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Ind. Eng. Chem. Res. 1998, 37, 4424-4433
Mechanistic Interpretation of Multiplicity in Hybrid Reactive Distillation: Physically Realizable Cases M. G. Sneesby, M. O. Tade´ ,* and T. N. Smith School of Chemical Engineering, Curtin University of Technology, Perth, Western Australia 6845, Australia
Reactive distillation columns can exhibit multiple steady states for the same feed and operating conditions. Two hybrid columns have been investigated here using both steady-state and dynamic simulations. In the first column, ETBE synthesis was considered and a physically realizable multiplicity was found for the case with a constant mass reflux rate. The highconversion steady state was associated with high concentrations of ethanol and high temperatures in the reaction zone. In the second column, MTBE synthesis was considered and a different type of multiplicity (with up to five possible steady states) was observed for the physically realizable case of constant reboiler duty. In this case, the high-conversion steady state was associated with low concentrations of methanol and low temperatures in the reaction zone. Mechanistic explanations for both multiplicities are presented using the simulation results, and these explanations are related to the fundamental and underlying causes of the multiplicities. Introduction Process design is traditionally undertaken in the steady state. That is, competing process designs and flow sheets are usually compared via steady-state designs (mostly produced by simulation) and justified on steady-state operation. From this perspective, hybrid reactive distillationsthe combination of reaction and distillation in a column with both reactive and nonreactive sectionsscan be an attractive technology for the synthesis of some chemicals, notably methyl tert-butyl ether (MTBE) production. Indeed, reactive distillation is now the preferred industrial synthesis route (Riddle, 1996) for MTBE and appears to have potential for ethyl tert-butyl ether (ETBE) production (e.g., Sneesby et al., 1997a,b). In both cases, the continuous recycling of reactants to the reactive section permits a conversion in excess of the equilibrium limit in a fixed-bed or stirred reactor. The transient operation of complex chemical processes is also very important and, in many instances, must be incorporated into any assessment of the process technology. For example, it is vital to consider the potential for reaction runaway during syntheses involving a strongly exothermic reaction, and there is increasing emphasis on the interaction between process design and controllability. Hybrid reaction distillation is a process with particularly complex transients. These columns can display unusual dynamic behavior and often have poor controllability characteristics so that the full (steady-state) benefits of using this technology might sometimes remain unrealized. One aspect of reactive distillation which contributes to this situation is the potential for multiple steady states since special control strategies and precautions are required (Sneesby et al., 1997c,d). Simulation evidence for multiplicity in hybrid reactive distillation has been described by several authors (e.g., Jacobs and Krishna, 1993; Nijhuis et al., 1993), but * To whom correspondence should be addressed (telephone, 618-9266-7704; fax, 618-9266-3554; email, tadem@che. curtin.edu.au).
conclusive experimental evidence has not yet been presented. Nevertheless, it appears that it is a real phenomenon that could affect operating columns. Jacobsen and Skogestad (1991) described two causes of multiple steady states in ideal binary distillation, and these causes are equally applicable to reactive distillation. Bekiaris and Morari (1993) described another cause of multiplicity with respect to azeotropic distillation. More recently, Gu¨ttinger and Morari (1997) extended these results to reactive distillation. That paper also introduced speculation of a fourth multiplicity mechanism that applies only to reactive distillation and is related to the complex interactions between reaction and separation. In addition to the above studies, which concern the fundamental causes of multiplicity, mechanistic explanations for multiplicity in hybrid columns have also been proposed (Hauan et al., 1995; Hauan and Lien, 1997). Such discussions are useful for elaborating the changes that occur within columns when undergoing transitions between parallel steady states. However, since these studies are necessarily based on simulation results, there is no requirement to model a physically realizable situation. An important example of this is the specification of a constant molar flow rate. It is not possible to physically control a molar flow rate, and a one-to-one relationship with the mass or volumetric flow is not guaranteed. In fact, the absence of a one-to-one relationship can be the very cause of the multiplicity being studied (Jacobsen and Skogestad, 1991). The term pseudomultiplicity is used to describe the circumstance where a multiplicity that is predicted by simulation cannot be demonstrated in practice due to a weakness in the underlying assumptions for the simulation. Clearly, pseudomultiplicities have few implications for the operation of real processes. Two examples of physically realizable multiplicities in hybrid reactive distillation, one in an ETBE column and one in a MTBE column, are presented here. In the first case, steady-state simulations were used to gain an understanding of the differences between the two stable steady states. In the second case, both steady-
10.1021/ie970938+ CCC: $15.00 © 1998 American Chemical Society Published on Web 10/02/1998
Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4425 Table 1. High- and Low-Conversion Steady States for an ETBE Column
property overall isobutylene conversion (mol %) bottoms ETBE purity (mol %) reboiler temperature (°C) reboiler duty boilup ratio (molar basis) bottoms yield (molar basis with respect to the feed) (%) reflux ratio
Figure 1. Hybrid reactive distillation column for ETBE production.
