Multiple Steady States in Homogeneous ... - ACS Publications

Ind. Eng. Chem. Res. 1996, 35, 4597-4611

4597

Multiple Steady States in Homogeneous Separation Sequences Thomas E. Gu 1 ttinger and Manfred Morari* Automatic Control Laboratory, ETH-Z, Swiss Federal Institute of Technology, CH-8092 Zu¨ rich, Switzerland

In this article, multiple steady states are studied in sequences of interlinked columns commonly used to separate azeotropes in ternary homogeneous distillation. More specifically, two major separation schemes were concerned: the intermediate entrainer scheme and the “boundary separation” scheme. On the basis of the results on multiplicities in single columns for the case of infinite reflux and infinite column length (infinite number of trays) similar criteria for separation sequences were developed. It is shown how to construct bifurcation diagrams on physical grounds with one product flow rate as the bifurcation parameter and how the product paths of a sequence can be located. Moreover, a necessary and sufficient condition for the existence of multiple steady states in column sequences is derived on the basis of the geometry of the product paths. The overall feed compositions that lead to multiple steady states in the composition space are also located. For the intermediate entrainer scheme multiplicities occur for all feed compositions. For the boundary separation scheme multiplicities may disappear when a single column is integrated into a sequence. By use of examples of the two separation schemes, it is shown that the prediction of the existence of multiple steady states in the ∞/∞ case has relevant implications for columns with finite length (finite number of trays) operated at finite reflux. 1. Introduction Azeotropic distillation is one of the most widely used and important separation processes in the chemical and the specialty chemical industries. Laroche et al. (1992b) have shown that azeotropic distillation columns can exhibit unusual features not observed in nonazeotropic distillation, the understanding of which is critical for proper column design, control and simulation. Multiple steady states have been discovered among the surprising features of azeotropic distillation columns. In the literature, the term multiple steady states (MSS) is used to describe various and sometimes different situations. In this article it is applied according to the following: Definition 1 (Multiple Steady States in Single Columns): By multiple steady states in a distillation column we refer to the general notion of multiplicities, i.e., that a system with as many parameters specified as there are degrees of freedom exhibits different solutions at steady state. For a given design, column pressure, feed flow rate, composition and phase we define the following: (1) output multiplicities, if there exist different product compositions and therefore different column profiles at steady state for the same set of operating parameters (2) input multiplicities, if we get the same product compositions for different values of the operating parameters In this article, the term “multiple steady states” refers to output multiplicities. The possible influence of these multiplicities on column design, operation and control is a topic of current investigations. Historically, the study of multiplicities started with Rosenbrock (1962), who investigated binary mixtures. Petlyuk and Avetyan (1971) first conjectured the possibility for multiple steady states in ternary homogeneous systems and Magnussen et al. (1979) presented simulation results showing three steady states for the heterogeneous mixture ethanol-water-benzene. (A * Author to whom correspondence should be addressed. Phone: +41 1 632-7626. FAX: +41 1 632-1211. E-mail: [email protected].

S0888-5885(96)00292-8 CCC: $12.00

mixture under consideration is called homogeneous if only one liquid phase exists throughout the composition space, and heterogeneous if two liquid phases exist for some compositions.) Furthermore, Jacobsen and Skogestad (1991) reported two different types of multiplicities in binary distillation columns with ideal vapor-liquid equilibrium. The existence of output multiplicities was studied thoroughly by Bekiaris et al. (1993) for ternary homogeneous and by Bekiaris et al. (1996) for ternary heterogeneous mixtures. A detailed review of the literature of the two cases was also provided in these publications. Bekiaris and Morari (1996) showed that output multiplicities may have numerous implications not only for distillation simulations but more importantly for the design, operation, and control of distillation columns. They also investigated quaternary systems. However, the previously mentioned work was limited to multiplicities in single, stand-alone distillation columns. One major application of azeotropic distillation is the separation of a binary azeotrope by the means of a third component, the entrainer. This process however, often referred to as extractive distillation, is carried out in two or more columns which enables the recycling of the entrainer (Stichlmair et al., 1989; Doherty and Caldarola, 1985). Laroche et al. (1992a) improved on the separability and flowsheet synthesis of such a homogeneous azeotropic distillation sequence. As a consequence, we now investigate the problem of multiple steady states in separation sequences consisting of two or more distillation columns, i.e., the columns form a separation train unlike interlinked columns (where multiplicities have also been reported by Chavez et al. (1986) and Lin et al. (1987)). First, the main results of Bekiaris et al. (1993) on multiple steady states in ternary homogeneous distillation are presented. Then, we inquire into the design of a homogeneous azeotropic distillation sequence and study the intermediate entrainer and the “boundary separation” schemes. Finally, the analytical results of Bekiaris et al. (1993) are extended to multicolumn separation sequences. © 1996 American Chemical Society

4598 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996

Figure 1. Residue curve diagram of a homogeneous ternary mixture belonging to the 201 class like ethanol (L)-ethyl propanoate (I)-toluene (H).

2. Preliminaries 2.1. Tools. The simple distillation residue curves (Doherty and Perkins, 1978) and distillation lines (Stichlmair et al., 1989) are two tools widely used for the description of distillation column profiles. A mixture’s residue curve (or distillation line) diagram is a drawing of several of the corresponding lines in the composition simplex. The pure components and the azeotropes form the singular points of both residue curve and distillation line diagrams, and they can be nodes (stable or unstable) or saddles. Bekiaris et al. (1996) extensively compared distillation lines and residue curves from an application point of view. A distillation region is defined as a subset of the composition simplex in which all residue curves originate from the same, locally lowest-boiling singular point and end at another, locally highest-boiling one. The curves which separate different distillation regions are called distillation boundaries. In this article, the term boundary is used for both residue curve boundaries (interior boundaries) and the edges of the composition simplex. Figure 1 illustrates a ternary mixture with two minimum boiling binary azeotropes (L-I and L-H). An interior boundary is connecting them and separates the composition space into two distillation regions. This class of mixtures, the 201 class (Matsuyama and Nishimura, 1977), will be used as an example later. At infinite reflux, the differential equations which describe packed columns become identical to the residue curve equations (Laroche et al., 1992a). Thus, the liquid composition profiles of packed columns operated at infinite reflux coincide exactly with the residue curves and not with the distillation lines which describe the profiles of tray columns at infinite reflux. Therefore, the residue curves are only an approximation to the profiles of tray columns at infinite reflux (Widagdo and Seider, 1996). There is one additional requirement in the special case of columns of infinite length: The column profile must contain at least one pinch point (singular point). More details on these issues can be found in Bekiaris et al. (1996). We refer to the case of columns with infinite length (or with an infinite number of trays) operated at infinite reflux as the ∞/∞ case. The following convention is used to refer to a given mixture: L corresponds to the component with the lowest boiling point, I to the intermediate, and H to the heaviest one. xPL, xPI , and xPH are used to denote the corresponding compositions in a stream P. The locations of the feed, distillate, and bottoms of a column in the composition triangle are referred to by F, D, and B,

Figure 2. Residue curve diagram of a homogeneous ternary mixture belonging to the 001 class like acetone (L)-benzene (I)heptane (H). Mass balance line of a feasible column design at infinite reflux.

