Azeotropic Distillation in a Middle Vessel Batch Column. 1. Model

1504

Ind. Eng. Chem. Res. 1999, 38, 1504-1530

Azeotropic Distillation in a Middle Vessel Batch Column. 1. Model Formulation and Linear 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

A mathematical model for the middle vessel batch distillation column (MVC) is developed using the concept of warped time analysis and used to study the qualitative dynamics of the MVC when it is used to separate multicomponent azeotropic mixtures. A limiting analysis is then developed for a MVC with an infinite number of trays, operated under infinite reflux/reboil ratios, under the assumption of linear separation boundaries. It is determined that, under limiting conditions, the distillate product drawn from the MVC is given by the R limit set of the MVC still pot composition, while the bottoms product drawn from the MVC is given by the ω limit set of the MVC still pot composition. The net product composition is determined by taking a convex combination of the two products. The notions of steering the still pot composition, the vector cone of possible motion for the still pot composition, and the equivalency of the MVC to the combined operation of a batch rectifier and a stripper are also explored. The definition of batch distillation regions for the MVC operated at a given value of the middle vessel parameter λ, and the bifurcation of these regions with the variation of λ, are investigated. Lastly, a mathematical model incorporating the concept of warped time is developed for a multivessel column. The MVC can be viewed as a specific case of the multivessel column. Introduction It has long been recognized that continuous distillation is much more energy-efficient and less laborintensive than batch distillation.1 However, batch distillation continues to be an important technology because of the greater operational flexibility that it offers, which makes it particularly suitable for smaller, multiproduct or multipurpose operations such as manufacturing in the pharmaceutical and speciality chemical industries where products are typically required in small volumes, and are subject to short product cycles and fluctuating demand. With the advancement of chemistry and biotechnology, the pharmaceutical and speciality chemical industries have grown in importance in recent years, resulting in a renewed academic interest in batch distillation processes. In particular, much attention has been devoted to a novel batch distillation column configuration, termed by various researchers as the complex batch distillation column,1,2-4 batch distillation column with a middle vessel,5,6 or just simply a “middle vessel column (MVC)”.7-10 In this paper, we will use the term middle vessel column (MVC) to denote this novel batch distillation configuration. Traditionally, the most common type of batch distillation columns were batch rectifiers or “regular” columns, for which the feed is charged into a large reboiler at the bottom of the rectifying column and the lighter components are removed from the top of the column. Less frequently used are batch strippers or “inverted” columns, where the feed is charged into a holdup tray at the top of the stripping column, and the heavier components are withdrawn from the bottom of the column. * Corresponding author. Phone: +1-617-253-6526. Fax: +1617-258-5042. E-mail: [email protected].

Recently, study on the feasibility of product sequences and optimal sequencing of columns has led researchers to reconsider a column configuration first proposed by Gilliland and Robinson in 1950.11 In this “novel” configuration, the feed from the vessel with large holdup is introduced in the middle of the column, such there is both a stripping section and a rectifying section in this column. Two variations have been proposed respectively by Hasebe et al.4 and Davidyan et al.5 Hasebe et al. proposed employing two batch rectifiers, with the distillate vapor outlet and liquid inlet in the first column (which acts as the stripping section) fed into/from the bottoms holdup vessel of the second batch rectifier which serves as the rectifying section of the middle vessel column. Alternatively, Davidyan et al. proposed introducing the liquid feed from a holdup tank into the middle of a traditional continuous column with a reboiler and a condenser, and with an option of introducing and removing heat from the holdup tank. Hasebe et al.’s configuration is helpful in a plant that already has existing batch rectifiers, whereas Davidyan et al.’s configuration can easily be applied to a modified continuous distillation column, as realized in the experiments of Barolo et al.2,3 Despite the large number of papers published on the middle vessel column, there has yet to be a paper which satisfactorily explains and characterizes the qualitative dynamics of the MVC completely, particularly for the separation of azeotropic mixtures. Davidyan et al.5 conducted a rigorous mathematical analysis of the MVC using a model assuming constant relative volatility, characterizing the dynamic behavior of the MVC. Meski and Morari6 then provided a limiting analysis of a mathematical model for the MVC, assuming negligible holdup on the trays and constant molar overflow. Their work was based on the model proposed by Devyatikh and Churbanov12 who proposed the use of the middle

10.1021/ie980469r CCC: $18.00 © 1999 American Chemical Society Published on Web 02/09/1999

Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1505

vessel column to achieve separations of higher purity. The possibility of azeotropes was however neglected in their analysis. The use of continuous entrainers in the MVC was explored by Safrit et al.,7,8 who showed that it was possible to “break” an azeotrope using a suitable entrainer added continuously over the entire operation of a MVC. Lelkes et al.13 showed that Safrit et al.’s7,8 qualitative explanation of feasibility for batch extractive distillation is incorrect, and provided a valid alternative explanation. In addition, Safrit et al. mentioned that it was possible to “steer” the still pot composition of the MVC. However, they stopped short of quantifying and qualifying the direction of this steerage. They also mentioned that the MVC had a characteristic distillation region which is a combination of the rectifying batch distillation region and the stripping batch distillation region, but it was not specified as to how these regions were related. In this paper, we will attempt to fill the gaps left by their work. Safrit and Westerberg9,10 also included the MVC in their algorithm for determining the optimal sequencing of batch distillation columns, but again stopped short of fully characterizing the MVC both mathematically and graphically. A critique of Safrit and Westerberg’s work, highlighting the shortcomings in their analysis, is provided in Cheong.14 Finally, Barolo et al.2,3 attempted an experimental characterization of a MVC using a holdup tank fed into an existing continuous distillation column (which usually has both a rectifying and a stripping section). Their results appear to be in good agreement with the theoretical models proposed thus far. A recent paper was also published on their attempts to experimentally characterize the performance of a MVC with respect to its operational parameters.15 Barolo et al. also built a computational model in which heat and tray holdup effects were considered and compared the results of the simulation with the experimental results. In summary, the existing literature has stopped short of fully developing a model suitable for characterizing the qualitative dynamics of the MVC: ideal mixtures were assumed,5 or the possibility of azeotropes were neglected,6 the use of continuous entrainers with the MVC has been explored,7,10 and the dynamic performance of the MVC investigated with respect to its operational parameters despite the less than satisfactory understanding of qualitative dynamics. Relatively rigorous models of the MVC have been built and simulated,15 but because of the complexity of the model, no attempt was made at a thorough qualitative theoretical analysis. It is thus the aim of this paper to bridge this gap, to gather the current work on MVCs, building on it to form a model of the MVC suitable for characterizing the qualitative behavior, especially in the presence of azeotropes. This paper is in three sections. In the first section, the mathematical model of the MVC is developed and a theoretical and graphical interpretation is provided for the model. The equivalence of the MVC to infinitesimal strippers and rectifiers is also explored. In the second section, a discussion about the similarities and differences of distillation lines versus residue curves is provided, followed by a limiting analysis, in which the operation of the MVC at infinite reflux/reboil ratios and an infinite number of trays is analyzed. A bifurcation analysis of the MVC batch distillation regions as a function of the middle vessel parameter λ is also

Figure 1. Schematic configuration of a MVC.

conducted, followed by a discussion on the equivalency of MVCs with the combined operation of strippers and rectifiers when both are operated under limiting conditions. The final section discusses a model of the multivessel column, which may also prove useful in the separation of multicomponent azeotropic mixtures. Basic Model of the MVC A coherent mathematical model is developed for the MVC, similar to previous work published on this subject by Davidyan et al.5 and Meski and Morari.6 It can thus be treated as an extension of the model developed by them. However, Davidyan et al. assumed ideal mixtures with constant relative volatility in their analysis, obtaining the solutions of the model for ideal mixtures. Meski and Morari included the following in their specifications: (1) the removal of only pure components as products, (2) specific analysis of only binary and ternary mixtures, and (3) the assumption of nonazeotropic mixtures. It was mentioned in their paper6 that their analysis should extend to azeotropic mixtures, but they did not quantify this statement mathematically and provided only a vague graphical representation. In this paper, the above assumptions will be relaxed. Development of the Model. A schematic configuration of the middle vessel column is shown in Figure 1. The middle vessel is actually located on ground level to avoid unnecessary structural difficulties in supporting a heavy vessel in midair. The schematic with the

