Page 353 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 339

plug-flow assumption. No flow maldistributions like bypass, leakage
streams, stagnant flow, and backmixing were considered. However, the
actual flow is very complicated, and various forms of maldistributions
may occur and degrade the thermal performance of exchangers, which
the plug-flow model fails to describe. In the cases of maldistributions, the
ideal plug flow greatly deviates from the real flow pattern. One way to cor-
rect this deviation is the application of dispersion models. Taylor (1954) may
have been the first to develop the dispersion model for mass transfer in tur-
bulent flow through a pipe. The parabolic dispersion model was mainly
based on a dispersed plug flow, that is, the main plug flow with longitudinal
dispersion, named backmixing by Mecklenburgh and Hartland (1975).In
the book Dynamic Behaviour of Heat Exchangers by Roetzel and Xuan
(1999), the axial dispersion models and their applications on the analysis
of heat exchanger dynamics were presented in detail. The parabolic model
(infinite propagation velocity) is recommended for transient processes (see
also Section 2.1.4).
In the parabolic dispersion model, an apparent axial heat conduction
term is introduced into the energy balance relationship, and the effect of
flow maldistribution is taken into account by this dispersion term in the
energy equation according to Fourier’s conduction law. For a shell-and-
tube heat exchanger, one is able to derive a set of energy balance equations
for both fluids and the tube wall and the shell as follows, in which the tube-
side flow is a plug flow and the shell-side flow can be considered as a disper-
sive flow due to flow nonuniformity:
2
C 1 ∂t 1 ∂t 1 ∂ t 1 ð αAÞ 1 ð αAÞ ws
Shell side : + _ C 1 ¼ A c,1 D + ð t w t 1 Þ + ð t ws t 1 Þ ¼ 0 (7.88)
L ∂τ ∂x ∂x 2 L L
C 2 ∂t 2 n ∂t 2 ð αAÞ 2
Tube side : + 1Þ _ C 2 ¼ ð t w t 2 Þ ¼ 0 (7.89)
ð
L ∂τ ∂x L
C w ∂t w ð αAÞ 1 ð αAÞ 2
Tubewall : ¼ ð t w t 1 Þ + ð t w t 2 Þ (7.90)
L ∂τ L L
C ws ∂t ws ð αAÞ ws
Shellwall : ¼ ð t 1 t ws Þ (7.91)
L ∂τ L
where n¼0 and n¼1 in Eq. (7.89) indicate the parallel flow and counter-
flow, respectively.
The transient problem of crossflow heat exchangers is more complex
because there are two spatial variables and one time variable in the simula-
tion of their transient processes. Dusinberre (1954) studied the transient
behavior of a crossflow exchanger, in which a general finite-difference