A fourth multiplicity mechanism, reaction hysteresis, is proposed here to explain that multiple steady states observed in simulations of the hybrid MTBE column described by Table 8.2. The multiplicity in this column denoted by D-E in Figure 8.9 cannot be adequately explained by any of the causes discussed in Sections 8.2.1-8.2.3.
This multiplicity is not caused by singularities in any mass-molar relationships as the multiplicity persists if molar units are considered. Neither is the multiplicity caused by energy balance effects since the multiplicity also persists if the energy balance is ignored. It is unclear whether azeotropes could be responsible for this multiplicity since the mechanism has only been explained for material-balance specifications (e.g. constant distillate or bottoms product rate) and an energy-balance specification was used in this case (i.e. constant reflux rate). However, the feed composition is outside the region which produces multiple steady states (as determined via an oo/oo analysis) so that an explanation based on the VLE characteristics only appears tenuous. Guttinger and Morari (1997) suggested that interactions between the reactive and non-reactive column sections might extend the boundaries of the multiplicity feed region but there appears to be little evidence for this speculation and the implications for energy-balance specifications (e.g. constant reboiler duty or reflux rate, as in this case) are unclear. Thus, an alternative explanation for the unusual behaviour of this column is required.
It is proposed that, in this case, the multiple steady states arise from interactions between the reactive and non-reactive column sections via the effect that the non-reactive sections have on the reaction zone conditions. This type of multiplicity is applicable to any hybrid column with any type of specifications, and is caused by a change in the reaction zone condition that is propagated and reinforced by the fractionation changes that concurrently resulted from the original disturbance. A very high process gain is possible from this combination of effects but, more importantly, the two effects may not reinforce each other in the opposite direction. This pattern describes a process hysteresis, which is a sufficient condition for multiplicity.
To understand the proposed mechanism, it is important to consider necessary and sufficient conditions for output multiplicity. Previous authors (e.g. Hauan et al., 1995; 1997) have discussed multiplicity in relation to strongly non-linear behaviour but non-linearity is only necessary and not sufficient for output multiplicity. Provided that a change in an operating condition is reversible, the bifurcation diagram for that parameter will be smooth and only one steady state will exist. A distinctive 'S' shape is seen in the bifurcation diagram where there are multiple steady states. The 'S' shape effectively defines a process hysteresis (i.e. a non-reversible change in operating conditions) which is both a necessary and sufficient condition for output multiplicity. One bifurcation branch is accessible only by increasing the bifurcation parameter while the other branch is accessible only by decreasing the bifurcation parameter.
The application of reaction hysteresis to this hybrid MTBE column is as follows. Where the reactive section of the column is cool and lean in methanol, increasing the boilup rate strips methanol from the bottoms product and promotes the MTBE synthesis reaction on the reactive stages. This is facilitated by the minimum-boiling azeotropes that form between methanol (the heaviest reactant) and the various C4 components. However, an increase in the boilup rate causes the phase equilibrium temperature to rise that tends to suppress the synthesis reaction due to its exothermic nature.
The duality of effects continues in this way with increasing boilup until a critical point is reached when the effect of the increasing temperature predominates and the decomposition reaction is favoured. This produces more methanol which propagates the trend of rising temperatures (the reactive residue curve terminates at methanol and not MTBE) and escalates the MTBE decomposition rate. The column profile then goes through a catastrophic change before stabilising to a new operating point with higher concentration of methanol in the reaction zone (and upper stripping section) and a lower overall conversion of reactants to MTBE. In reverse, a decrease in the boilup rate has only a slight effect on the temperature and composition profiles as the two effects no longer reinforce each other. The situation is somewhat analogous to a reaction runaway although fractionation effects trigger the runaway rather than kinetic effects.
As with multiplicities due to the presence of azeotropes in the VLE, the column configuration (i.e. the number of theoretical stages, and the internal vapour and liquid flow rates) can affect the presence of a reaction hysteresis With fewer stages and lower internal flows (i.e. low reflux and boilup ratios), composition differences between adjoining stages are lessened, and there is less scope for changes in the stage-to-stage temperatures and compositions to propagate and multiply.
Reaction hysteresis, as described above, is dependent on interactions between the reactive and non-reactive column sections, and is independent of reaction kinetics, the column energy balance and singularities in the mass-molar relationships. This type of behaviour exactly fits the simulation observations and could be responsible for the output multiplicity shown in Figure 8.9.
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