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Multiplicity and Pseudo-Multiplicity in MTBE and ETBE Reactive Distillation
0263±8762/98/$10.00+0.00 q Institution of Chemical Engineers Trans IChemE, Vol 76, Part A, May 1998
MULTIPLICITY AND PSEUDO-MULTIPLICITY IN MTBE AND ETBE REACTIVE DISTILLATION Â(MEMBER) and T. N. SMITH (FELLOW) M. G. SNEESBY (GRADUATE), M. O. TADE School of Chemical Engineering, Curtin University of Technology, Perth, Australia
M
ultiplicity of steady states can exist in three forms: input multiplicity, output multiplicity, and pseudo-multiplicity, which are described here. Input multiplicity is likely to be common in reactive distillation and places substantial restrictions on the selection of controlled variables. Output multiplicity is also possible in reactive distillation and also has less signi® cant implications for operation and control as it may in¯ uence the choices of control structure and the operating region. Pseudo-multiplicity parallels output multiplicity but is only present in terms of molar inputs. It has only minor implications for operation and control but is an important discovery as it reconciles recent reports of material balance output multiplicity with earlier conjecture which claimed that this was unlikely. Keywords: reactive distillation; input multiplicity; output multiplicity; pseudo-multiplicity
INTRODUCTION
variables are those which can be manipulated by controllers. For distillation processes, the primary input variables are the re¯ ux rate and reboiler duty (or boilup), but the distillate and bottoms rates and combinations of these (e.g. the re¯ ux ratio) can also be considered input variables as level controllers on the re¯ ux drum and reboiler sump transmit changes in the product rates directly to the column. Output variables are those which are either controlled or used to describe the conditions of the process (e.g. product or stage temperatures, compositions and yields). By convention, input variables are always plotted on the x-axis while output variables are plotted on the y-axis. One of the ® rst comprehensive discussion on multiplicity in distillation was provided by Jacobsen and Skogestad4 and identi® ed two fundamental causes of output multiplicity which were described as `input transformation’ and `energy balance interaction’ . Both effects could be found in ideal binary distillationÐ an important result which contradicted some earlier ® ndings. They derived several results analytically and showed that multiplicity is generally only likely with the re¯ ux rate-boilup (LV) control con® guration, and most likely with high re¯ ux ratios. Output multiplicity has been reported in MTBE reactive distillation by several researchers using simulation studies5- 9 and has also been con® rmed experimentally10 . However, the conditions required for multiplicity have not been de® ned and neither a consistent nor complete explanation has been proposed for the cause of the multiplicity. Several mechanisms have been discounted, including CSTR type multiplicity5 , crossing non-reactive distillation boundary via reaction5 , heat of reaction and multiple reactions11 . Several other proposed mechanisms can be disregarded including the effects of either normal or reactive azeotropes. It remains unclear whether multiplicities in the reactive distillation of MTBE, or other fuel ethers, are due to the chemical reaction or to the combination of components present.
Reactive distillation has importance for several industrial processes, including the etheri® cation of C4 and C5 isoole® ns with methanol or ethanol to produce commercially signi® cant fuel additives such as methyl tert-butyl ether (MTBE), ethyl tert-butyl ether (ETBE) and tert-amyl methyl ether (TAME). These ethers have applications as both octane enhancers and gasoline oxygenates and their use has grown rapidly during the 1980s and 1990s as environmental regulations concerning gasoline composition are tightened worldwide. There are signi® cant advantages to incorporating the reaction and separation stages of a process in a single unit operation. Most notably, the capital cost is reduced, energy integration is improved and the reactant conversion is increased for equilibrium limited reactions1 . However, the complexity of the combined process is known to create some unusual responses to changes in operating variables2,3 . Many of these effects are related to the phenomenon of multiplicity of steady states and have signi® cant implications for both the operation and control of reactive distillation columns. Multiplicity normally occurs as either input multiplicity or output multiplicity. Input multiplicity is associated with unusual, unexpected or inverse column responses. It occurs when two or more unique sets of input variables produce the same output condition. Output multiplicity occurs when one set of input variables results in two or more unique and independent sets of output variables. Figures 1 and 2 provide graphical distinctions between input and output multiplicity for arbitrary variables. Input multiplicity is present in the ® rst chart as input variable values of both a and b result in the same value of the output variable. Output multiplicity is present in the second case as an input value of c could result in output values of either d or e. The differentiation between input and output variables is best made with reference to control structures. Input 525
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Figure 1. Input multiplicity.
Figure 3. Reactive distillation column con® guration for ETBE synthesis.
Figure 2. Output multiplicity.
CHARAC TERISTICS OF REACTIVE DISTILLATION COLUM NS
Although analysis of conventional distillation4,12 suggests that output multiplicity is only likely for the LV con® guration, several researchers5- 7,9 have more recently reported reactive distillation output multiplicities using the re¯ ux ratio-bottoms rate (L/D:B) con® guration. Experimentally, the distillate rate-reboiler duty (DQR ) con® guration was found to yield unique solutions for MTBE reactive distillation while the re¯ ux rate-reboiler duty (LQR ) con® guration produced a multiplicity10 . Multiplicity in conventional and reactive distillation has been an active area of research in recent years. A full explanation of output multiplicity, which allows the prediction of instability early in the design phase (i.e. without requiring complete simulation models to be developed), is still forthcoming and research continues in this crucial area. Despite the recent activity, the signi® cance of multiplicity has attracted little attention (Jacobsen and Skogestad13 is a notable exception), although clearly important for operation and control of industrial columns, such as those used for etheri® cation. The distinction between input and output multiplicity has also often been neglected in many previous contributions, adding to the confusion and misunderstanding of multiplicity. This paper addresses the latter two issues in a practical manner and also contributes to the former area by de® ning necessary and suf® cient conditions for the existence of output multiplicity. The concept of pseudo-multiplicity is also introduced and this may be a key to solving the output multiplicity puzzle.
