Predicting Multiple Steady States in Equilibrium Reactive Distillation. 1

Ind. Eng. Chem. Res. 1999, 38, 1633-1648

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Predicting Multiple Steady States in Equilibrium Reactive Distillation. 1. Analysis of Nonhybrid Systems Thomas E. Gu 1 ttinger† and Manfred Morari* Automatic Control Laboratory, Swiss Federal Institute of Technology, ETHZ, CH-8092 Zu¨ rich, Switzerland

During the last few years, multiple steady states (output multiplicities) have been discovered for reactive distillation processes, for example, for the production of fuel ethers and for some esterification processes. Using the transformation of Ung and Doherty (Chem. Eng. Sci. 1995e, 50 (1), 23-48), a method is presented to predict the existence of output multiplicities based on the reactive vapor-liquid equilibrium for the limiting case of reactive columns of infinite length operated at infinite internal flows. This graphical method rests upon the ∞/∞ analysis for azeotropic distillation columns by Bekiaris et al. (Ind. Eng. Chem. Res. 1993, 32 (9), 20232038). It is directly applicable to systems where the reactions take place in the entire column (“nonhybrid” columns). When all possible profiles and products are located through a bifurcation analysis, qualitative and quantitative predictions are obtained. The region of feed compositions leading to multiple steady states can be constructed graphically by applying a necessary and sufficient geometrical condition. The prediction results are shown to carry over to finite columns by application to the methyl tert-butyl ether process and are verified by simulation. 1. Introduction During the last decade, the combination of reaction and separation in the same process unit, called reactive or catalytic distillation (RD), has gained increasing importance. Some processes in the chemical and petrochemical industries have been significantly improved by the use of RD technology, e.g., the production of fuel ethers (DeGarmo et al., 1992). In the end, the motivation for implementing RD processes in industry is always of an economical nature because capital and operating costs of a plant must be minimized. The synergies encouraging the combined use of reaction and separation can be assigned to either chemistry or separation in most cases. In terms of chemistry, conversion and yield of the desired product can often be improved by RD. This is achieved by carrying equilibrium-limited reactions to completion or by using the boiling point temperatures to shift the chemical equilibrium (Venkataraman et al., 1990; DeGarmo et al., 1992; Doherty and Buzad, 1992). Side reactions can be suppressed by feeding one reactant in excess (increasing yield), and unfavorable thermodynamics may be overcome (Doherty and Buzad, 1992; Isla and Irazoqui, 1996; Okasinski and Doherty, 1997a). Examples of processes which have profited from chemistry-based synergies are the production of MTBE (methyl tert-butyl ether) and esterification reactions (methyl or ethyl acetate). In terms of separation, close-boiling mixtures have been one of the driving forces in the development of RD processes, e.g., the separation of m- and p-xylene (Saito et al., 1971; Terrill et al., 1985). Additionally, these processes often allow one to overcome nonreactive azeotropes and distillation boundaries (Barbosa and Doherty, 1988c). The inherent direct heat integration * Author to whom correspondence should be addressed. Phone: +41 1 632-7626. Fax: +41 1 632-1211. E-mail: [email protected]. † Current address: Ciba Specialty Chemicals Inc., CH-4002 Basel, Switzerland.

of a distillation column utilizes the heat of reaction for evaporating the liquid phase (Grosser et al., 1987; Sundmacher et al., 1994). However, there are certainly other issues to consider in the evaluation of RD processes, e.g., control problems (Bartlett and Wahnschafft, 1997). The following concepts need to be defined. By equilibrium reactive distillation we denote reactive processes where the reactions, which are assumed to take place in the liquid phase, are infinitely fast. Thus, the system is able to reach chemical equilibrium on a distillation tray where vapor-liquid equilibrium (VLE) is established. For the purpose of this paper, simultaneous chemical and physical equilibrium is a justified assumption. However, this implies a restriction on the thermodynamic systems to be investigated in the analysis. Certainly, all RD processes are rate-based in nature, and kinetic considerations may have to be included in the design of processes undergoing slow reactions. Comparisons of the applicability of rate-based and equilibrium models to reactive distillation have been given by Serafimov et al. (1993) and Pilavachi et al. (1997). Hybrid columns are distillation columns with a reactive core; i.e., they are built from a reactive column section where the catalyst is present and one or more nonreactive sections (rectifying and/or stripping sections). Consequently, nonhybrid columns denote columns where all trays including the condenser and reboiler are reactive (Figure 1). Commonly, hybrid columns result from the use of a heterogeneous (solid) catalyst, whereas a homogeneous (liquid) catalyst would lead to a nonhybrid column design. The literature review in this paper will focus on the occurrence of multiple steady states in RD (next section). However, a few requisites from RD thermodynamics and column design will be mentioned here. Among others, Terrill et al. (1985) have surveyed many contributions starting from the early 1920s. Key industrial applications were studied in the review of Doherty and Buzad (1992) and by Sundmacher et al. (1994), who also attempted to classify RD processes.

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Figure 1. Nonhybrid (a) and hybrid (b) RD columns. The hybrid column corresponds to the setup used by Nijhuis et al. (1993).

Most of the pioneering work in the areas of equilibrium RD thermodynamics and design is based on transformations of the physical compositions (Zharov, 1970; Barbosa and Doherty, 1988c). Barbosa and Doherty (1988a) presented design and minimum reflux calculations for single-reaction systems. Using a generalized version of the composition transformations, Ung and Doherty (1995d) extensively analyzed the synthesis of RD systems with multiple reactions by graphical design methods based on reactive residue curves. Espinosa and co-workers studied the design of reactive columns under the aspect of inerts present and for columns with a reacting core, i.e., hybrid columns (Espinosa et al., 1995a,b, 1996). Similar developments have been published in the Russian literature simultaneously or prior and were reviewed by Serafimov et al. (1993) and Timofeev et al. (1994). 2. Multiple Steady States in Reactive Distillation 2.1. Definition and Physical Causes. Because of the occurrence of reactive azeotropes, many characteristics of RD processes are very similar to those observed in (nonreactive) azeotropic distillation. Among the unusual features of both azeotropic and reactive columns, multiple steady states (MSS) have been discovered. Definition 1. By MSS in a distillation column we refer to output multiplicities, i.e., that a column of a given design exhibits different column profiles (and therefore different product compositions) at steady state for the same set of inputs (feed composition, flow rate, and quality) and the same values of the operating parameters. Generally, the selection of the actual operating parameters from the set of possible parameters is referred to as the “configuration” of a distillation column. Configurations will be used for steady-state specifications and will be denoted by squared brackets. For example, [DM, Qr] is a common configuration where the mass flow rate of the distillate and the reboiler heat duty are specified. It is important to note that the definition given above implies that the existence of MSS may depend upon the configuration used. So far, three different physical phenomena have been found to give rise to output multiplicities in azeotropic and reactive distillation:

I. Jacobsen and Skogestad (1991) reported two different types of multiplicities in binary distillation columns with ideal VLE. (a) For constant molar overflow (CMO) and the [LM, VN] configuration, multiplicities can occur because of the nonlinear relationship between mass and molar flow rates (or volumetric and mass flow rates or heat duties and molar flow rates). This behavior is not expected for binary systems of configurations involving a product flow rate, e.g., [DM, VN]. Experimental studies have been reported by Kienle et al. (1995) and Koggersbøl et al. (1996). (b) Multiplicities can further be caused by the presence of energy balances in the [LN, VN] configuration of ideal binary columns. Type Ia multiplicities have also been shown for more complex distillation systems, e.g., multicomponent azeotropic mixtures (Gu¨ttinger and Morari, 1997). II. The existence of output multiplicities with molar specification of the flow rates was studied by Bekiaris et al. (1993, 1996) for ternary homogeneous and heterogeneous mixtures (∞/∞ analysis). These types of multiplicities even occur for CMO conditions and are based on the VLE. They have numerous implications not only for distillation simulations but more importantly for the design, operation, and control of distillation columns (Bekiaris and Morari, 1996). MSS of type II have been verified experimentally for a ternary azeotropic system by Gu¨ttinger et al. (1997). The experiments have been reproduced on a different pilot plant including stabilizing control (Dorn et al., 1998). By physical cause of multiplicities we refer to the system characteristics (e.g., thermodynamic properties, column design, or column configuration) which lead to MSS. 2.2. Literature Review. In this section, the literature on MSS in RD is studied. A detailed review on MSS in nonreactive distillation is beyond the scope of this paper, and the reader is referred to Gu¨ttinger (1998). The first report on MSS in RD was published by Pisarenko et al. (1987), who studied a single-product RD column. They were able to obtain three steady-state solutions for the same inputs and operating parameters by simulation; two of them were stable, and one was unstable. Recently, Karpilowski et al. (1997) continued their study and developed a mathematical analysis of single-product column multiplicities which is based on the limiting case of infinite reflux. However, their method is not likely to be extensible to classical twoproduct columns. It was successfully applied, however, to the production process of butyl acetate. In 1993, two simulation studies appeared at almost the same time, showing MSS for the RD process of MTBE and two similar industrial column setups (Nijhuis et al., 1993; Jacobs and Krishna, 1993). The columns differ only by one rectifying stage, and we will refer to both column setups as the “Nijhuis” column (Figure 1b). By variation of the methanol feed tray location, the authors were able to find several feed tray locations where two different steady states exist (using a steadystate process simulator). Further simulations for constant feed locations but varying reflux flow rate or varying catalyst amount confirmed the existence of the multiplicities. Both contributions made a first attempt to address the physical causes responsible for the existence of output multiplicities in this process. These arguments have been questioned by Gu¨ttinger and