respectively. Indices (D1, D2) are used to distinguish different columns of a sequence, the “)” sign to express equality of the molar flow rates and the “≡” for identical flow rates and compositions. Figure 2 illustrates the residue curve diagram of a ternary homogeneous mixture belonging to the 001 class (Matsuyama and Nishimura, 1977). In this article, this class and one of its representatives, the mixture acetone (L)-benzene (I)-heptane (H), will be used because it is the simplest class exhibiting multiple steady states (Bekiaris et al., 1993). The ABH mixture forms one azeotrope between the light and the heavy component (93% acetone and 7% heptane); all residue curves originate at the azeotrope (globally lowest-boiling point) and end at the heptane corner. Therefore, there exists one single distillation region and benzene can be seen as the entrainer that enables the separation of the acetone-heptane azeotrope. In all simulations in this article, mole fractions, a total condenser, a partial reboiler, and a tray efficiency of 1 were applied. The Appendix contains more information on the thermodynamic model used in the examples. 2.2. ∞/∞ Analysis for Single Columns. Bekiaris et al. (1993) developed the ∞/∞ analysis for single distillation columns and ternary homogeneous mixtures. In the following, we briefly review the main principles and assume that the reader is familiar with the issues mentioned here: ∞/∞ Bifurcation Analysis. Given any homogeneous mixture, its residue curve diagram, a distillation column of infinite length operated at infinite reflux and its feed composition, a bifurcation diagram can be constructed by the following steps (∞/∞ analysis): 1. Determine all feasible types of column profiles in the ∞/∞ case. These profiles must coincide with the residue curves and must contain at least one pinch point. Thus, all feasible column profiles contain either an unstable (type I) or a stable node (type II) or they run along the boundaries (interior boundaries and edges of the composition simplex) and contain at least one saddle point (type III) as shown in Figure 3 for a given feed F. 2. To locate the feasible product regions, some information about the equipment at the top and the bottom of the column is needed, i.e., the type of the condenser and the reboiler and the phases of the products. In the following, a total condenser, a partial reboiler, and all products taken out as liquids are assumed. Bekiaris et al. (1996) treated other possible types of equipment. The distillate and bottoms paths

Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4599

Figure 3. Feasible column profile types in the ∞/∞ case for a homogeneous mixture belonging to the 001 class. Figure 5. Bifurcation diagram of an homogeneous mixture belonging to the 001 class. a, b, and c correspond to the three steady state profiles of Figure 4.

Figure 4. Product paths in the ∞/∞ case of a homogeneous mixture belonging to the 001 class. The three profiles indicated by their mass balance lines a, b, and c correspond to the same product flow rate (multiple steady states).

for a 001 class mixture with given feed location F are shown in Figure 4. 3. Note that in the ∞/∞ case, given the feed composition xF and flow rate F, the only unspecified parameter of a column is one of the product flow rates, e.g., the distillate flow rate D. Using the lever rule, the product flow rates can be calculated for every pair of products on the path. Thus, a bifurcation diagram with one product flow rate as the bifurcation parameter can be constructed by performing a continuation of solutions along the feasible product path. Output multiplicities exist when the product flow rate varies non-monotonically along the continuation path (Figure 5). Necessary and Sufficient Geometrical Condition. Given any homogeneous mixture and its residue curve diagram, the feed composition region which leads to multiple steady states in the ∞/∞ case can be located using a necessary and sufficient condition based on graphical arguments. The reader is referred to Bekiaris et al. (1993) for further details and for studying the condition to determine the appropriate feed region. Predictions for the ∞/∞ Case. The following predictions contain the information that can be obtained from the ∞/∞ analysis: Existence of multiplicities: For any homogeneous mixture and column design, it can be predicted whether multiple steady states exist for some feed composition and product flow rate using the geometrical, necessary and sufficient, multiplicity condition. Feasible product regions and bifurcation diagram construction: For any homogeneous mixture, column design, and feed composition, the feasible product

regions can be located in the composition simplex and a bifurcation diagram can be constructed with one product flow rate as the bifurcation parameter. Limit points and product flow rate multiplicity range: From the bifurcation diagram of a system showing multiple steady states the location of the limit (turning) points and therefore the product flow rate multiplicity range is known. Multiplicity feed region: For any homogeneous mixture and column design, the feed composition region that leads to multiplicities can be located using the appropriate feed region condition. Predictions for “Finite” Columns. Bekiaris et al. (1993) demonstrated that the ∞/∞ multiplicity predictions carry over to the finite case and, moreover, that multiple steady states may exist at realistic operation conditions, i.e., for small reflux flows and a small number of trays. Note that the condition for the existence of multiplicities is only sufficient in the case of finite columns. Thus, there may be other causes for multiplicities in such columns not detected by the ∞/∞ analysis, e.g., those Jacobsen and Skogestad (1991). Finally, note that Bekiaris et al. (1996) extended the ∞/∞ analysis to ternary heterogeneous mixtures and demonstrated how it can be applied to different column designs. They further presented the fully detailed, accurate, and general geometrical multiplicity condition. 3. Problem Statement Given a homogeneous ternary azeotropic mixture exhibiting multiple steady states, the ∞/∞ analysis can be applied to obtain the predictions for a single column. Assume that there exists a feasible separation sequence that separates an azeotrope of that mixture into its pure components using more than one distillation column. We then can ask the following question: Does the use of a second column and the recycling of some product flows of that column affect the multiple steady states of the system? Or, more specifically, 1. Is the behavior of the system with recycles qualitatively different from the single-column case; i.e., do some of the multiple steady states disappear or new states appear? 2. Are there quantitative changes when closing the recycle flows: (a) Do the product regions change? (b) Does the product flow rate multiplicity range change? Before answering these questions, multiple steady states in separation sequences are defined:


Definition 2 (Multiple Steady States in Separation Sequences): By multiple steady states in a separation sequence of N columns we refer to the general notion of multiplicities, i.e., that a system with as many parameters specified as there are degrees of freedom exhibits different solutions at steady state. The definitions of output and input multiplicities can be adopted from definition 1. Again, we are interested in output multiplicities only. First, the feasibility of column sequences to separate an azeotrope into its pure components is investigated. Then, the ∞/∞ analysis of Bekiaris et al. (1993) is extended to the multiple column case. Finally, this method is illustrated using two examples: The first one is the separation of an azeotrope by an intermediate entrainer scheme using the 001 class mixture acetone (L)-benzene (I)-heptane (H). The second one is the separation of a saddle azeotrope by a boundary separation scheme using the 201 class mixture ethanol (L)ethyl propanoate (I)-toluene (H). (Ethyl propanoate means propanoic acid ethyl ester.) In the following, the terms feed and product are used in different ways: By products of a distillation column we refer to the distillate or the bottoms of a single column. However, external products are products of the sequence leaving the whole flowsheet (outputs) and thus column products as well. By feed we refer to any stream entering a distillation column, in contrast to the input stream of the whole flowsheet, which is called external feed and denoted by Ex. 4. Homogeneous Separation Sequences 4.1. Separability in ∞/∞ Columns. In recent years, many authors (Doherty and Caldarola, 1985; Stichlmair et al., 1989; Foucher et al., 1991) studied homogeneous azeotropic distillation and developed criteria for entrainer selection to separate minimum boiling binary azeotropes. In order to review these problems a definition of separability is repeated: Definition 3 (Feasibility of a Separation): An entrainer E makes the separation of an azeotrope A-B trough homogeneous azeotropic distillation feasible if there exists at least one separation sequence (with an arbitrary number of columns and recycles) which yields both A and B as pure products. Laroche et al. (1992b) showed that the condition of Doherty and Caldarola (1985) works well in the classical case of a heavy entrainer, but excludes many feasible entrainers and therefore is a sufficient condition. Stichlmair et al. (1989) developed conditions which proved to be neither necessary nor sufficient even though they work in most cases. This work was continued by Laroche et al. (1992a), who investigated the separability of a binary azeotrope at infinite reflux. They developed necessary and sufficient conditions for the separation of a binary azeotrope using one, two, or three distillation columns: 1. The separation in a simple column uses an entrainer whose boiling point is between the boiling points of the azeotropic components and which introduces no new azeotropes into the system. This case is referred to as the intermediate entrainer (IE) case. 2. Laroche et al. (1992a) showed that the separation using an intermediate entrainer is easier using two columns and that many binary minimum boiling saddle azeotropes can be separated in two columns, recycling one lighter boiling component or the azeotrope as

