Azeotropic Distillation in a Middle Vessel Batch Column. 2. Nonlinear

Ind. Eng. Chem. Res. 1999, 38, 1531-1548

1531

Azeotropic Distillation in a Middle Vessel Batch Column. 2. Nonlinear Separation Boundaries Weiyang Cheong and Paul I. Barton* Department of Chemical Engineering and Energy Laboratory, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

On the basis of the analytical tools developed for the middle vessel column (MVC) operated under limiting conditions, analysis of the qualitative dynamics of the MVC in separating an azeotropic mixture is extended to the more realistic case in which the separation boundaries are nonlinear. The differences between batch stripper pot composition boundaries and batch rectifier pot composition boundaries in the presence of curved separatrices results in the MVC still pot composition being able to cross these pot composition boundaries. On the basis of these insights, operating procedures are developed in which ternary azeotropic mixtures of acetone, benzene, and chloroform can be separated into their constituent pure components, a separation not achievable with either the batch stripper or the batch rectifier. The operating procedures suggested for separating the ternary azeotropic mixture of acetone, benzene, and chloroform in the MVC are then shown to be the time analogues of sequences of continuous distillation columns that achieve the same separation. On the basis of this space-time analogy, further analogies are developed between the MVC and a continuous column, and it is postulated that many complex separations currently achieved with sequences of continuous columns can also be achieved with a single MVC. Thus, the MVC represents the ultimate multipurpose solvent recovery technology, as it can handle, in a batch multipurpose mode, separations that will otherwise require a dedicated continuous distillation sequence. Finally, the characteristics of perfect MVC batch entrainers, which allow the complete separation of any azeotrope into its constituent pure components in a single MVC, are discussed. Introduction

(D M+ B) dt

(3)

dξ ) -d(ln M)

(4)

dξ )

In our previous article,1 a theoretical model of the middle vessel batch column (MVC) was developed, on the basis of the simplifying assumptions of (1) constant molar overflow (CMO) and (2) quasi steady state (QSS) in the column due to the negligible holdups assumed on the trays, in the reflux drum and in the reboiler. This model is of sufficient complexity to explore the qualitative dynamics of the MVC, especially in the presence of azeotropes, but still simple enough to remain tractable for analysis and simulation. The equation describing the change in the residual liquid composition in the MVC still pot was shown to be

dxM i D B ) xM i - λxi - (1 - λ)xi dξ

i ) 1...NC (1)

or in vector notation (where x’s are the NC vectors of composition):

dxM ) xM - λxD - (1 - λ)xB dξ

(2)

or alternatively as

such that ξ f ∞ as the still pot is boiled dry, and where λ ∈ [0,1], the middle vessel parameter was defined to be

λ)

D D+B

(5)

For the operation of the MVC under limiting conditions (infinite reflux/reboil ratios, infinite number of trays in each of the stripping and rectifying sections of the MVC), it was found that

xD(ξ) ) R(xM(ξ))

(6)

xB(ξ) ) ω(xM(ξ))

(7)

and

Superscripts indicate location of the composition: M for middle vessel, D for distillate product, B for bottoms product. ξ is the warped time variable (dimensionless time) defined in the interval [0,∞) as

where R(xM(ξ)) and ω(xM(ξ)) denote the R and ω limit sets,2 respectively, of the basic distillation region within which the current still pot composition given by xM(ξ) is located. Thus, eq 2 under limiting conditions can be written as

* Corresponding author. Phone: +1-617-253-6526. Fax: +1617-258-5042. E-mail: [email protected].

dxM ) xM(ξ) - xP(xM(ξ)) dξ

10.1021/ie980470q CCC: $18.00 © 1999 American Chemical Society Published on Web 02/09/1999

(8)

1532 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999

where xP, the net product drawn from the column, is defined by

xP ) λxD - (1 - λ)xB ) λR(xM) - (1 - λ)ω(xM)

(9) (10)

Hence, xP is strictly only a function of xM; thus, the current still pot composition xM determines completely the path of the change in residual liquid composition in the still pot. All of the theoretical analyses provided in the previous paper assumed straight line separatrices for the simple distillation residue curve maps, and hence linear pot composition boundaries for the MVC. However, most, if not all, azeotropic mixtures exhibit some curvature in the separatrices of the corresponding residue curve maps. Given that the net product xP is a convex combination of the distillate and bottoms products (xD and xB) drawn from the MVC, and that the MVC still pot composition motion is in a direction away from this net product, an analysis of azeotropic systems with nonlinear MVC pot composition boundaries (curved separatrices for the ternary mixture) is conducted in this paper. In the presence of nonlinear MVC pot composition boundaries, separations previously unattainable in the presence of linear boundaries can now be achieved. Thus, nonlinearity can actually result in the ability to split an azeotropic mixture in the MVC. In this paper, some interesting possibilities for the separation of azeotropic mixtures that exploit curved separatrices using a middle vessel batch distillation column (MVC) are explored by building on the tools developed in our analysis of the middle vessel batch distillation column.1 The ternary mixture of acetonebenzene-chloroform (A-B-C), which is an example of the inverse-020 system,3 will be studied in detail to illustrate these ideas. First, the implications of curved separatrices for pot composition boundaries in a MVC are explored, on the basis of the analysis developed in our previous paper1 regarding MVC pot composition boundaries. This leads to a simple operating procedure for breaking the AC azeotrope, which would allow us to charge the azeotrope or some composition in the ternary composition simplex into a MVC, mix it batchwise with pure entrainer (in this case benzene) and draw pure products (acetone, chloroform, and benzene) from the column without the need for a recycled azeotropic waste cut. This operating procedure will be validated via simulations using an ABACUSS [(Advanced Batch and Continuous Unsteady-State Simulator) process modeling software, a derivative work of gPROMS software, @1992 by Imperial College of Science, Technology and Medicine.] model of the MVC. The results of these simulations are presented in detail in our next paper.4 Ideas similar to those developed here were first postulated by Laroche et al.5 in the context of a continuous distillation column sequence. This was further formalized by Wahnschafft et al.,6,7 who illustrated the separation scheme suggested by Laroche et al. for an inverse-020 system3 with the acetonebenzene-chloroform system. In fact, the operating procedures suggested here can be interpreted as the time-domain analogues of the space-domain separation sequences suggested by Wahnschafft et al.6,7

Given the above analogies, a short discussion is thus provided regarding the equivalency of a continuous distillation column to that of a middle vessel batch distillation column. The limitations to this similarity, however, are also highlighted. Finally, on the basis of the operating procedure developed for the A-B-C mixture, an analysis regarding the perfect entrainer for a middle vessel batch distillation column is presented, although a comprehensive discussion of feasible entrainers is beyond the scope of this paper. An analysis of feasible entrainers was conducted by Laroche et al.5 for the separation of homogeneous azeotropes in continuous distillation columns. Some of the insights developed in their work can also be extended to the MVC, keeping in mind the minor differences between the middle vessel batch distillation column and a traditional continuous distillation column, thereby affording an analysis of feasible entrainers for the MVC. Pot Composition Boundaries for Strippers and Rectifiers Are Not Equivalent in the Presence of Curved Separatrices The analysis conducted in our previous paper1 was based completely on the assumption of straight line separatrices and planar (or polygonal for higher dimensional systems) pot composition boundaries. However, as explained by Reinders and De Minjer,8 separatrices can only be linear if the third component added to the first two components (which form the azeotrope) does not affect the relative volatility of the first two components. This would require a third component that interacts to exactly the same degree with each of the two components, which is unusual physically. Thus, some curvature can typically be expected of separatrices and pot composition boundaries formed from a bundle of trajectories (as is the case in systems of higher dimension, where the boundaries are formed from bundles of trajectories, some of which are separatrices). This nonlinearity renders some of our previous limiting analysis inaccurate, but it is curvature of the boundaries that allows the derivation of separation schemes that separates azeotropic mixtures completely and allows us to cross a surface that is theoretically a pot composition boundary under the assumption of linear separatrices. This further leads us to elucidate the characteristics of a perfect entrainer for breaking maximum-boiling azeotropes and the corresponding perfect entrainer for breaking minimum-boiling azeotropes. This is in addition to the entrainers proposed by Bernot et al. in their analysis of batch strippers for the separation of azeotropes.9 A summary of the azeotropic compositions for the three-component mixture of acetone, benzene, and chloroform used in this paper, together with the relevant ternary residue curve map, is provided in the Appendix. First, let us consider the topological structure of the (AB-C) residue curve map as given in Figure 1 for the mixture at atmospheric pressure. As calculated by the NRTL model, with parameters and equations provided by Aspen Plus,10 the separatrix in the acetone-benzene-chloroform system is highly curved and almost hugs the benzene-chloroform edge at xa ≈ 0.0, xb > 0.8, and xc < 0.2. As demonstrated in our next paper,4 using ABACUSS simulations of the MVC model presented in our previous paper,1 this extreme curvature of the separatrix allows us to sepa-

Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1533

Figure 1. Residue curves map for the acetone-benzenechloroform system at atmospheric pressure .