state and dynamic simulations were used to examine the internal column changes that occur during a transition between parallel steady states. Here, parallel steady states refer to different steady states (output conditions) that correspond to the same set of input conditions (i.e., feed rate and composition, column inventory, and the values of the free manipulated variables). These results were used to develop mechanistic explanations for the cause of the transitions and to form the basis of speculation concerning the fundamental causes of the multiplicities. Multiple Steady States in an ETBE Column Column Configuration. Figure 1 shows a hybrid reactive distillation column that could be used to synthesize ETBE from ethanol and a hydrocarbon stream containing some isobutylene. There are 28 ideal stages in the column (excluding the total condenser and partial reboiler), of which 7 are reactive. The column feed consists of a mixture of C4 hydrocarbons (25% isobutylene and 75% n-butylenes) and a stoichiometric amount of ethanol, which has been prereacted to 70% isobutylene conversion. A total of 100 kg/h of this mixture was specified to enter the column on the uppermost stripping stage (i.e., immediately below the reactive section). An overhead pressure of 800 kPa absolute was nominated. Bifurcation Diagram. A rigorous simulation model (see Sneesby et al., 1997a, for details of the model) of the above column was used to produce the bifurcation diagram shown in Figure 2. This plot represents the locus of all steady-state solutions for bottoms yields of 32-38% (mass-mass basis) and a reflux rate of 81.3 kg/h. There are three separate steady-state solutions for all bottoms yields between 35.2% and 36.6% of the feed rate. High- and Low-Conversion Steady States. Table 1 and Figures 3-5 indicate the differences between the high-conversion steady state (point A in Figure 2) and low-conversion steady state (point B) at a bottoms yield of 36.0 wt %. A simple comparison of these data yields
high-conversion low-conversion steady-state steady-state value (point A) value (point B) 90.7
84.1
90.4 144.7 0.914 4.67 20.1
79.6 135.7 0.920 4.33 21.1
1.272
1.271
the following striking contrasts: (a) the stripping section temperatures (Figure 3); (b) the stage-to-stage compositions in the stripping section (Figure 4); and (c) the reaction rate on the lowermost reactive stage (Figure 5). These differences are interpreted below. Mechanistic Interpretation. Figures 3 and 4 indicate that the rectifying section of the column is similar in both the high- and the low-conversion steady states. However, slight changes in the stage-to-stage compositions are magnified by the different reaction rate profiles (Figure 5) so that there are substantial differences in the temperature and compositions on the lower reactive stages. The effect of fractionation in the reaction zone is to concentrate ethanol (if present) near the lower reactive stages. This promotes the synthesis reaction due to the increased reactant availability but eventually initiates decomposition if the elevated phase temperature (caused by the increasing ethanol concentration and the formation of ETBE) reduces the reaction equilibrium constant too much. If there is no ethanol present in the reaction zone, the synthesis reaction is suppressed even though the reaction equilibrium constant is higher. Thus, the crucial consideration is the supply of ethanol to the reaction zone. This is dependent on the fractionation in the stripping section and the stage-tostage compositions in the stripping section. It is proposed that a high concentration of n-butylenes induces a pseudobinary separation between C4’s and ethanol/ ETBE, while a low concentration of n-butylenes allows a pseudobinary separation of ethanol (and all lighter components) and ETBE. It is important to realize that ETBE is the heavy boiler in this system even though the normal boiling point (NBP) of ethanol is higher than the NBP of ETBE, as the vapor pressure curves of ethanol and ETBE intersect at around 300 kPa (Sneesby et