Two reactive distillation columns have been studied. The ® rst column was for ETBE synthesis and had a con® guration which is described by Figure 3 (column) and Table 1 (input data). The second column is for MTBE synthesis. The MTBE column con® guration is similar to the ETBE column but it has ® ve more reactive stages and one extra recti® cation and stripping stage, which results in a con® guration that is similar to that used previously by other researchers. Its input is given in Table 2. The ETBE column was modelled using the equationbased process simulation package, SpeedUp14 and the equilibrium stage model for distillation systems, as discussed previously2,3 . The model incorporates rigorous expressions for the ETBE reaction equilibrium15 and kinetics that are valid at all ethanol concentrations equilibrium16 . The principal side-reaction involving the dimerization of isobutylene was modelled but other sidereactions, including the hydration of isobutylene to form isobutanol with the ingress of water, were assumed to be negligible. The UNIFAC model was used to predict component activities and recently published vapour pressure data17 was used in the phase equilibrium calculations. The MTBE process is similar to the ETBE process except that the synthesis reaction is less restricted by the chemical equilibrium. There is also a signi® cant azeotrope between methanol and the butylenes which contains up to 11% methanol. This has the effect of transferring unreacted methanol from the bottoms product (MTBE) to the distillate product and helps to increase the overall conversion of isobutylene. Reaction equilibrium was assumed for the
Table 1. ETBE reactive distillation column simulation input. Feed Conditions Temperature Rate Composition (mol%)
Overall Excess EtOH
Column Speci® cation 308 C 0.76 l/min ±1 29.1% ETBE, 9.1% EtOH, 7.3% i But, 54.5% nBut 5.0 mol%
Recti® cation Stages Reaction Stages Stripping Stages Total Stages Feed Stage Overhead Pressure Re¯ ux Rate
2 3 5 10 6 950 kPa 2.50 l/min ±1
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Table 2. MTBE reactive distillation column simulation input. Feed Conditions Temperature Rate Composition (mol%) Excess MeOH
Column Speci® cation 708 C 224 m3 hr±1 28.2% MeOH, 25.6% i But, 46.2% n C4 10.0 mol%
MTBE column rather than the kinetically controlled reaction model which was used for ETBE column. The dimerization side-reaction, which is normally also present in MTBE synthesis, was assumed to be negligible in this case. These assumptions do not indicate a lack of rigour but demonstrate that neither the side-reaction nor reaction kinetics are the sole cause of the observed multiplicities. The UNIFAC method was again used to predict liquid activities. The MTBE simulations were performed using a sequential modular simulation package, Pro/II18 . The use of a different simulation package provides further evidence that the phenomenon is not a result of the modelling techniques or mathematical anomalies. Reactive distillation requires that the operating conditions to be used for both reaction and separation should coincide. If the conditions required for good phase separation are such that temperatures are too high or too low for effective reaction, reactive distillation cannot be utilized successfully. This implies that the column operating conditions must be optimised with respect to both reaction and separation. For example, if the reboiler duty is too high, the composition pro® le shifts so that the concentration of ether in the reaction zone is such that the reverse (ether decomposition) reaction is favoured. Consequently, the isobutylene conversion is reduced signi® cantly and the bottom product becomes contaminated with alcohol. If the reboiler duty is too low, insuf® cient alcohol is recycled to the reaction zone and the isobutylene conversion is again low. In an ETBE column, the stripping section of the column (stages 6±10) separates ETBE from ethanol and C4 components, returning the lighter components to the reaction section where both ethanol and isobutylene react further to form more ETBE, and purifying the bottoms product which contains greater than 95% ETBE. The reaction section (stages 3±5) is packed with catalyst (Amberlyst 15y ) and operates at a temperature such that the forward reaction is favourable. The composition pro® le within the column is such that the ethanol:isobutylene ratio in the reaction section is much greater than one which further promotes the forward reaction to produce ETBE and increases the overall conversion of isobutylene. The recti® cation section (stages 1±2) removes ETBE from the distillate product and reduces the ethanol concentration of the distillate close to the azeotropic composition (approximately 1% ethanol and 99% butylenes). Only a limited number of recti® cation stages are used to ensure that unreacted isobutylene is recycled to the reaction zone. The functions of the three sections of MTBE column are essentially as described above for ETBE, except that more stripping stages can be tolerated as the minimum-boiling Trans IChemE, Vol 76, Part A, May 1998
Recti® cation Stages Reaction Stages Stripping Stages Total Stages Feed Stage Overhead Pressure Reboiler Duty
3 8 6 17 8 1100 kPa 80 MW
methanol-butylene azeotrope continues to recycle methanol to the reaction zone provided the methanol content of the feed does not exceed the `carrying capacity’ of the unreacted butylenes. INPUT MULTIPLICITY The need to optimize the heat input to the column is the source of a signi® cant input multiplicity in reactive distillation. This was demonstrated using simulation of the ETBE column. The results shown in Figure 4 were obtained by varying the reboiler duty while the re¯ ux rate was ® xed and show input multiplicities between the bottoms temperature, isobutylene conversion, ETBE product purity (all output variables) and the reboiler duty (an input variable). Reboiler duties of 8.20 kW and 8.97 kW both produce a bottoms temperature of 1508 C. Similarly, the same value of isobutylene conversion and ETBE product purity is obtained at two different values of the reboiler duty. Below the optimum reboiler duty (approximately 8.35 kW), the operating conditions are favourable for the ETBE reaction (as indicated by the relatively high conversions) but provide insuf® cient vapour-liquid traf® c for good separation (low ETBE purities). Above the optimum reboiler duty, the converse applies. For a small range of reboiler duties, both conversion and purity are high. This behaviour is common to MTBE reactive distillation columns and similar results can be produced with MTBE column simulations. It is important to note that these multiplicities are not present in conventional distillation, where increasing the reboiler duty normally always increases the bottoms temperature, but are essentially always present in reactive distillation. The two regions can be de® ned as separation controlled (reboiler duties
Figure 4. Input multiplicity in an ETBE reactive distillation column.
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below the optimum) and reaction controlled (reboiler duties above the optimum), respectively. OUTPUT MULTIPLICITY There is no evidence of output multiplicity in the ETBE column described above or in a similar column at reaction equilibrium. However, previous research has identi® ed this property in MTBE reactive distillation columns, which are potentially very similar. The MTBE column studied here was shown to produce a range of input multiplicities which were similar to those seen in the ETBE column, and was also found to give rise to several signi® cant output multiplicities. One such multiplicity is shown in Figure 5 which describes the response to varying the volumetric re¯ ux rate while the reboiler duty is constant. At re¯ ux rates between 995 and 1030 m3 hr- 1 there are three separate, unique solutions to exactly the same combination of reboiler duty and re¯ ux rate. Each solution corresponds to a different yield (bottoms product rate) of MTBE and conversion of isobutylene. Normally, increasing the re¯ ux rate at constant reboiler duty shifts the feed split to decrease the distillate yield. For some values of re¯ ux rate (between 995 and 1030 m3 hr- 1 ), the internal ¯ ows in the MTBE column described here conspire to cause the distillate yield to actually increase with increasing re¯ ux rate. This is described by condition (1) and appears to be the principal cause of the observed output multiplicity. Similarly, output multiplicity would also result if condition (2) was satis® ed.
¶Dv 0 > ¶Lv ¶Bv 0 > ¶QR
(1) (2)
The cause of output multiplicity in reactive distillation appears to be essentially the same as in conventional distillation. When condition (1) is satis® ed, the effect of re¯ ux on the mass balance via changes in the composition which are manifested through the energy balance are greater than the direct effect on the mass balance. This effect can be accentuated in reactive distillation by the effect of composition changes on the reaction rate. It might, therefore, be expected that output multiplicity would be more common in reactive distillation than conventional
Figure 5. Output multiplicity in a MTBE reactive distillation column.
Figure 6. Pseudo-multiplicity in an ETBE column.
distillation. In both cases, the necessary conditions for output multiplicity are described by the inequalities (1) and (2). PSEUDO-MULTIPLICITY A third type of multiplicity, designated pseudomultiplicity, is described here using the ETBE column shown in Figure 3. Essentially, pseudo-multiplicity is output multiplicity that is only seen in terms of molar inputs. If the molar boilup rate to the ETBE column is ® xed (at, say, 18.0 mol min- 1 ) and the molar re¯ ux rate is varied (LV con® guration), an apparent output multiplicity arises, as shown by Figure 6. However, if the reboiler duty is ® xed (e.g. 8.45 kW) and the volumetric re¯ ux rate is varied, the multiplicity disappears, as shown by Figure 7. This phenomenon appears to be caused by an output transformation which parallels the input transformation described by Jacobsen and Skogestad4 which was suggested as one of two principal causes of output multiplicity. Input transformation describes how a multiplicity might be transparent when considering molar inputs but real when considering volumetric or mass inputs. Here, the reverse is true. In both cases the transformation arises due to changing compositions and subsequent variation in the molar density. Both input and output transformations can be linked to a change in the sign of the gain between the molar re¯ ux rate and the volumetric re¯ ux rate, or between the molar boilup and reboiler duty, as shown in conditions (3) and (4):
¶Lm 0 < ¶Lv ¶Vm 0 < ¶QR
(3) (4)
Figure 7. LV output transformation (ETBE column).
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MULTIPLICITY AND PSEUDO-MULTIPLICITY IN MTBE AND ETBE REACTIVE DISTILLATION
Figure 8. Pseudo-multiplicity in a MTBE column.