Figure 6. Three possible separation sequences (splits) for the intermediate entrainer case.

entrainer. This last case was referred to as the light entrainer case. 3. Finally they proved that all binary minimum boiling saddle azeotropes can be separated using three columns and two recycles and stated the nonseparability of a node azeotrope in such a sequence. One major point Laroche et al. (1992a) proved was the infeasibility of the classical heavy entrainer separation scheme at infinite reflux, i.e., the separation of a binary azeotrope using a heavier entrainer which introduces no new azeotropes into the system (Doherty and Caldarola, 1985). Since the multiplicity analysis in this article is based on the predictions of the ∞/∞ case, we cannot treat multiple steady states in the heavy entrainer scheme. Laroche et al. (1991) also developed criteria to select an entrainer and to compare direct and indirect splits using equivolatility diagrams for the three cases (heavy, intermediate, and light entrainer). 4.2. Intermediate Entrainer Separations. First the definition of an intermediate entrainer (IE) separation is given: Definition 4 (Intermediate Entrainer Separation): Any minimum or maximum boiling binary azeotrope can be separated in an intermediate entrainer separation sequence if the boiling point of the entrainer (pure component or azeotrope) is between the boiling points of the two components forming the azeotrope, and if the entrainer does not show any singular points on the two boundaries connecting it to the pure components. Laroche and co-workers (1991, 1992a) treated the three possible splits of the IE separation schemes: the separation in one column, the direct split, and the indirect split. Figure 6 illustrates the corresponding column setups and profiles for the separation of a 001 class mixture. Moreover, the same split schemes can be applied in an analogous way to separate a maximum boiling azeotrope, e.g., in a 003 class mixture. Bekiaris et al. (1993) showed that, in the 001 class, multiple steady states exist in a single column for any ternary feed composition. Thus, if the entrainer recycle is not closed, multiple steady states are expected for the first column of the direct or the indirect split sequence (the other columns separate binary mixtures and cannot exhibit MSS in the ∞/∞ case). As mentioned above, the


Figure 7. Simulation setup to separate the acetone (L)-heptane (H) azeotrope using benzene (I) as entrainer in an intermediate entrainer separation sequence.

Figure 9. Residue curve diagram of the mixture ethanol (L)ethyl propanoate (I)-toluene (H) belonging to the 201 class.

Figure 8. Residue curve diagram of the mixture acetone (L)benzene (I)-heptane (H) belonging to the 001 class.

effect of closing the recycle flows on these multiplicities must be examined. In the following, we have chosen the direct split sequence although our argumentation can be applied to the indirect sequence as well. The design aspects of an IE direct split column sequence were treated by Laroche et al. (1991). The separation of the acetone (L)-heptane (H) minimum boiling azeotrope using benzene (I) as entrainer was chosen. The corresponding design setup is shown in Figure 7, the residue curve diagram is shown in Figure 8, and the thermodynamic data are listed in the Appendix. 4.3. Boundary Scheme Separations. Principles: The term boundary separation scheme (BS) is introduced to refer to the case Laroche et al. (1992a) have called light entrainer. They stated that any minimum boiling saddle azeotrope can be separated using a maximum of three distillation columns if it is connected to a curved boundary. However, a close look at example sequences shows that the third component does not need to be lighter than the two components forming the azeotropes. The correct condition is that the entrainer recycled must be lighter boiling than the saddle azeotrope to separate and that this entrainer can be any singular point (azeotrope or pure component) of the mixture. Our motivating example is the separation of the minimum boiling ethanol (L)-ethyl propanoate (I) azeotrope by adding toluene, which is heavier than the components forming the azeotrope. The residue curve diagram of this mixture (Figure 9) shows that all residue curves start at the E-T azeotrope (unstable node) and end either at the pure ethanol or the pure toluene corner. There is an interior boundary connect-

Figure 10. Feasible design of a boundary scheme separation sequence using three columns.

ing the E-T node azeotrope with the E-EP saddle azeotrope that we want to separate. The only singular point lighter boiling than the E-EP azeotrope is the E-T azeotrope and is therefore recycled as entrainer. The thermodynamic data of the mixture are listed in the Appendix. Feasible Design Using Two or Three Columns. The design of a boundary scheme separation using three distillation columns is illustrated in Figure 10. In the first column component I, which is on the concave side (distillation region 2) of the curved boundary, is isolated. The distillate of this column, D1, is located very close to the boundary. In the second column, the curvature of the boundary is used to have the mass balance line cross the boundary and split D1 into the entrainer, which is recycled, and a binary mixture of L and I. Thus, the profile of column two is located in distillation region 1 (Figure 10). This mixture is then separated in a third binary column into pure L and the L-I azeotrope which is recycled and mixed with the entrainer and the external feed.


Figure 11. Feasible design of a boundary scheme separation sequence using two columns and one side product.

Figure 12. Simulation setup of the boundary scheme separation sequence in the case of the 201 class mixture ethanol (L)-ethyl propanoate (I)-toluene (H).

The fact that the profiles of the second and the third column share a common part (from D3 to B2) offers the possibility to combine these two columns in one by drawing a side product at the original location of D3. The resulting sequence is shown in Figure 11. Since all feasible designs of the three-column scheme share some part of their profiles, this is always possible. Therefore, a two-column sequence can always be designed for this separation scheme by drawing some side product. Figure 12 shows the design setup of our motivating example ethanol (L)-ethyl propanoate (I)toluene (H). Introducing the “Boundary Scheme”. For this sequence the term light entrainer scheme seems to be fine because the recycled entrainer is lighter boiling compared to the saddle azeotrope to be separated. However, not only minimum boiling but also maximum boiling azeotropes can be separated using such a scheme. Imagine that all columns and the residue curve boundary in Figure 11 are reversed (distillates and bottoms exchanged and the column profiles run in opposite directions). The former minimum boiling saddle azeotrope is now maximum boiling and the entrainer is heavier boiling than the azeotrope to be

separated. Because there is no light entrainer, the term “boundary scheme” is introduced: Definition 5 (Boundary Scheme): Any minimum or maximum boiling azeotrope can be separated by means of a boundary scheme separation sequence using two columns if the azeotrope is a saddle in the corresponding residue curve diagram and the boundary running through this saddle is curved. Note that the first column of the sequence must always be on the concave side of the curved boundary to make the separation feasible. Heuristic Optimal Design. In the first column of Figure 11 the distillate can be placed at any point of the curved boundary by design and still the bottoms is obtained at the pure I component corner and the second column is crossing the boundary. Note that the composition of the sum of the feeds into the first column, xF, cannot be fixed because the compositions of the recycles are not yet known. The question is where to place D1 to get an economically optimal design in the ∞/∞ case. Obviously there is not much sense in looking at the influence of the number of trays or the reflux flow rate to the economics of the column sequence in the ∞/∞ case. Assume that all flows are scaled by the sum of the feeds into the first column (F is constant). Then, the external feed flow rate processed by the equipment is to be maximized. Or for given flow rates, the maximum possible amount of feed azeotrope is to be separated into pure components, i.e., the fraction Ex/F is to be maximized. At the design point, pure products L and I are obtained. Thus, the corresponding external product flow rates B1 and B2 (Figure 12) are given by the amounts of these components in the external feed flow, xEx L and xEx I . For the same reason the ratio of the external product flows is specified by the external feed composition, in this case by the composition of the saddle azeotrope (assuming that there is almost no heavy component in the external feed, xEx H ≈ 0):