al., 1995). Although the azeotrope between ethanol and ETBE creates a distillation boundary where pure ethanol is the stable node in one distillation region, in this instance, both steady states (and the third, unstable, steady state) are in the other distillation region (i.e., where pure ETBE is the stable node). Thus, the main effect of the ethanol-ETBE azeotrope is to increase the curvature of the distillation lines in the vicinity of the azeotrope. There is no evidence to suggest that the azeotrope causes the multiplicity. The two steady states arise since the additional production rate of ETBE which results from maintaining an adequate supply of ethanol to the reaction zone can exactly balance the molar flow of ethanol that would otherwise leave the column with the bottoms product. That is, there is a singularity in the relationship between the bottoms mass flow and the bottoms molar flow. If the results are presented in terms of the molar
4426 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998
Figure 2. Bifurcation diagram for ETBE column with fixed mass reflux rate.
Figure 3. Temperature profiles in the ETBE column.
bottoms yield (Figure 6), only one steady state is evident. Note that the data presented in Figures 2 and 6 are identical and that the numerical values of yield differ only due to the differences in molecular weights of the distillate and bottoms products. It can be argued that the multiplicity is a manifestation of the necessity to control mass or volumetric flows (rather than molar flows) and that multiple steady state would not be seen if a molar flow controller could be implemented. This type of anomaly was designated an input transformation multiplicity by Jacobsen and Skogestad (1991) with respect to nonreactive distillation. Multiple Steady States in a MTBE Column Column Configuration. The column structure shown in Figure 1 can also be applied to MTBE production. In this case, a column with a total of 17 stages (a total condenser, 2 rectifying stages, 8 reactive
stages, 5 stripping stages, and a partial reboiler) was considered. The feed to this column enters on the lowest reactive stage and is composed of 30 mol % isobutylene, 33% methanol (a stoichiometric excess of 10%), and 37% n-butane. Each stage operates at 1100 kPa absolute. Bifurcation Diagram. There are 2 degrees of freedom in the above MTBE column. If the reboiler duty is fixed (using 1 degree of freedom), any other parameter can be varied to produce a locus of steadystate solutions. The bifurcation diagram that is obtained by plotting the mass reflux rate against a characteristic output parameter (i.e., the reboiler temperature) is shown in Figure 7 for a restricted range of reflux rates. There are at least three (and up to five) steady states for all reflux rates between 15980 and 16820 kg/h. High- and Low-Conversion Steady States. As with the multiplicity that was detected in the ETBE
Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4427
Figure 4. Selected composition profiles in the ETBE column.
Figure 5. Reaction rate profiles in the ETBE column.
column described above, there is a substantial variation in the operating conditions for the high-conversion steady state (point X in Figure 7) and the low-conversion steady state (point Y). Table 2 indicates the differences in the key outputs. Figure 8 compares the temperature profiles and highlights the significant differences on stages 6-10 (lower reactive section) and stages 14-17 (stripping section). The reaction zone temperatures in the low-conversion steady state are up to 20 °C higher, while the stripping section temperatures are up to 20 °C lower. Figure 9 indicates that these differences are predominantly due to the prevalence of MTBE in the reactive section and methanol in the stripping section in the low-conversion steady state. There are also important differences in the reaction rate profile (Figure 10): the low-conversion steady state is characterized by high reaction rates on the upper reactive stages and MTBE decomposition on the lower reactive stages.