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Morari (1997), where a more detailed analysis of the physical causes can be found. Ciric and Miao (1994) turned the attention to the irreversible production of ethylene glycol in a singleproduct reactive column. Their bifurcation calculations showed a large regime with three steady states and a small region of switchback multiplicities. Extensive case studies showed that the existence of MSS for the glycol process is quite robust against variations of the feed location, the holdup distribution, the presence of side reactions, and the heat of reaction. Then the attention turned back to the MTBE process where Hauan et al. (1995) were able to reproduce the results of Jacobs and Krishna (1993) under the assumption of chemical reaction equilibrium. They presented a very detailed mechanistic explanation of the phase behavior at the stable steady states for feed tray locations in the interval showing the multiplicity. Moreover, they rigorously explained the resulting behavior on the two steady-state branches but were not able to predict why a steady-state multiplicity exists in this process. At the same time, Sundmacher and Hoffmann (1995) observed oscillations in their laboratory column operating on the MTBE reactive distillation system (the “Clausthal” column). Surprisingly, even the nonreactive binary mixture methanol-isobutene showed similar oscillations, and the authors associated the thermodynamics of this binary mixture with the occurrence of the oscillations. Oscillations for feed composition steps and steady-state multiplicities for the reactive MTBE system have further been demonstrated by dynamic simulations of the Nijhuis column (Schrans et al., 1996; Hauan et al., 1997). Schrans et al. (1996) first reported that there are three steady states possible for the MTBE system, two of them stable and one unstable. Recently, Mohl et al. (1997) did bifurcation calculations for the Nijhuis column and showed that MSS disappear at low reflux values but still exist at high reflux. However, no oscillations were observed in the dynamic transitions between the different steady states in the multiplicity interval. This contradicts the previous results because both studies used step disturbances in the feed composition. MSS have further been demonstrated for the industrial production of TAME (Bravo et al., 1993; Mohl et al., 1997). The effect of different thermodynamic models and parameters on the existence of MSS for the Nijhuis and Clausthal columns was studied by Pilavachi et al. (1997). Accordingly, MSS in the MTBE process depend critically on the correct representation of the reactive equilibrium but are more robust against variations in the VLE model; see also Okasinski and Doherty (1997b). After those simulation studies, two approaches have been published for the prediction of MSS in RD. Gehrke and Marquardt (1997) presented a numerical method to calculate column multiplicities based on a singlestage model by a singularity theory approach. Qualitatively correct predictions were obtained for the ethylene glycol column studied by Ciric and Miao (1994). However, their approach is computationally intensive, especially if high-order singularities must be found. Alternatively, Gu¨ttinger and Morari (1997) presented a first approach to extend the ∞/∞ analysis of Bekiaris et al. (1993) to RD systems. By the use of reactive residue curves and the transformation of Ung and Doherty (1995e), they were able to predict the existence

of MSS by graphical methods. They demonstrated that the multiplicities in the MTBE process can be of either type Ia or type II (physical causes). The ∞/∞ singularity analysis provides the necessary tools for determining the influence of the column configuration on the existence of MSS. The existence of type Ia MSS in the MTBE process has further been shown by Sneesby et al. (1997b). Bartlett and Wahnschafft (1997) studied the control of an MTBE column operating in the multiplicity interval. The current paper will start with the description of the thermodynamic requisites for the reactive ∞/∞ analysis. In a first step, the methods sketched by Gu¨ttinger and Morari (1997) will be extended and presented in detail for nonhybrid columns. In a second step, prediction methods for hybrid columns will be developed (Gu¨ttinger and Morari, 1998). By the ∞/∞ case we refer to distillation columns of infinite length (with an infinite number of equilibrium stages), operated at infinite internal flows (Petlyuk and Avet’yan, 1971; Bekiaris et al., 1993). For reactive ∞/∞ columns there is the additional requisite of infinite catalyst activity on each tray of the reactive section to achieve equilibrium reactions. It will be shown that the ∞/∞ analysis can be fully expanded to reactive systems, and prediction tools for the occurrence of multiplicities will be developed. 3. Reactive VLE and Column Profiles 3.1. Reactive Composition Transformation. In reactive equilibrium of a c component mixture, the reachable compositions form a subspace of lower dimension in the physical composition space, the “reaction space” (Bessling et al., 1997). The dimension of the subspace is given by the degrees of freedom from the extended Gibbs phase rule (Smith et al., 1996):

degrees of freedom ) 2 - π + c - r

(1)

where π is the number of phases and r the number of reactions present. Thus, a homogeneous nonreactive system in vapor-liquid equilibrium has c degrees of freedom. Each linearly independent reaction reduces the dimension of the reaction space by 1. Bessling et al. (1997) presented illustrations of the physical space and the reaction space for a variety of topologically different reactive systems. The basic idea to obtain a convenient description of the reactive VLE is to find a set of transformed composition variables X which can be used as a basis for the reaction space. The new variables should exhibit properties similar to those of the mole fractions in the physical composition space (e.g., sum to unity). Such a transformation has first been published by Zharov (1970) for isothermal open-evaporation processes. The transformations were rediscovered by Barbosa and Doherty (1988c) and refined by Ung and Doherty (1995e) and Espinosa et al. (1995a). We will base this work on the transformation of Ung and Doherty (1995e) which can be applied to any number of reactive and inert components and to any number of reactions r:

D: [0...1]c f R c-r Xi :)

xi - νTi (νref)-1xref 1 - νTtot(νref)-1xref

(2)


In a similar way, molar flow rates can be transformed (P denotes any of the streams D, B, Fi, L, or V; see Figure 1):

PD ) PN(1 - νTtot(νref)-1xPref)

(3)

The transformed compositions X exhibit the following properties (Ung and Doherty, 1995c,e): 1. X expresses the conservation of mass for any value of the extent of reaction; they are reaction-invariant compositions. Therefore, X takes the same value for any reaction extent of a given liquid mixture. 2. The X vector sums to unity. 3. The transformed representation is canonical in the sense that it does not depend on the combination of the reaction equations chosen (as long as this combination is linearly independent). 4. Reactive azeotropes occur if and only if X ) Y (the transformed liquid composition is equal to the transformed vapor composition). A physical explanation of reactive azeotropes was given by Barbosa and Doherty (1988b). 5. The reaction space is always a convex polyhedron but not necessarily a simplex. Zharov (1970) and Ung and Doherty (1995a) derived the equations describing the simple distillation (open evaporation) of an equilibrium reactive system. The reactive residue curves are obtained by solving the following differential equation in transformed compositions:

dXi ) Xi - Yi dτ

i ) 1, ..., c - r

(4)

where τ is the “warped time”. Criteria on the choice of reference components to obtain a well-defined description are given in the appendix of work by Ung and Doherty (1995a). Equation 4 is identical with the residue curve equation for nonreactive systems if the transformed compositions are replaced by molar ones. Therefore, the properties of reactive and nonreactive residue curves are very similar (Ung and Doherty, 1995a). The singular points with X ) Y correspond to reactive azeotropes, “surviving” nonreactive azeotropes, or pure components. They can be either stable or unstable nodes or saddles. In addition, RD regions, reactive boundaries, and RD lines can be defined in an analogous manner (Bessling et al., 1997). 3.2. Reactive ∞/∞ Profiles. The following feasibility analysis of profiles in the ∞/∞ case is based on a nonhybrid RD column (Figure 1a). Ung and Doherty (1995d) derived the equations describing the operating lines (material balances) of such a nonhybrid column. They showed that the equations for the rectifying and stripping sections are identical to those for nonreactive columns if all molar compositions and flow rates in the latter are replaced by their transformed analogues. Moreover, the overall material balances maintain their form in transformed coordinates (the unit base of flow will be denoted by superscript D for transformed, N for molar, M for mass, or V for volumetric flow rates):