B1 ) xEx I Ex and B2 )

xEx L Ex

B2 xEx L w ) ) K (constant) (1) B1 xEx I

Using this equation and the overall mass balance Ex ) B1 + B2, the heuristic optimality condition takes the form:

Ex B1 + B2 B1(1 + K) ) ) F F F

(2)

Since K and F are constant, the flow rate of B1 should be maximized. By the lever rule the ratio of the distances D1F to B1F is to be maximized. Looking at Figure 11, we see that the maximum B1 is obtained when D2 and Az are recycled at the compositions of the azeotropes and D1 is placed to exploit the maximum curvature of the boundary. Thus, the compositions of all flows have been specified. Using the lever rule:

D1 B1F Az + Ex D2F and ) ) B1 D F D2 AzF

(3)

1

The scaled flows of all streams in the flowsheet are


obtained from the material balances:

F ) D1 + B1 ) Ex + D2 + Az

(4)

B1 1 ) F 1 + (D1/B1)

(5)

D1 B1 D1 ) F F B1

(6)

B2 Ex B1 ) F F F

(7)

(

)

Az + Ex F - D2 D2 Az + Ex ) 1+ ) w D2 D2 F D2

-1

(8)

Now, the composition and flow rates of all streams are specified for the ∞/∞ case and all three material balances (one for each column and one for the mixing point of the recycles) are fulfilled. Since the distillates and bottoms of the columns are connected by residue curves, the design is feasible. 5. ∞/∞ Analysis 5.1. Column Sequence Design. The basic idea behind the ∞/∞ analysis for column sequences is that we can follow the corresponding analysis of Bekiaris et al. (1993) for single columns and replace the singlecolumn feed by the external feed of the sequence and the single-column products by the external products of the sequence. To observe multiple steady states according to definition 2, the design of the column sequence is fixed. This means that the flow rate and composition of the external feed and the flow rates of all recycle streams are fixed at the values calculated in the design procedure. For illustration the degrees of freedom of a twocolumn sequence with c components are examined. Without any specification or connection a sequence of two columns has 2(c + 2) degrees of freedom (c + 2 for each column: c for the feed composition and flow rate and 2 operating parameters). After specification of the external feed composition (c - 1 degrees), flow rate (1 degree) and the recycle flow rate (1 degree), there are 3 degrees remaining (note that the feed of the second column is a product of the first one and thus another c degrees are fixed). Furthermore, in the ∞/∞ case the reflux of each column is specified (to be infinite). Therefore, only one degree of freedom is left and is assigned to be the bifurcation parameter. In this article, the flow rate of the external product of the first column is used for this purpose. 5.2. Configurations of Column Sequences. It was shown that the boundary scheme and intermediate entrainer separations can be carried out in two distillation columns. Therefore, the formulation of the ∞/∞ analysis is based on the case of two columns where the feed of the second column is either the distillate or the bottoms of the first column (D1 or B1). Additionally, the feed of the first column is a mixture of the external feed Ex, the recycle of the distillate or bottoms of the second column (D2 or B2), and the side product drawn from the second column (Az, if there is one). Thus, one product stream of each column is a product stream of the sequence and the remaining product streams are connected to the other column.

By combination four possible configurations are possible: (D,B) Configuration: This is the configuration shown in Figure 7 and used to separate a minimum or maximum boiling azeotrope using an intermediate entrainer sequence and a direct split. It is one of the two cases of main concern in the subsequent analysis. (B,D) Configuration: The separation of an azeotrope using an IE separation sequence and the indirect split requires a (B,D) configuration. Since the analysis of this case is analogous to the one of the (D,B) case, details are not of concern. (B,B) Configuration: To separate a minimum boiling azeotrope applying a boundary scheme, the column sequence must be configured as in Figure 12. This (B,B) case is treated in detail in the analysis. (D,D) Configuration: The analysis of the (D,D) configuration is analogous to the one of the (B,B) case and we are not concerned about the details. 5.3. ∞/∞ Bifurcation Analysis for Column Sequences. In order to find whether multiple steady states occur in a column sequence (i.e., whether different column profiles correspond to the same set of inputs and parameters), all possible product locations and column composition profiles are found by tracking the external products in the composition triangle. That is, we perform a bifurcation study (continuation of solutions) using one external product flow rate as the bifurcation parameter. This product flow rate is referred to by P1 and the other product flow rate by P2 ) Ex - P1 (from the overall material balance). The continuation path is defined as the path generating all possible and feasible column profiles starting from the profiles with P1 ) 0 and ending at the profiles with P1 ) Ex. Multiple steady states exist in a separation sequence if the external product flow rate varies non-monotonically along the continuation path, i.e., when P1 decreases on some segment along the path. Some important results of the thorough analysis of Bekiaris et al. (1993) can be easily adapted for separation sequences: Along the continuation path, P1 increases monotonically as we track all type I and II column profiles, i.e., profiles where one of the two external product compositions is that of a stable or unstable node in the residue curve diagram. Therefore, a decrease in P1 can only occur as type III column profiles are tracked, where the external products are not located at nodes of the mixture (but may be located at saddle singular points). Given a feasible column design for the ∞/∞ case, the external feed flow rate, and the composition, we can formulate the following procedure to perform a bifurcation analysis for any configuration of a two column separation sequence: 1. First, one of the external product flow rates is set to zero, P1 ) 0 (P2 ) Ex), and the corresponding steady state is calculated. The resulting column profiles form the initial solution of the continuation path. 2. Then, the same external product flow rate is set to the external feed flow rate (P1 ) Ex) and the last steady state of the continuation path is calculated. 3. Finally, the external product flow rate P1 is varied, starting at the initial state until the final profiles are reached. Thus, all possible column profiles and therefore all possible external product locations are tracked. It is shown in the following sections that all the steady states along this continuation path correspond to feasible sequence designs.


Figure 14. Initial profiles of the bifurcation analysis at D1 ) 0 for the ∞/∞ case of a 001 class mixture (intermediate entrainer scheme).

Figure 13. Geometrical multiplicity condition: (I) not satisfied (P1 increases along the continuation path); (II) satisfied (P1 decreases along the continuation path).