Similar behaviour is evident for the MTBE column when considering the LB con® guration. If the molar bottoms ¯ ow is ® xed (674 kmol hr- 1 ) and the molar re¯ ux rate is varied, an output multiplicity arises (Figure 8). However, if the volumetric bottoms ¯ ow is ® xed (e.g. 75 m3 hr- 1 ) and the volumetric re¯ ux rate is varied, the multiplicity disappears (Figure 9). Again, an output transformation appears to be the cause of this behaviour. In this case, the transformation is between the molar bottoms rate and the volumetric or mass bottoms rate, but the multiplicity is still associated with a change of sign in the gain:
¶Bm 0 < ¶Bv
(5)
It is important to recognize that previous reports of output multiplicity in reactive distillation for material balance control con® gurations5- 7,9 , that is con® gurations such as the DV, LB or (L/D)B schemes where either the distillate or bottoms rate are manipulated directly to control composition, are actually examples of pseudo-multiplicity. Thus, previous conjecture that output multiplicity is only likely for energy balance control con® gurations (i.e. schemes where composition control is achieved indirectly by manipulation of the internal column ¯ ows) remains sound and convincing. CAUSES OF MULTIPLICITY Input multiplicity is caused by two (or more) con¯ icting effects operating via the same manipulated variable. For example, increasing the reboiler duty in an ETBE column while the re¯ ux rate is held constant improves the separation between ETBE and the butylenes (a favourable effect) but also increases the reaction zone temperature which
Figure 9. LB output transformation (MTBE column).
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encourages the decomposition reaction to take place and reduces the overall conversion of isobutylene (a detrimental effect). As reactive distillation usually always involves a compromise between reaction and separation effects, there are many variables and operating conditions that have the potential to result in input multiplicities. The cause(s) of output multiplicity is (are) much more complex. It is important to realize that in any system which displays output multiplicity there appears to always be at least one manipulated variable which maintains a oneto-one (or many-to-one) mapping with the output variable that describes the output multiplicity. For example, if a particular MTBE column shows an output multiplicity between the re¯ ux rate and the bottoms purity (with constant reboiler duty), each value of the bottoms rate (or another variable) will still produce one and only one value of bottoms purity. Once a variable is found that does not show any output multiplicity, it is possible to create all solutions and plot the results in terms of any input variable so that output multiplicities (including unstable solutions) can be revealed. The principal cause of the output multiplicity lies in the behaviour of manipulated variable which does not show output multiplicity. Either condition (1) or (2) must be satis® ed for the column to produce an output multiplicity. In fact, the inequalities (1) and (2) represent necessary and suf® cient conditions for output multiplicity. The physical interpretation of conditions (1) and (2) is that changes in the mass balance which result from the effects of composition changes on the energy balance are more signi® cant than the direct effect on the mass balance. Jacobsen and Skogestad4 derived similar results for conventional binary distillation and demonstrated analytically that instability (which indicates multiplicity) can generally only arise by manipulating a variable that does not affect the feed split directly (i.e. the re¯ ux rate and reboiler duty but not the distillate or bottoms rates). Similar conditions cannot easily be derived for multi-component distillation but the results would appear to be extendable at an empirical level. A consequence of this result is that, even in columns which display output multiplicity in one or more input variables, output multiplicity in terms of the distillate or bottoms rate is unlikely. This provides a mechanism for ® nding the output multiplicities described above and also a means of limiting their impact on column operation and control. Output multiplicity is possible in non-reactive distillation4,12 and has been demonstrated10 . However, the presence of a chemical reaction within the distillation system allows the composition pro® le to be shifted more rapidly for relatively small changes in the input variables (both `feedsplit’ variables and `energy-balance’ variables). The effect of the energy balance on the mass balance via the composition pro® le can, therefore, be magni® ed so that conditions (1) and (2) are more easily satis® ed. This result predisposes reactive distillation columns to the phenomenon of output multiplicity and, perhaps, accounts for the growing number of reports of output multiplicity in a wide variety of reactive distillation columns. It is more dif® cult to ® nd a physical interpretation of pseudo-multiplicity as the phenomenon does not correspond to a `real’ situation. As with output multiplicity,
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pseudo-multiplicity arises through interaction between the mass and energy balances and is manifested through composition changes. Pseudo-multiplicity requires that changes in the product composition(s) which result from changes in the molar column inputs have a more signi® cant effect on the mass balance than direct manipulation of the mass balance via the molar inputs. This is described by conditions (3), (4) and (5) which are necessary and suf® cient conditions for pseudo-multiplicity. Figures 8 and 9 demonstrate that, pseudo-multiplicity can occur with material balance control schemes (i.e. where a `feed-split’ variable is the primary manipulated variable), unlike output multiplicity. This can be the result of a nonequimolar reaction or another property of the system. Where a non-equimolar reaction is present, the reaction changes the total number of moles present in the system. This effectively changes the feed split without changing the molar product rate by forcing the bottoms (or distillate) mass yield to change. However, this is not the only cause of material balance pseudo-multiplicity as control schemes which manipulate the molar yield of either the distillate or bottoms or schemes which ® x the molar reaction rate can also lead to multiplicities. As described above for output multiplicity, the presence of chemical reaction in reactive distillation provides a means of rapidly changing product compositions without requiring signi® cant changes in the feed-split or the energy balance. This property is a crucial requirement for pseudomultiplicity as it provides a means for energy balance effects to predominate over feed split effects so that one or more of conditions (3) to (5) can be satis® ed. In conventional distillation, the size of the two effects normally differs by an order of magnitude. Furthermore, the effects are usually essentially independent of whether mass or molar inputs are considered. Consequently, pseudomultiplicity is unlikely for non-reactive distillation, at least in terms of the `feed-split’ variables. OPERATION AND CONTRO L The presence of input multiplicity in reactive distillation has two signi® cant implications for column operation and control. Firstly, the narrow range of conditions where phase and chemical equilibrium intersect favourably necessitates tight control of the primary operating variable (e.g. reboiler duty, bottoms rate, etc.). Manual control of a reactive distillation column is, therefore, unlikely to provide adequate control of the heat input and could lead to either the isobutylene conversion or ETBE purity being compromised. The necessity for tight control could even in¯ uence the equipment design as some types of reboiler (e.g. integrated exchanger networks) are inherently harder to control than others (e.g. ® red heater). The need for tight reboiler control cannot be avoided by selecting a material balance control con® guration (e.g. DV or LB con® guration) as the performance of these controllers is intimately linked to the performance of the level control loops and tight reboiler control would still be required. Secondly, the input multiplicity prevents the bottoms temperature (or composition) being directly used as a controlled variable, even though it correlates well with the principal operating objectives of high isobutylene conversion and high ETBE purity. The use of either the bottoms