rate a ternary mixture of acetone, benzene, and chloroform into pure components (with purities of >99%), thereby overcoming the binary azeotrope acetonechloroform (AC) (xa ≈ 0.35; xc ≈ 0.65). More importantly, in the presence of linear separatrices, the B-AC line segment would have been interpreted as being a pot composition boundary for both the stripper and rectifier and for all values of 0 < λ < 1. Hence, on the basis of our earlier definition of pot composition boundaries in MVCs, it would also have been a pot composition boundary for the MVC, as it is a boundary common to both the stripper and the rectifier which exists for all values of λ as it varies between 0 and 1. It should be noted, however, as is the case here, that the boundary that exists at different values of λ need not be the same entity. For 0 < λ e 1, the B-AC line segment exists as a boundary because it is a stable separatrix. A stable separatrix exists as a pot composition boundary for all values of λ except λ ) 0. However, when λ ) 0 and the “separatrix” pot composition boundary1 ceases to exist, a “mass-balance” pot composition boundary moves into the line segment B-AC at exactly λ ) 0. Thus, even though these pot composition boundaries are not the same entity, because there exists a pot composition boundary at the line segment B-AC for all values of A, B-AC will be a pot composition boundary for a MVC even if it is free to operate at all values of 0 e λ e 1. However, because of the curvature of the separatrix, it turns out that the rectifier boundary and the stripper boundary actually do not coincide. Thus, the line segment B-AC does not form a pot composition boundary for the middle vessel column which operates at varying λ, and we are able to cross this “middle vessel pot composition boundary” by a clever manipulation of the operating profile of the parameter λ. To illustrate that the stripper and rectifier boundaries are not equivalent, consider the products that will be drawn from the batch stripper and batch rectifier, respectively. For the batch stripper, it is the ω limit set of the current still pot composition that is drawn as the product. From Figure 1, the ω limit set of all points interior to the composition simplex is given by the stable node of the ternary system, namely, pure benzene. Thus,

Figure 2. Batch distillation regions/pot composition boundaries for a batch stripper in the acetone-benzene-chloroform system.

any initial still pot composition interior to the composition simplex would draw pure benzene as the product in a batch stripper. This is shown in Figure 2a, where all initial still pot compositions will move directly away from the fixed point of pure benzene as governed by the following equation:

dxM ) xM - xB dξ

(11)

Thus, the interior of the composition simplex is divided into two batch stripper regions, η and κ as shown in Figure 2b, with the pot composition boundary given by the straight line segment B-AC. Initial still pot compositions in the region η draw the product sequence of (B,AC,C), while initial pot compositions in the region κ draw the product sequence (B,AC,A). For the case of a batch rectifier, the R limit set of the current still pot composition will be drawn as the distillate product, and the direction of motion of the still pot is governed by the following equation:

dxM ) xM - xD dξ

(12)

However, as mentioned in our previous paper,1 all stable separatrices are pot composition boundaries for a batch rectifier configuration (when λ ) 1). For region µ as shown in Figure 3a, the distillate product would be C, and the still pot composition of any


Figure 3. Batch distillation regions/pot composition boundaries for a batch rectifier in the acetone-benzene-chloroform system.

initial point in this region would move in a straight line away from the fixed point of pure C. Thus, the still pot composition will eventually encounter the curved stable separatrix connecting the fixed points B and AC. When the still pot composition enters the curved separatrix, the rectifier will start drawing the R limit set of the separatrix, which is the fixed point AC. Because of the curvature of the separatrix, drawing of product AC will cause the still pot composition to move off the separatrix back into the region marked µ, and hence change its R limit set back to pure C. Drawing pure C moves the still pot composition back onto the separatrix, which then changes the R limit set back to AC, which in turn pushes the still pot composition back into region µ. This seesawing of composition of the product drawn as distillate from the rectifier will ensure that the still pot composition stays on the curved separatrix, and follows it to the fixed point B, without ever crossing into region ν. Region µ thus forms a single batch rectifier region with product sequence given by (C, AC-C mixture, B). Compositions originating in region v, on the other hand, will draw A (its R limit set) as the distillate product until it encounters the separatrix between B and AC. The still pot composition will enter the separatrix, change its R limit set to AC, and start drawing AC as its distillate product. However, when it draws AC, the still pot composition crosses the separatrix again, and a new R limit set of pure C will result. Pure C will be drawn as a distillate product, but this will force the still pot composition back onto the separatrix, and

the see-sawing which had occurred around the separatrix for an initial still pot composition in region µ will also occur here. The still pot composition thus traces out the separatrix to the fixed point of pure B. As before, the separatrix forms a pot composition boundary for the region ν, and any still pot composition that starts in region ν is unable to enter region µ. The product sequence would consequently be given by (A, AC-C mixture, B). Although the stripper and rectifier pot composition boundaries both serve to separate the interior of the composition simplex into separate batch distillation regions, it should be noted that the nature of these boundaries are qualitatively different. The boundary for the stripper is not really a constraint to motion, in that the still pot composition does not attempt to “cross” this boundary. Rather, it is merely by mass conservation that an initial still pot composition which starts on one side of the boundary (B-AC straight line) and draws only pure B from the column does not cross this B-AC line. This type of pot composition boundary was defined as “mass-balance” pot composition boundary in our previous paper.1 In contrast, the pot composition boundary for the batch rectifier, as given by the separatrix between the fixed points B and AC, constrains the motion, in that a still pot composition does actually enter the pot composition boundary and is unable to cross the boundary, because of a change in the R or ω limit set of the still pot composition once the pot composition boundary (also a separatrix) is entered. This type of pot composition boundary was defined as a “separatrix-type” pot composition boundary also in our previous paper.1 The product sequence switches as the boundary is infinitesimally crossed, and this forces the pot composition back onto the boundary, thereby constraining the motion of the still pot of a batch rectifier within its own batch distillation region. It should also be noted as an aside that, in the presence of linear separatrices (Figure 4), the pot composition boundary for the rectifier would have been the same as that for the batch stripper, (i.e., given by the straight line segment B-AC). An initial pot composition starting in region µs will enter the pot composition boundary given by the straight separatrix and draw the azeotrope AC as the new R limit set. The pot composition will stay in the batch distillation region given by the pot composition boundary of the straight line segment B-AC, until pure B is left in the still pot. Similarly, an initial pot composition starting in region νs will draw A as a distillate product, enter the straight separatrix, and draw AC followed by B. The product sequences for µs and νs are thus given by (C,AC,B) and (A,AC,B), respectively. This is not significantly different from the product sequence in the presence of curved boundaries. The major difference is that the pot composition boundaries for the stripper (Figure 2) and rectifier (Figure 4) are both now in common, and the pot composition boundary given by the straight line B-AC will be a pot composition boundary for the MVC. With the assumption of linear separatrices, the pot composition boundary of the rectifier (λ ) 1) and that of the stripper (λ ) 0) would be the same for the A-B-C system. Further, the pot composition boundary for the MVC operating at variable λ are the pot composition boundaries that do not transform as λ varies between 0 and 1. For the A-B-C system, in the presence of straight line separatrices, the line segment B-AC will


Figure 4. Batch distillation regions/pot composition boundaries for a batch rectifier in the presence of straight separatrices.

Figure 6. Operating procedure for crossing the rectifier and stripper pot composition boundaries in the presence of separatrix curvature. Figure 5. Pot composition boundary for the acetone-benzenechloroform system in a middle vessel column. Table 1. Product Sequences for Regions µi for i ) 1...4, Straight Line Separatrices region

first cut

second cut

third cut

µ1 µ2 µ3 µ4

[C,B] [C,B] [C,B] [C,B]

[AC,B] [AC,B] [C,AC] [C,AC]

[B,B] [AC,AC] [AC,AC] [C,C]

Table 2. Product Sequences for Regions νi for i ) 1...4, Straight Line Separatrices region

first cut

second cut

third cut

ν1 ν2 ν3 ν4

[A,B] [A,B] [A,B] [A,B]

[AC,B] [AC,B] [A,AC] [A,AC]

[B,B] [AC,AC] [AC,AC] [A,A]

be a pot composition boundary for the MVC (variable λ) as illustrated in Figure 5. This implies that a mixture starting at any point in regions µi, (where i ) 1...4) in Figure 5, can never draw pure A as one of its products. The sequence of products for each of the regions µ1-µ4 is listed in Table 1. Similarly, any initial composition within any of the regions νi, (where i ) 1...4) in Figure 5 can never draw pure C as one of its products. The corresponding sequence of products is in Table 2. Thus, as illustrated by the sequences in Tables 1 and 2, any initial composition point in the composition

simplex of the acetone-benzene-chloroform system can only draw either (B,C) as pure products or (A,B) as pure products in the presence of straight line separatrices. This, however, is not the case in the presence of curved separatrices, in which the pot composition boundary of the MVC for the A-B-C system undergoes a discrete change as λ is varied. The pot composition boundary for the MVC exists as the curved separatrix which connects AC and B for all values of 0 < λ e 1 and switches to the straight line segment between AC and B at λ ) 0. Any pot composition boundary, for a MVC at a fixed λ, which moves as λ varies will not be a pot composition boundary for a MVC free to operate at different values of λ. Ultimately, in a MVC, there is the operational freedom to choose a value of λ at a given point in time (or warped time ξ), which will allow the shifting of boundaries in the appropriate direction, such that the pot composition barrier which existed at the original λ is moved. To illustrate this point, consider the following operating procedure. Starting with a point σ1 in the region ν, we operate the MVC as a rectifier drawing pure A as the product and cross the straight line segment B-AC as it is not a pot composition boundary for the rectifier (see Figure 6a). After crossing the line segment B-AC under batch rectification, the pot composition is now in the batch stripper region given by η as given by point


Figure 7. Stable separatrix in the residue curves map of the acetone-benzene-chloroform system.

