Page 105 - Design and Operation of Heat Exchangers and their Networks

Steady-state characteristics of heat exchangers 93

with the boundary condition
x ¼ 0 : t h ¼ 1; y ¼ 0 : t c ¼ 0 (3.127)

in which t c is only the function of y, t c ¼ t c yðÞ.
Integration of Eq. (3.125) yields

t h x, yÞ ¼ t c yðÞ +1 t c yðފe x (3.128)
ð
½
Substituting Eq. (3.128) into (3.126), we obtain the following dimen-
sionless temperature distributions:

x 1 e NTU h Þy=NTU h
t h ¼ 1 1 e e ð (3.129)
ð
t c ¼ 1 e 1 e NTU h Þy=NTU h (3.130)
The dimensionless outlet temperatures are
1 e NTU h Þy=NTU h
00
t ¼ 1 1 e NTU h e ð (3.131)
h
00
ð
t ¼ 1 e 1 e NTU h ÞR h (3.132)
c
The dimensionless mean outlet temperature of hot fluid is derived by
integration of Eq. (3.131) from 0 to NTU c , which yields
h i
ð
t 00 ¼ 1 1 e 1 e NTU h ÞR h (3.133)
h,m =R h
According to Eq. (3.101), the dimensionless temperature changes are
obtained as
h i
ð
ε h ¼ 1 t 00 ¼ 1 e 1 e NTU h ÞR h =R h (3.134)
h,m
ð
00
ε c ¼ t ¼ 1 e 1 e NTU h ÞR h (3.135)
c
which are suitable for rating the exchangers and for sizing the exchangers by
rewriting Eq. (3.135) in

1
NTU h ¼ ln 1 + ln 1 R h ε h Þ (3.136)
ð
R h
3.3.3 Crossflow with both fluids mixed
The dimensionless energy equations for the crossflow with both fluids lat-
erally well mixed are presented by
ð
dt h xðÞ 1 NTU c
¼ ½ t c y ðÞ t h x ðފdy (3.137)
dx NTU c 0