Page 106 - Design and Operation of Heat Exchangers and their Networks
P. 106
94 Design and operation of heat exchangers and their networks
ð NTU h
dt c yðÞ 1
¼ ½ t h xðÞ t c yðÞdx (3.138)
dy NTU h 0
with the boundary condition
x ¼ 0 : t h ¼ 1; y ¼ 0 : t c ¼ 0 (3.139)
Ð NTU c 1 Ð NTU h
1
Let C 1 ¼ t c yðÞdy and C 2 ¼ t h xðÞdx; the
NTU c 0 NTU h 0
integration of Eqs. (3.137), (3.138) yields the dimensionless temperature
distributions as
x
t h xðÞ ¼ C 1 +1 C 1 Þe (3.140)
ð
y
t c yðÞ ¼ C 2 1 e (3.141)
with
ð
½ 1 1 e NTU c Þ=NTU c 1 e NTU h Þ=NTU h
ð
C 1 ¼ (3.142)
1 1 1 e NTU h Þ=NTU h 1 1 e NTU c Þ=NTU c
ð
½
½
ð
ð 1 e NTU h Þ=NTU h
C 2 ¼ (3.143)
1 1 1 e NTU h Þ=NTU h 1 1 eð½ NTU c Þ=NTU c
½
ð
The dimensionless outlet temperatures can be obtained by setting
x ¼ NTU h in Eq. (3.140) and y ¼ NTU c in Eq. (3.141), respectively, from
which the effectiveness can be derived as (Smith, 1934)
1 1 R 1
¼ + (3.144)
ε 1 e NTU 1 e RNTU NTU
which can be respect either to the hot fluid or to the cold one.
Eq. (3.144) can be directly used for a rating problem. For sizing prob-
lems, we can use a solver to find the root of Eq. (3.144) for a given value
of ε. The initial value of kA can be evaluated with Eqs. (3.123), (3.124),
for which (Roetzel and Spang, 2010, 2013, Table 1)
a ¼ 0:251, b ¼ 2:06, c ¼ 0:677, and d ¼ 0:5
3.3.4 Multipass crossflow heat exchangers
If two or more crossflow units are coupled together, it may form a variety of
different flow arrangements. Such multipass crossflow heat exchangers have
beenoftenusedinindustriestomeetspecialstructuralandthermalrequirements.
For two-pass crossflow heat exchangers, Baclic (1990) summarized
totally 72 flow arrangements and their corresponding ε-NTU relationships
