Singularity Analysis of a One-Stage Column
A model of a one-stage distillation process for ethylene glycol synthesis is reported by Gehrke and Marquardt . Singularity analysis reveals a codimension 4 singularity for a Damkohler number Da = 0.0247, a heat duty Q/(AHVF) = 0.2871 and feed concentrations xW = 0.4597, xEG = 0.072, xEO = 0.3961, xDEG = 0.072. However, there is no indication yet that there will not exist any higher codimension singularity in addition. Fig. 10.7, left, shows the regions of multiplicity in the vicinity of the codimension 4 singularity in a cross section of the space of parameters. The typical butterfly type singularity can be observed giving rise to a maximum of five steady states. A bifurcation diagram is displayed in Fig. 10.7, right. The bifurcation diagram shows largely the large-scale three branch multiplicity indicating a saddle-node bifurcation, but with some perturbation close to a turning point to result in a region with five steady states. The region of five steady states is however confined to a fairly small parameter interval.
An analysis of the physical reasons for this behavior can be carried out by studying different kinds of models accounting for different physicochemical phenomena. For example, if all the consecutive reactions are neglected, a codimension 4 singularity can also be identified. Hence, the consecutive reaction(s) are not necessary to devise five steady states. They may rather give rise to higher codimension singularities indicating an even more degenerate behavior. This conjecture is
I I maximum of 5 steady-state solutions I I maximum of 3 steady-state solutions I I unique steady-state solution
backed by the fact that there are up to seven steady states for two consecutive exothermic reactions in a CSTR whereas only three steady states can coexist if just a single exothermic irreversible reaction is considered .
The physical reasons for the observed steady state multiplicity can be traced back to the interplay between chemical reaction and vapor-liquid equilibrium if an analysis of the one-stage column is carried out just with the main reaction. The multiplicity is mainly caused by a self-accelerating behavior due to an increasing rate of reaction with increasing conversion. This is due to the increasing boiling temperature of the mixture with increasing conversion, since TBEO < TBW < TBEG. Since only three steady states occur for a single exothermic reaction in a CSTR, the more complex behavior can be attributed to the cooperative effects of reaction and separation. Surprisingly, the heat of reaction does only contribute to the nonlinear behavior to a lesser extent. An analysis with a very small (positive) heat of reaction or even with a negative heat of reaction does not change the qualitative nonlinear behavior.
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