Page 90 - Design and Operation of Heat Exchangers and their Networks

78 Design and operation of heat exchangers and their networks

where d i are the constants to be determined by the boundary conditions. For
_
_
the counterflow heat exchanger with C h ¼ C c , the analytical solution
becomes
_
t h t 0 C h + kA 1 zð Þ
c
¼ (3.42)
t t 0 _
0
h c kA + C h
t c t 0 kA
c
¼ ð 1 zÞ (3.43)
0
t t 0 _
h c kA + C h
The outlet temperatures of hot and cold fluids are
00
t t 0 _
h c C h
¼ (3.44)
t t 0 _
0
h c kA + C h
t t 0 kA
00
c c
¼ (3.45)
t t 0 _
0
h c kA + C h
The heat load is
_
kAC h
0
Q ¼ _ t t c 0 (3.46)
h
kA + C h
In practice, the multiple root of the eigenvalue can be avoid by slightly
∗
_
_
change in the thermal capacity rates, that is, by setting C ¼ 1:00001C h ,so
h
that Eqs. (3.34)–(3.38) can still be used, but the calculating results will not be
affected.
3.2 Rating and sizing problems
The heat exchanger design problems are often treated as the rating problems
and sizing problems. For an existing exchanger, the performance evaluation
problem is referred to as the rating problem. The purpose of rating is either
to verify vendor’s specifications or to determine the performance at the
off-design conditions. To design a new exchanger for the specified perfor-
mance within known constraints is referred to as the sizing problem or a
design problem.
3.2.1 Logarithmic mean temperature difference
In Section 3.1, we have already obtained the logarithmic mean temperature
difference for parallel-flow heat exchangers (Eq. 3.27) and counterflow heat
exchangers (Eq. 3.40). Here, we derivate the logarithmic mean temperature
difference in an easy way.