temperature or an analyser measuring the bottoms composition would require the controller gain to vary through zero to avoid closed-loop instability. This requirement arises from the uncertainty of whether to increase or decrease the manipulated variable to move towards the controlled variables set point (see Figure 4). This restriction favours the use of an alternative controlled variable, such as a temperature midway up the stripping section, which displays no input multiplicity with potential manipulated variables3 . It is also important to know which operating region (reaction controlled or separation controlled) an etheri® cation column is operating in order to understand how the column will respond to changes in operating variables. Equivalent bottoms temperatures exist in the two regions so that some other means of distinguishing the regions must be determined. The temperature difference across the reaction zone can provide this information. A low temperature difference (say, less than 58 C) implies that the column in operating in the separation controlled region and that the reboiler duty needs to be increased to increase both the isobutylene conversion and ether purity. A high temperature difference (say, greater than 58 C) implies that the column is operating in the reaction controlled region and that the reboiler duty needs to be decreased to increase the conversion and purity. The implications of output multiplicity for operation and control are not as substantial as might initially be thought. Firstly, the output multiplicity illustrated here, and other reported multiplicities4,11 , only occur with very high vapourliquid traf® c inside the column and, therefore, do not represent probable operating points as such columns designs are likely to be uneconomical. Secondly, there is no evidence of output multiplicity occurring with material balance control con® gurations (e.g. DV or LB) unless the associated level control is very poor12 so that the possibility of multiplicity can generally be avoided with an appropriate control scheme design. However, the possibility exists that process disturbances could result in a shift to an undesirable steady state without either the reboiler duty or re¯ ux rate being changed. This provides an incentive to avoid manual control. A closed-loop controller would detect the shift in the operating region (if appropriate controlled variables have been selected) and try to manipulate either the reboiler duty or re¯ ux rate to return the controlled variables to their set points. The dif® culty here arises with associated input multiplicities that make the selection of appropriate controlled variables critical. However, constraints imposed on the product yields (or reboiler duty) could be useful to detect a possible transition between parallel states and trigger alarms or prevent excessively large control moves. If an undesirable steady state is reached during startup or otherwise, the transition to the preferred steady state can be made by temporarily decommissioning the control scheme and manually increasing or decreasing the product yield to force the feed split close to the desired level. Once this has been achieved, the control scheme can be recommissioned. Pseudo-multiplicity has less signi® cant implications for operation and control as molar ¯ ow controllers are not physically realizable. However, it demonstrates the need to work only in terms of mass or volume inputs (rather than Trans IChemE, Vol 76, Part A, May 1998
MULTIPLICITY AND PSEUDO-MULTIPLICITY IN MTBE AND ETBE REACTIVE DISTILLATION molar inputs which are still commonly employed) when using simulations to model an actual process or design. Most signi® cantly, the discovery of pseudo-multiplicity reconciles recent reports5- 7,9 with the previously established result that output multiplicity is unlikely with control con® gurations other than the LV scheme. This is a crucial ® nding as the latter result has been demonstrated by analytical studies4,12 and experimental work10 . Essentially, it allows some alarming results of hysteresis to be ignored for design, operation and control of `real’ columns. CONCLUSIONS Input multiplicity occurs where more than one set of input conditions (e.g. re¯ ux rate and reboiler duty) can be used to produce the same output condition (e.g. bottoms temperature or purity). The presence of input multiplicity creates a change of sign in the process gain which in¯ uences the selection of control con® guration and restricts the range of variables that can be used to infer the state of the column. With no control system (column being operated in manual), the operator still requires additional information to determine which operating region the column is in, so that appropriate changes to the manipulated variables can be made to move the column towards its optimum operating point. Output multiplicity occurs where a single set of inputs results in two or more unique outputs. This results from unusual relationships between either the distillate and re¯ ux rates or between the reboiler duty and the bottoms rate (conditions 1 and 2). The state of the column can, therefore, not be determined solely from the inputs and some other measure is required to provide suf® cient input to a control system to allow stable and appropriate control moves to be made. Hysteresis is also possible where more than one steady state solution exists. Pseudo-multiplicity occurs where molar inputs (rather than mass or volumes inputs that would result from control valves) produce an output multiplicity. This can only be observed via simulation and is not associated with actual operating columns. The key implication of this phenomenon is that it creates a requirement to conduct simulation studies (for design, control or otherwise) in terms of mass or volume inputs only. The ® nding also reconciles some recent contributions with earlier studies and experimental work in the same area.
n But nC4 Vm QR
531
n-butylenes n-butane molar boilup rate reboiler duty
REFERENCES 1. Sneesby, M. G., TadeÂ,M. O. and Datta, R., 1995, tert-Butyl ethersÐ A comparison of properties, synthesis techniques and operating conditions for high conversions, Devel Chem Eng & Mineral Processing, 3: 89±116. 2. Sneesby, M. G., TadeÂ, M. O., Datta, R. and Smith, T. N., 1996, ETBE synthesis by reactive distillation. 1. Steady state simulation and design aspects, Ind Eng Chem Res, 36(5): 1855±1869. 3. Sneesby, M. G., TadeÂ, M. O., Datta, R. and Smith, T. N., 1996, ETBE synthesis by reactive distillation. 2. Dynamic simulation and control aspects, Ind Eng Chem Res, 36(5): 1870±1881. 4. Jacobsen, E. W. and Skogestad, S., 1991, Multiple steady states in ideal two-product distillation, AIChEJ, 37(4): 499±511. 5. Nijhuis, S. A., Kerkhof, F. P. J. M. and Mak, A. N. S., 1993, Multiple steady states during reactive distillation of methyl tert-butyl ether, Ind Eng Chem Res, 32: 2767±2774. 6. Jacobs, R. and Krishna, R., 1993, Multiple solutions in reactive distillation for methyl tert-butyl ether synthesis, Ind Eng Chem Res, 32: 1706±1709. 7. Hauan, S., Hertzberg, T. and Lien, K. M., 1995, Why methyl tert-butyl ether production by reactive distillation may yield multiple solutions, Ind Eng Chem Res, 34: 987±991. 8. Perez-Cisneros, E., Schenk, M., Gani, R. and Pilavachi, P. A., 1996, Aspects of simulation, design and analysis of reactive distillation operations, Comp Chem Eng, 20(Suppl): S267±S272. 9. Schrans, S., de Wolf, S. and Baur, R., 1996, Dynamic simulation of reactive distillation: An MTBE case study, Comp Chem Eng, 20(Suppl): S1619±S1624. 10. Sundmacher, K. and Hoffman, U., 1995, Oscillatory vapor-liquid transport phenomena in a packed reactive distillation column for fuel ether production, Chem Eng J, 57: 219±228. 11. Ciric, A. R. and Miao, P., 1994, Steady state multiplicity in an ethylene glycol reactive distillation column, Ind Eng Chem Res, 33: 2738±2748. 12. Jacobsen, E. W. and Skogestad, S., 1994, Instability of distillation columns, AIChEJ, 40(9): 1466±1478. 13. Jacobsen, E. W. and Skogestad, S., 1995, Multiple steady states and instability in distillation. Implications for operation and control, Ind Eng Chem Res, 34: 4395±4405. 14. Aspen Technology Inc., 1993, The SpeedUp User’ s Manual, (Cambridge, Massachusetts, USA). 15. Jensen, K. L. and Datta, R., 1995, Ethers from ethanol 1. Equilibrium thermodynamic analysis of the liquid phase ethyl tert-butyl ether reaction, Ind Eng Chem Res, 34: 392. 16. Jensen, K. L. and Datta, R., 1996, Ethers from ethanol 7. Transition state theory analysis of the kinetics of liquid phase ethyl tert-butyl ether synthesis reaction, submitted to Ind Eng Chem Res. 17. KraÈhenbuÈhl, M. A. and Gmehling, J., 1994, Vapor pressures of methyl tert-butyl ether, ethyl tert-butyl ether, isopropyl tert-butyl ether, tertamyl methyl ether and tert-amyl ethyl ether, J Chem Eng Data, 39: 759. 18. Simulation Sciences Inc. (1994) Pro/II Keyword Input Manual (Brea, California, USA).
NOMENCLATUR E Bm Bv EtOH ETBE Dv iBut Lm Lv MeOH MTBE
molar bottoms rate volumetric bottoms rate ethanol ethyl tert-butyl ether volumetric distillate rate isobutylene molar re¯ ux rate volumetric re¯ ux rate methanol methyl tert-butyl ether
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ADDRESS Correspondence concerning this paper should be addressed to Dr Moses TadeÂ, School of Chemical Engineering, Curtin University of Technology, Perth, GPO Box U1987, WA 6845, Australia. (E-mail: [email protected]). The manuscript was communicated via our International Editor for Australia, Professor G. J. Jameson. It was received 29 November 1996 and accepted for publication 2 December 1997.