FD ) DD + BD FDXF ) DDXD + BDXB

(5)

Thus, most of the standard tools for column design and

sequencing can be applied directly to equilibrium reactive systems. A reactive distillation column profile is feasible if the feed, distillate, and bottom compositions, in transformed coordinates, all lie on a straight line. This necessary condition corresponds to the well-known lever rule. At infinite reflux, the composition profile of a packed nonreactive distillation column must coincide with some part of a nonreactive residue curve, the profile of a tray column with some part of a nonreactive distillation line (Sarafimov et al., 1973; Bekiaris et al., 1993, 1996; Po¨llmann and Blass, 1994). Because all column material balances have the same form as that for nonreactive columns, the following important result carries over by analogy: The transformed composition profiles of packed reactive columns coincide with the reactive residue curves (Barbosa and Doherty, 1988a), and those of reactive tray columns coincide with reactive distillation lines. Moreover, the results of Bekiaris et al. (1996) can be easily adapted to reactive systems to study the influence of the condenser and reboiler types (partial or total). As Widagdo and Seider (1996) in their review indicate, residue curves are a good approximation for distillation lines and the subsequent analysis is based upon the former. If a distillation column has infinite length, i.e., an infinite number of equilibrium stages, its column profile must contain at least one pinch point (Petlyuk and Avet’yan, 1971). From the analogy described above, it is obvious that this fact does carry over to reactive columns. Thus, the transformed composition profile of a reactive column (section) of infinite length must contain at least one pinch point. The only candidate pinch points for reactive columns at infinite reflux are the singular points of the reactive residue curve diagram as shown for nonreactive distillation by Bekiaris et al. (1993). Therefore, the following conditions are necessary and sufficient for a feasible reactive column profile in the ∞/∞ case of a nonhybrid column: (1) the transformed profile coincides with a part of a reactive residue curve, (2) the transformed profile contains at least one pinch point (reactive singular point), and (3) the transformed lever rule (5) holds. Note that these conditions are necessary but not sufficient for hybrid columns (Gu¨ttinger and Morari, 1998). Because the conditions derived above are completely analogous to those of nonreactive columns, the description method for nonreactive column profiles of Bekiaris et al. (1993) can be applied directly to reactive profiles. In the ∞/∞ case of a nonhybrid column, all feasible column profiles (fulfilling the conditions above) belong to one of the following three types: (I) the light-boiling end (distillate) of the reactive profile is located at a reactive unstable node, (II) the heavy-boiling end (bottoms) of the reactive profile is located at a reactive stable node, or (III) the reactive profile contains at least one reactive saddle singular point (but no nodes). It was shown above that the lever rule holds in the transformed composition space (5). Alternatively, one may ask if there are other units of flows and compositions for which the overall column balances take the form of a lever rule for a nonhybrid reactive column, i.e., in which the balances do not contain an explicit reaction term. Obviously, such a lever rule does not hold in molar units because the number of moles of the components are altered by the reactions. Even the total number of moles will change in the general case


of non-equimolar reactions, νtot * 0:

FN ) DN + BN - νTtot FNxF ) DNxD + BNxB - ν

(6)

The overall material balance in mass units can be written without a reaction term as mass is conserved:

FM ) DM + BM

(7)

However, the mass fractions will be altered by the reactions as the mole fractions and the component balances in mass units contain a reaction term. Hence, there is no lever rule in mass units for reactive columns. In conclusion, the only considered units where the lever rule holds are the transformed compositions and flow rates. These are the units of choice for the description of nonhybrid reactive distillation columns in the ∞/∞ case. 4. Reactive ∞/∞ Analysis 4.1. ∞/∞ Bifurcation Analysis. In this section it is demonstrated that the nonreactive prediction tools of Bekiaris et al. (1993) can be extended to nonhybrid reactive distillation columns. The only requisites for the existence of an analogous theory are the availability of a lever rule and a method to describe all possible composition profiles in the ∞/∞ case of a reactive distillation column. Actually, the conditions for feasible column profiles derived in the previous section fulfill these requirements. It is important to note that the analysis presented here is only applicable to nonhybrid reactive columns (Figure 1a), a restriction which will be relaxed later. A homogeneous two-product distillation column has two operational degrees of freedom for a given design and given feed composition, flow rate, and quality. In the ∞/∞ case, the reflux is specified to be infinite, and as a result, all internal flows are infinitely large. Thus, there is 1 degree of freedom remaining which is chosen to be a product flow rate, e.g., the distillate flow rate. Because transformed units are the basis for the description of nonhybrid columns, the transformed distillate flow rate DD is chosen (BD is determined by eq 5). The ∞/∞ bifurcation analysis discussed here is used to predict output multiplicities of type II. By adaption of the definition of the MSS types, these multiplicities are now caused by the reactive VLE. First, the limiting case of a distillation column with zero distillate flow is examined, D ) 0 (DD ) DN ) DM ) 0). By eqs 5-7, the following statements about the bottoms flow can be derived: BD ) FD, BM ) FM, but BN * FN. In all cases, no distillate is taken out and all feeds entering will be leaving in the bottoms after they have reached chemical equilibrium in the reactive column. Therefore, the transformed bottoms composition XB will be equal to the transformed feed composition XF (by eq 5). In molar units, the bottoms composition will not be that of the feed but of the feed mixture in chemical equilibrium. For a feasible reactive profile the distillate must be located on the low-boiling part of the reactive residue curve containing the bottoms (and the feed) such that the profile also contains a reactive pinch point. Thus, the distillate is located at the lightest-boiling reactive node of the distillation region containing the feeds similar to that for nonreactive columns (Bekiaris et al., 1993). Hence, the initial column profile is of type I.

Second, for the limiting case of zero bottoms flow, the following equations must hold: BD ) BN ) BM ) 0, and DD ) FD, and DM ) FM. The bottoms composition XB is that of the heavy-boiling reactive node of the distillation region containing the feed, and the transformed distillate composition XD is equal to XF, the composition of the feed mixture in chemical equilibrium. The resulting column profile is of type II. Third, a bifurcation study (continuation of solutions) is performed by tracking the distillate and bottoms compositions in the transformed composition space, starting from the profile with DD ) 0 and ending at the profile with DD ) FD. By this procedure, all feasible transformed composition profiles are obtained for the ∞/∞ case of a nonhybrid reactive column. This continuation of solutions can be constructed based on physical arguments only, i.e., applying the feasibility conditions for reactive profiles and using the reactive residue curve diagram. The continuation path is defined as the path generating all possible column profiles. The distillate and bottoms compositions along this path form the distillate product path and the bottoms product path. Each pair of distillate and bottoms locations, XD and XB, is assigned to a value of the transformed flow rate DD by the lever rule (5). Therefore, the distillate and bottoms product paths are parametrized by a product flow rate, XD(DD) and XB(DD). When the transformed compositions XP are plotted against the transformed distillate flow rate DD, a series of bifurcation diagrams can be constructed. If the transformed distillate flow rate is varying nonmonotonically along the continuation path, MSS exist for the ∞/∞ case of a nonhybrid column and the transformed feed composition analyzed. Precisely, MSS will then exist for all feeds resulting in the same composition in chemical equilibrium, i.e., having the same value XF. Because of the analogy described above, the following result from Bekiaris et al. (1993) is straightforward for nonhybrid columns. Starting at the initial point (type I profile), the distillate flow rate DD is monotonically increasing as all type I profiles are tracked. For the same reason, DD is also monotonically increasing as all type II profiles are tracked, ending at the final point with DD ) FD. Therefore, a decrease of DD can only occur for type III column profiles. Furthermore, if DD is varying nonmonotonically along the product path, the bottoms flow rate BD will also vary nonmonotonically (by eq 5). Note again that the whole bifurcation analysis must be done using transformed compositions and flow rates. To obtain the product paths in molar compositions and flow rates, the inverse of the transformation (“backtransformation”) must be applied. Unfortunately, this is a procedure involving numerical iteration, but the effort can often be avoided as demonstrated in Appendix C. Importantly, the existence of MSS in transformed flow rates has not been shown yet to be necessary and sufficient for multiplicities to exist in other units, i.e., molar or mass flow rates. However, the predictions can be extended to cover other units by applying the ∞/∞ singularity analysis described below. 4.2. Geometrical MSS Feed Region Condition. MSS exist if the transformed distillate flow rate, DD, is decreasing somewhere along the continuation path. A decrease in DD, i.e., the existence of MSS, can be detected by the geometrical, necessary, and sufficient multiplicity condition. Essentially, this condition takes