σ2 in Figure 6b. It is now possible to switch the operation of the column from that of a rectifier (λ ) 1) to that of a stripper (λ ) 0), move the still pot composition across the separatrix (which is not a pot composition boundary for the stripper configuration), and enter the batch rectifier region given by µ, to a point such as σ3 (see Figure 6c). Effectively, the still pot composition crossed a batch stripper pot composition boundary given by the straight line between B-AC (moving from σ1 to σ2), and it also crossed a batch rectifier pot composition boundary as given by the curved separatrix between B and AC (moving from σ2 to σ3). This follows the prediction in our previous paper,1 where it was stated that it was possible to cross the stripper or rectifier pot composition boundaries with a MVC, when the stripper and rectifier regions are not equivalent, as is the case here in the presence of curved separatrices. It should be noted that while it is possible to move from σl via σ2 to σ3, it is not possible to move in the reverse direction. That is to say, it is not possible to move the still pot composition from region µ to region ν, even though it was possible to move the still pot composition from ν to µ. Similarly, it is not possible to move the still pot composition from region η into κ, even though it was possible for the still pot composition to move from κ to η. Thus, even though differences in the stripper and rectifier regions offered some additional separation possibilities in a middle vessel column, the possibilities are not unlimited. Operating Procedures Applicable for Breaking Azeotropes In the presence of curved separatrices, as is the case for the A-B-C system, the boundaries of the stripper and the rectifier no longer coincide. Furthermore, on a closer observation of the curvature of the stable separatrix connecting the fixed points AC to B, we see that the separatrix almost touches the composition simplex edge given by line segment B-C (see Figure 7). On the basis of only the curved separatrix, it is possible to design an operating procedure (mode a) which involves the recycle of an azeotropic offcut to the original feed to the still pot, provided that the initial still pot composition lies in the region given by ν. This is as follows: (1) Assume some original composition of the charge to be separated, which is designated as F in Figure 8.

Figure 8. Operating procedure for separating an acetonebenzene-chloroform mixture with recycle of azeotropic offcut.

Mix this with the azeotropic offcut from the last batch to obtain a point M in the composition space. This is the initial still pot composition. (2) Operate the MVC as a rectifier (with λ ) 1) such that pure A is drawn and the still pot composition moves directly away from the fixed point of A. Continue this operation until the still pot composition encounters the separatrix at point D (the separatrix forms a pot composition boundary ∀ λ ∈ (0,1]) and the rectifier product starts changing from pure A to a mixture of AC-C. (3) Since pure cuts are desired, at this point, the operation of the MVC will be switched to that of a pure stripper (with λ ) 0), such that the pot composition boundary is shifted from that of the separatrix to that of the straight line segment between AC and B. The still pot composition is thus able to cross the separatrix, and the column is operated as a stripper, drawing pure B as the bottoms product of the column. The still pot composition moves directly away from B and encounters the C-AC edge at point E after all the benzene has been removed from the column. (4) At point E, the R and ω limit sets of the still pot composition are given by C and AC, respectively. Since pure products are of interest, the operation of the column will now be switched to that of a rectifier again (λ ) 1), such that pure C is drawn as the distillate product. The azeotropic composition of AC will remain in the still pot after all pure C has been drawn. This still pot residue is then mixed with the next batch of mixture to be charged to the pot. Note that a sharp split is not really necessary in this stage. (5) Alternatively, at point E (Figure 8), instead of using step 4, it is also possible to operate the MVC at a value of λ ) l2/(l1 + l2) such that the column operates as a quasi static operation, with the column holdup composition, distillate and bottoms product composition, all remaining constant until the still pot runs dry (i.e., as ξ f ∞). This operating policy utilizes the full capability of the MVC (its ability to draw up to two products at any time). It is not necessarily beneficial, however, as the azeotropic offcut will be recycled and mixed. The purity of the azeotropic offcut is therefore not of great importance, and as such, recycling of the still pot residue as in step 4 will typically be sufficient. Even though the above operating procedures allow us to extract all three components in the ternary mixture (F) as pure products, it is unsatisfactory for a few reasons. First, it precludes the separation of


an original composition that is in the region µ, as any composition that begins in µ cannot be mixed with the azeotropic offcut to produce an initial still pot composition which lies in region ν. It is impossible to draw pure A with a MVC from a still pot composition starting in region µ, since it is possible to cross from region ν to µ but not the reverse. Second, a batch distillation system is usually used in preference to a continuous distillation system because of small processing amounts or infrequent or discontinuous processing demands. As such, there might not be a “next batch” with which to mix the azeotropic offcut into. This azeotropic offcut may have to be discarded. It would thus be desirable to obtain a separation sequence in which no azeotropic offcuts are drawn nor recycled to the next batch. Last, but not least, it precludes the separation of a mixture that is completely made up of the azeotrope AC, which is the most important mixture within the composition simplex; any point in the composition simplex can theoretically be separated into the pure components and the azeotropic offcut with traditional batch distillation columns (either the stripper or the rectifier). Thus, if the azeotropic offcut can be separated, all cuts can be separated. Furthermore, as the interest is in separating azeotropes, it would seem appropriate that the azeotropic mixture should be separable by the suggested operating strategy. A better separation scheme for a mixture of acetone, benzene, and chloroform, which addresses the problems stated above, can then be synthesized based on these observations: (1) A still pot composition which lies along the B-C edge of the composition simplex when separated in a MVC operating as a stripper would draw pure B as the product, with pure C as the last cut from the column. However, the curved separatrix lies so closely to the B-C edge that it is almost on the B-C line segment around the region F as shown in Figure 7. If it is possible to steer the still pot composition into the separatrix around the region F, it would then be possible to separate the remaining B and C in the still pot into pure B and almost pure C. It should be noted that a trickle of A might exist as impurity in the cut for C. (2) In order for any mixture of A, B, and C to be separated completely, it implies that pure A, pure B, and pure C must all be drawn as products from the MVC. Pure A is the R limit set of the region ν as shown in Figure 7 and can only be drawn as the distillate product of the MVC if the still pot composition is in the region ν. Pure B, being the ω limit set, is drawn as the bottoms product for region ν. If it is possible to steer the still pot composition into the separatrix in the vicinity of F, it would then be possible to draw C as the last cut, as mentioned above, without drawing and recycling any azeotrope AC at all. (3) However, it should be noted that the movement of the still pot composition is restricted by the vector cone of possible motion for the MVC. In region v, this is given by the two direction vectors, one of which points through the still pot composition away from the ω limit set (pure B), the other which points through the still pot composition away from the R limit set (pure A). However, this means that only a limited subset of composition points in the simplex can be steered into the desired location of the separatrix in the vicinity of F. This subset of still pot compositions which can be

Figure 9. Quasi static operating procedure for separating the acetone-benzene-chloroform mixture with addition of benzene as an “Entrainer”.

steered into the separatrix in the vicinity of F is represented by the region σ in Figure 7. (4) Furthermore, a composition point in the region µ would not be able to draw pure A, regardless of the operating procedure. It can only draw pure B, pure C, and a mixture of the azeotrope AC and C (as the still pot composition follows the separatrix). As such, the region µ is of little interest in the attempt to separate a ternary mixture of A, B, and C. (5) If pure benzene is drawn as one of the cuts, it can be recycled and mixed with the next batch of ternary mixture to be separated. Since pure benzene will again be drawn as pure product in this next batch operation in the limit of sharp splits, all the benzene that was added into the system will be recovered as pure benzene without any loss of benzene in the process (i.e., no or minimal make-up benzene is necessary). (6) Any composition in the acetone-benzene-chloroform composition simplex can be mixed with benzene to obtain a still pot composition for the separation different from the original composition of the mixture to be separated. The new initial composition is then related to the original mixture composition and the pure benzene fixed point by an appropriate tie line. By varying the amount of benzene added, we can manipulate the position of the initial still pot composition, with the limits being amount of benzene added f 0

M xinitial f xoriginal

M amount of benzene added f ∞ xinitial f pure B

(13)

It is thus theoretically possible to move any original composition point in the composition simplex into the region given by σ in Figure 7, with an appropriate amount of benzene added to the original mixture to be separated. The resulting operating procedure (mode b) would then be given by the following steps, and it is illustrated in Figure 9. This can be achieved with any of the three categories of original mixture compositions, namely, that of (1) a composition in region µ (δ1 in Figure 9), (2) a composition in region ν (δ2), and (3) the azeotropic composition AC (δ3). (a) Add an appropriate amount of pure benzene to the original batch of ternary mixture (of A, B, and C) to be separated, such that the initial composition of the mixture to be separated in the MVC is moved into region σ as denoted in Figure 9. This moves each of the original