MULTIPLICITY AND PSEUDO-MULTIPLICITY IN MTBE AND ETBE REACTIVE DISTILLATION Â(MEMBER) and T. N. SMITH (FELLOW) M. G. SNEESBY (GRADUATE), M. O. TADE School of Chemical Engineering, Curtin University of Technology, Perth, Australia
M
ultiplicity of steady states can exist in three forms: input multiplicity, output multiplicity, and pseudo-multiplicity, which are described here. Input multiplicity is likely to be common in reactive distillation and places substantial restrictions on the selection of controlled variables. Output multiplicity is also possible in reactive distillation and also has less signi® cant implications for operation and control as it may in¯ uence the choices of control structure and the operating region. Pseudo-multiplicity parallels output multiplicity but is only present in terms of molar inputs. It has only minor implications for operation and control but is an important discovery as it reconciles recent reports of material balance output multiplicity with earlier conjecture which claimed that this was unlikely. Keywords: reactive distillation; input multiplicity; output multiplicity; pseudo-multiplicity
INTRODUCTION
variables are those which can be manipulated by controllers. For distillation processes, the primary input variables are the re¯ ux rate and reboiler duty (or boilup), but the distillate and bottoms rates and combinations of these (e.g. the re¯ ux ratio) can also be considered input variables as level controllers on the re¯ ux drum and reboiler sump transmit changes in the product rates directly to the column. Output variables are those which are either controlled or used to describe the conditions of the process (e.g. product or stage temperatures, compositions and yields). By convention, input variables are always plotted on the x-axis while output variables are plotted on the y-axis. One of the ® rst comprehensive discussion on multiplicity in distillation was provided by Jacobsen and Skogestad4 and identi® ed two fundamental causes of output multiplicity which were described as `input transformation’ and `energy balance interaction’ . Both effects could be found in ideal binary distillationÐ an important result which contradicted some earlier ® ndings. They derived several results analytically and showed that multiplicity is generally only likely with the re¯ ux rate-boilup (LV) control con® guration, and most likely with high re¯ ux ratios. Output multiplicity has been reported in MTBE reactive distillation by several researchers using simulation studies5- 9 and has also been con® rmed experimentally10 . However, the conditions required for multiplicity have not been de® ned and neither a consistent nor complete explanation has been proposed for the cause of the multiplicity. Several mechanisms have been discounted, including CSTR type multiplicity5 , crossing non-reactive distillation boundary via reaction5 , heat of reaction and multiple reactions11 . Several other proposed mechanisms can be disregarded including the effects of either normal or reactive azeotropes. It remains unclear whether multiplicities in the reactive distillation of MTBE, or other fuel ethers, are due to the chemical reaction or to the combination of components present.
Reactive distillation has importance for several industrial processes, including the etheri® cation of C4 and C5 isoole® ns with methanol or ethanol to produce commercially signi® cant fuel additives such as methyl tert-butyl ether (MTBE), ethyl tert-butyl ether (ETBE) and tert-amyl methyl ether (TAME). These ethers have applications as both octane enhancers and gasoline oxygenates and their use has grown rapidly during the 1980s and 1990s as environmental regulations concerning gasoline composition are tightened worldwide. There are signi® cant advantages to incorporating the reaction and separation stages of a process in a single unit operation. Most notably, the capital cost is reduced, energy integration is improved and the reactant conversion is increased for equilibrium limited reactions1 . However, the complexity of the combined process is known to create some unusual responses to changes in operating variables2,3 . Many of these effects are related to the phenomenon of multiplicity of steady states and have signi® cant implications for both the operation and control of reactive distillation columns. Multiplicity normally occurs as either input multiplicity or output multiplicity. Input multiplicity is associated with unusual, unexpected or inverse column responses. It occurs when two or more unique sets of input variables produce the same output condition. Output multiplicity occurs when one set of input variables results in two or more unique and independent sets of output variables. Figures 1 and 2 provide graphical distinctions between input and output multiplicity for arbitrary variables. Input multiplicity is present in the ® rst chart as input variable values of both a and b result in the same value of the output variable. Output multiplicity is present in the second case as an input value of c could result in output values of either d or e. The differentiation between input and output variables is best made with reference to control structures. Input 525
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Figure 1. Input multiplicity.
Figure 3. Reactive distillation column con® guration for ETBE synthesis.
Figure 2. Output multiplicity.
CHARAC TERISTICS OF REACTIVE DISTILLATION COLUM NS
Although analysis of conventional distillation4,12 suggests that output multiplicity is only likely for the LV con® guration, several researchers5- 7,9 have more recently reported reactive distillation output multiplicities using the re¯ ux ratio-bottoms rate (L/D:B) con® guration. Experimentally, the distillate rate-reboiler duty (DQR ) con® guration was found to yield unique solutions for MTBE reactive distillation while the re¯ ux rate-reboiler duty (LQR ) con® guration produced a multiplicity10 . Multiplicity in conventional and reactive distillation has been an active area of research in recent years. A full explanation of output multiplicity, which allows the prediction of instability early in the design phase (i.e. without requiring complete simulation models to be developed), is still forthcoming and research continues in this crucial area. Despite the recent activity, the signi® cance of multiplicity has attracted little attention (Jacobsen and Skogestad13 is a notable exception), although clearly important for operation and control of industrial columns, such as those used for etheri® cation. The distinction between input and output multiplicity has also often been neglected in many previous contributions, adding to the confusion and misunderstanding of multiplicity. This paper addresses the latter two issues in a practical manner and also contributes to the former area by de® ning necessary and suf® cient conditions for the existence of output multiplicity. The concept of pseudo-multiplicity is also introduced and this may be a key to solving the output multiplicity puzzle.
