Page 55 - Design and Operation of Heat Exchangers and their Networks

42 Design and operation of heat exchangers and their networks

2.1.3 Consideration of temperature-dependent heat transfer
coefficients

For highly temperature-dependent heat transfer coefficients, as may occur
with viscous liquids and with radiation, the common calculation method
with arithmetic mean of inlet and outlet temperatures as reference tem-
peratures may lead to undesirable errors in design. For such cases, the
two-point method (Roetzel, 1969; Roetzel and Luo, 2011; Roetzel
and Spang, 2010, 2013) can be applied in which two special pairs of ref-
erence temperatures are used for the calculation of two overall heat trans-
fer coefficients and their effective mean value. The method is valid for
counterflow and parallel flow but canalsobeadapted to otherflow
arrangements.
In the two-point method, the heat transfer coefficients are calculated at
two reference points using two special pairs of reference temperatures for
the fluids. At the reference points i¼1 and i¼2, the reference temperatures
t h,i and t c,i are determined for counterflow as follows:

00
0
t h,i ¼ t + ψ t t 00 (2.84)
h i h h

0
00
t c,i ¼ t + ψ t t 0 (2.85)
c i c c
where
θ 1
m i
ψ ¼ (2.86)
i
θ 1
t t 00
0
θ ¼ h c (2.87)
00
t t 0 c
h
1 1 p ffiffiffi
m 1,2 ¼ 3 (2.88)
2 6
_
_
For a balanced counterflow heat exchanger (C h ¼ C c ), we have ψ i ¼m i .
0 00
For parallel-flow heat exchanger, the same equations can be used if t c and t c
are exchanged.
The heat transfer coefficients and overall heat transfer coefficients at the
two reference points, that is, (kA) 1 and (kA) 2 , are calculated in the usual way.
The effective mean value (kA) m is calculated from

1 1 1 1
¼ + (2.89)
ð kAÞ 2 ð kAÞ ð kAÞ
m 1 2
Eqs. (2.84)–(2.86) are valid for constant heat capacities of the fluids. Var-
iable heat capacities can be approximately replaced by constant mean values