Page 48 - Design and Operation of Heat Exchangers and their Networks

Basic thermal design theory for heat exchangers 35

The substation of Eq. (2.71) into Eq. (2.68) yields the mean overall heat
transfer coefficient as
A
Z
kt h t c ÞdA
ð
0
A
k m ¼ Z (2.72)
ð t h t c ÞdA
0
Eqs. (2.66)–(2.68) are three basic equations for heat exchanger design.
However, by using Eq. (2.68), the mean temperature difference Δt m should
be known or needs to be determined. Fig. 2.7 shows the temperature
variations in two typical heat exchangers: parallel-flow heat exchanger
and counterflow heat exchanger. For both flow arrangements, the mean
temperature difference is equal to the logarithmic mean temperature
difference:

Δt 1 Δt 2
Δt m ¼ Δt LM ¼ (2.73)
ln Δt 1 =Δt 2 Þ
ð
in which Δt 1 is the temperature difference at one end of the exchanger and
Δt 2 is that at the other end.
A special case is Δt 1 ¼Δt 2 . It will happen in a counterflow heat
exchanger if the thermal capacity rates of the two streams are the same, that
_
_
is, C h ¼ C c . In such a case, Eq. (2.73) cannot be used. An expression was
proposed by Chen (1987):
0:3275 0:3275 1=0:3275
Δt LM Δt + Δt =2 (2.74)
1 2
which is a good approximation even for large values of Δt 1 /Δt 2 . A practical
method is to use Eq. (2.75) for Δt LM :
8
Δt 1 Δt 2
< 6
, j Δt 1 Δt 2 j > 10
Δt LM ¼ ln Δt 1 =Δt 2 Þ (2.75)
ð
: 6
j
ð Δt 1 + Δt 2 Þ=2, Δt 1 Δt 2 j 10
Another special case is that the temperature of one fluid remains
00
unchanged in the exchanger, t(z)¼t ¼t . For example, in the two-phase
0
heat transfer region of a condenser or an evaporator, the fluid temperature
maintains at its saturation temperature. In such a case, Eq. (2.73) is valid
not only for counterflow and parallel-flow but also for crossflow. This con-
clusion can be extended to other types of heat exchangers.
A correction factor for the logarithmic mean temperature difference of
counterflow can be introduced, which is defined by