the same form as that developed for single azeotropic columns and column sequences if the reactive continuation path and transformed compositions for the feed and the products are used (Bekiaris et al., 1996; Gu¨ttinger and Morari, 1996). The reader is referred to Appendix A for a detailed description of this condition. From the bifurcation procedure and the geometrical condition, it is clear that the existence of MSS depends strongly on the feed composition XF. All possible transformed feed locations leading to MSS in the ∞/∞ case of a nonhybrid column can be determined by the MSS feed region condition. The condition is also adapted from the nonreactive ∞/∞ analysis (Bekiaris et al., 1996). The detailed procedure to construct the MSS feed region is given in Appendix B. It will be demonstrated later by application to the MTBE process. Note that the MSS feed region is defined in the transformed composition space and is suitable to test a candidate feed composition xF with respect to the occurrence of MSS. With eq 2 the transformed feed composition XF(xF) is obtained, and one can easily check if it is located in the MSS feed region. However, sometimes it may be necessary to represent the feed region in the physical composition space. Because the transformed feed composition XF is invariant for all mixtures reaching the same composition in chemical equilibrium, eq 2 must be fulfilled for all physical compositions xF leading to XF. These c - r - 1 linearly independent restrictions on the c independent physical compositions xF leave c - 1 - (c - r - 1) ) r degrees of freedom in xF. By multiplication with the denominator of (2), a system of equations linear in x is obtained. Because all of these equations are coupled by the reference components, it is always possible to reduce the system by substitution of some x. It can be shown that the system can be reduced to one single linear equation relating r + 1 of the remaining mole fractions xF and containing the known transformed composition XF. Therefore, for each transformed composition XF, this equation describes a linear geometrical object of dimension r (equal to the degrees of freedom obtained before). A straight line is obtained for r ) 1, a plane for r ) 2, etc. Hence, the MSS feed region in the physical composition space is built from straight lines (or planes), each of those belonging to one transformed composition XF contained in the feed region. From the properties of these stoichiometric lines, it can be concluded that it is sufficient to evaluate XF on the boundary of the MSS feed region to obtain the convex hull of the feed region in physical coordinates. This procedure will be demonstrated in the MTBE application section. 4.3. ∞/∞ Singularity Analysis. The ∞/∞ singularity analysis was first published by Gu¨ttinger and Morari (1997), and its application to reactive columns will be demonstrated here in detail. From the ∞/∞ analysis, the product paths as functions of the product flow rates are obtained in transformed variables, XD(DD) and XB(DD), and the existence of MSS can be predicted for transformed flow rates. Even though transformed units can hardly be implemented in a real distillation column, this result is still useful. In the ∞/∞ case, multiplicities of type II (caused by the reactive VLE) are mainly governed by the material balances, i.e., by the lever rule. For reactive columns, the lever rule is only applicable in the transformed space, and thus, transformed units are the units of choice to appropriately describe a reactive ∞/∞ column. Currently, only the analysis in the

transformed space allows one to predict whether MSS of type II are possible for a given feed (MSS feed region condition). To avoid MSS, one can possibly shift the feed location out of the MSS feed region. However, if the feed cannot be changed for some reason, if MSS are unavoidable to match the product specifications, or if MSS are economically attractive, it would be more important to predict the existence of MSS for practical relevant specifications, e.g., for mass or volumetric flow rates. This is where the singularity analysis comes in. Without loss of generality, any of the distillate and bottoms products will be denoted by P and the corresponding product path XP(PD). Starting from the results of the bifurcation analysis with respect to PD, it is possible to predict MSS for molar (PN), mass (PM), or volumetric flow rates (PV) for the ∞/∞ case of a reactive column. This means that the ∞/∞ predictions are extended to cover also type Ia multiplicities, i.e., singularities in the input relationships (Jacobsen and Skogestad, 1991). The following equations are obtained from the transformation (3) or are the standard relations between molar, mass, and volumetric flows:

PN ) (1 - νTtot(νref)-1xPref)-1PD PM ) W(xP)PN PV ) PM/F(xP,T P) To obtain the product singularity functions, the derivatives are taken:

∑

N,D )

dPN ) (1 - νTtot(νref)-1xPref)-1 dPD PD d T (νtot(νref)-1xPref) (8) (1 - νTtot(νref)-1xPref)2 dPD

∑

M,N )

dPV

P dPM P N dW(x ) ) W(x ) + P dPN dPN

PM

∑V,M ) dPM ) F(xP,T P)-1 - F(xP,T P)2

dF(xP,T P) dPM

(9) (10)

Note that the complexity of expression (8) is caused by its general formulation, and the equation is much simpler for a specific transformation. In Figure 2 four possible types of singularities are illustrated. On the left-hand side, the product path of a product P, which was obtained from the predictions, is plotted versus two flow rates of the same flow in different units, P1 and P2. For example, P1 and P2 could be DD and DN, respectively, to depict the distillate product path XD(DD) versus the two flow rates. The derivative of this curve is the “singularity function” (8), which is shown on the right-hand side as ∑1,2. Now, each of the product paths can be analyzed with respect to specification of either the flow rate P1 or P2 and four cases can be distinguished (Figure 2): (a) No singularities occur and no MSS exist for specification of either P1 nor P2. The qualitative behavior remains unchanged. The singularity function ∑1,2 does not intersect zero. (b) A singularity occurs and MSS exist for P1 but not for P2. The multiplicities disappear or reappear depend-


Figure 2. Illustration of the four possible types of singularities including example singularity functions ∑.

ing on the direction of the input transformation. The singularity function crosses zero at the limit points ∧. (c) Singularities occur and MSS exist for both specifications but with different limit points * and ∧. The singularity function crosses zero four times: at the limit points *, it changes from +∞ to -∞, and at the limit points ∧, it is continuous and intersects zero. The multiplicities are qualitatively different in units of P1 and P2 because the limit points do not coincide. (d) Degenerated case c with identical limit points and MSS existing for both flow rates. The singularity function is not intersecting zero but jumping from the positive to the negative side and vice versa. The multiplicity intervals are identical from a physical point of view. These results lead to the question of how the singularity analysis would be performed in practice. To obtain the ∞/∞ predictions for specifications involving a molar product flow rate, ∑N,D has to be evaluated. However, the evaluation of eq 8 requires the knowledge of the physical compositions x and, hence, the backtransformation has to be carried out (see Appendix C). This is a major drawback of the singularity analysis for reactive systems. This problem does not occur for nonreactive systems because a transformation is not necessary; see Gu¨ttinger and Morari (1997). After the predictions for molar flow rates are obtained, it is very simple to evaluate the other singularity functions, ∑M,N and ∑V,M, because x and the boiling-point temperature T are known (either by design or from the backtransformation). One singularity function will be evaluated after the other in the order given above, (8)-(10), and zero intersections will be determined. Each of the resulting singularity functions will show the behavior of one of the four cases a-d, and MSS as well as the product flowrate interval where they occur can be predicted. The following remarks are important: First, it is often difficult to detect the zero intersection where the