still pot compositions (δ1-δ3) to the initial still pot compositions for separation in the MVC, as given by points γ1-γ3, respectively. Note that γ1-γ3 all lie on the edge of region σ because this represents the least amount of benzene added to each of initial compositions δ1-δ3 to move these points into the region σ. (b) With the initial still pot composition in the region σ, operate the MVC as a rectifier (with λ ) 1) so as to steer the still pot composition to enter the separatrix in region F where the separatrix is as near to the B-C edge as possible, hence minimizing the amount of impurity (component A) in the final cut of C from the MVC. (c) After the still pot composition encounters the separatrix at point G, the R limit set of the current still pot composition will change to that of the fixed point of AC. However, AC is not a desired product; hence, the product flow rate of the rectifying section of the column is shut off at this time. Instead, the product flow rate for the stripping section of the column is turned on, such that the MVC now operates as a stripper. λ has thus changed from 1 to 0 in this operating step, and pure B is now drawn as bottoms product from the column. As mentioned earlier, the separatrix forms a pot composition boundary for the rectifier only, and as such does not constitute a pot composition boundary for the column operating as a stripper. The still pot composition is thus able to cross the separatrix. (d) Continue to draw pure B as product until all of the benzene has been removed from the column. The still pot composition will encounter the C-AC edge at this point in time, and the residue in the still pot will be almost pure C with a trickle of A as impurity. If this purity is reasonable, the still pot can be collected as pure C product. This purity should be easily more than 99%. (e) If the purity does not meet the specifications, the MVC can be further operated as a rectifier (λ ) 1) such that the R limit set of the still pot composition, namely, pure C, is drawn as distillate product from the column. The residue that remains will be the azeotrope AC, but it will be a trickle, as determined by the amount of A that remains in the column, which will be significantly less than 1% of the original charge to be separated. Alternatively, the column could also be operated as a stripper for a short period of time to remove the azeotrope offcut as bottoms product, leaving C in the still pot. (f) Alternatively, we could again substitute steps d and e with the following procedure which utilizes the full capability of the column. For this case, the MVC is operated as a stripper such that the still pot composition crosses the separatrix. Once the MVC composition crosses the separatrix, the R limit set of the still pot composition will be changed to that of pure C. It is possible to change the operation of the MVC from a stripper to employ a quasi static mode of operation at this point in time with the appropriate value of A. However, as there is a trickle of acetone left in the pot (the separatrix hugs the B-C edge, but is not on the B-C edge), drawing B and C simultaneously right after the still pot composition has crossed the separatrix would result in the still pot moving back onto the separatrix after only a relatively short period of time. When this occurs, we would no longer be able to draw both benzene and chloroform at arbitrarily high purities. Hence, to delay the motion of the still pot back onto the separatrix, the column is operated until a point where

the amount of benzene in the MVC equals the amount of chloroform in the MVC. At this point, a quasi static operation of the MVC with λ ) l3/(l3 + l4) ) 1/2 is employed so as to draw the distillate product C and bottoms product B while keeping the pot composition unchanged (and hence the R and ω limit sets unchanged, which results in unchanged product compositions). This utilizes the complete capacity of the column to draw two products at a time, while maintaining the required purity of the products (by maintaining the reflux and reboil ratio). The final still residue in the pot, when the still pot composition moves back onto the separatrix, would be a trickle of some mixture of A, B, and C, which can be discarded or mixed with a subsequent batch. With the above operating scheme, all the benzene that was added to the system is recovered and can be recycled or resold. The waste cut is negligible and can be essentially considered as nonexistent. Note, however, that there will always be material left in the pot at the end of a batch distillation (i.e., at least the material accumulated on the trays/packing). The separation scheme above thus essentially allows us to separate any mixture of acetone, chloroform, and benzene into its individual components at purities higher than 99%. An actual simulation of each of these operating policies was conducted using the ABACUSS model of the MVC and the results are summarized in our next paper.4 Finally, it should be noted that, ultimately, if there were a sufficient number of batches of mixture to be separated (such that a recycle of the azeotropic offcut was feasible) and the original composition was in region ν, there would be a trade-off between the use of the first operating procedure (mode a: recycle of azeotrope, no addition of benzene) against the use of the second operating procedure (mode b: addition of benzene, no recycle of azeotrope). In mode a operating procedure, without recycle of azeotropic offcut, a potentially large portion of the original mixture is recovered as an azeotropic offcut which is undesirable and represents both a waste disposal cost and a raw material cost (due to pure components not recovered). However, the operating load on the column is smaller (i.e., less charge per batch/less batches of charge), and hence, energy costs are lower. Alternatively, following mode a of operation, the azeotropic offcut can be recycled, resulting in larger batches, higher energy costs, but complete recovery of the pure components, with no waste disposal costs and lower raw material costs. Conversely, mode b of the operation could be used to recover almost all of the mixture as pure components (or within purity specifications), with a negligible waste cut (and hence minimal loss in revenue). However, the addition of benzene as an entrainer does result in a much larger initial charge into the still pot, which in turn may represent higher capacity requirement and energy cost. As such, it is not immediately clear which of the three suggested operating procedures are preferable. However, with the current emphasis on zero-waste production technologies for the specialty chemical and pharmaceutical industries, the use of the second and third procedures would seem more attractive, as they are zero-effluent technologies in which all of the waste cut is recycled to the next batch. Indeed, it is possible that a hybrid of the operating strategies (i.e., mixing with pure benzene combined with a recycle of an azeotropic offcut) could be optimal. This is an exercise in open-loop optimal control which will depend largely on the specification


Figure 10. Dependency of recycle cut size with the amount of benzene added as an “Entrainer”.

of the problem (such as cost of raw materials, amount of material, capital cost of equipment, energy costs, and waste disposal costs). It should also be noted that the size of the recycle cut is a direct function of the amount of benzene added, and hence, choosing the amount of benzene added directly determines the amount of azeotropic offcut obtained. For a given mixture to be separated, the addition of more benzene results in a still pot composition encountering the separatrix (pot composition boundary) at a point nearer to the region F (for example, H in Figure 10), pure benzene is then drawn from the column until the pot composition encounters point K on the C-AC edge. As can be seen from the diagram, the proportion of C to AC drawn from the still pot is thus given by the lever ratio of point K between C and AC, which would mean that a fraction (l6+l7)/l5+l6+l7) of the C in the mixture, prior to the addition of benzene, will be recovered as pure C, while l5/(l5+l6+l7) of C has to be recycled or is lost in the AC cut. In comparison, if less benzene were added, and the pot composition encounters the separatrix at a point such as J instead (see Figure 10), the resulting intersection of the still pot composition after drawing all the benzene in a stripper configuration will be at L, and only l7/(l5+l6+l7) of the original C can be recovered as pure C, while (l5+l6)/(l5+l6+l7) of the C is lost as AC. As can be seen, a larger amount of benzene added corresponds to a smaller amount of AC recycled (or discarded), and a direct relationship between the amount of benzene added, the intersection point of the still pot composition with the separatrix B-AC, and correspondingly the fraction of C recovered as pure C or recycled as the azeotropic offcut AC, can be obtained. There is thus only 1 degree of freedom in the selection of an optimal policy (i.e., a cost minimization problem with respect to one variable) for this separation process. Comparison to Wahnschafft et al.’s Continuous Column Sequences for Separating the Acetone-Benzene-Chloroform Mixture The above operating procedures for the separation of acetone, benzene, and chloroform in a middle vessel batch distillation column were formulated on the basis of an appreciation of the behavior of a MVC which was developed from our limiting analysis of the MVC. The spatial analog of these operating procedures in a continuous distillation column were, however, first proposed for the separation of the acetone-benzenechloroform mixture by Wahnschafft et al. in two sepa-

Figure 11. Continuous distillation analogue of the first operating procedure with no addition of benzene and recycle of azeotropic offcut.

rate papers regarding separability of azeotropes in a continuous distillation column,6 and the synthesis of separation processes with recycle streams for continuous systems.7 An analogue to the first operating procedure described earlier (mode a) was proposed in their work on the separability of an azeotrope in a continuous column.6 On the basis of the analysis of Laroche et al.5 who highlighted the separability of an inverse-020 system using multiple columns, Wahnschafft et al. proposed an analogous sequence of continuous distillation columns for the acetone-benzene-chloroform system (an inverse020 system) and a depiction of the location of the cuts and recycle streams on the composition simplex as illustrated in Figure 11. Their sequence of columns is exactly analogous to our first operating procedure (mode a) in the middle vessel batch distillation column in the following manner: (1) The first step involves the mixing of an azeotropic cut (B3, the bottoms product of their third distillation column) with the original feed F, such that an initial feed composition into the first column is obtained at M, much like our initial mixing of the azeotropic cut from the previous batch with a new batch of material with feed composition F to obtain a new initial composition (see Figure 8). (2) The first column then draws pure acetone (D1) as the distillate, much like the first operating step in our first operating procedure, where pure acetone (D1) is drawn as a distillate product from the MVC operating as a rectifier (λ ) 1). (3) The second continuous column takes as feed the bottoms product of the first column, which has a composition that lies on the separatrix at B1. B1 lies collinearly with D1 and M, a consequence of the mass balance around the column at steady state. This is an


