Page 342 - Design and Operation of Heat Exchangers and their Networks

328 Design and operation of heat exchangers and their networks

Eqs. (7.19), (7.20) contain 12 transfer functions for the outlet tempera-
ture responses of two fluids to the disturbances in the inlet temperatures,
thermal capacity flow rates, and overall heat transfer coefficient and the
switch from the initial steady state to the new mean operating condition.
The temperature responses in the real-time domain can be obtained by
the inverse Laplace transform (see Chapter 2).
For the startup problem, both fluids have an initial steady-state temper-
ature t 0 . In this case, we have ^ t h ¼^ t c ¼ t 0 for the solutions (7.19) and (7.20).

7.1.2 Distributed parameter model
The distributed parameter model can represent the dynamic behavior of heat
exchangers more accurately. By distributed parameter approaches, variations
of the temperatures with both time and space variables were taken into
account and therefore more closely coincide with the real operation than
the lumped parameter model. The main assumption for the distributed
parameter model is that the fluid flow in the heat exchanger is a nondisper-
sive plug flow.
Since the governing equations of the distributed parameter model are
partial differential equations that containtimeand spacevariables,numer-
ical methods had to be used to obtain the dynamic behavior of heat
exchangers (Forghieri and Papa, 1978; Tan and Spinner, 1984; Ontko
and Harris, 1990; Lakshmanan and Potter, 1994; Abdelghani-Idrissi
et al., 2001). Meanwhile, more efforts were put on the analytical solutions
of the distributed parameter models. With distributed parameter model
and Laplace transform, Romie (1984, 1985) obtained the analytical
solutions of the real-time temperature responses of counterflow and
parallel-flow heat exchangers to a unit step change in an inlet fluid tem-
perature. By neglecting the wall heat capacity, Li (1986) solved the real-
time dynamics of parallel-flow heat exchangers by means of the Laplace
transform. Gvozdenac obtained the real-time solutions of the heat
exchanger dynamics for parallel-flow heat exchangers considering the
thermal capacities of the wall and fluids (Gvozdenac, 1990)and for
parallel-flow and counterflow heat exchangers neglecting the fluid thermal
capacities (Gvozdenac, 1987). Roetzel and Xuan (1992a) used the Laplace
transform and numerical inverse algorithm to calculate the real-time
temperature dynamics of parallel-flow and counterflow heat exchangers.
With this method, the temperature dynamics of shell-and-tube heat
exchangers can also be obtained (Roetzel and Xuan, 1993a). They further