Output Multiplicity due to Unit Singularities

Jacobsen and Skogestad (1991) presented an example of an ideal distillation process that yields multiple steady states (i.e. it exhibits output multiplicity) when mass units were used but only one steady state when molar units were used. They argued that the change of units created the multiplicity as the partial derivative of the mass flow with respect to the molar flow was not necessarily always positive. They proposed the term input transformation to describe this behaviour. Later, Guttinger and Morari (1997) considered singularities in the mass-molar relationships with respect to output multiplicity in reactive distillation and proposed a geometrical technique to analyse singularities which occur with external specifications (i.e. product flow rates). However, they were unable to provide an example of a reactive distillation system where multiplicity was directly attributable to a singularity.

A more general designation of this condition is proposed here: unit singularities, referring to any stationary point in an otherwise smooth relationship between input variables. This description is preferred to others since it may pertain to a singularity in dissimilar variables (e.g. the relationship between the molar boil up and the reboiler duty). This is important since these singularities are equally capable of causing multiplicity. A specific example of a singularity in the mass-molar relationship causing an output multiplicity in a reactive distillation column is also provided.

The column considered was a 30 stage ETBE reactive column with the configuration given by Table 8.3. The output multiplicity is shown in Figure 8.3, which is part of a bifurcation diagram for this system (i.e. the locus of steady state solutions for a set of inputs that differs by one parameter only - the independent variable) for the conditions given in Table 8.3. This data was obtained from a series of Pro/II simulations. There are three separate steady state solutions for all bottoms yields between 35.2% and 36.6% of the feed rate. The salient features of the upper and lower branches are indicated in Table 8.4, which pertains to points A and B on Figure 8.3.

Table 8.3 - 30 Stage ETBE Column Characteristics

Design Parameter


Rectifying stages (including total condenser)


Reactive stages


Stripping stages (including partial reboiler)


Feed stage

uppermost stripping stage

Hydrocarbon feed composition (mol %)

25% isobutene, 75% n-butenes

Stoichiometric ethanol excess (mol %)


Total feed rate (kmol/hr)

100 kg/hr

Overhead pressure

700 kPa-g

Reflux rate

81.3 kg/hr

Partial Reboiler
Figure 8.3 - Output Multiplicity due to a Unit Singularity in a Reactive ETBE Column
Table 8.4 - High and Low Conversion Steady States for ETBE Column

High Conversion

Low Conversion


Steady State Value

Steady State Value

(Point A)

(Point B)

Overall isobutene conversion (mol %)



Bottoms ETBE purity (mol %)



Reboiler temperature (°C)



Reboiler duty



Boilup ratio (molar basis)



Bottoms yield (molar basis w.r.t. feed)



Reflux ratio



The cause of the output multiplicity in this column is a singularity in the relationship between the molar bottoms flow (B) and the mass bottoms flow (BjJ, as shown in Figure 8.4. The stationary points from this plot satisfy condition (8.4), which is sufficient for multiplicity. Equation (8.5) implies a unit singularity between the molar boilup rate (V) and the reboiler duty (Qr) and, although not present in this column, such a singularity is also sufficient for output multiplicity. Indeed, any singularity between a molar input (i.e. molar reflux, boilup, distillate and bottoms rates) and the actual input (i.e. volumetric or mass reflux, distillate and bottoms rates, and the reboiler duty) will cause an output multiplicity.

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