analogue of our first operating step which draws pure acetone and moves the still pot composition in a straight line directly away from the acetone fixed point until it encounters the separatrix, with a still pot composition given by a point such as B1. (4) With B1 as its feed, the second continuous column then draws the bottoms product of pure benzene (B2) and sends the distillate with composition given by D2 into the third column. This is again exactly analogous to our operation of the MVC as a stripper (λ ) 0) in our second operational step which drew pure benzene as the bottoms product. As before, B2, B1, and D2 lie collinearly to each other, which is an analogue of the movement of the still pot composition directly away from the fixed point of pure benzene, resulting in a still pot composition such as D2 at the end of the operating step. (5) Finally, the distillate of the second column D2 is fed to the third column, where pure chloroform (D3) is drawn as the distillate product, with the azeotrope of AC (given by B3) drawn as the bottoms product for recycle back to the first column. This is again analogous to our third operating step, where pure chloroform was drawn from the MVC operating as a pure rectifier (λ ) 1), and the azeotrope is left as the residue in the still pot (step 4, mode a operating procedure). In fact, it is even more similar to the alternative operating step proposed (step 5, first operating procedure) where pure C was drawn as distillate products and the azeotrope AC was drawn as bottoms product, to be recycled into the next batch of mixture to be separated. It should be noted that this recycle of the azeotropic offcut back to the initial column/operating step is also similar across the two distillation columns: continuous and middle vessel batch. Thus, the main difference between the two processes is that in the continuous column sequence the separation was achieved using three columns (with two sections, rectifying and stripping), at the same point in time (i.e., once the continuous sequence is operating at steady state, pure acetone, benzene, and chloroform will all be drawn at the same point in time), whereas for the operating procedure proposed for the middle vessel batch distillation column, the separation was achieved using one column (again with both a rectifying and a stripping section), but over a period of time with three different operating steps. The proposed separation scheme by Wahnschafft et al. was achieved by spreading out the process in space domain (spatially distributed into three columns), as opposed to our proposed operating procedure spread out in the time domain (varying the operating procedure over time in the same column). A spatially distributed operating sequence similar to the second operating procedure (mode b) was also proposed by Wahnschaft et al. in their discussion of algorithms for synthesizing complex separation processes with recycle.7 Figure 12, illustrates their sequence of continuous distillation columns with the corresponding distillate, feed, and bottoms composition on a composition simplex. On the composition simplex, Wahnschafft et al. designated a region of compositions “satisfying mixing goal”, similar to region σ in Figure 7. The operating procedure is specified as follows: (1) As with the first step of the mode b operating procedure, the first “step” of Wahnschafft et al.’s proposed sequence was the mixing of a benzene recycle flow (from the bottoms product of the second continuous

Figure 12. Continuous distillation analogue of the second operating procedure with the addition of benzene and no recycle of azeotropic offcut.

column) with the original feed to the separation train F, to obtain a continuous feed of composition M into the first column. To minimize the amount of benzene recycled (and hence minimize the energy consumption/ size requirement of the columns), the feed composition was also moved onto the edge of the region of compositions which “satisfy the mixing goal” (corresponding to σ). (2) This mixed feed is then introduced into the first continuous column where pure acetone is drawn as the distillate product, while the bottoms product corresponds to a point on the residue curve separatrix in a region where it hugs the B-C edge. By mass balance, the feed to column M, distillate (D1), and bottoms (B1) product composition are all collinear. This is again similar to the first operational step in our operating procedure in which pure acetone is drawn as the distillate product, resulting in a still pot composition similar to the point B1 (which is collinear with the initial still pot composition M and the distillate product drawn D1). This bottoms product in then used as a feed to the next continuous column, much like the still pot residue in the middle vessel column which is the charge for the separation in the next operational step. (3) With the feed on a point nearly on the B-C edge, Wahnschafft et al. pointed out that it was possible to draw pure benzene and 99% pure chloroform as products from the second continuous column. This is again analogous to our analysis that it is possible to separate the resulting still pot composition on the separatrix edge and B-C edge, by either operating the column as a stripper (in which case pure B is obtained as a distillate product, with the 99% purity chloroform left as the still pot residue) or with a quasi static mode of operation. Note that while it is possible to draw both pure benzene and pure chloroform as the distillate and bottoms


product, respectively, by operating the MVC under a quasi static operation policy, such an operational policy is not possible with the continuous column. Thus, as before, the structure of column sequence proposed by Wahnschafft et al. is closely analogous to the time-domain operating policy discussed for the MVC. In this case, Wahnschafft et al.’s two-column sequence of continuous columns with all three pure products drawn at the same point in time is the spacedomain analogue of the one-column two-operational step policy in which the products are drawn over different periods of time. A notable difference between the MVC operating policy and the continuous distillation column sequences is that the MVC offers the possibility of drawing arbitrarily pure chloroform as one of the product cuts and recycle/discard the azeotropic offcut which is approximately 35% chloroform but of a negligible quantity, whereas by mass balance, the continuous column has to draw chloroform of 99% purity. It should be noted that it is, however, possible to put this chloroform of 99% purity into another continuous column to obtain chloroform of close to 100% purity, but that would require an additional column for separation. Hence, the fact that the distillate, bottoms, and still pot compositions in a MVC need not be collinear to each other, results in a greater flexibility in separations with a MVC as compared to a continuous column. From the close parallel of the separation possibilities afforded by sequences of continuous columns versus that of the middle vessel batch distillation column, it is not difficult to realize that the two types of columns are closely related in their behavior. In the next section, we will briefly discuss their similarities and differences.

Figure 13. Continuous distillation column operating at nonlimiting conditions.

A Discussion about the Equivalence of the MVC versus a Continuous Distillation Column In this section, the similarities and differences are enumerated between a continuous column versus a MVC operated in a quasi static state and a MVC operated with an open-loop control policy. In a continuous column, the distillate and bottoms product compositions must lie collinearly with the feed composition. The distillate and bottoms products must also lie on the same column composition profile. For a MVC operated at an infinite reflux/reboil ratio, the composition profile of trays in any column would be approximately traced out by the residue curve that passes through the feed tray composition (i.e., the composition profile of the tray column are discrete points very close to the residue curve), whereas at lower/finite reflux/reboil ratios, the composition profile of the column tends to have a greater curvature when compared to that of the residue curves. Thus, assuming nonlimiting conditions (finite trays, finite reflux/reboil ratios), a continuous column processing a feed which is a mixture of acetone, benzene, and chloroform could have a feed F, bottoms product B1, and distillate product D1 as shown in Figure 13 with the appropriate number of trays in each of the rectifying and stripping sections.6 As seen in Figure 13, the feed composition in a continuous column (F) need not correspond to that of the feed tray composition (M). The feed composition must, however, be collinear to the composition of the two products. This is a fundamental restriction on the operating possibilities of the continuous column that does not exist in the MVC. An example of the effect of this restriction was mentioned above, where chloroform

Figure 14. Middle vessel batch distillation column operating at nonlimiting conditions.

of close to 100% purity could be drawn from the MVC but not from the continuous column. To maintain material balance over the column, the ratio of the D1 flow rate to the B1 flow rate drawn from the column is then given by l1:l2. The tray compositions are given by discrete points on the column composition profile curve. It is then our claim that a separation such as that illustrated in Figure 13 that is achievable at steady state in the continuous column will also almost always be achievable in a middle vessel column. “Almost always” is used because it is recognized that intrinsic differences between the two columns will mean that there will be examples in which a separation is achievable by a continuous column but not by a middle vessel batch column. Using the same feed and product composition points (as that of Figure 13), a middle vessel batch column analogue of the continuous column separation is illustrated in Figure 14. The first significant restriction on the MVC, which does not apply to a continuous column, is that, while the feed/charge composition need not lie collinearly on the composition simplex with the bottoms and distillate compositions, they must, however, lie on the same distillation line (or column profile). It seems, however, possible in most cases to vary the number of trays in each of the stripping and rectifying sections of the column and vary the reflux/reboil ratios for the column, such that the product points of D1 and B1 can be


achieved as before. Operating the column under a quasi static operating policy, the ratio of D1 to B1 flow rates is again given by l1:l2, or in other words, λ ) l1/(l1+l2). This quasi static operating policy completely reproduces the separation achieved by the continuous column, in that with λ ) l1/(l1+l2), the net product drawn from the column is effectively situated at point M. That is to say, the net vector of still pot composition motion as given by (xM - xP) is of zero length and hence nondirectional. The still pot composition will not move as long as λ is unchanging. Conversely, the composition of the products will not vary as long as M remains stationary (because xD ) xD(xM) and xB ) xB(xM)), and the reflux and reboil ratios are maintained in the rectifying and stripping sections of the column, respectively. Thus, with a clever manipulation of the operating parameters of the MVC, a separation that is achievable with a continuous column can also be conducted in a MVC. The only difference is that while the continuous column will operate until t f ∞, the middle vessel batch column can only operate until ξ f ∞; that is to say, the operation has to end in finite time. Other than that, the feed/ charge and products are of the same composition and remain at the same composition, throughout the operation of both columns. The distinction between this quasi static operating policy and total reflux operation of the MVC (e.g., see refs 11 and 12) should be noted. In the former case products are withdrawn continuously during the course of the operation, whereas in the latter case no products are withdrawn during the entire operation. Operation of the MVC with a quasi static operating policy requires the specification of the distillate and bottoms product compositions in some manner and the value of λ. The distillate and bottoms product compositions must be specified such that the middle vessel composition is a convex combination of them, which then allows us to draw a net product that coincides exactly with the current still pot composition by choosing the appropriate value of λ. Since in a ternary system the convex combination requirement implies two nonredundant constraints, we require 3 degrees of freedom in the operation of the MVC to maintain a quasi static condition. Normally, only 2 degrees of freedom remain in a continuous distillation column once pressure and inventory control have been established. On the other hand, as we shall show, the MVC does indeed have 3 degrees of freedom remaining once pressure and inventory control have been established. For example (Figure 15), if we consider the column configuration suggested by Hasebe et al.,13 Davidyan et al.,14 and Barolo et al.,15 where the vapor flow rate in the rectifying and stripping sections remains the same because of the absence of a heat exchanger in the middle vessel, we can choose to manipulate (1) the reboiler heat duty Q˙ R (controls V), (2) the distillate flow (D), and (3) the flow from the middle vessel Lb to achieve our operational objectives. This statement at first appears puzzling because this choice of instrumentation implies that the middle vessel is a pure integrator (i.e., the liquid level in the middle vessel is not controlled). However, in the MVC, this result is exactly what we want to achieve because the level in the middle vessel is intended to drop continuously over the operation of the MVC, and there is thus no concern about the middle vessel overflowing. Thus, all that remains to be shown is that the level in the middle vessel will drop at a rate