Transition between Steady States. Operating a nonreactive column with constant reboiler duty and constant reflux rate (i.e., no automatic composition control) is often effective if composition disturbances are unlikely, as the feed split will remain essentially the same for all feed rates. Since the feed split has a much larger effect on the product compositions than fractionation, adequate performance can be expected. However, this type of operating policy is not recommended for reactive distillation columns, especially if multiple steady states are possible. Figures 11-13 show the effect of a feed rate perturbation (a 20% increase followed by an equal and opposite decrease) on the temperatures in the reboiler and stages 10 and 14 (Figure 11), the bottoms product rates (Figure 12), and the reaction rate on stage 10 (Figure 13). The net effect of the perturbation is to cause the operating
4428 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998
Figure 6. Bifurcation diagram for ETBE column presented in terms of bottoms molar yield (cf. Figure 2).
Figure 7. Bifurcation diagram for MTBE column with fixed reboiler duty. Table 2. High- and Low-Conversion Steady States for a MTBE Column
property overall isobutylene conversion (mol %) bottoms ETBE purity (mol %) reboiler temperature (°C) boilup ratio (molar basis) bottoms yield (mass basis with respect to the feed) (%) bottoms yield (molar basis with respect to the feed) (%) reflux ratio
high-conversion steady-state value (point X)
low-conversion steady-state value (point Y)
98.3
69.0
99.9 152.2 8.74 53.2
65.4 133.1 6.47 44.4
41.9
39.9
7.05
5.94
point to shift from the initial, high-conversion steady state to the low-conversion steady state. Mechanistic Interpretation. As with the ETBE column, the rectifying section of the MTBE column is
similar in both the high- and low-conversion steady states, but differences emerge in the reactive section. The high-conversion steady state corresponds to low concentrations of methanol and low temperatures. It appears that this combination prevents or restricts the decomposition reaction on the lower reactive stages. Although some decomposition was not necessarily detrimental in the ETBE column, it resulted in a much lower overall conversion of isobutylene in this case. Once again, the supply of reactants to the reactive section of the column is vital in determining the extent of the reaction and the steady state that the column settles to. The MTBE system includes azeotropes between methanol and isobutylene and between methanol and n-butane, and these play an important role. Within the stripping section, a pseudobinary separation of MTBE and the methanol-C4 azeotropes occurs. At
Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4429
Figure 8. Temperature profiles in the MTBE column.
Figure 9. Selected composition profiles in the MTBE column.
a constant reboiler duty, the reflux rate determines the split between the pseudocomponents. A very high MTBE purity is possible if the feed split is such that almost all the unreacted methanol is returned to the reaction zone to form more MTBE. However, if MTBE is destroyed on the lower reactive stages, the additional methanol is actually driven away from the reactive section to maintain the same feed split and the bottoms product will become contaminated. Thus, changing the feed split (via a change in the reflux rate) can either increase or decrease the bottoms product purity. This duality is the key mechanism in interpreting the multiplicity. The dynamic transition between the high- and lowconversion steady states can be explained via the same mechanism. The effect of the initial increase in the feed rate is to reduce fractionation and starve the reactive section of methanol and isobutylene. The change in reaction conditions initiates the decomposition reaction
and increases the phase equilibrium temperature. The higher temperature reduces the reaction equilibrium constant and propagates the decomposition reaction. The methanol that is formed in this way inundates the stripping section and changes the feed split to favor the bottoms product. Thus, the new steady state is characterized by higher reaction zone temperatures, a larger bottoms rate, and more methanol (and less MTBE) in the stripping section and reboiler. After the feed rate is decreased again, fractionation is increased and more methanol is returned to the reactive section. However, the phase temperatures remain high in the reactive section and the synthesis reaction is not favored. The bottoms rate decreases but the reaction zone temperatures do not return to the initial state due to the high methanol concentrations. In this case, it is not easy to assign a cause for the observed multiplicity from the range of causes that have been elaborated in the literature. None of the explana-
4430 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998
Figure 10. Reaction rate profiles in the MTBE column.