Recall once more that multiple steady states exist in a separation sequence when the external product flow rate varies non-monotonically along the continuation path, i.e., when P1 decreases on some segment along the path. 5.4. Geometrical, Necessary and Sufficient Multiplicity Condition. Using the results of the analytical procedure above, the geometrical, necessary and sufficient condition for the existence of multiple steady states of Bekiaris et al. (1996) was adapted to the case of column sequences: Let the external product continuation path be given as the path generating all possible column profiles of the sequence. Pick a feasible pair of external product compositions P1 and P2 on this path which will both be located on some distillation region boundary (Figure 13). Now pick P1′ and P2′ sufficiently close to P1 and P2 and such that the column profiles containing P1′ and P2′ correspond to “later” profiles along the continuation path. For the existence of multiple steady states it is required, as we move along the continuation path from (P1,P2) to (P1′,P2′), that the line passing from P1 parallel to P2 P2′ crosses the P1′P2′ line segment (at S). 5.5. External Feed Multiplicity Region. The above procedure leads to the fact that multiple steady states depend on the external feed composition Ex. Thus, there is the need for a procedure to determine the appropriate external feed region which leads to multiple steady states in a separation sequence. Pick an external product P1. Then find the set containing all other product compositions (P2) so that the geometrical condition is satisfied for the picked P1. Name this set SP(P1). Note that SP(P1) is always part of a distillation region boundary and that, in some rare cases, SP(P1) may contain an inflection point and/or it may consist of more than one nonconnected boundary segments. Draw the straight lines connecting P1 with the end points of each boundary segment belonging to SP(P1). For the P1 chosen, the appropriate feed composition is the union of the areas enclosed by each boundary segment that belongs to SP(P1) and the corresponding straight line segments connecting P1 with the end points of the boundary segments of SP(P1). Name this union of areas AP(P1), pick another P1, and repeat. Finally, for any given mixture, the external feed

compositions that lead to output multiplicities lie in the union of all the areas AP(P1), i.e., in the union of all areas enclosed by each boundary segment that belongs to some SP(P1) and the corresponding straight line segments connecting the external product P1 associated to SP(P1) with the end points of the boundary segments of SP(P1). We will provide an illustration using the motivating examples later. 6. Intermediate Entrainer Scheme 6.1. Initial and Final Profiles. Given a feasible column sequence design, the first and the last steady state in the ∞/∞ bifurcation analysis is to be calculated for the intermediate entrainer scheme. The corresponding configuration is shown in Figure 7 with D1 and B2 being the external products (outputs) of the sequence. Remember that the recycle flow rate D2 was fixed by design. If one product flow rate is specified, e.g., D1 ) 0, all flow rates of the sequence can be calculated using the overall mass balance as well as the material balances of the two columns. The difficulty is to find the compositions of all flows of the sequence. To search for the first possible profiles of the columns, D1 is set to zero and B2 is equal to Ex by the overall mass balance. Moreover, B2 ≡ Ex; i.e., the composition of B2 is equivalent to the one of the external feed. Then, using the first column mass balance, B1 ≡ F. This situation is sketched in Figure 14 for a 001 class mixture. To locate the distillate compositions D1 and D2, we have to remember that in the ∞/∞ case a column profile must follow the residue curves and must contain at least one pinch point (singular point in the composition diagram). In the second column, the bottoms B2 has the same composition as the external feed Ex and the distillate has to lie somewhere on the residue curve running through B2. More specifically, D2 has to be on the lighter boiling segment of that curve, i.e., between the L-H azeotrope and B2. Since the azeotrope is the only candidate pinch point on this segment, D2 has this azeotrope’s composition. Since D2 is recycled, the feed of the first column, F, is a mixture of D2 and the external feed Ex. For the same reason as in the second column, the distillate of the first column also has azeotropic composition. Finally, all streams of the sequence have been located and a unique solution has been found. Very similar arguments can be used to locate all streams for the last possible profiles at D1 ) Ex which are illustrated in Figure 15. Here, D1 ≡ Ex and B1 must be at a singular point on the heavier boiling segment of the residue curve through Ex, i.e., B1 ≡ H. In the second column, B1 is split into D2 and B2; this implies B2 ≡ H.


Figure 15. Final profiles of the bifurcation analysis at D1 ) Ex for the ∞/∞ case of a 001 class mixture (intermediate entrainer scheme). Figure 17. ∞/∞ predictions of the D1 compositions along the bifurcation path for a 001 class mixture (intermediate entrainer scheme).

Figure 16. ∞/∞ predictions of the column product paths for a 001 class mixture (intermediate entrainer scheme).

Since D2 ) B1 and B2 ≡ H, D2 also has the composition of the heavy node H. 6.2. ∞/∞ Theoretical Predictions. Now, the ∞/∞ analysis is applied to the illustrative example for the intermediate entrainer separation scheme, where benzene (I) acts as the entrainer to separate the minimum boiling acetone (L)-heptane (H) azeotrope. We start with the initial profiles at D1 ) 0 (Figure 14 and I of Figure 16) where both distillates are located at the L-H azeotrope, B2 is located at the external feed Ex, and B1 is located somewhere between (given by design). As D1 is increased, all type I profiles are tracked; i.e., the distillate D1 remains at the azeotrope and the bottoms B2 starts to move on the straight line D1Ex away from Ex. This behavior is easy to understand when compared to the single-column case where the same is happening to the column products (Bekiaris et al., 1993). The only difference is that now the external products of the sequence track the type I profiles. Since the second column profile has to contain a pinch point, the distillate D2 has to remain at the azeotrope and therefore, the input into the first column remains constant (compare Figure 7). In this column, the compositions of D1 and F do not change, but the flow rate of D1 is increasing. Thus, B1 has to move away

from F toward the external feed composition Ex (on the straight line D1Ex). These moves are denoted by “a” and the corresponding product paths are shown in I of Figure 16. The compositions of acetone and heptane in D1 are plotted versus the distillate flow D1 in Figure 17 (in a bifurcation diagram). As B2 approaches the I-H boundary, the first limit point is reached. Its distillate flow rate, Lp1, can easily be calculated from the lever rule applied to the overall mass balance. Now, the profile of the second column runs through the pure L and I component corners and there is no need that the distillate D2 remains at the azeotrope to have the profile contain a singular point (see II). With this value of the distillate flow rate remaining almost constant, D1 ≈ Lp1, the second column distillate is switching to the I component corner which is the new singular point in that column profile. Think of this event in the sense that the column is “shortening” its profile to make the separation easier. The feed of the first column is now a mixture of Ex and pure I and the distillate D1 is located at the azeotrope. Thus, the bottoms B1 is given by the material balances and lies close to the I corner. These changes of the product compositions were indicated by “b” in the figures. For a further change of the distillate flow rate, D1 cannot stay at the azeotrope anymore since we have tracked all possible type I profiles. The only possibility to move is toward the pure L component (because a residue curve has to connect D1 and B1) and, by the overall material balance, B2 is moving toward the heavy component (“c” in III of Figure 16). The necessity of containing a pinch point in the column profile demand D2 to remain at the intermediate component corner. Therefore, F, the feed of the first column, is not moving either and the first column material balance makes B1 move away from the intermediate corner accordingly. At the end of this move “c” the design point and the second limit point have been reached. Its distillate flow rate Lp2 can be predicted again applying the overall mass balance (Figure 17). At this steady state all the entrainer makeup is leaving the system in the bottoms B2. Applying the lever rule, it can easily be checked that the distillate flow rate D1 is decreasing along this segment of the continuation path, which will lead to multiple steady states. Next, the distillate flow rate D1 increases again as D1 moves toward I and B2 is approaching H (move “d”