singularity function rises to infinity and switches to minus infinity, e.g., at * in case c. However, by evaluation of the inverse ∑N,M ) ∑-1 M,N, these limit points will take the simpler form of the zero intersections at ∧ in case b. Therefore, it is numerically advantageous to simultaneously evaluate ∑M,N and ∑N,M to correctly identify the type of the singularity. Second, the singularity analysis has to be done for each product flow independently to obtain the predictions for specifications involving molar flow rates, because the reaction term in eq 6 can cause MSS to exist in one product flow but not in the other. This is also true for volumetric flow rates but not for mass flow rates, where the total balance holds as for transformed compositions; see eqs 5 and 7. Third, the ∞/∞ singularity analysis described here can be applied to nonhybrid and hybrid columns as it only depends on the product paths and is independent of the prediction method used to generate these paths. Last, there is an alternative procedure without direct evaluation of the singularity functions. With an effort similar to that necessary to obtain the singularity functions, the product path XP(PD) can be recalculated to get XP(PN) by backtransformation, and the diagram on the left of Figure 2 can be drawn. Then, the multiplicity behavior can be evaluated graphically. Of course, the existence of MSS for different types of specifications can be directly studied by simulation, but at the cost of much more data and effort needed. The ∞/∞ singularity analysis is based solely on the prediction results obtained by geometrical methods. 4.4 ∞/∞ Predictions: Inputs and Results. Prior to summarizing the results obtained from the ∞/∞ bifurcation and singularity analyses, the necessary information to obtain the ∞/∞ predictions is discussed. Basically, the information needed is of the same type as that for the nonreactive analysis, i.e., the boiling points and locations of the azeotropes and the distillation boundaries (if there are any). However, in contrast to the widely available databases for nonreactive azeotropes, the existence and location of reactive azeotropes are rarely known in advance, because they depend not only on the VLE but also on the reaction equilibrium (Okasinski and Doherty, 1997b). If a reliable VLE and reaction equilibrium model is available, a reactive residue curve diagram can be calculated for the mixture of interest. Alternatively, reactive residue curves could be measured in an experiment. Then, a composition transformation has to be found for the stoichiometry of the reactions in the mixtures. After the differential-algebraic equation system is solved, all necessary information can be read from the reactive residue curve diagram. It is important to note that the predictions depend only mildly on the specific column design. The only information needed is the types of the column internals (trays or packing) as well as the types of condenser and reboiler to appropriately select reactive distillation lines or residue curves (Bekiaris et al., 1996). From the ∞/∞ analysis for nonhybrid reactive systems, the following information can be obtained: Existence of Multiplicities. The existence of MSS caused by the reactive VLE can be predicted in the ∞/∞ case of a nonhybrid RD column and a given feed by the geometrical, necessary, and sufficient multiplicity condition. MSS Feed Region. For any nonhybrid system, the region of feeds that lead to output multiplicities in the ∞/∞ case can be predicted in the transformed composi-


tion space. Determining the corresponding region in the physical composition space is straightforward and does not require backtransformation. Quantitative Predictions. For any nonhybrid system and a given feed, all possible column profiles can be located in the transformed composition space by performing a continuation of solutions. From the product paths obtained by the continuation, bifurcation diagrams can be constructed and the limit points as well as the product flow-rate multiplicity range (the interval of the product flow rate where MSS exist) can be determined. Singularity Analysis. When a series of singularity analyses are performed, the predictions obtained for transformed flow rates are extended to specifications involving molar, mass, or volumetric flow rates. Qualitative and quantitative changes in the multiplicity behavior are detected. This means that the ∞/∞ predictions are expanded to also cover the type Ia multiplicities studied by Jacobsen and Skogestad (1991). However, because the analysis requires that one product flow rate is used for the specification, the [L,V] configurations studied by Jacobsen and Skogestad (1991) cannot be taken into account. Hence, the methods presented allow the simultaneous prediction of output multiplicities caused by the reactive VLE (type II) and by the nonlinearities in the flow-rate relationships (type Ia). 4.5. Finite Columns. The ∞/∞ case is the limiting case of high reflux, fast reactions, and a large number of equilibrium stages. Therefore, if MSS are predicted by the geometrical condition for a given feed, these multiplicities should also exist for columns with sufficiently large reflux, sufficiently fast reactions, and a large number of stages. However, the “critical design” where MSS disappear cannot be predicted. Because the geometrical condition is only sufficient for finite columns, there may be multiplicities in finite columns which cannot be predicted using the ∞/∞ analysis, e.g., those related to heat effects (type Ib). Similarly, the MSS feed region is affected for finite columns. From our experience, the feed region for finite but large columns is smaller than that in the ∞/∞ case. If the reflux and the number of equilibrium stages are decreased, the MSS feed region collapses and the multiplicities disappear (Bekiaris and Morari, 1996). These observations have the following consequences if the existence of MSS is questioned for a given finite nonhybrid column and a given feed. First, if the feed lies inside the MSS feed region, MSS will exist for transformed flow rates and a sufficiently large column operated at large internal flows. In principle, they could disappear for molar or mass flow rates by singularities in the corresponding relationships although this has not been observed in any of our studies. Second, MSS of type II will not exist for a feed outside the MSS feed region (in transformed coordinates). However, singularities in the relationship between transformed and molar flows seem to play an important role if the feed is located close to the MSS feed region and can result in a MSS feed region in molar coordinates different from the transformed one. Even though they are theoretically possible, singularities between molar and mass flows have not been reported for points outside the MSS feed region (in transformed coordinates) and nonhybrid columns. Nevertheless, the existence of MSS of type Ia strongly depends on the column configuration.

Finally, all of the prediction results described in this paper with the exception of the singularity analysis are obtained for nonhybrid columns only and are directly useful for processes where nonhybrid columns and homogeneous catalysts are used, e.g., for esterification and hydration reactions. Moreover, it will be shown that these predictions are useful for the understanding of the multiplicity phenomenon in hybrid columns (Gu¨ttinger and Morari, 1998). 5. Application to the MTBE Process 5.1. MTBE: System Thermodynamics. Even though MSS have been reported for some other reactive distillation systems, the application presented here focuses on the production of MTBE. The reason is, first of all, that MTBE is produced in huge amounts as an antiknock agent in gasoline, and the manufacturing process involving reactive distillation is of great importance (Smith and Huddleston, 1982; Sneesby et al., 1997a). Furthermore, the multiplicities for the MTBE system have been studied extensively by various researchers. MTBE (index 3) is the product of the reaction of isobutene (index 1) with methanol (index 2). The reaction can either be heterogeneously catalyzed by a strongacid ion-exchange resin, e.g., Amberlyst 15, or homogeneously catalyzed by sulfuric acid. The reaction, possible side reactions, and strategies on how to avoid them have been published on the basis of experimental work (Rehfinger and Hoffmann, 1990; Isla and Irazoqui, 1996). Sneesby et al. (1997b) confirmed that the side reactions do not significantly influence the multiplicity phenomena for this system. There is some disagreement if the reaction approaches chemical equilibrium closely in a RD column. Even though Sundmacher et al. (1994) claim to operate in a kinetically controlled regime with their catalyst, Abufares and Douglas (1995) validated the chemical equilibrium assumption for the same thermodynamic and kinetic data. Further evidence was provided by Isla and Irazoqui (1996), and MSS have been shown under the assumption of chemical equilibrium (Hauan et al., 1995; Gu¨ttinger and Morari, 1997). In an industrial situation, the isobutene source is normally originating from either an FCC cracker or a steam cracking unit and, thus, the feed contains a varying amount of different butanes and butenes. Because the reaction is highly selective for isobutene, the other organic compounds of the isobutene feed stream, whose properties are all very similar, can be lumped together as n-butane (index 4) for the purpose of system analysis (Abufares and Douglas, 1995). In the following, the studies of DeGarmo et al. (1992) and Nijhuis et al. (1993) form the basis of our investigations, which are all done at a pressure of 11 atm. The sources of the thermodynamic data (chemical and VLE) are given in Appendix D. The nonreactive residue curve diagram of the quaternary system is shown in Figure 3. There are three minimum-boiling azeotropes in the system: IM between isobutene and methanol (8.3 mol % methanol boiling at 346.9 K), NM between n-butane and methanol (12.2 mol % methanol boiling at 355.0 K), and MM between methanol and MTBE (57.7 mol % methanol boiling at 406.9 K). The three azeotropes represent the corners of a nonreactive distillation boundary plane which divides the composition space into a MTBE-rich and a methanol-rich distillation region (not shown in the figure). All residue curves start


Figure 3. Nonreactive residue curves of the quaternary MTBE system at a pressure of 11 atm.

