Analysis of a Column for Multiple Steady States

A priori knowledge of the likelihood for multiple steady states in the operation of a column will be valuable for the process operators. An understanding of the cause of any multiplicities will also help to avoid or manage operating difficulties that arise and help to define the conditions that lead to multiple steady states.

The principal tool for analysing columns for multiple steady states is the bifurcation study. Bifurcation diagrams are constructed by determining the steady state solution defined by fixing one operating parameter at a constant value and varying another parameter between the values that correspond to nil distillate and nil bottoms. The requisite data can be generated with any simulation model and the resulting data set can be plotted against any variable, in molar, mass or volumetric units. Many simulations could be required to fully define the bifurcation curve where complex behaviour is present (e.g. approximately 70 points needed for Figure 8.9) but these can be completed relatively quickly with advanced simulation tools.

It is desirable to complete bifurcation studies with a rigorous model and with a CMO model, and to prepare bifurcation diagrams in both molar and mass units in order to determine the principal cause of any observed multiplicities. There is no physical significance to the simulations completed with either the CMO model or with constant molar flow rates since both of these cases are based on incorrect assumptions (i.e. constant molar overflow, and a one-one relationship between molar flows and mass or volumetric flows). However, the comparison of all four sets of results points to the fundamental cause of the multiplicity and this could provide further insight into the physical process.

Table 8.5 indicates how the bifurcation results should be collectively interpreted. Cause I refers to unit singularities, cause II refers to energy balance multiplicities, cause III refers to the azeotropic mechanism and cause IV is reaction hysteresis. These results also provide the data necessary to determine whether multiple steady states are physically realisable or due to a pseudo-multiplicity: this distinction has significant implications for the operation and control of the process.

Table 8.5 - Types and Causes of Multiplicity

Multiple Steady States Detected?

Model includes energy CMO cai.c

Cause

Type

balances Molar units Mass units

Molar units

Mass units

x S

X

I

real

S x

V

X

I

pseudo

V V

X

X

II

real

x S

X

X

II

real

✓ X

X

X

II

pseudo

s

III or IV

real

The distinction between multiplicity due to azeotropes and reaction hysteresis is best made using the reactive qo/qo technique (Giittinger and Morari, 1997). Simulations are not required and, where an oo/qo analysis has been completed previously (e.g. the MTBE analysis appears in the literature), only the feed composition in transformed coordinates is required to determine whether multiple steady states are possible. However, as indicated previously, there is uncertainty in the results of an oo/qd analysis for finite columns and hybrid columns, and the analysis is only applicable for internal specifications. Nevertheless, oo/oo predictions are frequently accurate and the technique is useful to eliminate azeotropes as a possible source of multiple steady states.

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