4606 Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 Table 1. Comparison of Some Specific D1 Flow Rates of an Intermediate Entrainer Sequence Separating Acetone (L)-Benzene (I)-Heptane (H) D1

predicted

simulation

Lp1 Lp2 Le

96.10 90.00 90.90

96.10 90.55 91.13

in Figures 16 and 17). For the same reason as above, D2 remains at I and B1 moves toward H. At the end of that move, all the entrainer makeup is leaving the sequence in D1 and thus, D1 cannot move further toward the I corner. Again, the corresponding flow rate Le can be determined. Additionally, B2, the other external product, has reached the heavy node (type II profile) and therefore, all type III profiles have been tracked. D2 is not forced to stay at I any longer and moves quickly to H (the second column is filled up with heavy component and is not doing a separation anymore). By the first column material balance, B1 is moving to H too (due to the recycle composition changes). All the movement “e” happens at an approximately constant distillate flow rate D1 ≈ Le. Finally, D1 is approaching the external feed Ex and all products end at the compositions shown in Figure 15. At the end of “f” (IV in Figure 16), all possible column profiles have been tracked and the corresponding bifurcation diagram is shown in Figure 17. All the arguments above depend neither on the external feed location Ex, as long as it contains all three components of the mixture, nor on the composition of the azeotrope. Therefore we conclude that these ∞/∞ predictions are applicable to any 001 class system. 6.3. Simulation Results and Comparisons. The simulation setup used to calculate the bifurcation path for the mixture acetone (L)-benzene (I)-heptane (H) is shown in Figure 7. The reflux flow rates were fixed at a high value of 5000 (R/F ≈ 25), and the Auto Wilson model according to the Appendix was used. The corresponding results are shown in Figure 18 together with the results of a single column using the appropriate setup and model. (We used the same thermodynamics and a feed F of the single column that was identical to the first column inlet of the sequence at the design point.) The excellent agreement of theoretical predictions (Figure 17) and simulation results (Figure 18) can be seen not only graphically, but even more by comparing the predicted and simulated values of the distillate flow rates shown in Table 1. Local stability of the solutions was determined by looking at the eigenvalues of the linearized system along the bifurcation path. It turned out that the two branches between D1 ) 0 and Lp1 and between Lp2 and D1 ) F were stable and the solutions between the limit points were unstable, which is the same result as for a single column. Moreover, the results were validated using a full, rigorous model (Aspen Wilson according to the Appendix) and the steady state simulator Aspen Plus V.9. As shown in Figure 18, we were able to reproduce the two stable branches of the bifurcation path. Remarkably, the heat balances had only a minor effect. (The deviation of the azeotropic composition is due to a slight difference in the thermodynamics.) Note that the heat balances are responsible for a different type of multiplicity reported by Jacobsen and Skogestad (1991). In this 001 class mixture, the existence of multiple steady states does neither depend on the external feed location Ex nor on the location of the azeotropesthis is

Figure 18. Simulation results of the 001 class mixture acetone (L)-benzene (I)-heptane (H) for a single column and the column sequence using different models.

Figure 19. External product paths of the intermediate entrainer separation scheme (001 class mixture).

the same result as in the single-column case of Bekiaris et al. (1993) and can easily be proved by applying the condition for the appropriate external feed multiplicity region. Thus, all mixtures of this class will exhibit multiplicities in single columns and sequences. Furthermore, the external product paths shown in Figure 19 are identical to the product paths of a single column operated with a feed Ex (external feed of the sequence, compare Figure 4). Note that they are not identical with the paths of the single column in Figure 18 since that column was fed with a composition F, the feed of the first column of the sequence at the design point, which is different from Ex. Finally, the questions in our problem statement can be answered: In an intermediate scheme separation sequence there are no qualitative changes compared to the single-column case. The distillate flow rate multiplicity range (Figure 18) also did not change by much, except the changes


Figure 20. Initial profiles of the bifurcation analysis at B1 ) 0 for the ∞/∞ case of a 201 class mixture (boundary separation scheme).

Figure 21. Final profiles of the bifurcation analysis at B1 ) Ex for the ∞/∞ case of a 201 class mixture (boundary separation scheme).

expected when simulating with a finite number of trays. The product paths of the external products are different from those of the single column, but they are predictable since they correspond to the product paths of a single column with an appropriate feed composition. 7. Boundary Scheme Separation 7.1. Initial and Final Profiles. As in the first example, the first and the last steady states in the ∞/∞ bifurcation analysis of the boundary scheme are calculated for a feasible column sequence design. The corresponding configuration is shown in Figure 12 with B1 and B2 being the external products (outputs) of the sequence. Remember that the flow rates of the recycles D2 and Az were fixed by design. If one product flow rate is specified, e.g., B1 ) 0, all flow rates in the sequence can be calculated applying the overall mass balance as well as the material balances of the two columns. Again, the difficulty is to find the compositions of all flows of the sequence. As B1 is set to zero to get the initial profiles, the overall mass balance implies that B2 ≡ Ex, and by the first column material balance D1 ≡ F. This situation is sketched in Figure 20 for a 201 class mixture. Since there must be at least one singular point in an ∞/∞ profile, the distillate of the second column, D2, must have the composition of the L-H azeotrope. Thus, the feed of the second column, D1 (and F ≡ D1) lies on the straight line between D2 and B2. Again, the ∞/∞ conditions force the bottoms of the first column, B1, to be at the H corner. The side product composition has to lie on the residue curve segment between D2 and B2 and is given by the second column material balance, i.e., by the design chosen. In the design of Figure 12, Az is located very close to the external feed Ex. Now, it is briefly described how to find the final profiles of the continuation path where B1 ≡ Ex and B2 ) 0 as shown in Figure 21. Again, D1 has the composition of the L-H azeotrope and is fed into the second column. Since there is no lighter point on any residue curve through D1 than the L-H azeotrope itself, D2 ≡ D1. This implies B2 to have pure L composition and, thus, Az to lie somewhere on the distillation region boundary between the two azeotropes (according to the design). 7.2. ∞/∞ Theoretical Predictions. The boundary separation scheme of the 201 class mixture ethanol (L)ethyl propanoate (I)-toluene (H) is studied. For illustration purposes the azeotropes were repositioned

Figure 22. ∞/∞ predictions of the column product paths for a 201 class mixture (boundary scheme).

and the curvature was exaggerated compared to the original system used for the simulations (Figure 9). We start with the initial profiles (B1 ) 0, Figure 20, and I in Figure 22) where B1 is located at the pure heavy component, B2 at the external feed Ex, D2 is located at the node azeotrope (L-H), and D1 is a mixture of the external feed and the two recycles given by design. As B1 is increased, all type I profiles are tracked; i.e., B2 moves on the straight line B1Ex toward the L-I boundary and B1 itself, whose composition along the bifurcation path is shown in Figure 23, remains at H. The ∞/∞ conditions force D2 to stay at the node azeotrope, and by the change in the compositions of the recycles D1 moves to the saddle azeotrope. To track all type III profiles, B1 must move away from the H corner toward the I corner (the other possibility is not feasible since no residue curve is connecting the ends of the first column). By the external mass balance B2 moves toward L and the B1 flow rate is still increasing (by the lever rule). Since both column profiles now contain the saddle azeotrope as a pinch point, D2 must not stay at the unstable node and is moving along the curved boundary; D1 takes a move in the opposite direction. As the B1 flow rate is further increased (in III), B1 is moving on toward I and B2 approaches the L component corner. Both distillate


Figure 23. ∞/∞ predictions of B1 compositions along the bifurcation path for a 201 class mixture (boundary scheme).

compositions D1 and D2 move back along the boundary to the node azeotrope. Now B2 (external product) has reached the L component corner and the type II profiles are to be tracked (IV). B1 moves toward the external feed Ex and D1 is staying at the node azeotrope so that the first column profile still contains that pinch point. Finally, we have reached B1 ) Ex and therefore tracked the complete continuation path. Since the bottoms flow rate B1 never decreased along this path (Figure 23), multiple steady states do not exist in this specific column sequence. By construction, the column sequence is feasible for any given value of the bifurcation parameter. All the above arguments are independent of the feed location Ex as long as this point is contained in the distillate region 2 of Figure 11. 7.3. Simulation Results and Comparisons. In the simulations the external feed Ex is located very close to the L-I saddle azeotrope, and thus, the movements I and II of Figure 22 happen in a very small interval of the bifurcation parameter B1. This heavily affects the bifurcation plot of the simulation results of our 201 class example mixture ethanol (L)-ethyl propanoate (I)toluene (H). The corresponding simulation setup is shown in Figure 12, the reflux flow rates were fixed at 5000 (R/F ≈ 50), and the Auto Wilson model according to the Appendix was used. The resulting continuation of solutions is shown in Figure 24 for the B1 compositions, and all solutions are known to be locally stable (by looking at the eigenvalues of the linearized model along the path). Again, a validation using a rigorous Aspen Model was done and the corresponding results were plotted on the same figure. The good agreement of prediction and simulation can be seen comparing Figures 23 and 24. In Figure 25, the simulation results of a single column (using a feed composition equal to that of the first column of the sequence at the design point) and the sequence were compared. Although the single column shows multiplicities, the sequence does notsa qualitative change in the occurrence of multiple steady states was found for this specific case of a boundary scheme separation. Obviously, the product paths did also change significantly. Moreover, these facts depend neither on the location of the azeotropes nor on the curvature of the interior boundary of this 201 class mixture, because the external product paths are inde-

Figure 24. Simulation results of the 201 class mixture ethanol (L)-ethyl propanoate (I)-toluene (H) for the column sequence using different models.