Figure 11. Selected temperatures during a transition between steady states.
tions fit the observations. Jacobsen and Skogestad (1991) described the multiplicities caused by input transformations and energy balance effects. However, 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 (i.e., the constant molar overflow case). Gu¨ttinger and Morari (1997) developed an excellent tool for the analysis of reactive VLE to determine whether the mechanism that was originally described by Bekiaris and Morari (1993) for azeotropic distillation can cause multiplicity in a column with a given feed composition. In this case, the feed is outside the region that produces multiple steady states. However, there is uncertainty with this tool when considering finite or hybrid columns, and Gu¨ttinger and Morari have suggested that interactions
between the reactive and nonreactive column sections could extend this region. Thus, it remains unclear whether azeotropes can account for the unusual behavior in this case. The complexity of the reactive distillation environment suggests that an alternative mechanism might also be possible. A new mechanism, reaction hysteresis, is proposed as follows: composition changes can be propagated and reinforced by separate effects in the reactive and nonreactive column sections, but these changes cannot necessarily be reversed after they have been initiated. To understand the mechanism, it is important to consider necessary and sufficient conditions for output multiplicity. Previous authors (e.g., Hauan et al., 1995; Hauan and Lien, 1997) have discussed multiplicity in relation to strongly nonlinear behavior, but nonlinearity is only necessary and not sufficient for output multiplic-
Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4431
Figure 12. Mass and molar bottoms rates during a steady-state transition.
Figure 13. Reaction rate on stage 10 during a steady-state transition.
ity. 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 nonreversible change in operating conditions) which is both necessary and sufficient 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. Reaction hysteresis can occur in the hybrid reactive distillation of MTBE. 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 minimumboiling 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, which 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 favored. 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 stabilizing to a new operating point with a 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
4432 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998
Figure 14. Analogy between reaction runaway and reaction hystersis within a reactive distillation column.
“runaway” rather than kinetic effects, as shown in Figure 14. A reaction runaway is driven by the interaction between temperature and the rate constant in a no-reversible equilibrium reaction. Here, a temperature change initiates a similar sequence of events where the reaction rate is driven by changes in the composition (i.e., the availability of reactants). It is proposed that this effect occurs only over a narrow range of temperatures and compositions (the multiplicity interval). To further test this hypothesis within the simulation, the recycle loop was broken by altering the relationship between the reaction rate and reaction zone composition. This was achieved by making the equilibrium constant, independent of temperature. The bifurcation analysis was repeated for this condition, and the multiplicity was found to disappear. 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 vapor 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 the interactions between the reactive and nonreactive column sections and is independent of reaction kinetics, the column energy balance, and singularities in the mass-molar relationships. This type of behavior exactly fits the simulation observations and could be responsible for the output multiplicity shown in Figure 7. Conclusions Two hybrid reactive distillation columns have been examined for the presence of physically realizable multiplicities. In the first column, ETBE synthesis was considered and multiple steady states were found for the case where the mass reflux rate was constant. MTBE synthesis was considered in the second column, and multiple steady states were found for a given reboiler duty. There are at least three (and possibly more) known causes of output multiplicity in reactive distillation. These causes can be manifested in a variety of ways, but a process hysteresis (i.e., a nonreversible change in