Two reactive distillation columns have been studied. The ® rst column was for ETBE synthesis and had a con® guration which is described by Figure 3 (column) and Table 1 (input data). The second column is for MTBE synthesis. The MTBE column con® guration is similar to the ETBE column but it has ® ve more reactive stages and one extra recti® cation and stripping stage, which results in a con® guration that is similar to that used previously by other researchers. Its input is given in Table 2. The ETBE column was modelled using the equationbased process simulation package, SpeedUp14 and the equilibrium stage model for distillation systems, as discussed previously2,3 . The model incorporates rigorous expressions for the ETBE reaction equilibrium15 and kinetics that are valid at all ethanol concentrations equilibrium16 . The principal side-reaction involving the dimerization of isobutylene was modelled but other sidereactions, including the hydration of isobutylene to form isobutanol with the ingress of water, were assumed to be negligible. The UNIFAC model was used to predict component activities and recently published vapour pressure data17 was used in the phase equilibrium calculations. The MTBE process is similar to the ETBE process except that the synthesis reaction is less restricted by the chemical equilibrium. There is also a signi® cant azeotrope between methanol and the butylenes which contains up to 11% methanol. This has the effect of transferring unreacted methanol from the bottoms product (MTBE) to the distillate product and helps to increase the overall conversion of isobutylene. Reaction equilibrium was assumed for the
Table 1. ETBE reactive distillation column simulation input. Feed Conditions Temperature Rate Composition (mol%)
Overall Excess EtOH
Column Speci® cation 308 C 0.76 l/min ±1 29.1% ETBE, 9.1% EtOH, 7.3% i But, 54.5% nBut 5.0 mol%
Recti® cation Stages Reaction Stages Stripping Stages Total Stages Feed Stage Overhead Pressure Re¯ ux Rate
2 3 5 10 6 950 kPa 2.50 l/min ±1
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Table 2. MTBE reactive distillation column simulation input. Feed Conditions Temperature Rate Composition (mol%) Excess MeOH
Column Speci® cation 708 C 224 m3 hr±1 28.2% MeOH, 25.6% i But, 46.2% n C4 10.0 mol%
MTBE column rather than the kinetically controlled reaction model which was used for ETBE column. The dimerization side-reaction, which is normally also present in MTBE synthesis, was assumed to be negligible in this case. These assumptions do not indicate a lack of rigour but demonstrate that neither the side-reaction nor reaction kinetics are the sole cause of the observed multiplicities. The UNIFAC method was again used to predict liquid activities. The MTBE simulations were performed using a sequential modular simulation package, Pro/II18 . The use of a different simulation package provides further evidence that the phenomenon is not a result of the modelling techniques or mathematical anomalies. Reactive distillation requires that the operating conditions to be used for both reaction and separation should coincide. If the conditions required for good phase separation are such that temperatures are too high or too low for effective reaction, reactive distillation cannot be utilized successfully. This implies that the column operating conditions must be optimised with respect to both reaction and separation. For example, if the reboiler duty is too high, the composition pro® le shifts so that the concentration of ether in the reaction zone is such that the reverse (ether decomposition) reaction is favoured. Consequently, the isobutylene conversion is reduced signi® cantly and the bottom product becomes contaminated with alcohol. If the reboiler duty is too low, insuf® cient alcohol is recycled to the reaction zone and the isobutylene conversion is again low. In an ETBE column, the stripping section of the column (stages 6±10) separates ETBE from ethanol and C4 components, returning the lighter components to the reaction section where both ethanol and isobutylene react further to form more ETBE, and purifying the bottoms product which contains greater than 95% ETBE. The reaction section (stages 3±5) is packed with catalyst (Amberlyst 15y ) and operates at a temperature such that the forward reaction is favourable. The composition pro® le within the column is such that the ethanol:isobutylene ratio in the reaction section is much greater than one which further promotes the forward reaction to produce ETBE and increases the overall conversion of isobutylene. The recti® cation section (stages 1±2) removes ETBE from the distillate product and reduces the ethanol concentration of the distillate close to the azeotropic composition (approximately 1% ethanol and 99% butylenes). Only a limited number of recti® cation stages are used to ensure that unreacted isobutylene is recycled to the reaction zone. The functions of the three sections of MTBE column are essentially as described above for ETBE, except that more stripping stages can be tolerated as the minimum-boiling Trans IChemE, Vol 76, Part A, May 1998
Recti® cation Stages Reaction Stages Stripping Stages Total Stages Feed Stage Overhead Pressure Reboiler Duty
3 8 6 17 8 1100 kPa 80 MW
methanol-butylene azeotrope continues to recycle methanol to the reaction zone provided the methanol content of the feed does not exceed the `carrying capacity’ of the unreacted butylenes. INPUT MULTIPLICITY The need to optimize the heat input to the column is the source of a signi® cant input multiplicity in reactive distillation. This was demonstrated using simulation of the ETBE column. The results shown in Figure 4 were obtained by varying the reboiler duty while the re¯ ux rate was ® xed and show input multiplicities between the bottoms temperature, isobutylene conversion, ETBE product purity (all output variables) and the reboiler duty (an input variable). Reboiler duties of 8.20 kW and 8.97 kW both produce a bottoms temperature of 1508 C. Similarly, the same value of isobutylene conversion and ETBE product purity is obtained at two different values of the reboiler duty. Below the optimum reboiler duty (approximately 8.35 kW), the operating conditions are favourable for the ETBE reaction (as indicated by the relatively high conversions) but provide insuf® cient vapour-liquid traf® c for good separation (low ETBE purities). Above the optimum reboiler duty, the converse applies. For a small range of reboiler duties, both conversion and purity are high. This behaviour is common to MTBE reactive distillation columns and similar results can be produced with MTBE column simulations. It is important to note that these multiplicities are not present in conventional distillation, where increasing the reboiler duty normally always increases the bottoms temperature, but are essentially always present in reactive distillation. The two regions can be de® ned as separation controlled (reboiler duties
Figure 4. Input multiplicity in an ETBE reactive distillation column.
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below the optimum) and reaction controlled (reboiler duties above the optimum), respectively. OUTPUT MULTIPLICITY There is no evidence of output multiplicity in the ETBE column described above or in a similar column at reaction equilibrium. However, previous research has identi® ed this property in MTBE reactive distillation columns, which are potentially very similar. The MTBE column studied here was shown to produce a range of input multiplicities which were similar to those seen in the ETBE column, and was also found to give rise to several signi® cant output multiplicities. One such multiplicity is shown in Figure 5 which describes the response to varying the volumetric re¯ ux rate while the reboiler duty is constant. At re¯ ux rates between 995 and 1030 m3 hr- 1 there are three separate, unique solutions to exactly the same combination of reboiler duty and re¯ ux rate. Each solution corresponds to a different yield (bottoms product rate) of MTBE and conversion of isobutylene. Normally, increasing the re¯ ux rate at constant reboiler duty shifts the feed split to decrease the distillate yield. For some values of re¯ ux rate (between 995 and 1030 m3 hr- 1 ), the internal ¯ ows in the MTBE column described here conspire to cause the distillate yield to actually increase with increasing re¯ ux rate. This is described by condition (1) and appears to be the principal cause of the observed output multiplicity. Similarly, output multiplicity would also result if condition (2) was satis® ed.
¶Dv 0 > ¶Lv ¶Bv 0 > ¶QR
(1) (2)
The cause of output multiplicity in reactive distillation appears to be essentially the same as in conventional distillation. When condition (1) is satis® ed, the effect of re¯ ux on the mass balance via changes in the composition which are manifested through the energy balance are greater than the direct effect on the mass balance. This effect can be accentuated in reactive distillation by the effect of composition changes on the reaction rate. It might, therefore, be expected that output multiplicity would be more common in reactive distillation than conventional
Figure 5. Output multiplicity in a MTBE reactive distillation column.
Figure 6. Pseudo-multiplicity in an ETBE column.
distillation. In both cases, the necessary conditions for output multiplicity are described by the inequalities (1) and (2). PSEUDO-MULTIPLICITY A third type of multiplicity, designated pseudomultiplicity, is described here using the ETBE column shown in Figure 3. Essentially, pseudo-multiplicity is output multiplicity that is only seen in terms of molar inputs. If the molar boilup rate to the ETBE column is ® xed (at, say, 18.0 mol min- 1 ) and the molar re¯ ux rate is varied (LV con® guration), an apparent output multiplicity arises, as shown by Figure 6. However, if the reboiler duty is ® xed (e.g. 8.45 kW) and the volumetric re¯ ux rate is varied, the multiplicity disappears, as shown by Figure 7. This phenomenon appears to be caused by an output transformation which parallels the input transformation described by Jacobsen and Skogestad4 which was suggested as one of two principal causes of output multiplicity. Input transformation describes how a multiplicity might be transparent when considering molar inputs but real when considering volumetric or mass inputs. Here, the reverse is true. In both cases the transformation arises due to changing compositions and subsequent variation in the molar density. Both input and output transformations can be linked to a change in the sign of the gain between the molar re¯ ux rate and the volumetric re¯ ux rate, or between the molar boilup and reboiler duty, as shown in conditions (3) and (4):
¶Lm 0 < ¶Lv ¶Vm 0 < ¶QR
(3) (4)
Figure 7. LV output transformation (ETBE column).
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MULTIPLICITY AND PSEUDO-MULTIPLICITY IN MTBE AND ETBE REACTIVE DISTILLATION
Figure 8. Pseudo-multiplicity in a MTBE column.
