Page 338 - Design and Operation of Heat Exchangers and their Networks

324 Design and operation of heat exchangers and their networks

Historical reviews of early investigations on modeling the dynamics of heat
exchangers were given by Kanoh (1982). Using the lumped parameter
model, Khan et al. (1988) obtained the transfer functions of the outlet fluid
temperatures to the mass flow rate variations in a plate heat exchanger. The
same model was used by Al-Dawery et al. (2012). The lumped parameter
model was developed by Cai and his coworkers (Li, 1995; Du, 1996; Du
et al., 1996a,b; Xu and Cai, 1998), which provides the dynamic behavior
of the apparatus without dealing with detailed local temperature distribu-
tions. They compared different forms of lumped parameter model with
experimental results and found that the model using the outlet temperatures
as lumped parameters and the logarithmic mean temperature differences as
the driving forces would give the best results (Li, 1995). The coefficients
appearing in the transfer functions were determined by experiments. Ye
(1998) further obtained the outlet fluid temperature responses by inversing
the transfer functions into the real-time domain. The coefficients of the
transfer functions were determined by parameter matching between the ana-
lytical outlet temperature variations and the numerical results obtained from
a distributed parameter model in which the transient component of the heat
flux in fins is neglected (Du, 1996; Xu and Cai, 1998). Although their
research objects were plate-fin heat exchangers, the conclusions are useful
for the analysis of the transient behavior of parallel-flow and counterflow
heat exchangers.
For a two-stream heat exchanger, if the heat losses to the surroundings
are negligible and there are no heat sources in the fluids and the solid wall,
the energy equations (2.9) and (2.10) can be reduced to

dt h
ð
M h c p,h ¼ _m h c p,h t h,in t h,out Þ α h A h Δt m,h (7.1)
dτ
dt c
ð
M c c p,c ¼ _m c c p,c t c,in t c,out Þ α c A c Δt m,c (7.2)
dτ
dt w
M w c w ¼ α h A h Δt m,h + α c A c Δt m,c (7.3)
dτ
where
ð
1
Δt m,h ¼ ð t h t w ÞdA (7.4)
A h A h
ð
1
Δt m,c ¼ ð t c t w ÞdA (7.5)
A c A c