1506 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999

middle vessel suspended between the stripping and rectifying sections is only used for ease of representation. However, it should be noted that to maintain symmetry of the model, we specify that vapor from the stripping section of the column is bubbled into the holdup vessel, where it is equilibrated with the liquid in the holdup vessel, before it is fed into the rectifying section of the column. This is in contrast to the MVC configurations proposed by Hasebe et al.4 and Davidyan et al.,5 where the vapor stream from the stripping section bypassed the holdup vessel and is introduced directly into the rectifying section of the column. This results in differences in the column composition profile between our MVC configuration and that of Hasebe et al. and Davidyan et al., but will not affect the qualitative conclusions. As with Davidyan et al. and Meski and Morari, the model assumes (1) constant molar overflow (CMO) and (2) quasisteady state (QSS) in the column due to negligible holdup of liquid and vapor in the stages, and is based on a differential model of the rate of composition change in the middle vessel. The column has ND trays and a total condenser in the rectifying section and NB trays and a total reboiler in the stripping section. A total reboiler is not usually used in industry, but for the purpose of symmetry in the model, we will assume its use. The use of a typical kettle reboiler/partial reboiler will result in the equivalent of one extra stage of separation in the stripping section, but will not affect the nature of our results. The mixture to be distilled has NC components and is characterized with nonideal vapor-liquid equilibrium (VLE) models such as the nonrandom two liquid (NRTL) or Wilson local-composition activity coefficient models. Total molar holdup in the middle vessel is M, and two different vapor and liquid flow rates exist respectively in the stripping and rectifying sections. In the rectifying section, the vapor and liquid flow rates are Vd and Ld, respectively, while in the stripping section, the corresponding vapor and liquid flow rates are Vb and Lb, respectively. Note that, other than the reboiler and condenser heat duties, there is also an optional heat exchanger (heating/cooling) at the middle vessel so as to allow differing vapor flow rates (Vd and Vb) in the two sections of the column. Distillate is drawn at a flow rate of D from the total condenser, and a bottoms product is drawn at a flow rate of B from the total reboiler. Finally, we complete the preliminaries by defining our dimensionless “middle vessel parameter” as λ ∈ [0, 1], where

λ)

D D+B

(1)

As the product withdrawal rates in a batch distillation column, D and B, can vary with time, it should be noted that λ may also be a function of time. Considering the overall and component mass balances around the whole column, and introducing a dimensionless warped time ξ in the spirit of Bernot et al.16 as

dξ )

(D M+ B) dt

(2)

we obtain for i ) (1, ..., NC)

dxM i D B ) xM i - λxi - (1 - λ)xi dξ

(3)

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

dxM ) xM - λxD - (1 - λ)xB dξ

(4)

Superscripts indicate location of the composition, M for middle vessel, D for distillate product, and B for bottoms product. The warped time can also be re-expressed as

dξ ) -d(ln M)

(5)

The detailed derivation of these equations is provided in Appendix A. Note that we make no assumptions about the actual composition of the products drawn from the distillate, xD or the bottoms, xB. On the basis of the QSS assumption, the instantaneous compositions of these products are a function of the operating parameters of the column (number of trays ND, NB; pressure, P in the column which may or may not be a function of ξ) and are related to the composition of the still pot, at a given warped time ξ, by static mass-balance and phase-equilibrium relationships which involve the vapor and liquid flow rates in both sections of the column, Vd(ξ), Ld(ξ), Vb(ξ), and Lb(ξ). xD and xB can thus be expressed in a simplified form as the composite functions:

xD(ξ) ) xD(P(ξ),ND,NB,Vd(ξ),Ld(ξ),Vb(ξ),Lb(ξ),xM(ξ)) (6) and

xB(ξ) ) xB(P(ξ),ND,NB,Vd(ξ),Ld(ξ),Vb(ξ),Lb(ξ),xM(ξ)) (7) In the presence of finite reflux and reboil ratios, eqs 6 and 7 can be further simplified by introducing a reflux ratio Rd(ξ) for the rectifying section and a reboil ratio Rb(ξ) for the stripping section, so that xD and xB are re-expressed as

xD(ξ) ) xD(P(ξ),ND,NB,Rd(ξ),Rb(ξ),xM(ξ))

(8)

xB(ξ) ) xB(P(ξ),ND,NB,Rd(ξ),Rb(ξ),xM(ξ))

(9)

and

The details of the formulation for the set of equations, (6) and (7) or for (8) and (9) are encapsulated in the algebraic mass balances which can be written for the trays in the column. Details of these equations are included in Appendix A. As formulated, this model treats Rb, Rd, and λ as independently specifiable parameters. An analysis presented in part 217 shows that if heat is not added/ removed from the middle vessel, then for non-zero flows to exist in the MVC, these parameters must satisfy the following relation:

(Rb + Rd + 1)λ ) Rb

(10)

However, as Rb and Rd tend toward large values (e.g., in the case of the limiting analysis presented in this paper), they may be chosen to yield any desired value for λ without significantly altering the composition of the top and bottom products. It is also shown in part


Figure 2. Vector cone of possible still pot composition movement at a given point in time.