Figure 15. Operational degrees of freedom for the MVC operated with a quasi static policy. Column with adiabatic middle vessel.

determined by the amount of material drawn from the column, as denoted by D + B. From an overall mass balance around the boundary shown in Figure 15, we realize that as in Barolo et al.15 the liquid flow rate into the middle vessel (Ld) is always smaller than the liquid flow rate out of the middle vessel (Lb). First,

Lb ) Ld + B + D

(14)

and given that the product flow rates cannot be negative:

B+Dg0

(15)

Lb g Ld

(16)

then results in

In deriving this result we have assumed that there is no significant accumulation of mass within the massbalance boundary during the course of the operation. This is reasonable since the tray sections are selfcompensating because of the overflow weirs on each individual tray, and we have also assumed that inventory control has been established for each of the reboiler and condenser holdups (which can be achieved locally via valves f and valve a, respectively) and the pressure in the column (via valve b). This then leaves valves c-e available as degrees of freedom for controlling the operation of the MVC; the level in the middle vessel will take care of itself. However, even given 3 degrees of freedom, it remains to be shown that two purity specifications and a specification on λ can be met simultaneously. Overall mass balances (again assuming inventory control) around


the stripping and rectifying sections respectively yield

D ) V - Ld

(17)

B ) Lb - V

(18)

and the following modified definitions can be adopted, where we avoid the need to assume CMO by defining the reboil/reflux ratio with respect to a vapor/liquid flow rate at the midsection of the MVC rather than the vapor flow rate at the bottom of the stripping section and the liquid flow rate at the top of the rectifying section (which are the conventional definitions):

Rd )

Ld D

(19)

Rb )

V B

(20)

λ)

D D+B

(21)

which yields five equations in eight unknowns. Note that the only assumption used to derive these equations is adequate inventory control. Since setting valves c-e can be interpreted as fixing D, Lb, and V, respectively, it is evident from eqs 17-21 that fixing these flow rates implies unique and feasible product flow rates, reflux and reboil ratios, and λ (provided Lb g V g Ld). Alternatively, an inspection of the system of linear equations (17-21) yields in matrix notation the following homogeneous system:

[

1 -1 A1x1 ) 0 -1 0

-1 0 Rd 0 λ-1

0 1 0 0 0

-1 0 -1 0 0

0 -1 0 Rb λ

][ ] [ ] V 0 D 0 Lb ) 0 Ld 0 0 B

(22)

and the determinant of A1 is given by

det A1 ) (Rb + Rd + 1)λ - Rb

(23)

If this determinant is not equal to zero, it implies that A1 is nonsingular, and the only internal and product flow rates that satisfy the system is the trivial solution x1 ) 0 where there are zero flow rates everywhere. Hence, only when the determinant is equal to zero and the matrix is singular, that is

λ)

Rb

(24)

(Rb + Rd + 1)

can non-zero internal and product flow rates be found. This can only be achieved by specifying Rb, Rd, and λ such that they satisfy (24). When eq 24 is satisfied, elementary row operations yield

[

1 0 U1 ) 0 0 0

-1 -1 0 0 0

0 1 -1 0 0

-1 -1 0 -1 - Rd 0

0 -1 Rb + 1 RbRd 0

]

(25)

which indicates that the matrix will have rank 4 for all nonnegative (i.e., physical) values of Rd. Thus, by

Figure 16. Operational degrees of freedom for the MVC operated with a quasi static policy. Column with heat addition/removal from the middle vessel.

specifying an appropriate (extensive) flow variable (B, D, Ld, Lb, or V), the other four flow variables are completely determined provided that eq 24 is satisfied by the appropriate choice of the three intensive variables Rb, Rd, and λ. In essence, this specification of an extensive variable fixes the basis/net production rate for the operation, which may be picked arbitrarily provided (24) is satisfied. While investigating the operational characteristics of a constant vapor flow rate MVC, Barolo et al.15 also highlighted that there were 3 degrees of freedom available in the control of a MVC. However, they did not provide an analysis to justify this statement, nor did they highlight the significance of these 3 degrees of freedom in allowing the MVC to mimic the operation of a continuous column. With this constant vapor flow rate MVC, the fact that we are only able to specify two out of the three intensive variables (Rb, Rd, and λ) for non-zero flow rates implies that we might not be able to mimic all the separation achievable using a continuous column if we are unable to vary the number of trays in each of the sections in the column. However, if we consider a MVC where the vapor flow rate in the rectifying section need not be equal to the vapor flow rate in the stripping section (Figure 16), this problem does not exist. A simple analysis of the overall mass balances (again assuming adequate inventory control) around the stripping and rectifying sections in the system yields the following equations much like the constant vapor flow rate MVC, except that there are now two distinct vapor flow rates:

B ) Lb - Vb

(26)

D ) Vd - Ld

(27)

As before, we avoid the need to assume CMO by defining the reboil/reflux ratio with respect to the vapor/liquid


flow rates in the middle of the column:

Rb )

Vb B

(28)

Rd )

Ld D

(29)

λ)

D D+B

(30)

which yields five equations in nine unknowns, namely, Vd, Ld, Vb, Lb, D, B, Rd, Rb, and λ. If we assume that we are able to set values for Rd, Rb, and λ as desired, these values become parameters, and the resulting homogeneous linear system is

[

1 0 A2x2 ) -1 0 0

0 1 0 0 0

-1 0 0 0 0

0 -1 0 -1 0

1 0 Rb 0 -λ

0 -1 0 Rd 1-λ

][ ] [ ] ][ ] [ ] Vb 0 Vd 0 Lb ) 0 Ld 0 B 0 D

(31)

industries. It also seems that almost any sequence of continuous distillation columns used for the separation of azeotropic mixtures can be reproduced as a sequential operation in a single MVC; thus, the MVC can serve to replace more than one continuous distillation column, as illustrated in our example of separating the azeotropic mixture of acetone, benzene, and chloroform. Finally, it should be noted that although the MVC offers the added flexibility of still pot composition motion and the added flexibility in that the feed and product compositions need not be collinear, there is also the additional constraint that the feed composition must lie on the same distillation line (or column profile). Thus, while the MVC may offer some additional operating flexibilities, there may also be additional operating constraints which would normally not apply to a continuous column. This could then possibly preclude some of the separations achievable in a continuous column. Nevertheless, this quasi equivalency of the MVC with the continuous column or continuous column sequence could be useful in some cases to industries with low and/ or intermittent volume of chemical output, in helping them formulate separation policies for their products or wastes.

A single elementary row operation yields

[

Vb 1 0 -1 0 1 0 0 Vd 0 1 0 -1 0 -1 0 Lb Rb + 1 0 ) 0 U2x2 ) 0 0 -1 0 Ld Rd 0 0 0 0 -1 0 B 0 0 0 0 0 -λ 1-λ D

(32)

indicating that this matrix will have full row rank for all λ, independently of both Rb and Rd. Hence, after choosing independent values for Rb, Rd, and λ, we can still specify any one of the six flow rates Vb, Vd, Lb, Ld, B, and D, and the other five flow rates will be scaled accordingly to the value of the specified flow rate. Control of these variables is easily achieved with the use of the appropriate valves in the MVC as illustrated in Figure 16. As explained earlier, the level in the middle vessel does not have to be controlled; hence, only two level controllers (reboiler and reflux drum) are required. This is achieved using valve a (reflux drum) and valve f (reboiler). The pressure in the column is controlled by valve b through varying the condenser load. This leaves 4 degrees of freedom (valves c-e and g) to meet specifications on Rb, Rd, λ and one extensive flow rate (to fix the net production rate). In conclusion, if we allow for different vapor flow rates in the stripping and rectifying sections of the MVC, it is possible to achieve many desired continuous column separations in an MVC. It is also possible to scale the separation arbitrarily by specifying an appropriate value for one of the flows in the column (within physical limits such as column flooding). This could prove extremely useful for pharmaceutical and specialty chemicals industries, where a separation which is currently achievable with a continuous column is not conducted because of the small capacity and infrequent need for this separation. With the possibility of using a MVC to achieve the same separation for batches (rather than a continuous flow of feed), less waste and more recycling of chemicals may be achieved economically by these

A Discussion on the Perfect Entrainer The characteristics of a perfect entrainer for separating an azeotrope in a MVC was also elucidated from the above analysis. A perfect entrainer should enable the separation of a given azeotropic feed such that no or negligible azeotropic offcuts are drawn or recycled, and all products drawn from the MVC have purity that meet reasonable specifications (for example, purity >99%). The need for no recycle of the azeotropic offcuts stems from the fact that there is sometimes no “subsequent” batch to which to recycle the azeotropic cut to, and this azeotropic offcut would have to discarded. One such example of a perfect entrainer for a binary azeotrope was provided by Bernot et al. in their discussion on the behavior of batch strippers.9 It was not stated explicitly in their work that such an entrainer was “perfect”, but they proposed the addition of an intermediate boiling entrainer to a binary mixture XiYi which exhibits a binary azeotrope XiYi so as to produce either a 001 system with a minimum-boiling azeotrope or an inverse-001 system with a maximumboiling azeotrope. Addition of these entrainers to the azeotrope would then allow us to draw pure Xi, Yi, and entrainer Ei using a MVC without the need for recycle of the azeotropic off-cut. As highlighted by Bernot et al., the minimum-boiling azeotrope X1Y1 which forms a 001 system with the entrainer can then be separated completely into its pure components by a stripper, while the maximum-boiling azeotrope X2Y2 which forms an inverse-001 system can then be separated completely into its pure components by a rectifier. The principles behind these separations are based on the fact that a stripper draws as its product the ω limit set of its current pot composition, while a rectifier draws as its product the R limit set of its current pot composition. In fact, entrainers which allow perfect separations of a given azeotrope in a stripper or in a rectifier would also serve as perfect entrainers for a MVC. This is illustrated in Figures 17 and 18. For the 001 system in Figure 17, after the initial still pot composition of M1 is obtained via mixing the azeotrope with the entrainer, the column is operated at λ ) 0 as a pure stripper,


Figure 17. Complete separation of a minimum-boiling azeotrope in a MVC by the addition of entrainers to form a 001 system.