Figure 25. Simulation results of the 201 class mixture ethanol (L)-ethyl propanoate (I)-toluene (H) for a single column and the column sequence using the Auto Wilson model.

pendent of these properties in the ∞/∞ case. This can be seen in part I of Figure 26. However, it cannot be concluded that there are no boundary separation schemes showing multiple steady states, and we will treat this issue in the next section. Note that one can still locate the design point at the intersection of the two bifurcation paths at B1 ≈ 0.63 (close to the lower limit point on the lower stable branch of the single-column path in Figure 25). If we look at the input-output behavior of the two-column sequence, i.e., we consider the external feed to be separated (very close to but not at the L-I azeotrope) and the external product paths depicted in the upper part of Figure 26,

Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4609 Table 2. Overview of Multiple Steady States in Single Columns and Column Sequences According to Topological Classification

Figure 26. Examples of a 201 class (I) and a 241-M class (II) mixture with the continuation paths of the external products of a boundary scheme separation sequence. The shaded region corresponds to external feed locations that lead to multiple steady states.

the whole system behaves like a single column separating an ideal (000 class) mixture. The distillate path of such a replacement column would correspond to B2 and the bottoms path to B1 of the sequence. Thus, this particular boundary scheme sequence behaves similarly as a single column separating a mixture without the azeotropes and the boundary. 8. Separability and Multiple Steady States In this section an overview on multiple steady states in the 113 topological classes of ternary homogeneous mixtures according to Matsuyama and Nishimura (1977) is provided. Furthermore, the geometrical multiplicity condition and the procedure to determine the external feed region which leads to output multiplicities in separation sequences are applied. Note that there are few classes which show indeterminacy (Foucher et al., 1991) and that Baburina et al. (1988) questioned the existence of some classes. We rely on the topological analysis of Doherty and Perkins (1979) and use all 113 topological consistent classes for the overview. Looking at multiple steady states in single columns, the statistics of Bekiaris et al. (1993) is extended in Table 2: Six of the 113 classes do not show output multiplicities for any feed composition (classes without MSS). In 30 classes, a very highly curved boundary can induce MSS, but this does not seem to be a very common phenomenon (classes with pseudo MSS). Forty of the remaining classes always show multiple steady states, i.e., for any feed composition, azeotrope location and boundary curvature (classes always showing MSS). For the remaining 37 classes, MSS depend critically on the feed and azeotropes positions or on the boundary curvature (classes where MSS depend). Note that no direct conclusion on the frequency of occurrence of MSS

MSS in Single Columns classes without MSS classes with pseudo MSS classes where MSS depend classes always showing MSS

6 30 37 40

MSS in Column Sequences BS sequences without MSS BS sequences where MSS disappear BS sequences with MSS IE sequences (always MSS)

26 55 49 24

can be made because some classes may contain many mixtures and some only a few; see Bekiaris et al. (1993) on this issue. Compare the two classes of systems in Figure 26: System I is the 201 class example that was used in the analysis and where multiplicities do not exist in the column sequence. Looking at the product paths of the external products B1 and B2, the appropriate external feed region procedure is applied. For any B2 picked, the geometrical condition is not fulfilled for all points of the B1 path and therefore, SP(B2) and the union of all AP(B2) are empty. Thus, no multiplicities exist in this class for any feasible external feed point. As a second system, the 241-M class is concerned where one additional maximum boiling binary azeotrope between the I and H component and a globally maximum boiling ternary azeotrope exist (Figure 26). (We know that this may be a rare class; it is used here as an illustration.) Basically, the B1 external product path now contains one additional segment along the boundary between the I-H and the ternary azeotrope. Applying the geometrical multiplicity condition for any point on the B2 product path, it can be seen that all points between I-H and the ternary azeotrope fulfill the condition and belong to SP(B2). The union of all the resulting areas AP(B2) is shaded in the figure and corresponds to the external feed locations Ex that lead to multiple steady states in such a system. Therefore in a BS scheme, the occurrence of MSS critically depends on the shape of the boundaries which are part of the external product paths. Still, the occurrence of MSS is predictable using the geometrical condition described earlier. Finally, separability and multiplicities are considered in Table 2. In the 113 classes there are 26 BS separation sequences where no MSS exist for the single column and the sequence. In 55 BS sequences, MSS disappear when closing the recycle flows, and in 49, they still exist in the resulting sequences. (Note that all these 49 sequences belong to classes doubted by Baburina et al. (1988).) In all 24 possible IE sequences MSS occur in single columns and column sequences. Note that the existence of multiple steady states in a separation sequence implies multiple steady states in single columns for some feed location independent of the boundary geometries. Since MSS depend on the product paths and these paths do not differ for single columns and column sequences in the IE case (see section 6.3), this is certainly true for all IE sequences. Recall that the product paths are always part of the boundaries. A BS column sequence can never have a larger path than a single column with an appropriate feed location. Not all product locations of the single column correspond to a feasible sequence, and thus the sequence will normally


have a shorter path. For example, compare the single column path in II of Figure 26, starting at L-H over L-I, I, I-H to Max, with the sequence path of B1 between L-I and Max. Because the product paths are reduced by the separability requirement, MSS can only vanish but no new multiplicities can appear when we go from a single column to a column sequence. Thus, MSS in a BS sequence imply MSS in a single column in the ∞/∞ case. 9. Conclusions In this article, the existence of multiple steady states in separation sequences was studied in detail for ternary homogeneous azeotropic mixtures. First, heuristic criteria for an optimal design of a boundary scheme separation sequence involving columns with infinite length (infinite number of trays) operated at infinite reflux (∞/∞ case) were sketched. Then, a general and systematic procedure to predict multiple steady states in sequences of interlinked columns for any given residue curve diagram and feed composition was derived. The necessary and sufficient geometrical condition for the existence of multiple steady states was extended and applied to separation sequences. The condition an external feed must satisfy to lead to multiple steady states in a column sequence was derived. Furthermore, these conditions were applied to intermediate entrainer and boundary scheme separations: Intermediate Entrainer Scheme. It is shown that as in the single-column case multiple steady states exist in a separation sequence for all topological classes with a feasible sequence. The quantitative changes to expect, i.e., the differences in the product paths and thus in the product flow rate multiplicity range, are minor. Boundary Scheme. In separations using this scheme the existence of multiple steady states in a sequence depends on the geometry of the product paths and therefore on the geometry of the boundaries of the mixture. Using the external feed multiplicity condition, it is possible to determine the region of external feeds leading to multiple steady states. For the BS, multiplicities can disappear when closing the recycle flows of a sequence. However, no new multiplicities can occur for both schemes studied. Finally, we have shown by simulations that the predictions from the ∞/∞ case carry over to the case of finite columns. This is analogous to the single-column case where Bekiaris et al. (1993) argued that multiple steady states in ∞/∞ columns imply multiple steady states in finite columns of sufficient length operated at a sufficiently high reflux. Acknowledgment We thank Prof. Michael Doherty and Jeffrey Knapp (University of Massachusetts, Amherst) for providing us thermodynamic data and subroutines. We also thank Prof. Michael Michelsen (Danish Technical University) for his enlightened comments about selecting the appropriate thermodynamic model. Appendix. Thermodynamic Models and Mixture Data This appendix contains information on the thermodynamic model and on the simulation models used in this article. Vapor-liquid equilibrium calculations are