217 that if heat can be added/removed from the middle vessel, the constraint given by eq 10 no longer needs to be satisfied. However, when applying this simplified model to a MVC with no heat addition/removal from the middle vessel, it is important to keep in mind that this implicit constraint exists on the specifications that may be made. It should be noted that the model described here is a generalization of previous batch distillation models, since it encompasses both the batch rectifier (λ ) 1) and the batch stripper (λ ) 0) as special cases. A Graphical Interpretation of the Model. An interpretation of eqs 4-7 is as follows: The still pot composition in the MVC moves away from the instantaneous top and bottom product compositions, in a direction that lies in a vector cone swept out by two vectors: one connecting the distillate composition xD to the still pot composition xM, and one connecting the bottoms composition xB to the still pot composition xM. The actual direction is given by the relative weights on these vectors, given by λ and 1 - λ, respectively. This concept is illustrated in Figure 2. This interpretation is easily obtained via an elementary rearrangement of eq 4, which gives

dxM ) (λ)(xM - xD) + (1 - λ)(xM - xB) dξ

(11)

where the direction of change in the pot composition dxM/dξ is given by the direction of vector (xM - xD) (which points in the direction away from the distillate product composition toward the still pot composition) proportionally weighted with λ, and the vector (xM xB) (which points in the direction away from the bottoms product composition toward the still pot composition) proportionally weighted with 1 - λ. This concept is graphically illustrated in Figure 3 in three forms. In Figure 3a, the appropriate weight is applied to each of the vectors, (xM - xD) (with weight λ) and (xM - xD) (with weight (1 - λ)). These two weighted vectors were added vectorially to obtain the direction and magnitude of motion for the still pot composition. In an alternative representation, Figure 3b shows the application of the ratio theorem to the vector cone emanating from xM. The direction of motion is given by the appropriate division of the line segment Rβ connecting the heads of the two vectors emanating from the still pot composition, in the ratio of λ and 1 - λ at the point γ. γ - xM then gives the direction of motion of the still pot.

Finally, in Figure 3c, a third interpretation that is based on the form of eq 4 is shown. The still pot composition xM moves directly away from the net product composition point xP given by the weighted average of the two product compositions: xP ) λxD + (1 - λ)xB. Equivalently, xP gives the instantaneous composition of the combined product (distillate and bottoms) drawn from the column. Thus, by varying the value of λ (the MVC parameter) a whole range of possible still pot composition paths, as swept out by the vector cone between xM - xD and xM - xB, are possible. By varying the value of λ with time, we are also able to “steer” the composition in the still pot in different directions as we progress in time. This was an idea first mentioned by Safrit et al.,7-10 but it was not elucidated how this steering could be achieved. A graphical interpretation of this behavior is shown in Figure 4. We should note that xD and xB both vary with time when there are finite number of trays and finite reflux and reboil ratios. Thus, a clever manipulation of λ(ξ), keeping in mind this variation of xD(ξ) and xB(ξ), would result in the ability to steer the still pot composition in a desired manner as shown in Figure 4. Determining the operating profile of λ(t) to obtain the desired still pot composition path in a column with a finite number of trays and finite reflux/reboil ratios poses a challenging open-loop optimal control problem, and will not be covered in detail in this paper. However, an extension of this analysis to the limiting behavior in the presence of an infinite number of trays and an infinite reflux/reboil ratio will be presented later in this paper. The above analysis is also applicable to higher dimensional systems and systems with azeotropes. To illustrate this point, Figures 2-4 are reproduced for a generic quaternary system with azeotropes, in Figures 5-7. As we can see from Figure 5, the concept of the vector cone between (xM - xD) and (xM - xB) is NOT affected by the presence of separatrices in the composition space. As before, the still pot composition can move in any direction within the vector cone. However, bundles of trajectories containing these separatrices form the boundaries of the basic distillation regions and in some cases these boundaries may be the pot composition boundaries for the MVC. In the presence of linear boundaries, this results in a limited variety of separations as the movement of the still pot composition is restricted to lie within the region bounded by the pot composition barrier.18 These pot composition boundaries and barriers were defined by Ahmad and Barton, and their characteristics are explained in detail in their work.18 The actual direction of motion is still given by the weighted average of the two vectors xM - xD and xM xB, unaffected by the presence of azeotropes or the move to a higher dimension, as illustrated in Figure 6. The presence of azeotropes only serves to distort the residue curve map by introducing pot composition boundaries (composed of bundles of trajectories that include separatrices) into the composition space, which affect the range of possible values of xD and xB, but once these values are determined, the direction of motion of the still pot is dictated by eq 11. The concept of steering the still pot composition is also unaffected by the move