Figure 18. Complete separation of a maximum-boiling azeotrope in a MVC by the addition of entrainers to form an inverse-001 system.

drawing the first ω limit set (pure X1) as a product until the still pot composition encounters the E1-Y1 simplex edge. The column can then be operated at λ ) l1/(l1+l2), such that the column draws the new R and ω limit sets of pure Y1 and pure E1 as the distillate and bottoms product, respectively. Thus, entrainer E1 does indeed

allow complete separation of the minimum-boiling azeotrope X1Y1 in a MVC. Similarly, an analysis of the inverse-001 system in Figure 18 shows that it is possible to draw pure X2 by operating the MVC, at λ ) 1 (as a rectifier), on the mixed charge M2 until it encounters the E2-Y2 edge. At this point, λ is switched to λ ) l4/ (l3+l4) such that the column operates in a quasi static mode, and pure E2 is drawn as the distillate, while pure Y2 is drawn as the bottoms product. Thus, E2 is also a perfect entrainer for the separation of the maximumboiling azeotrope X2Y2 in a MVC. It should be noted as an aside that the use of a quasi static mode of operation might not be optimal in terms of product recovery because the still pot residue remaining will not be pure at the end of the operation and the still pot cannot be physically operated until it is empty. This is in contrast to the rectifier/stripper configurations, where the column can be operated until virtually all the light/heavy desired components have been drawn from the column. However, the use of a quasi static mode of operation results in time savings, as two products are drawn at the same time from the column versus the one product drawn at any point in time from the rectifier/stripper configurations. Alternatively, the MVC could be operated in a quasi static mode for most of the step, finishing with a short period of operation as a rectifier or stripper. There is thus a trade-off between higher product recovery versus shorter operating time. This trade-off is explored in detail in our next paper,4 when the MVC operated without a quasi static step is simulated and compared to the simulation results of an identical operation with a quasi static step. The above form of entrainers arise from the fact when a dimension is added to the composition simplex, the pot composition boundary (represented by the azeotropic point XiYi in the original one-dimensional composition simplex of line segment Xi-Yi) does not extend into the two-dimensional composition simplex. Hence, we are able to go around the boundary (rather than across it). Addition of the Bernot et al. “perfect entrainer” has joined the two distillation regions (line segments XiXiYi and XiYi-Yi) via the composition simplex edge of Ei-Yi. Bernot et al.’s approach, however, is limited to residue curve map topologies which do not contain separatrices internal to the composition simplex. The physical likelihood of such intermediate boiling entrainers is typically low. Thus, we would like to highlight another class of perfect entrainers in which separatrices do exist, but are curved. Since curvature is to be expected of most separatrices,8 it should be much easier to find such perfect entrainers of the second class when compared to the ones suggested by Bernot et al. This other class of perfect entrainers is a higher boiling liquid (than either of the two pure components X2 or Y2 and the azeotrope) which forms an inverse-020 system with the maximum-boiling binary azeotrope X2Y2 with a very curved stable separatrix as illustrated by the example of the acetone-chloroform azeotrope, with benzene as an entrainer. Alternatively, for a minimum-boiling azeotrope such as X1Y1, the entrainer would have to be a lower boiling liquid (than either X1 or Y1 and the azeotrope) and form a 020-system with a very curved unstable separatrix in order for complete separation to occur. The underlying factor for the usefulness of these entrainers is the curvature of the separatrix formed. Just as the Bernot et al. “perfect


Figure 19. Operating procedure for separating a minimumboiling azeotrope by adding a lowest boiling entrainer which forms a highly curved unstable separatrix.

entrainer” joins the two distillation regions (line segments Xi-XiYi and XiYi-Yi) via the composition simplex edge Ei-Yi, the curved separatrices in the inverse-020 and 020 systems also joins the two regions which contain each of the desired pure products Xi and Yi via the simplex edge Ei-Yi. However, because of the presence of the separatrix between Ei and XiYi, the MVC still pot composition can only approach the Ei-Yi edge (it does not quite reach it) in the region where the separatrix hugs the Ei-Yi edge (region F in Figure 7). This allows pure Yi to be drawn from the MVC. The separation scheme for the inverse-020 system was already explained in our example with the acetonebenzene-chloroform system (with benzene as the entrainer). The separation scheme for the 020 system is then the reverse of the operating procedure outlined for the A-B-C mixture. This operating procedure is illustrated in Figure 19. An example of such a system is given by the ternary mixture of ethanol, water, and methanol, where methanol is the lighter boiling entrainer which serves to separate the minimum-boiling azeotrope of ethanol and water. Even though the curvature of the unstable separatrix in this mixture is much less pronounced than that shown in Figure 19 (and hence the separation achievable would require even more methanol as the entrainer), this is a particularly interesting process because there are many applications in which methanol is a much more attractive entrainer than the classical heterogeneous entrainer benzene. Addition of batch entrainer Ei must make the azeotrope XiYi a saddle in the ternary residue curve map. Consequently, the entrainer itself must either be a stable node (inverse-020 system) or an unstable node (020 system) when the azeotropes are maximum-boiling and minimum-boiling, respectively. This is to ensure that it is possible to avoid drawing the azeotrope XiYi as one of the products of the MVC at all times, thereby allowing the complete separation of the azeotrope and entrainer into the constituent pure components only. Further, having a separatrix that hugs the edge will allow us to achieve separations with a negligible amount of waste (azeotropic) offcut and is thus highly desirable. However, it is not absolutely necessary, in that the azeotrope can be broken without the separatrix being extremely close to the composition simplex edge. Recovery per batch, however, would be less than perfect. As illustrated in Figure 19, the basic operating procedure of the MVC for separating the minimum-

boiling azeotrope X1Y1 with a lowest boiling point entrainer such as E1 is as follows: (1) Add sufficient entrainer E1 to the original feed F1 such that the desired region (a) is attained at the initial composition of M1. (2) Operate the column as a stripper (λ ) 0), drawing pure X1 as bottoms product, until the E1-Y1 edge/ separatrix is reached. (3) Continue operating the column in a quasi static operation, by setting λ ) l2/(l1+l2), thereby drawing almost pure Y1 as bottoms product and pure E1 as distillate product until the still pot runs dry (ξ f ∞). The above examples of perfect entrainers were focused on the separation of binary azeotropes. However, it is possible that the rationale can be applied to azeotropes of higher dimensionality. The main characteristic of the perfect entrainer is that it connects the otherwise separate distillation regions which contain the desired pure products that make up the azeotrope. This can be achieved via an entrainer which does not add new separation boundaries (such as the Bernot et al. “perfect entrainer”) or an entrainer which forms a separation boundary that is so curved that it is almost touches one of the edges or faces of the composition simplex. This allows different regions to be joined, and separation of the azeotrope into its constituent pure components is thus achievable. The point, however, is to recognize that there is a possibility that the addition of the correct entrainers in appropriate phases of the operating procedure could result in the complete separation of an azeotrope into its constituent pure components without the need for an azeotropic recycle. Comparison to Batch Extractive Distillation The use of batch extractive distillation (BED) for breaking azeotropic mixtures was first proposed by Yatim et al.16 and its basic operational policy was detailed by Lang et al.17 Further work on its ability to break azeotropic mixtures in a batchwise manner has also been conducted.18-20 In a BED process, the entrainer is added continuously into the column throughout the duration of the operation. However, there are several reasons that make the MVC with a batchwise entrainer superior in terms of flexibility when compared to the BED process with a continuous entrainer. Feasibility of the BED process is based on a specific residue curve map topology and the use of nonlimiting operating conditions in the rectifying section of the column,20 such that rather than drawing the R limit set of the basic distillation region (a minimum-boiling azeotrope) as the product, some other composition which is closer in composition to a pure product is drawn as the distillate. This requires an exact determination of the number of trays required in the column to achieve that particular desired composition.20 If there are too few trays, the azeotropic mixture cannot be sufficiently separated (the required purity is not achieved), and if there are too many trays, the azeotropic mixture is separated “too well”, such that the composition of the distillate product drawn out of the column moves closer to that of the minimum-boiling azeotrope instead of the desired nearly pure product. Because of this need for an exact calculation of the number of trays required for a given mixture, the BED column required for a given mixture would not necessarily be suitable for another mixture.


