Multiplicity and Oscillations in Chemical Process Systems
A process model of any chemical process system is given by a system of differential-algebraic equations, which depend on some parameters. The steady state solution branches can be traced out in the parameter space. An exemplary situation is shown in Fig. 10.1 where some norm of the steady states x is plotted above the plane spanned by two selected parameters p; and pj. In the triangular shaped region in the parameter space, three steady states can coexist for the same set of para-
meters (e.g., at point P in the figure). In the remainder of the parameter space, a unique steady state exists (e.g., point Q in the figure). There is a distinct point (point O in the figure) that is most degenerate in the following sense: All different types of qualitative behavior of the system can be observed in the vicinity of O, which is therefore sometimes called the organizing center. In addition to steady state multiplicity, dynamic bifurcations may occur leading to regions in parameter space with self-sustained oscillations. An example is shown in Fig. 10.1 at point R where sustained oscillations are present in a certain interval of parameter pj for fixed p;. The filled symbols indicate the amplitude of the oscillations around a locally unstable steady represented by the dotted part of the upper branch of the steady-state solution plot. A detailed introduction into nonlinear dynamics may be found in textbooks [30, 81, 94].
The multiplicity in the vicinity of P in Fig. 10.1 is termed output multiplicity, since it occurs when a set of given inputs (parameters or manipulated variables) results in different values of the output variables (measured quantities) corresponding to different steady states. In RD, inputs are, for example, the reflux rate at the top of the column or the reboiler duty at the bottom of the column, whereas outputs are usually some temperatures inside the column or the product compositions at the bottom or the top, respectively. In case of output multiplicity, usually some of the steady states are open loop stable, while others are open loop unstable. The latter are indicated by the dashed line in the inserts of Fig. 10.1. Depending on the history (e. g., startup strategy) the column settles either down to the upper stable steady state or to the lower stable steady state for parameters at point P. This implies that a suitable startup strategy is required to operate the column into the desired steady state. Further, when operating the column in one of the stable steady states, the column may 'jump' onto the other, undesired stable steady state branch after some disturbance. This is easily avoided by some suitable control for disturbance rejection. Moreover, it is also possible to operate the column at an unstable steady state, if desirable, when stabilizing control is applied. Finally, it is worth noting that output multiplicity and open loop stability do not depend on the choice of output or measured variables.
In contrast, input multiplicity  can occur when different sets of input variables produce the same set of output variables. This input multiplicity will depend on the choice of output or measured variables. It is associated with the so-called zero dynamics of the system, which can be observed by an unexpected inverse response of the outputs after a step change has been applied to the inputs. Therefore, it can have severe implications for closed loop control . We will come back to this point in Section 10.8.
The analysis of nonlinear dynamics of chemical process systems has a long tradition. Most emphasis has been on chemical reactors initiated by the seminal work of Bilous and Amundson , van Heerden , and Aris and Amundson . Comprehensive reviews have been given by Razon and Schmitz  or Elnashaie and Elshishini . Multiplicity analysis of non-RD can be traced back to the paper of Rosenbrock  where stability and hence uniqueness of steady-states of a binary distillation column is demonstrated under quite general assumptions.
The numerical study of Magnussen et al.  has raised a lot of attention and triggered the analysis of multiplicity in distillation [8, 17, 46-48, 54, 55]. Reviews on dynamics and control of non-RD columns can be found in the literature [50, 60, 95],
RD columns share some common features with chemical reactors on the one hand and with distillation columns on the other hand. The behavior of these multifunctional processes may be either close to that of non-RD columns or to chemical reactors. Further, new patterns of behavior can be introduced by the superposition of reaction and separation in a single processing unit. Hence, another interesting question that will be addressed in this chapter, is under what conditions and in what sense is the dynamic behavior of an RD column similar to that of a chemical reactor or to that of a non-RD column.
Most emphasis has been on output multiplicity as well as on sustained oscillations in chemical process systems. The role of input multiplicity compared to output multiplicity has been treated for RD processes in [26, 98], and the notion of pseudomultiplicity was introduced in . This corresponds to a situation where '...molar inputs (rather than mass or volume inputs that would result from control valves) produce an output multiplicity'. Since this behavior can only be observed via simulation and is not associated with actually operating columns we will focus in this chapter on input and output multiplicity and do not treat pseudomultiplicity.
In order to facilitate the analysis and to gradually build up our understanding of the causes for the observed phenomena with respect to the chemical system and the process configuration, we study different model problems. The most simple is comprised by a continuous stirred tank reactor. If a reboiler and a reflux condenser is added, we obtain a one-stage column, where the whole column is lumped into one stage. Another simple configuration is a very long column section with a large number of trays. Such a configuration will be most favorable for the analysis of wave propagation in Section 10.7. Closely related is a column with an infinite number of trays operated under total reflux: an assumption that simplifies the (so-called <x>/<x>) analysis tremendously. Finally, we study a full-scale RD column with various kinds of mathematical models differing in the level of detail.
The mathematical tools we are going to employ are of various types, too. We employ singularity theory  and attempt to directly compute the most degenerate point (i.e., the organizing center) of the steady state manifold [22, 23]. Due to the computational complexity, this method is still restricted to fairly small-scale problems . ^/^-analysis originally introduced by Petlyuk and Avetyan  and further developed by Bekiaris et al.  is a simple method to study the qualitative behavior of complete non-reactive and RD columns under some limiting assumptions. Numerical bifurcation analysis by means of continuation and an integrated analysis of the stability behavior of the steady state solutions  is used to study the behavior of the large-scale differential-algebraic models [53, 63] describing full scale RD columns. Last but not least, wave front analysis [50, 65] will be employed to study the spatiotemporal patterns in column sections.
Was this article helpful?