Figure 3. Three ways of representing the weighted average motion.

to a higher dimensional system and the presence of azeotropes, as seen in Figure 7. Finally, following the argument presented above, this analysis also applies to n-component systems with n > 4, but because of difficulties in representing this on twodimensional paper, systems of higher dimension are not represented geometrically in this paper. It should be noted, however, that there is one major difference between a quaternary system and a ternary system. The vector cone of possible motion in the quaternary system remains as a two-dimensional cone, despite a three-dimensional composition space for a quaternary mixture (as was illustrated in Figure 7). The MVC still pot composition motion is restricted to lie in the plane x1 + x2 + x3 ) 1 in the ternary case, but the motion of the MVC still pot composition in higher dimensions is not restricted to a two-dimensional plane. However, as only two products are drawn at any time, there are only 2 degrees of freedom for the composition

motion, resulting in a two-dimensional vector cone. Thus, the possible directions of motion for the MVC still pot composition lies in a two-dimensional plane, as defined by the three points xB, xD, and xM. In the ternary case, this plane is redundant with the summation of mole fractions. From this analysis, we conjecture that it may be useful to have a third product stream drawn from the column in a quaternary system, which would result in a three-dimensional vector cone of possible motion for the still pot composition, thereby removing any restrictions in the dimensionality of the motion. This provides an impetus for recognizing the usefulness of multivessel columns for systems of higher dimension, where the composition space is (n-1)-dimensional (n > 4), and restrictions in the dimensionality of the motion can be removed by drawing n - 1 streams from each of the n - 1 holdup trays in the multivessel column. A brief note


Figure 4. Dynamic steerage of still pot composition by varying λ(t).

Figure 5. Vector cone of possible still pot composition movement, four-component system.

regarding the multivessel column will be presented in the last section of this paper. Equivalence of MVC with Infinitesimal Rectifiers and Strippers. The MVC can be interpreted as

being equivalent to an operating schedule in which the still pot holdup is continuously transferred between a batch rectifier and a batch stripper, and operated for infinitesimally short periods of time in both the rectifier and the stripper. To illustrate this idea, consider the ternary composition space with no azeotropes as given in Figure 8a, with the middle vessel composition path given by a value of λ ) 0.5, resulting in the instantaneous direction given by the vector (λ)(xM - xD) + (1 - λ)(xM - xB). However, the direction vector of a mixture being distilled in a rectifier would have been given by (xM - xD), while the direction vector of a mixture distilled from the still pot of a stripper would have been given by (xM - xB). Thus, the still pot in the middle vessel is fractionally behaving like a rectifier with ND trays and fractionally behaving like a stripper with NB trays. Therefore, we could interpret the direction of motion for the MVC still pot composition as being an infinitesimal move in the direction of xM - xD (i.e., acting as a rectifier with ND trays) followed by an infinitesimal


Figure 8. Pot composition path using a MVC vs a stripper and a rectifier.

Figure 6. Three ways of representing the weighted average motion, four-component system.

Figure 7. Dynamic steerage of still pot composition by varying λ(t), four-component system.

move in the direction of xM - xB (i.e., acting as a stripper with NB trays), with the weights of each move given

by λ and 1 - λ, respectively. This representation is shown in Figure 8b. It should be noted that, after the initial infinitesimal batch rectification move, the still pot composition will be on a new residue curve; thus, the product composition of the infinitesimal batch stripping move will differ from the bottoms product obtained from the MVC. However, since the initial rectification move is infinitesimal, these bottoms products will be equivalent to a first approximation. As the magnitude of this infinitesimal move tends toward zero, the product composition drawn from the stripping operation will be exactly that of the bottoms product in the MVC. Equivalently, the order of the infinitesimal moves could be reversed (distillation in a stripper, followed by distillation in a rectifier) with no loss of generality. This two-step operation will again be equivalent to that of a MVC, with the composition of the products drawn from the distillation operation approaching that of the distillate product drawn from the MVC as the infinitesimal move approaches zero. This problem of having a slightly different product in the second operation (of the equivalent two-step operation) will not occur as the reflux/reboil ratio and the number of stages approaches infinity. This will be explained in detail when we consider the limiting behavior of the MVC. With this theoretical equivalency of the MVC and the combined operation of the batch stripper/batch rectifier at limiting conditions, it appears that the MVC is perhaps irrelevant. However, this is not the case, as there appears to be advantages (increased separation possibilities) with operating a MVC at finite reflux/reboil ratios and finite number of trays.17 Further, the MVC would be needed to avoid an infinity of transfers. The advantages and disadvantages of using a MVC will also be explored in greater detail in a later section.