Table 3. Antoine Coefficients of the Components Used in This Article (Sources Referenced in the Appendix) component

A

B

C

acetone benzene n-heptane ethanol ethyl propanoate toluene

21.3099 20.7936 20.7664 23.5807 20.6470 20.9064

-2801.53 -2788.51 -2911.32 -3673.81 -2667.60 -3096.52

-42.875 -52.360 -56.514 -46.681 -79.527 -53.668

Table 4. Binary Wilson Parameters of the Components Used in This Article (Sources Referenced in the Appendix) acetone benzene n-heptane

acetone

benzene

n-heptane

1.0000 0.5528 0.2740

1.0985 1.0000 0.5290

0.5100 1.1750 1.0000

ethanol ethyl propanoate toluene

ethanol

ethyl propanoate

toluene

1.0000 0.4087 0.4280

0.8571 1.0000 0.3264

0.1830 1.9373 1.0000

based on the following equation:

yiP ) xiPsat i (T) γi(T,x)

(9)

where P is atmospheric pressure. x and y are mole fractions of the liquid and the vapor phase, Psat is the i vapor pressure, and γi are the liquid activity coefficients. Vapor pressures were computed by the Antoine equation:

ln Psat i ) Ai +

Bi T + Ci

(10)

where T is in K and Psat in Pa. Table 3 contains the i Antoine coefficients from Thermopack of the components used. Thermopack are thermodynamic subroutines provided by Prof. M. Doherty and J. Knapp (University of Massachusetts, Amherst). The liquid activity coefficients were computed using the Wilson model; the corresponding coefficients are listed in Table 4. The coefficients for the binaries containing heptane and for ethanol-toluene are taken from Thermopack. The acetone-benzene data and the binary data for the mixtures containing ethyl propanoate are taken from the Aspen Plus V.9 library (Aspen, 1995). In this article, two different types of simulation models are used: Auto Wilson. Simulation of the distillation columns was performed using the material balances without computing heat balances. Auto from Doedel and Wang (1994) was used to perform the bifurcation computations using this model. Aspen Wilson. Rigorous distillation simulations were performed using models including material and heat balances. “Radfrac” from Aspen Plus was used for this purpose (Aspen, 1995). Literature Cited Aspen. Aspen Plus Release 9 Reference Manual: Physical Property Methods and Models; Aspen Technology Inc.: Ten Canal Park, Cambridge, MA, 1995. Baburina, L. V.; Platonov, V. M.; Slin’ko, M. G. Classification of Vapor-Liquid Phase Diagrams for Homoazeotropic Systems. Theor. Found. Chem. Eng. (UDSSR) 1993, 22, 390. Bekiaris, N.; Morari, M. Multiple Steady States in Distillations∞/∞ Predictions, Extensions and Implications for Design, Synthesis and Simulation. Ind. Eng. Chem. Res. 1996, in press.

Ind. Eng. Chem. Res., Vol. 35, No. 12, 1996 4611 Bekiaris, N.; Meski, G. A.; Radu, C. M.; Morari, M. Multiple Steady States in Homogeneous Azeotropic Distillation. Ind. Eng. Chem. Res. 1993, 32, 2023. Bekiaris, N.; Meski, G. A.; Morari, M. Multiple Steady States in Heterogeneous Azeotropic Distillation. Ind. Eng. Chem. Res. 1996, 35 (1), 207. Chavez, C. R.; Seader, J. D.; Wayburn, T. L. Multiple Steady-State Solutions for Interlinked Separation Systems. Ind. Eng. Chem. Res. 1986, 25, 566. Doedel, E. J.; Wang, X. AUTO94: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations; Computer Science Department of Concordia University: Montreal, Canada, 1994. Doherty, M. F.; Perkins, J. D. On the Dynamics of Distillation Processes I: The Simple Distillation of Multicomponent NonReacting Homogeneous Liquid Mixtures. Chem. Eng. Sci. 1978, 33, 281. Doherty, M. F.; Perkins, J. D. On the Dynamics of Distillation Processes III: The Topological Structure of Ternary Residue Curve Maps. Chem. Eng. Sci. 1979, 34, 1401. Doherty, M. F.; Caldarola, G. A. Design and Synthesis of Homogeneous Azeotropic Distillations 3: The Sequencing of Columns for Azeotropic and Extractive Distillations. Ind. Eng. Chem. Fundam. 1985, 24 (4), 474. Foucher, E. R.; Doherty, M. F.; Malone, M. F. Automatic Screening of Entrainers for Homogeneous Azeotropic Distillation. Ind. Eng. Chem. Res. 1991, 30, 760. Jacobsen, E. W.; Skogestad, S. Multiple Steady States in Ideal Two-Product Distillation. AIChE J. 1991, 37 (4), 499. Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morari, M. Homogeneous Azeotropic Distillation: Comparing Entrainers. Can. J. Chem. Eng. 1991, 69, 1302. Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morari, M. Homogeneous Azeotropic Distillation: Separability and Flowsheet Synthesis. Ind. Eng. Chem. Res. 1992a, 31 (9), 2190.

Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morari, M. The Curious Behavior of Homogeneous Azeotropic DistillationsImplications for Entrainer Selection. AIChE J. 1992b, 38 (9), 1309. Lin, W. J.; Seader, J. D.; Wayburn, T. L. Computing Multiple Solutions to Systems of Interlinked Separation Columns. AIChE J. 1987, 33 (6), 886. Magnussen, T.; Michelsen, M. L.; Fredenslund, A. Azeotropic Distillation using UNIFAC. In Third International Symposium on Distillation; Institution of Chemical Engineers Symposium Series 56; The Institution of Chemical Engineers: London, 1979; p 1. Matsuyama, H.; Nishimura, H. Topological and Thermodynamic Classification of Ternary Vapor-Liquid Equilibria. J. Chem. Eng. Jpn. 1977, 10 (3), 181. Petlyuk, F. B.; Avetyan, V. S. Investigation of Three Component Distillation at Infinite Reflux (in Russian). Theor. Found. Chem. Eng. 1971, 5 (4), 499. Rosenbrock, H. H. A Lyapunov Function with Applications to Some Nonlinear Physical Systems. Automatica 1962, 1, 31. Stichlmair, J.; Fair, J. R.; Bravo, J. L. Separation of Azeotropic Mixtures via Enhanced Distillation. Chem. Eng. Prog. 1989, 85, 63. Widagdo, S.; Seider, W. D. Azeotropic Distillation. AIChE J. 1996, 42 (1), 96.

Received for review May 28, 1996 Revised manuscript received August 28, 1996 Accepted August 30, 1996X IE9602926

X Abstract published in Advance ACS Abstracts, October 15, 1996.

Multiple Steady States in Homogeneous ... - ACS Publications

Recommend Documents