Figure 4. Reactive residue curves of the MTBE system at a pressure of 11 atm. “RSA” is the reactive saddle azeotrope.

at the IM azeotrope (unstable node) and end at either the methanol or the MTBE pure-component corner (stable nodes). All other singular points (pure components and azeotropes) are saddles. Next, the mixture is represented in the transformed composition space. Because there are c ) 4 components and the system is undergoing a single reaction (r ) 1), the transformed composition space has the dimension c - r - 1 ) 2. The transformation equations are taken from Ung and Doherty (1995e), and the three transformed compositions X1, X2, and X3 sum to unity:

X1 ) X3 )

x1 + x3 1 + x3

x4 1 + x3

X2 )

x2 + x3 1 + x3

PD ) PN(1 + xP3 )

(11)

When this transformation is applied to the liquid and vapor compositions, the reactive residue curve equations (4) can be solved together with the reaction (14) and the vapor-liquid equilibrium (15) to obtain a reactive residue curve diagram at 11 atm (Figure 4). The reactive residue curves originate either from the NM azeotrope, which has survived the reaction, or from the isobutene pure-component corner (unstable nodes). All reactive curves end at pure methanol, the stable node. There is

a quaternary reactive saddle azeotrope (RSA) close to pure n-butane which introduces four reactive distillation boundaries: one from NM to the RSA, one from isobutene to the RSA, one from the RSA to methanol, and one from the RSA to n-butane. For the purpose of the analysis, the marginal part of the composition space between isobutene, the RSA, the NM azeotrope, and n-butane can be neglected. The remaining composition space (methanol-NM-RSA-isobutene-MTBE) is divided into a methanol-rich and an isobutene-rich RD region by the RSA-methanol reactive boundary. The diagram in Figure 4 is identical to that used by Ung and Doherty (1995a), and the reader is referred to their study for additional representations at different pressures. 5.2. MTBE: Columns and Assumptions. There are two different ways of feeding an MTBE reactive distillation column. In some practical cases, the isobutene and the methanol feed streams are fed to a reactor before entering a finishing reactive column for economic reasons (Isla and Irazoqui, 1996; Bartlett and Wahnschafft, 1997). Because the reaction is exothermic, most of the heat is released in the reactor and cannot be used directly for distillation (only by heat integration). The resulting column feed is mostly liquid below its boiling point, contains a small amount of isobutene, and is fed at the lower end of the reactive section. A second feed strategy is to distribute the vaporized isobutene feed around the lower end of the reactive section (to avoid hot spots) and let liquid methanol enter the column at about the same location (Nijhuis et al., 1993). Three different columns are of main use in the MTBE RD studies. The Nijhuis column depicted in Figure 1b is a hybrid reactive column, where the rectifying section has three theoretical stages and a total condenser, the reactive section consists of eight theoretical stages, and the stripping section has six theoretical stages including a partial reboiler (Nijhuis et al., 1993). The column was originally operated with a reflux ratio of L/D ) 7, the liquid methanol feed enters on stage 11, and the vapor isobutene feeds are distributed above stages 10-12 (second feed strategy from above). An almost identical column was also studied by Jacobs and Krishna (1993), but with one less rectifying tray. However, this difference has no qualitative influence on the multiplicity phenomenon, as can be shown by comparison of Nijhuis’ results with those obtained for the shorter column (Jacobs and Krishna, 1993; Hauan et al., 1995). Sundmacher and Hoffmann (1995) described a laboratory column with two sections only, a reactive and a stripping section (Clausthal column). Their column and that of Sneesby et al. (1997b), which is rather short (3 reactive trays and 10 trays in total), are not studied here, but the Nijhuis column is used as the basis of the analysis. Because nonhybrid columns are investigated in the current step of the analysis, the reaction is assumed to take place in the entire column. However, there is one important difference from the nonhybrid column depicted in Figure 1a. The total condenser is assumed to be nonreactive for technical reasons, which causes small deviations in the column behavior shown later. The resulting column, however, behaves approximately like a nonhybrid one and will be denoted by the “pure” column. The simulations presented for the MTBE system are based on the following two column models: Aspen Model. Aspen Plus was used for the simulations, i.e., the standard Radfrac model including a block


Figure 5. Construction of the region of feeds leading to MSS in the ∞/∞ case of a nonhybrid MTBE column.

for equilibrium chemical reactions. Radfrac solves the MESH equations ( Mass balances, Equilibrium and Sum equations, and Heat balances) for the specified steady state (Aspen Plus, 1995). Auto Model. This is a CMO model (neglecting all heat balances) implemented in the bifurcation solver Auto97 (Doedel et al., 1997). Its simplicity helped to isolate the physical cause leading to MSS. Note that the ∞/∞ model is the limiting case of a distillation column and does not depend on the column dimensions. 5.3. MTBE: MSS Feed Region. As a first step in the analysis of the MTBE system, the region of feed compositions leading to MSS will be constructed for the ∞/∞ case of a nonhybrid reactive column. The application of the MSS feed region condition described in Appendix B is shown in Figure 5. There are two RD regions of interest, one rich in methanol and the other rich in isobutene. If the feed is located in the isobutenerich region, two product paths are possible for ∞/∞ profiles (from Figure 4): from isobutene over MTBE to methanol and from isobutene along the distillation boundaries over the RSA (close to n-butane) to methanol. If the feed is located in the methanol-rich region, the two possible paths are as follows: from the NM azeotrope to methanol and from the NM azeotrope along the boundaries over the RSA to methanol. However, because the profiles with the products on the NMmethanol path are not of type III (they require a binary feed of n-butane and methanol and no reaction could occur), this path is not of interest. The same argumentation applies to the isobutene-MTBE-methanol path where the feed would not contain any inert n-butane. Hence, the two paths remaining are NM-RSA-methanol and isobutene-RSA-methanol. We start with a methanol-rich feed and the path from the NM azeotrope over the RSA azeotrope to pure methanol (NM-RSA-methanol). The first distillate D chosen has a composition XD equal to that of the lightest-boiling node of the corresponding distillation region, i.e., the NM azeotrope (step 1 in Appendix B). At point a, the tangent to the boundary is parallel to the tangent at the NM azeotrope. By application of the geometrical multiplicity condition described in Appendix A, all compositions XB on the RSA-methanol boundary between a and pure methanol belong to the set S(XD) (Figure 5 and step 2). The union of all lines connecting

Figure 6. Region of feeds leading to MSS in the ∞/∞ case of a nonhybrid MTBE column. Note that the boundary line of the feed region is not a straight line in this case.

D with the points of the set S(XD) is the area enclosed by the straight line NM-a, the boundary a-methanol, and the edge methanol-NM (steps 3 and 4). All feeds belonging to this area will lead to MSS. Next, a distillate at point b is analyzed (step 5). The set S(XD) of XB product locations again contains the points on the boundary from methanol up to a point very close to a. The distillate location closest to the RSA for which the geometrical condition is fulfilled is connected to point c, and a tangent to the boundary is parallel to the methanol-MTBE edge. When this procedure is repeated for all distillate locations on the boundary between NM and c, all sets S(XD) are determined and the corresponding areas are constructed as the unions of the lines connecting each D with its corresponding set S(XD). Finally, the feed region leading to MSS in the ∞/∞ case is constructed by joining all of these resulting areas and is depicted in Figure 6. For any feed with transformed composition inside this feed region, output multiplicities will occur for some product flow-rate interval in the ∞/∞ case of a nonhybrid column. If a candidate feed composition is given in molar units, its transformed composition is calculated from eq 11, and it can be graphically or numerically evaluated if this feed belongs to the MSS feed region and will lead to MSS. Alternatively, the MSS feed region in molar compositions can be calculated from the transformed feed region, and the result can be depicted in the tetragonal composition space. The locus of all molar compositions leading to the same transformed composition has been studied in a previous section. For the MTBE system with c ) 4 components and r ) 1 reaction, the locus has dimension r ) 1 and is a straight line, the stoichiometric line. Because the transformation equations (11) are coupled by x3, the system of these equations can be reduced to obtain a description of the stoichiometric lines. By elimination of x3, the linear independent equations for X1 and X2 from eq 11 can be rewritten as

x1 )

x2(X1 - 1) + (X1 - X1X2 + X2) X2 - 1

(12)

