), (3.118) are easy to be calculated and converge rapidly. The
dimensionless outlet temperature distributions of hot and cold fluids are
∞ n k
y
X n X NTU
00
t yðÞ ¼ e NTU h y h (3.119)
h n! k!
n¼1 k¼0
∞ n n 1
x
X ð R h NTU h Þ X k
00
t xðÞ ¼ e x R h NTU h (3.120)
c n! k!
n¼1 k¼0
The mean dimensionless outlet temperature of hot fluid can be derived
by integrating Eq. (3.119) from 0 to R h NTU h as
∞ n 1 n 1 k
X ð R h NTU h Þ X NTU
ð
t 00 ¼ e 1+ R h ÞNTU h ð n kÞ h (3.121)
h,m n! k!
n¼1 k¼0
According to Eq. (3.101), the dimensionless temperature change ε h can
be expressed as
∞ n 1 n 1 k
X ð R h NTU h Þ X NTU
ð
ε h ¼ 1 t 00 ¼ 1 e 1+ R h ÞNTU h ð n kÞ h
h,m n! k!
n¼1 k¼0
(3.122)
A MatLab program containing the functions crossflow_t_h (x, y), cross-
flow_t_c (x, y), and crossflow_t_h_m (x, y) for calculating Eqs. (3.119),
(3.121), respectively, can be found in the MatLab program “crossflow.m”
in the appendix.