In contrast, for the MVC, in which a limiting analysis is employed to determine feasibility, addition of more trays would only make the separation sharper but not change the qualitative results. Thus, a given MVC with a sufficiently large number of trays can be used in batchwise entrainer mode for the separation of almost any azeotropic mixture (given a suitable entrainer), making it much more flexible than the BED process. Further, the limiting analysis of the MVC developed in this series of papers is ideal for identifying separation opportunities in a multipurpose solvent recovery facility in which the azeotropic mixtures to be separated are constantly changing. Finally, because of the fact that feasibility of BED separation is based on nonlimiting conditions in the BED column, it is usually difficult to obtain products of arbitrarily high purity.20 This is in contrast to the separation offered by the MVC in batch entrainer mode, for which the pure products (R and ω limit sets) can always be drawn at 100% purity in the limit of an infinite number of trays and infinite reflux/reboil ratios. While the flexibility of the BED does not compare to that of the MVC, it should be noted that there are advantages associated with using a BED. Use of a MVC with a continuous entrainer feed will result in a smaller required middle vessel capacity when compared to a batchwise entrainer. Furthermore, the BED can be applied to an existing rectifier by modifying the rectifier to accept the entrainer feeding. However, if there are no existing rectifiers, and a plant is to be built from scratch,1 the MVC will probably still be preferable as it is obviously more flexible than the BED in the separation of azeotropic mixtures. Conclusions In this paper our theoretical analysis of the separation of azeotropic mixtures in a MVC1 was extended to mixtures with curved separation boundaries. It was shown that, in the presence of curvature in the separatrices (which form MVC batch distillation boundaries for some values of λ), MVC batch distillation boundaries that exist under the assumption of straight line separatrices may no longer be boundaries. This resulted in added separation possibilities not previously thought to be possible under the assumption of straight line separatrices. One such example was illustrated for the mixture of acetone, benzene, and chloroform. The possibility of separating any composition of this mixture into its individual pure components was then interpreted as the use of benzene as a batchwise entrainer to split the binary azeotrope of acetone and chloroform. It was then highlighted that the operating procedure of using benzene as a batchwise entrainer in the MVC to split the acetone-chloroform azeotrope was a time-domain analogue to the continuous distillation column sequence proposed by Wahnschafft et al.6,7 (which was the spacedomain equivalent). The separation proposed in this paper, to be conducted in three operational steps over time in a single MVC column, was proposed analogously by Wahnschafft et al. in the space domain via a sequence of three continuous columns operating at the same time. Building on the analysis of the operating procedure for separating the acetone and chloroform azeotrope in a MVC using benzene as a batchwise entrainer, a short discussion regarding the characteristics of a perfect

Table 3. Fixed Points of the Acetone-Benzene-Chloroform System fixed point

xAcetone

xBenzene

xChloroform

characteristic

A B C AC

1.0 0.0 0.0 0.3455

0.0 1.0 0.0 0.6545

0.0 0.0 1.0 0.0

unstable node stable node unstable node saddle point

entrainer (for use in a MVC) was provided. It was shown that, in addition to the Bernot et al. “perfect entrainer”, there are also the perfect entrainers which form inverse020 and 020 systems with maximum- and minimumboiling binary azeotropes, respectively. These two classes of “perfect entrainers” would allow us to separate the azeotrope and entrainer completely into their pure constituent components, without the need of an azeotropic offcut recycle. This would prove especially useful for the class of batch processes in which only a few batches are made, and a substantial recycle is undesirable. It was also shown that the MVC in batchwise entrainer mode is more flexible and offers higher product purity than a BED process. Finally, we suggest some directions for future work on the topic of the middle vessel column. An in-depth study on the behavior of the MVC under nonlimiting conditions should be conducted using the ABACUSS model developed for the MVC.4 As shown by Wahnschafft et al.6 and supported by Stichlmair et al.21 and Laroche et al.,5 it is often beneficial to operate a column at finite reflux ratios as compared to infinite reflux ratios; thus, the MVC operated at finite reflux ratios could also offer a wider variety of separations. An in-depth study of the optimal control problems associated with the operation of the MVC either to mimic an operation usually conducted in a continuous column/sequence of continuous columns or to optimize the separation in an MVC with respect to minimum cost is also worthwhile. Openloop optimal control policies can thus be developed for the MVC on the basis of the understanding of the qualitative dynamics of the MVC developed in this paper. Last, a study of feasible entrainers for the separation of azeotropes in a MVC should be conducted because a perfect entrainer is not always available for a given azeotrope to be separated. An understanding of the characteristics of a feasible entrainer would ensure a higher probability that an entrainer can be found for all azeotropic mixtures to be separated in a MVC. Acknowledgment This work was supported by the U. S. Department of Energy under Grant DE-FG02-94ER14447 and the MIT Undergraduate Research Opportunities Program. Appendix Data for the Ternary System of Acetone, Benzene, and Chloroform. The residue curve map for the acetone, benzene, and chloroform system at atmospheric pressure was presented in Figure 1. There is only one azeotrope exhibited by this ternary system, which is given by the binary mixture of acetone and chloroform. The composition of this binary azeotrope was calculated to be (0.3455, 0.0, 0.6545). This is in close agreement with the experimental values obtained for the composition of the acetone-chloroform azeotrope, which was found to range between (0.3400, 0.0, 0.6600) and (0.3607,


0.0, 0.6393).22,23 The stability and character of each of the fixed points in the A-B-C system are summarized in Table 3. Literature Cited (1) Cheong, W.; Barton, P. I. Azeotropic distillation in a middle vessel batch column. 1. Model formulation and linear separation boundaries. Ind. Eng. Chem. Res. 1999, 38, 1504-1530. (2) Ahmad, B. S.; Barton, P. I. Homogeneous multicomponent azeotropic batch distillation. AIChE J. 1996, 42, 3419-3433. (3) 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, 474-485. (4) Cheong, W.; Barton P. I. Azeotropic distillation in a middle vessel batch column. 3. Model validation. Ind. Eng. Chem. Res. 1999, 38, 1549-1564. (5) Laroche, L.; Bekiaris, N.; Andersen, H. W.; Morari, M. Homogeneous azeotropic distillation: Separability and flowsheet synthesis. Ind. Eng. Chem. Res. 1992, 31, 2190- 2209. (6) Wahnschafft, O. M.; Koehler, J. W.; Blass, E.; Westerberg, A. W. The product composition regions of single-feed azeotropic distillation columns. Ind. Eng. Chem. Res. 1992, 31, 2345-2362. (7) Wahnschafft, O. M.; Le Rudulier, J. P.; Westerberg, A. W. A problem decomposition approach for the synthesis of complex separation processes with recycles. Ind. Eng. Chem. Res. 1993, 32, 1121-1141. (8) Reinders, W.; De Minjer, C. H. Vapor liquid equilibrium in ternary systems. Recl. Trans. Chim. 1940, 59, 207-230. (9) Bernot, C.; Doherty, M. F.; Malone, M. F. Feasibility and separation sequencing in multicomponent batch distillation. Chem. Eng. Sci. 1991, 46, 1311-1326. (10) Aspen Technology. ASPEN PLUS User Manual Release 9; Aspen Technology Inc.: Cambridge, MA, 1995. (11) Skogestad, S.; Wittgens, B.; Litto, R.; Sorensen, E. Multivessel batch distillation. AIChE J. 1997, 43, 971-978. (12) Barolo, M.; Botteon, F. Simple method of obtaining pure products by batch distillation. AIChE J. 1997, 43 (10), 2601-2604.

(13) Hasebe, S.; Kurooka, T.; Aziz, B. B. A.; Hashimoto, I.; Watanabe, T. Simultaneous separation of light and heavy impurities by a complex batch distillation column. J. Chem. Eng. Jpn. 1996, 29, 1000-1006. (14) Davidyan, A. G.; Kiva, V. N.; Meski, G. A.; Morari, M. Batch distillation in a column with a middle vessel. Chem. Eng. Sci. 1994, 49, 3033-3051. (15) Barolo, M.; Guarise, G. B.; Rienzi, S. A.; Trotta, A. Understanding the dynamics of a batch distillation column with a middle vessel. Comput. Chem. Eng. 1998, 22, S37-S44. (16) Yatim, H.; Moszkowicz, P.; Otterbein, M.; Lang, P. Dynamic simulation of batch extractive distillation process. Comput. Chem. Eng. 1992, 17 (S), S57-S62. (17) Lang, P.; Yatim, H.; Moszkowicz, P.; Otterbein, M. Batch extractive distillation under constant reflux ratio. Comput. Chem. Eng. 1994, 18, 1057-1069. (18) Safrit, B. T.; Westerberg, A. W.; Diwekar, U.; Wahnschafft, O. M. Extending continuous conventional and extractive distillation feasibility insights to batch distillation. Ind. Eng. Chem. Res. 1995, 34, 3257-3264. (19) Safrit, B. T.; Westerberg, A. W. Improved operational policies for batch extractive distillation columns. Ind. Eng. Chem. Res. 1997, 36, 436-443. (20) Lelkes, Z.; Lang, P.; Benadda, B.; Moszkowicz, P. Feasibility of extractive distillation in a batch rectifier. AIChE J. 1998, 44, 810-822. (21) Stichlmair, J.; Fair, J. R.; Bravo, J. L. Separation of azeotropic mixtures via enhanced distillation. Chem. Eng. Prog. 1989, 85, 63-69. (22) Horsley, L. H. Azeotropic Data; American Chemical Society: Washington, DC, 1952. (23) Horsley, L. H. Azeotropic Data; American Chemical Society: Washington, DC, 1962.

Received for review July 17, 1998 Revised manuscript received December 17, 1998 Accepted December 18, 1998 IE980470Q

Azeotropic Distillation in a Middle Vessel Batch Column. 2. Nonlinear

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