## Source of Pseudo Multiplicity

Mass-molar singularities are responsible for pseudo-multiplicity and sufficient conditions for pseudo-multiplicity are found from the inverses of equations (8.4) and (S.5).

Equations (8.6) and (8.7) are also sufficient for pseudo-multiplicity if they are evaluated in molar units. If mass or volumetric units are used, the conditions are sufficient for physically realisable multiple steady states.

The transformation mechanism (from one steady state in realisable units to multiple steady states in molar units) is demonstrated in Figures 8.16 and 8.17, which pertain to the pseudo-multiplicities shown in Figure 8.14 and 8.15, respectively. In each case, the locus of points that satisfy the molar constraint comprises two or more points from the curves for realisable constraints. Since the molar constraints and the realisable constraints intersect at more than one point, there must exist points that satisfy equation (8.14) (Figure 8.17) and equation (8.15) (Figure 8.16).

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Reflux Rate (m3/hr)

Figure 8.17 - Bottoms Rate Transformation m the 17 Stage MTBE Column

Jacobsen and Skogestad (1991) conjectured that output multiplicity was only likely for internal specifications. Recent results (e.g. Schrans, 1996) create doubt in this hypothesis as multiple steady states were reported for fixed bottoms rates in hybrid reactive distillation. However, these results were only obtained by assuming an unrealisable condition (e.g. a constant molar flow) so that the original conjecture remains sound and convincing.

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Reflux Rate (m3/hr)

Figure 8.17 - Bottoms Rate Transformation m the 17 Stage MTBE Column

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