The other feed compositions are determined by rewriting the second equation of eq 11 as


x3 )

x2 - X2 X2 - 1

x4 ) 1 - x1 - x2 - x3

(13)

Given a transformed feed composition X ) XF, a line of molar feed compositions parametrized by x2 is obtained (1 degree of freedom). Hence, the MSS feed region in the physical composition space can be built from the straight lines obtained for each transformed feed location of the MSS feed region. Finally, the second product path from isobutene over the RSA to methanol has to be considered. If the same analysis is performed for this path, it can be shown that no feed location in the isobutene-rich distillation region will lead to MSS. Therefore, the MSS feed region depicted in Figure 6 contains all feeds leading to output multiplicities for ∞/∞ nonhybrid columns and the MTBE system. 5.4. MTBE: Bifurcation Analysis. After the MSS feed region is constructed, a bifurcation analysis is performed for two selected feed locations. The first feed is the one used by Nijhuis et al. (1993). When its transformed composition is calculated and drawn in Figure 6, it can be seen that this feed is not expected to show MSS in the ∞/∞ case of a nonhybrid reactive column. Alternatively, a second feed inside the MSS feed region was selected and denoted by the “MSS Feed”. Because both feed locations are contained in the methanol-rich distillation region, the ∞/∞ bifurcation analysis is similar. In the following, the MSS Feed was chosen for the demonstration. In this paragraph, the general procedure described in the section ∞/∞ Bifurcation Analysis is carried out, and the results are depicted in Figure 7. In order to determine the initial steady state, the transformed distillate flow rate is set to zero, DD ) 0. The distillate is located at the unstable node of the distillation region containing this feed point, the NM azeotrope. The transformed bottoms composition is equal to that of the feed. First, all type I profiles are followed and, hence, DD is increasing. As for nonreactive distillation, the distillate stays at the light node, the NM azeotrope, and the bottoms moves on the straight line through the feed and the NM azeotrope, until it reaches the distillation boundary (Figure 7a). The feasibility conditions for reactive ∞/∞ profiles are fulfilled for all profiles on this part of the continuation path: they contain a singular point, the NM azeotrope; a residue curve is connecting D and B, and the lever rule is fulfilled. As the bottoms has reached the boundary, the profile now contains the RSA as a singular point and, hence, the distillate is allowed to leave the NM azeotrope. The only path which it can follow and still have a residue curve connecting D and B is the reactive boundary toward the RSA. Correspondingly, the bottoms moves along the reactive boundary toward methanol (Figure 7b). The residue curves connecting D and B on this part of the continuation path are the boundaries themselves (type III profiles), and by the lever rule (or by the geometrical condition from Appendix A), the distillate flow rate DD is decreasing on this segment of the path. Then, the situation is considered where D has reached the RSA, a ternary saddle point in the transformed composition space. There are three candidate boundaries for the distillate to follow (excluding the one which was already used). However, if D would move toward n-

Figure 7. Steps in the ∞/∞ bifurcation analysis of a nonhybrid MTBE column and the “MSS Feed”.

Figure 8. Product paths of the ∞/∞ predictions for a nonhybrid MTBE column and the “MSS Feed”.

butane or follow the boundary to methanol, there would either be no residue curve connecting D and B or no singular point in the column profile. The only possibility to fulfill the feasibility conditions is to have the distillate moving along the reactive boundary from the RSA to isobutene. Note that this boundary almost coincides with the edge in Figure 7c. Therefore, the distillate D moves toward isobutene until the bottoms has reached the heavy node, pure methanol (Figure 7c). The profiles of all steady states on this section of the continuation path follow the reactive boundaries and contain the RSA (type III profiles). By the lever rule, the distillate flow rate DD is increasing again. Finally, all type II profiles are followed with the bottoms at the heavy node (methanol) and the distillate moving toward the feed (Figure 7d). DD is increasing again for type II profiles. By this analysis, the continuation path has been constructed for all possible distillate flow rates DD between zero and the feed flow rate FD. The corresponding product paths are depicted in Figure 8. By the lever rule (5), the value of the distillate flow rate DD is calculated for each steady state on the continuation path


Figure 9. Comparison of ∞/∞ predictions and bifurcation calculations with the Auto model for the “MSS Feed” (bottoms compositions).

Figure 10. Comparison of ∞/∞ predictions and bifurcation calculations with the Auto model for the “Nijhuis” feed (bottoms compositions).

and a bifurcation diagram can be drawn. In Figure 9, the resulting diagram for the MSS Feed is shown in transformed compositions and the interval of the product flow rate where MSS exist can easily be identified (for the x axis, note that BD ) FD - DD). Again, backtransformation is needed to obtain the ∞/∞ predictions in molar compositions. These results are not shown here because they do not provide additional insight. The analysis for the Nijhuis feed can be performed in a manner analogous to that for the MSS Feed. However, the distillate flow rate is now increasing for the distillate locations between the NM and RSA azeotrope and, hence, MSS do not exist for this feed and an ∞/∞ nonhybrid column; see Figure 10. Finally, the ∞/∞ predictions are verified by performing bifurcation calculations for the pure column, both feed locations, and a finite reflux of LN ) 3500 (FN ) 755). In Figures 9 and 10, the simulation results are compared with the ∞/∞ predictions. The qualitative agreement (existence of MSS) is excellent. However, there are some quantitative differences in the distillate flow rate interval where MSS exist. They can be explained as follows: 1. The pure column used for the bifurcation calculations is not really a nonhybrid column because it operates with a nonreactive condenser (as explained

earlier). Even though the condenser is total, the distillate composition leaving the condenser is not in chemical equilibrium, because it is equal to the composition of the vapor leaving the top tray. Using a reactive condenser, the distillate composition would reach chemical equilibrium. This fact has quite a significant effect because the products lie on curved boundaries for those steady states where the deviations are significant. The composition of the vapor in equilibrium with a liquid mixture lying on a reactive residue curve (on the boundary) does not lie on the same curve. It is located on the line of compositions which are in vapor-liquid equilibrium with the reactive residue curve boundary in this case. 2. Another reason for the deviations is that the pure column simulated is fairly small (17 equilibrium stages plus condenser) and thus far from the ∞/∞ limiting case. Therefore, some differences caused by finite column effects must be expected. 3. Nevertheless, the occurrence of MSS in this case is “robust” in the sense that the qualitative behavior is unchanged for rather large variations of the column size, the reflux flow rate, and even the feed location (as long as it belongs to the MSS feed region). From the simulations the stability of the steady states can be determined using local linearization and Lyapunov’s direct method. The steady states between the limit points are unstable, and all other steady states are stable. By simulation using the Aspen model, the steady states have been verified and the results agree well with those of the bifurcation calculations. Similar results using Aspen have been reported by Vadapalli and Seader (1997). Moreover, a singularity of type d, Figure 2, has been found in the relation between transformed and molar compositions for both feeds (∞/∞ singularity analysis). Because the limit points are physically identical, neither the existence nor the extent of the multiplicities are altered by this. There are also type d singularities in the mass-molar relationship in the ∞/∞ case. For the finite column, however, it degenerates to a singularity of type c and causes the multiplicity interval to be significantly smaller when measured in mass units. The discussion of the results as well as a detailed study of the physical causes leading to MSS in the MTBE process is presented in the succeeding paper (Gu¨ttinger and Morari, 1998). 6. Conclusions The ∞/∞ analysis to predict MSS in equilibrium RD has been developed for columns where the reactions take place in the whole column including condenser and reboiler (nonhybrid columns). The existence of output multiplicities caused by the reactive vapor-liquid equilibrium can be predicted by a necessary and sufficient condition for the ∞/∞ limiting case of equilibrium reactive columns of infinite length, operated at infinite internal flows. Through the ∞/∞ singularity analysis, the predictions have been extended to cover multiplicities introduced by the nonlinear relationships between transformed, molar, mass, and volumetric flow rates. The predictions have been shown to carry over to finite columns by application to the MTBE process. The region of feed compositions which will lead to MSS in a nonhybrid MTBE column has been constructed. The


