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 and py In the triangular shaped region in the parameter space, three steady states can coexist for the same set of para- 

In contrast, input multiplicity [46] 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 [46]. 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 [10], van Heerden [109], and Aris and Amundson [3]. Comprehensive reviews have been given by Razon and Schmitz [86] or Elnashaie and Elshishini [19]. Multiplicity analysis of non-RD can be traced back to the paper of Rosenbrock [91] 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. [61] 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. 

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