Similar behaviour is evident for the MTBE column when considering the LB con® guration. If the molar bottoms ¯ ow is ® xed (674 kmol hr- 1 ) and the molar re¯ ux rate is varied, an output multiplicity arises (Figure 8). However, if the volumetric bottoms ¯ ow is ® xed (e.g. 75 m3 hr- 1 ) and the volumetric re¯ ux rate is varied, the multiplicity disappears (Figure 9). Again, an output transformation appears to be the cause of this behaviour. In this case, the transformation is between the molar bottoms rate and the volumetric or mass bottoms rate, but the multiplicity is still associated with a change of sign in the gain:
¶Bm 0 < ¶Bv
(5)
It is important to recognize that previous reports of output multiplicity in reactive distillation for material balance control con® gurations5- 7,9 , that is con® gurations such as the DV, LB or (L/D)B schemes where either the distillate or bottoms rate are manipulated directly to control composition, are actually examples of pseudo-multiplicity. Thus, previous conjecture that output multiplicity is only likely for energy balance control con® gurations (i.e. schemes where composition control is achieved indirectly by manipulation of the internal column ¯ ows) remains sound and convincing. CAUSES OF MULTIPLICITY Input multiplicity is caused by two (or more) con¯ icting effects operating via the same manipulated variable. For example, increasing the reboiler duty in an ETBE column while the re¯ ux rate is held constant improves the separation between ETBE and the butylenes (a favourable effect) but also increases the reaction zone temperature which
Figure 9. LB output transformation (MTBE column).
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encourages the decomposition reaction to take place and reduces the overall conversion of isobutylene (a detrimental effect). As reactive distillation usually always involves a compromise between reaction and separation effects, there are many variables and operating conditions that have the potential to result in input multiplicities. The cause(s) of output multiplicity is (are) much more complex. It is important to realize that in any system which displays output multiplicity there appears to always be at least one manipulated variable which maintains a oneto-one (or many-to-one) mapping with the output variable that describes the output multiplicity. For example, if a particular MTBE column shows an output multiplicity between the re¯ ux rate and the bottoms purity (with constant reboiler duty), each value of the bottoms rate (or another variable) will still produce one and only one value of bottoms purity. Once a variable is found that does not show any output multiplicity, it is possible to create all solutions and plot the results in terms of any input variable so that output multiplicities (including unstable solutions) can be revealed. The principal cause of the output multiplicity lies in the behaviour of manipulated variable which does not show output multiplicity. Either condition (1) or (2) must be satis® ed for the column to produce an output multiplicity. In fact, the inequalities (1) and (2) represent necessary and suf® cient conditions for output multiplicity. The physical interpretation of conditions (1) and (2) is that changes in the mass balance which result from the effects of composition changes on the energy balance are more signi® cant than the direct effect on the mass balance. Jacobsen and Skogestad4 derived similar results for conventional binary distillation and demonstrated analytically that instability (which indicates multiplicity) can generally only arise by manipulating a variable that does not affect the feed split directly (i.e. the re¯ ux rate and reboiler duty but not the distillate or bottoms rates). Similar conditions cannot easily be derived for multi-component distillation but the results would appear to be extendable at an empirical level. A consequence of this result is that, even in columns which display output multiplicity in one or more input variables, output multiplicity in terms of the distillate or bottoms rate is unlikely. This provides a mechanism for ® nding the output multiplicities described above and also a means of limiting their impact on column operation and control. Output multiplicity is possible in non-reactive distillation4,12 and has been demonstrated10 . However, the presence of a chemical reaction within the distillation system allows the composition pro® le to be shifted more rapidly for relatively small changes in the input variables (both `feedsplit’ variables and `energy-balance’ variables). The effect of the energy balance on the mass balance via the composition pro® le can, therefore, be magni® ed so that conditions (1) and (2) are more easily satis® ed. This result predisposes reactive distillation columns to the phenomenon of output multiplicity and, perhaps, accounts for the growing number of reports of output multiplicity in a wide variety of reactive distillation columns. It is more dif® cult to ® nd a physical interpretation of pseudo-multiplicity as the phenomenon does not correspond to a `real’ situation. As with output multiplicity,
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pseudo-multiplicity arises through interaction between the mass and energy balances and is manifested through composition changes. Pseudo-multiplicity requires that changes in the product composition(s) which result from changes in the molar column inputs have a more signi® cant effect on the mass balance than direct manipulation of the mass balance via the molar inputs. This is described by conditions (3), (4) and (5) which are necessary and suf® cient conditions for pseudo-multiplicity. Figures 8 and 9 demonstrate that, pseudo-multiplicity can occur with material balance control schemes (i.e. where a `feed-split’ variable is the primary manipulated variable), unlike output multiplicity. This can be the result of a nonequimolar reaction or another property of the system. Where a non-equimolar reaction is present, the reaction changes the total number of moles present in the system. This effectively changes the feed split without changing the molar product rate by forcing the bottoms (or distillate) mass yield to change. However, this is not the only cause of material balance pseudo-multiplicity as control schemes which manipulate the molar yield of either the distillate or bottoms or schemes which ® x the molar reaction rate can also lead to multiplicities. As described above for output multiplicity, the presence of chemical reaction in reactive distillation provides a means of rapidly changing product compositions without requiring signi® cant changes in the feed-split or the energy balance. This property is a crucial requirement for pseudomultiplicity as it provides a means for energy balance effects to predominate over feed split effects so that one or more of conditions (3) to (5) can be satis® ed. In conventional distillation, the size of the two effects normally differs by an order of magnitude. Furthermore, the effects are usually essentially independent of whether mass or molar inputs are considered. Consequently, pseudomultiplicity is unlikely for non-reactive distillation, at least in terms of the `feed-split’ variables. OPERATION AND CONTRO L The presence of input multiplicity in reactive distillation has two signi® cant implications for column operation and control. Firstly, the narrow range of conditions where phase and chemical equilibrium intersect favourably necessitates tight control of the primary operating variable (e.g. reboiler duty, bottoms rate, etc.). Manual control of a reactive distillation column is, therefore, unlikely to provide adequate control of the heat input and could lead to either the isobutylene conversion or ETBE purity being compromised. The necessity for tight control could even in¯ uence the equipment design as some types of reboiler (e.g. integrated exchanger networks) are inherently harder to control than others (e.g. ® red heater). The need for tight reboiler control cannot be avoided by selecting a material balance control con® guration (e.g. DV or LB con® guration) as the performance of these controllers is intimately linked to the performance of the level control loops and tight reboiler control would still be required. Secondly, the input multiplicity prevents the bottoms temperature (or composition) being directly used as a controlled variable, even though it correlates well with the principal operating objectives of high isobutylene conversion and high ETBE purity. The use of either the bottoms