investigations will continue by presenting the prediction methods for hybrid columns in a second paper. Notation a ) activity of a component B ) bottoms flow rate or product location c ) number of components in the system D ) distillate flow rate or product location ) vector of the extents of reaction F ) overall feed flow rate or stream location γ ) activity coefficient of a component K ) chemical equilibrium constant L ) liquid reflux entering the top tray Λ ) binary Wilson coefficient ν ) stoichiometric coefficient matrix νtot ) sum vector of stoichiometric coefficients for each reaction νref ) stoichiometric coefficient reference matrix P ) any product flow D or B of the column p ) system pressure p0 ) vapor pressure of a pure component π ) number of phases in a system Qr ) reboiler heat duty r ) number of equilibrium reactions in a system F ) density of a mixture ∑ ) product singularity function T ) boiling point temperature τ ) warped time for reactive residue curves V ) vapor boilup leaving the reboiler W ) molar weight of a mixture x ) liquid mole fractions xref ) reference mole fractions X ) transformed liquid compositions y ) vapor mole fractions Y ) transformed vapor compositions Superscripts on flow rates indicate the units used for measuring that stream: N is used for molar flow rates, M for mass flow rates, V for volumetric flow rates, and D for transformed flow rates. Superscripts on compositions denote their location, e.g., D for distillate. Subscripts are used to indicate the component index. Superscript T denotes transposition of a vector. Abbreviations CMO ) constant molar overflow MSS ) multiple steady states (output multiplicities) MTBE ) methyl tert-butyl ether RD ) reactive distillation RSA ) reactive saddle azeotrope VLE ) vapor-liquid equilibrium

Supplementary material is available under Technical Report AUT98-06 on http://www.aut.ethz.ch. Appendix A. Geometrical Multiplicity Condition In this section the geometrical, necessary, and sufficient multiplicity condition is adapted from Bekiaris et al. (1996). The extension of the condition to reactive systems is straightforward if the transformed continuation paths and the transformed feed composition are used. If a continuation path is defined as the path generating all possible column profiles starting at DD ) 0 and ending at DD ) FD, MSS result when the transformed distillate flow rate is decreasing somewhere along this path. Note that subscripts are used for indexing different steady states in this case.

Figure 11. Illustration of the geometrical, necessary, and sufficient multiplicity condition for RD columns.

Pick a pair of transformed product compositions XD 1 and XB1 from the path which is connected by a type III profile; i.e., the profile runs over at least one reactive saddle singular point. Thus, these product locations will be located on a reactive distillation boundary or on the edges of the transformed composition space (Figure 11). B D B Now pick XD 2 and X2 sufficiently close to X1 and X1 such that the new column profile corresponds to a “later” profile on the continuation path. For the existence of MSS it is required, as we move along the continuation path from the product locations 1 to 2, that the line B B D B passing from XD 1 parallel to X1 X2 crosses the X2 X2 line segment at I. Interestingly, this same formulation of the geometrical multiplicity condition can be applied to four entirely different column types with minor modifications: homogeneous and heterogeneous azeotropic columns, column sequences, and nonhybrid as well as hybrid reactive distillation columns (Bekiaris et al., 1996; Gu¨ttinger and Morari, 1996). Appendix B. MSS Feed Region Construction The construction procedure for the MSS feed region was adapted from Bekiaris et al. (1996). From all possible product paths in the ∞/∞ case, the sections for which the products are connected by a type III profile are investigated. It is sufficient to consider type III profiles only for the construction of the feed region, because the product flow rates can only vary nonmonotonically for such profiles. For each possible section of the path of type III profiles, the procedure is performed as follows: 1. Pick a transformed product location from a product path, e.g., XD. 2. Find the set containing all other product compositions XB such that the geometrical condition is satisfied for the picked XD, and name this set S(XD). Note that S(XD) is always part of a RD boundary or the edges of the transformed composition space and that it may contain an inflexion point and/or it may consist of more than one nonconnected boundary segment. 3. Draw the straight lines connecting XD with the end points of each of the boundary segments belonging to S(XD).


4. For the XD chosen, the appropriate feed composition is the union of the areas enclosed by each boundary segment that belongs to S(XD) and the corresponding straight lines connecting the end points of the segment with XD. Name this union A(XD). 5. Choose another XD and repeat from 1 for any XD on the continuation path. 6. Finally, the transformed feed compositions XF that lead to output multiplicities lie in the union of all of the areas A(XD), i.e., in the union of all areas enclosed by the boundary segments that belong to some S(XD). Appendix C. Backtransformation of the ∞/∞ Predictions The nonhybrid ∞/∞ theoretical predictions are obtained in transformed compositions and flow rates, XP(PD) (where P stands for any product flow D or B). To obtain the predictions using molar compositions, XP has to be transformed back. This means that the c - r - 1 independent transformation equations (2) have to be solved simultaneously with the r chemical equilibrium equations. In general, the latter take the form (for each reaction) c

K(T) )

aνi ) f (xi,γ(xi,T)) ∏ i)1 i

(14)

Table 1. Pure-Component Molar Weights and Antoine Parameters for the MTBE System

name

index

molar weight (g/mol)

isobutene methanol MTBE n-butane

1 2 3 4

56.1084 32.0424 88.1508 58.1243

component

Antoine parameters A

B (K)

C (K)

20.64556 23.49989 20.71616 20.57097

-2125.74886 -3643.31362 -2571.58460 -2154.89730

-33.160 -33.434 -48.406 -34.420

Table 2. Binary Parameters for the Wilson Activity Coefficient Model (17) Di, j isobutene

methanol

MTBE

n-butane

0.0 0.7420 -0.2413 0.0

-0.7420 0.0 -0.9833 -0.81492

0.2413 0.9833 0.0 0.0

0.0 0.81492 0.0 0.0

isobutene methanol MTBE n-butane

Ei, j isobutene methanol MTBE n-butane

isobutene

methanol

MTBE

n-butane

0.0 -1296.719 -136.6574 0.0

-85.5447 0.0 204.5029 -192.4019

30.2477 -746.3971 0.0 0.0

0.0 -1149.286 0.0 0.0

the vapor pressures in pascal, p0i , from the temperatures in kelvin:

ln(p0i ) ) A +

where f is nonlinear. This equation is problematic because the boiling point temperature T is involved in an implicit and nonlinear manner. T is determined by a bubble-point calculation at a given pressure p which involves x and γ, is nonlinear, and is implicit in T:

∑i xi p0i (T) γi(x,T) ) p

(

(15)

Therefore, it is obvious that the backtransformation problem can only be solved numerically for each XP. If the equilibrium constant K is not temperature-dependent and the relationship (14) does not involve the activity coefficients, then an analytical solution would be possible. Yet, there is a simple trick on how the backtransformation can often be avoided. For type III profiles it was shown that the products lie either on the edges of the triangle or on interior boundaries. Normally, the backtransformation of compositions lying on the edges of the transformed space is rather simple because some components are missing. Moreover, the interior boundaries are reactive residue curve boundaries which are obtained by integration of eq 4. In order to get Y(X), a bubble-point calculation has to be done and, hence, the molar compositions x and y are known. When x is stored together with the transformed compositions on the boundary, the backtransformation can be simplified using a look-up table and interpolation methods. Appendix D. Thermodynamic Models and Data The thermodynamic data used in this paper to describe the MTBE system were taken from Ung and Doherty (1995e) including corrections (Ung and Doherty, 1995b). The pure-component data are shown in Table 1. The Antoine equation was used to calculate

(16)

The Wilson activity coefficient model (with parameters from Table 2) was used to describe the multicomponent interactions. Flato and Hoffmann (1992) demonstrated why the Wilson model should be preferred for this mixture.

c

find T such that

B T+C

Λi, j ) exp Di, j + c

ln(γi) ) 1 - ln(

)

Ei, j T c

xkΛk,i

xjΛi, j) - ∑ ∑ c j)1 k)1 xjΛk, j ∑ j)1

(17)

Vapor-liquid calculations assumed ideal vapor phase and were carried out according to eq 15. The chemical equilibrium constant was calculated by

ln(K) )

4254.05 - 10.0982 + 0.2667 ln(T) (18) T

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Received for review May 26, 1998 Accepted December 8, 1998 IE980327X

Predicting Multiple Steady States in Equilibrium Reactive Distillation. 1

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