the column conditions) is always required to produce the distinctive S shape which is seen on the bifurcation diagrams. The underlying physical mechanism of a process hysteresis is an inverse effect that can be interpreted as a nonintuitive response to a change in an operating variable. The inverse effect that was seen in the ETBE column was an increase in the bottoms mass rate following an increase in the reboiler duty. The bottom molar rate always decreased in response to an increase in reboiler duty, but singularities in the mass-molar relationship resulted in the inverse effect that was observed with respect to the bottoms mass rate. From a mechanistic perspective, the supply of ethanol was crucial. The supply of methanol to the reactive section of the MTBE column was seen to be equally as important. Mechanistically, methanol can only be supplied to the reaction zone advantageously if the reaction zone temperatures are low and the synthesis reaction is favored. Otherwise, the additional methanol promotes the decomposition reaction. The inverse effect that was observed in this case was an increasing bottoms product purity in response to an increasing reflux rate (with constant reboiler duty). Two explanations fit this behavior: interactions between the reactive and the nonreactive column sections expand the feed region, which results in multiple steady states due to azeotropes (as described by Gu¨ttinger and Morari, 1997); and reaction hysteresis (as described above). In both of the columns that were examined, it was necessary for the decomposition reaction to occur on some reactive stages in one of the steady states. However, it is not possible to generalize that either decomposition or multiple reactive stages are necessary requirements for multiplicity. Nevertheless, the steadystate and dynamic simulations were both useful to reveal the mechanistic behavior of columns that exhibit physically realizable multiple steady states. Literature Cited Bekiaris, N.; Morari, M. Multiple Steady States in Homogeneous Azeotropic Distillation. Ind. Eng. Chem. Res. 1993, 32, 2023. Gu¨ttinger, T. E.; Morari, M. Predicting Multiple Steady States in Distillation: Singularity Analysis and Reactive Systems. Comput. Chem. Eng. 1997, 21 (Supp), S995. Hauan, S.; Lien, K. M. Dynamic Evidence of the Multiplicity Mechanism in Methyl tert-Butyl Ether Reactive Distillation. Ind. Eng. Chem. Res. 1997, 36, 3995. Hauan, S.; Hertzberg, T.; Lien, K. M. Why Methyl tert-Butyl Ether Production by Reactive Distillation May Yield Multiple Solutions. Ind. Eng. Chem. Res. 1995, 34, 987. Jacobs, R.; Krishna, R. Multiple Solutions in Reactive Distillation for Methyl tert-Butyl Ether Synthesis. Ind. Eng. Chem. Res. 1993, 32, 1706. Jacobsen, E. W.; Skogestad, S. Multiple Steady States in Ideal Two-Product Distillation. AIChE J. 1991, 37 (4), 499. Nijhuis, S. A.; Kerkhof, F. P. J. M.; Mak, A. N. S. Multiple Steady States during Reactive Distillation of Methyl tert-Butyl Ether. Ind. Eng. Chem. Res. 1993, 32, 2767. Riddle, L. HPI Construction Boxscore. Hydrocarbon Process. 1996, 75 (10B). Sneesby, M. G.; Tade´, M. O.; Datta, R. Tert-Butyl EtherssA Comparison of Properties, Synthesis Techniques and Operating Conditions for High Conversions. Dev. Chem Eng., Mineral Process. 1995, 3 (2), 89. Sneesby, M. G.; Tade´, M. O.; Datta, R.; Smith, T. N. ETBE Synthesis by Reactive Distillation. 1. Steady-State Simulation and Design Aspects. Ind. Eng. Chem. Res. 1997a, 36, 1855.
Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4433 Sneesby, M. G.; Tade´, M. O.; Datta, R.; Smith, T. N. The Design of Reactive Distillation Processes for ETBE Synthesis. Submitted to J. Inst. Eng. Singapore 1997b. Sneesby, M. G.; Tade´, M. O.; Smith, T. N. Implications of Reactive Distillation Multiplicity for Operation and Control of Etherification Columns. Proceedings of Distillation and Absorption ‘97, Maastricht, 1997c. Sneesby, M. G.; Tade´, M. O.; Smith, T. N. Steady-State Transitions in the Reactive Distillation of MTBE. Comput. Chem. Eng. 1997d, in press.
Sneesby, M. G.; Tade´, M. O.; Smith, T. N. Multiplicity and PseudoMultiplicity in MTBE and ETBE Reactive Distillation. Trans. Inst. Chem. Eng. 1997e, in press.
Received for review December 30, 1997 Revised manuscript received June 8, 1998 Accepted August 10, 1998 IE970938+
