Dyanmic compartmental models for separation processes

Abstract

A low-order modeling technique for separation processes is developed by considering a staged column as a compartmental system in which a number of stages are lumped to form an equivalent stage. This method leads to low-order models of separation processes directly and without linearization. Moreover, the resulting models have state variables and parameters that are physically significant. In contrast t o alternative model reduction methods, compartmental analysis guarantees preservation of both material balances and steady states for arbitrary changes in the input variables.A comparison of compartmental analysis to a recently proposed technique based on orthogonal collocation, both methods incorporating an equimolal overflow assumption, shows the efficiency and robustness of the compartmental method.A. Benallou SCOPELow-order, dynamic models of separation processes are desirable for a number of reasons including the evaluation of alternative control strategies, the development of on-line applications of advanced control schemes, and the training of plant personnel.Most of the available methods for building low-order models of separation processes are deficient because they are so abstract and complicated, require linearization of the system equations, or yield models with parameters or state variables having no physical significance.The most recent of these methods uses orthogonal collocation to derive reduced models (Wong and Luus, 1980;Cho and Joseph, 1980, 1983;Stewart et al., 1981). This approach introduces mathematical complications and leads to state variables that may not necessarily correspond to tray locations.Another approach, linearization techniques combined with model reduction, can lead to low-order models in a state space or transfer function form (Mockzek et al.. 1965;Wahl and Harriott, 1970;Weigand et al., 1972; Waller, 1972 Waller, , 1979Crockett, 1978). This class of methods offers a simpler alternative, but leads to models in which the parameters have no physical significance.By considering a separation column as a compartmented system, it is possible to derive low-order models that avoid these deficiencies. At the same time, steady state characteristics of high-and low-order models can be matched exactly, and transient characteristics of the low-order models can closely approximate the high-order system dynamics. Moreover, the model parameters retain their physical significance, permitting broad use of the model, especially when operating conditions change. In this development of the basic theory, several simplifying assumptions are made, including equimolal overflow, in order to compare results with those from collocation methods. Benallou (1 982) presents extensions of the method to more realistic systems. CONCLUSIONS AND SIGNIFICANCECompartmental modeling techniques are based on the physical characteristics of separation processes. They are suitable for developing low-order models for AIChE JournalJuly 1986 Vol. 32, No. 7 1067 simulation or control purposes that are valid, with no fur...

Quality control of binary distillation columns via nonlinear aggregated models

“…Our method is of the same spirit as Benallou et al (1986) Benallou et al (1986) to the 5 compartments case as displayed on Fig. 5, we obtain a reduced model whose structure does not remain tridiagonal whereas our aggregated model does.…”

“…5, we obtain a reduced model whose structure does not remain tridiagonal whereas our aggregated model does. This results from the application of the equations (24) to (32) in Benallou et al (1986). One can verify that the differential equation corresponding to the feed compartment depends, in Benallou et al (1986), on x~ and x,, the product compositions, whereas, in our aggregated model (18) it does not.…”

“…This results from the application of the equations (24) to (32) in Benallou et al (1986). One can verify that the differential equation corresponding to the feed compartment depends, in Benallou et al (1986), on x~ and x,, the product compositions, whereas, in our aggregated model (18) it does not. On the other hand, as in Benallou et al (1986), the steady states of the outputs y~ =xl and y2=Xn are preserved whatever the inputs L, V, F and zf are.…”

“…More precisely, this can be done by splitting the column into a given number of sections of consecutive trays (called compartments), and by aggregation of each section separately. We now present the aggregation method on a given section of trays which gives an alternative model to the compartmental models of Benallou et al (1986) or of Espafia and Landau (1978).…”

“…Since the early study of Rosenbrock (1962), who established most of the qualitative results concerning such models, little theoretical progress has been observed. Other modeling ideas have been developed, for example by Espafia and Landau (1978) and more recently by Benallou et al (1986). They propose compartmental approximation techniques to obtain a simpler model of reduced order.…”