temperature or an analyser measuring the bottoms composition would require the controller gain to vary through zero to avoid closed-loop instability. This requirement arises from the uncertainty of whether to increase or decrease the manipulated variable to move towards the controlled variables set point (see Figure 4). This restriction favours the use of an alternative controlled variable, such as a temperature midway up the stripping section, which displays no input multiplicity with potential manipulated variables3 . It is also important to know which operating region (reaction controlled or separation controlled) an etheri® cation column is operating in order to understand how the column will respond to changes in operating variables. Equivalent bottoms temperatures exist in the two regions so that some other means of distinguishing the regions must be determined. The temperature difference across the reaction zone can provide this information. A low temperature difference (say, less than 58 C) implies that the column in operating in the separation controlled region and that the reboiler duty needs to be increased to increase both the isobutylene conversion and ether purity. A high temperature difference (say, greater than 58 C) implies that the column is operating in the reaction controlled region and that the reboiler duty needs to be decreased to increase the conversion and purity. The implications of output multiplicity for operation and control are not as substantial as might initially be thought. Firstly, the output multiplicity illustrated here, and other reported multiplicities4,11 , only occur with very high vapourliquid traf® c inside the column and, therefore, do not represent probable operating points as such columns designs are likely to be uneconomical. Secondly, there is no evidence of output multiplicity occurring with material balance control con® gurations (e.g. DV or LB) unless the associated level control is very poor12 so that the possibility of multiplicity can generally be avoided with an appropriate control scheme design. However, the possibility exists that process disturbances could result in a shift to an undesirable steady state without either the reboiler duty or re¯ ux rate being changed. This provides an incentive to avoid manual control. A closed-loop controller would detect the shift in the operating region (if appropriate controlled variables have been selected) and try to manipulate either the reboiler duty or re¯ ux rate to return the controlled variables to their set points. The dif® culty here arises with associated input multiplicities that make the selection of appropriate controlled variables critical. However, constraints imposed on the product yields (or reboiler duty) could be useful to detect a possible transition between parallel states and trigger alarms or prevent excessively large control moves. If an undesirable steady state is reached during startup or otherwise, the transition to the preferred steady state can be made by temporarily decommissioning the control scheme and manually increasing or decreasing the product yield to force the feed split close to the desired level. Once this has been achieved, the control scheme can be recommissioned. Pseudo-multiplicity has less signi® cant implications for operation and control as molar ¯ ow controllers are not physically realizable. However, it demonstrates the need to work only in terms of mass or volume inputs (rather than Trans IChemE, Vol 76, Part A, May 1998
MULTIPLICITY AND PSEUDO-MULTIPLICITY IN MTBE AND ETBE REACTIVE DISTILLATION molar inputs which are still commonly employed) when using simulations to model an actual process or design. Most signi® cantly, the discovery of pseudo-multiplicity reconciles recent reports5- 7,9 with the previously established result that output multiplicity is unlikely with control con® gurations other than the LV scheme. This is a crucial ® nding as the latter result has been demonstrated by analytical studies4,12 and experimental work10 . Essentially, it allows some alarming results of hysteresis to be ignored for design, operation and control of `real’ columns. CONCLUSIONS Input multiplicity occurs where more than one set of input conditions (e.g. re¯ ux rate and reboiler duty) can be used to produce the same output condition (e.g. bottoms temperature or purity). The presence of input multiplicity creates a change of sign in the process gain which in¯ uences the selection of control con® guration and restricts the range of variables that can be used to infer the state of the column. With no control system (column being operated in manual), the operator still requires additional information to determine which operating region the column is in, so that appropriate changes to the manipulated variables can be made to move the column towards its optimum operating point. Output multiplicity occurs where a single set of inputs results in two or more unique outputs. This results from unusual relationships between either the distillate and re¯ ux rates or between the reboiler duty and the bottoms rate (conditions 1 and 2). The state of the column can, therefore, not be determined solely from the inputs and some other measure is required to provide suf® cient input to a control system to allow stable and appropriate control moves to be made. Hysteresis is also possible where more than one steady state solution exists. Pseudo-multiplicity occurs where molar inputs (rather than mass or volumes inputs that would result from control valves) produce an output multiplicity. This can only be observed via simulation and is not associated with actual operating columns. The key implication of this phenomenon is that it creates a requirement to conduct simulation studies (for design, control or otherwise) in terms of mass or volume inputs only. The ® nding also reconciles some recent contributions with earlier studies and experimental work in the same area.
n But nC4 Vm QR
531
n-butylenes n-butane molar boilup rate reboiler duty
REFERENCES 1. Sneesby, M. G., TadeÂ,M. O. and Datta, R., 1995, tert-Butyl ethersÐ A comparison of properties, synthesis techniques and operating conditions for high conversions, Devel Chem Eng & Mineral Processing, 3: 89±116. 2. Sneesby, M. G., TadeÂ, M. O., Datta, R. and Smith, T. N., 1996, ETBE synthesis by reactive distillation. 1. Steady state simulation and design aspects, Ind Eng Chem Res, 36(5): 1855±1869. 3. Sneesby, M. G., TadeÂ, M. O., Datta, R. and Smith, T. N., 1996, ETBE synthesis by reactive distillation. 2. Dynamic simulation and control aspects, Ind Eng Chem Res, 36(5): 1870±1881. 4. Jacobsen, E. W. and Skogestad, S., 1991, Multiple steady states in ideal two-product distillation, AIChEJ, 37(4): 499±511. 5. Nijhuis, S. A., Kerkhof, F. P. J. M. and Mak, A. N. S., 1993, Multiple steady states during reactive distillation of methyl tert-butyl ether, Ind Eng Chem Res, 32: 2767±2774. 6. Jacobs, R. and Krishna, R., 1993, Multiple solutions in reactive distillation for methyl tert-butyl ether synthesis, Ind Eng Chem Res, 32: 1706±1709. 7. Hauan, S., Hertzberg, T. and Lien, K. M., 1995, Why methyl tert-butyl ether production by reactive distillation may yield multiple solutions, Ind Eng Chem Res, 34: 987±991. 8. Perez-Cisneros, E., Schenk, M., Gani, R. and Pilavachi, P. A., 1996, Aspects of simulation, design and analysis of reactive distillation operations, Comp Chem Eng, 20(Suppl): S267±S272. 9. Schrans, S., de Wolf, S. and Baur, R., 1996, Dynamic simulation of reactive distillation: An MTBE case study, Comp Chem Eng, 20(Suppl): S1619±S1624. 10. Sundmacher, K. and Hoffman, U., 1995, Oscillatory vapor-liquid transport phenomena in a packed reactive distillation column for fuel ether production, Chem Eng J, 57: 219±228. 11. Ciric, A. R. and Miao, P., 1994, Steady state multiplicity in an ethylene glycol reactive distillation column, Ind Eng Chem Res, 33: 2738±2748. 12. Jacobsen, E. W. and Skogestad, S., 1994, Instability of distillation columns, AIChEJ, 40(9): 1466±1478. 13. Jacobsen, E. W. and Skogestad, S., 1995, Multiple steady states and instability in distillation. Implications for operation and control, Ind Eng Chem Res, 34: 4395±4405. 14. Aspen Technology Inc., 1993, The SpeedUp User’ s Manual, (Cambridge, Massachusetts, USA). 15. Jensen, K. L. and Datta, R., 1995, Ethers from ethanol 1. Equilibrium thermodynamic analysis of the liquid phase ethyl tert-butyl ether reaction, Ind Eng Chem Res, 34: 392. 16. Jensen, K. L. and Datta, R., 1996, Ethers from ethanol 7. Transition state theory analysis of the kinetics of liquid phase ethyl tert-butyl ether synthesis reaction, submitted to Ind Eng Chem Res. 17. KraÈhenbuÈhl, M. A. and Gmehling, J., 1994, Vapor pressures of methyl tert-butyl ether, ethyl tert-butyl ether, isopropyl tert-butyl ether, tertamyl methyl ether and tert-amyl ethyl ether, J Chem Eng Data, 39: 759. 18. Simulation Sciences Inc. (1994) Pro/II Keyword Input Manual (Brea, California, USA).
NOMENCLATUR E Bm Bv EtOH ETBE Dv iBut Lm Lv MeOH MTBE
molar bottoms rate volumetric bottoms rate ethanol ethyl tert-butyl ether volumetric distillate rate isobutylene molar re¯ ux rate volumetric re¯ ux rate methanol methyl tert-butyl ether
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ADDRESS Correspondence concerning this paper should be addressed to Dr Moses TadeÂ, School of Chemical Engineering, Curtin University of Technology, Perth, GPO Box U1987, WA 6845, Australia. (E-mail: [email protected]). The manuscript was communicated via our International Editor for Australia, Professor G. J. Jameson. It was received 29 November 1996 and accepted for publication 2 December 1997.