Continuation Analysis of Industrial Size Distillation Column

The findings from singularity analysis of the one-stage column will be checked for their relevance for the nonlinear behavior of an industrial size RD column by means of numerical bifurcation analysis using DIVA [53, 63]. The model used is of moderate complexity. The major assumptions are constant liquid holdup on every tray, negligible vapor holdup, constant heat capacities and heats of vaporization of all the species, ideal gas phase, almost ideal liquid phase, perfect mixing on trays, and kinetically controlled reaction and mass transfer. As a case study, we consider the column design reported by Ciric and Gu [14]: The column has 10 stages, water is added on stage 10 at the top, ethylene oxide is added on stage 5. The holdups of the reactive stages 5-10 have been set according to [14]. The column is operated at total reflux and at a boilup ratio of 0.958. A ternary mixture of water and two of the glycols (EG, DEG) is withdrawn at the bottom.

Fig. 10.8 shows the multiplicity behavior of the distillation column in comparison to the multiplicity of the one-stage column. The production rate of ethylene glycol is plotted as a function of the residence time, which has been changed by multiplying the holdups on every tray by a constant continuation parameter. The qualitative behavior of both, the one-stage and the ten-stage column, is remarkably similar. In both cases the overall behavior is dominated by a large-scale three branch bifurcation diagram with three steady states. There is, however, a small

Fig. 10.8 Bifurcation diagrams for ethylene glycol synthesis: one-stage column (bottom) and ten-stage distillation column (top)

region with a more complex multiplicity pattern for low residence times in both cases. Hence, numerical bifurcation analysis of a ten-stage distillation column largely confirms the findings from singularity analysis of the one-stage column. Therefore, we may conclude that the multiplicity behavior is largely determined by the properties of the chemical system and that it is influenced to a much lesser extent by the details of column design. Though, due to the potential stage-wise multiplication of the multiplicity found for a single tray, much higher order multiplicity may be faced for a distillation column.

Our findings are confirmed by the studies of other authors. Ciric and Miao [15] have carried out intensive numerical investigations using a homotopy continuation technique to trace out the parameter dependent steady state solution branches. Their model assumes phase equilibrium on the trays and simplifies the energy balance; it is therefore slightly simpler as the model used in our studies. However, their findings largely coincide with ours indicating that the intricate multiplicity behavior with seven steady states is most likely not an artifact of certain modeling assumptions. They conclude that the large-scale three solution branch of the bifurcation diagram is sensitive to vapor-liquid equilibrium and the bimolecular irreversible reaction. Further, these authors also showed that it is not due to the consecutive reactions or to the exothermicity of the main reaction. Also, the influence of the details of the column design (e.g., the number and location of feed streams) on the qualitative nonlinear behavior is negligible. On the other hand, the details in the bifurcation diagram giving rise to higher order multiplicity is sensitive to the reaction mechanism and kinetics. Further studies of this RD process by Kumar and Daoutidis [58] confirm these results. They show, that the solution curve between the upper and lower stable branches changes completely, if the reactive trays are also extended into the stripping section.

The three solution branch dominating the nonlinear behavior of the process can also be observed in a conventional process with a reactor-separator recycle as recently reported by Blagov et al. [12]. This flowsheet is analogous to a column with one reactive and one non-reactive column section with a single product stream, the multiplicity behavior of which has been studied Cerafimov and coworkers [49, 83]. In these papers it is shown that this type of output multiplicity is a generic phenomenon for reaction systems with competing irreversible reactions and a similar distribution of volatilities between reactants and products.

However, a careful analysis is always recommended if the behavior of a concrete process is of interest. Baur et al. [5], for example, show in a recent study, that multiplicity will most likely not occur in this particular example column for ethylene glycol synthesis, since only a single steady state complies with the fluid dynamics operating envelope of a tray column. This is due to the fact that the flow rates inside the column differ significantly for the different steady states. It is however expected that this behavior only occurs for highly exothermic reactions like the ethylene glycol reaction. In these cases, the flow rates strongly depend on the reaction rates because of the exothermicity of the reaction. The findings in [5] have only been possible through continuation analysis with a very detailed non-equilibrium cell model capturing the details of the tray design.

Fig. 10.9 Cell model to represent cross-flow pattern on distillation tray

So far, our bifurcation study has been assuming ideal mixing on the individual trays of the RD column. However, the fluid dynamic behavior of real trays with large holdups (as they are required for ethylene glycol production) will significantly differ from an ideally mixed state. Since residence time distribution of the tray will influence the selectivity of a complex reaction network, we might expect also an influence on the multiplicity behavior. For this purpose a single tray has been studied employing a cell model [25] to represent the cross-flow pattern on a real tray (compare Fig. 10.9). A common backmixing model [69] has been employed to account for dispersion between adjacent cells. This model does not capture fluid dynamics and heat and mass transfer at the level of detail presented later in [6], but covers the most significant effects.

The results of a numerical continuation study for a single tray are shown in Fig. 10.10. The top row of diagrams presents a comparison of a well-mixed and a cross-flow situation for the ethylene glycol system. In case of perfect mixing, three steady states may coexist for a certain holdup region, whereas up to eight steady can occur for a cross-flow tray modeled by four cells (corresponding to a Peclet number of about 8). Further study reveals an increasing number of steady states with an increasing number of cells to approximate the cross-flow pattern. Hence, there is evidence that much more complicated nonlinear behavior may arise for real flow situation as compared to the case of an ideally mixed tray as usually assumed in simulation studies.

The bottom row of diagrams in Fig. 10.10 show the results of a comparable study for methyl formate synthesis. There is always a unique steady state, regardless the assumptions on the flow pattern on a tray. Hence, the flow pattern on a tray seems to influence the qualitative behavior of a column tray only in those cases, when complex reaction networks with strongly nonlinear reaction models are considered.

The results of a similar continuation study for a complete RD column with perfectly and imperfectly mixed trays is shown in Fig. 10.11 for ethylene glycol synthesis. In contrast to a single tray, where the qualitative behavior for perfect and imperfect mixing differs significantly (compare Fig. 10.10), roughly the same behavior is observed for the column as displayed in the two diagrams on the left of Fig. 10.11. A multiplicity of order three occurs in a large region of the parameter

Fig. 10.10 Cell model to represent cross-flow pattern on distillation tray. Ethylene glycol synthesis (top row), methyl formate synthesis (bottom row). Well-mixed liquid (left) and liquid plug flow with dispersion for four cells (right)

space regardless of mixing. In contrast to this structurally stable situation, the multiplicity of order seven in a small region of the parameter space vanishes if imperfect mixing is taken into account.

Stability and the dynamic behavior of an ethylene glycol column has been studied in the region of multiplicity by means of dynamic equilibrium models by several authors [26, 58, 70]. In addition to the steady state bifurcation discussed so far, self-sustained oscillation also may occur in the ethylene glycol process if both consecutive reactions are considered. Kienle et al. [56] not only reported multiple steady states of order three for a column design with a distributed feed [15] but also found a region with a Hopf bifurcation. Fig. 10.12 shows an exemplary result. These oscillations seem to be closely related to the energy balance since no oscillatory behavior could have been identified if the energy balance is neglected. A more systematic study aiming to identify the causes of the oscillations and the sensitivity to modeling assumptions is however largely missing.

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