Theoretical Analysis of the Limiting Behavior of the MVC Model In the presence of a finite number of trays and finite reflux and reboil ratios, the dynamic variation of the product compositions result in a model that is impossible to analyze theoretically. A theoretical analysis on the behavior of the still pot composition in the MVC can, however, be conducted for the limiting case of infinite reflux and reboil ratios and infinite number of trays. Our analysis builds on the work of Van Dongen and Doherty19 on the similarity of simple distillation residue curve maps with distillation lines for batch rectification, under the limiting conditions of large N (infinite number of trays) and large r (infinite reflux ratio). Bernot et al. also examined the dynamics of the holdup vessel composition in a batch rectifier16 and a batch stripper20 using the same method of analysis. This section is in five parts: the first part clarifies the differences between residue curves and distillation lines (or total reflux column profiles) and justifies the use of a residue curve as an approximation to the column composition profile at high reflux/reboil ratios (as suggested by Van Dongen and Doherty19). Next, the basic tools developed by Van Dongen and Doherty19 and Bernot et al.16,20 are introduced, and an analysis is conducted on a simple nonazeotropic ternary system, and a complex ternary system of acetone-chloroformmethanol which exhibits multiple binary and ternary azeotropes. The third part of this section extends the analysis to higher dimensional systems, with the number of components n g 4. Systems with n g 5 are hard to represent geometrically on two-dimensional paper and are not represented in this paper, but the analysis will extend accordingly to them. The fourth part explores the definition of an Ewell and Welch-type “batch distillation region”21 in the context of a MVC, which will be a function of the MVC parameter λ. A bifurcation analysis is also conducted to identify the transformation of these batch distillation regions, and the nature of these transformations, as a function of the parameter λ. It should be noted that the analyses conducted in this paper are based on the assumption of straightline separatrices which greatly simplifies the actual analysis. However, almost all separatrices are curved as Reinders and De Minjer22 pointed out. In fact, as shown by Cheong and Barton,17 it is precisely this curvature that gives rise to interesting possibilities in the separation of azeotropic mixtures in a MVC. Lastly, the equivalency between the MVC and the combined operation of rectifier and strippers is examined under the limiting conditions of operation, and the limits of this equivalency and the advantages and disadvantages of each approach are discussed. Nonequivalence of Residue Curves and Distillation Column Profiles. Van Dongen and Doherty19 examined the dynamics of the batch rectification process, comparing them with the traditional simple distillation residue curve maps and came to the conclusion that, at very high reflux ratios (r g 7), there is very little error in the approximation that xD and xM both lie on the same simple distillation residue curve and that the simple residue curve approximates the distillation composition profile in a packed column. Since Van Dongen and Doherty’s definitive work, many researchers have used the residue curves and the composition profile in distillation columns interchangeably. They often point to the similarity of the equation

describing the simple distillation processes

dxM ) xM - yM(xM) dξ

(12)

to that of the batch distillation process

dxM ) xM - xD dξ

(13)

and claim that, at total reflux, the simple distillation residue curve describes the column profile in the column. They then go on to use the residue curves as an approximation of the column composition profile even at high (but not infinite) reflux ratios (r > 7). However, as highlighted by Widagdo and Seider,23 the simple distillation residue curve and the actual composition profile in the column are never equivalent, even if they may asymptotically approach each other under limiting conditions. Widagdo and Seider then clarified the definition of distillation lines as lines which describe the composition profile in a distillation column under total reflux conditions. For packed bed columns, the distillation line is a continuous line profile (because of the continuous nature of the column) which starts off at the feed zone composition and extends as a function of the height of the column, while for tray columns (for which a discrete composition exists for each tray) the distillation line is a set of discrete points starting at the feed tray composition and each related to the next tray composition by the vapor-liquid equilibrium on each tray and the mass-balance relationship between passing streams. It was then pointed out in their paper that distillation lines for a continuous column are always more curved than the residue curve that passes through the feed tray composition. The above observation was confirmed by Wahnschafft et al.24 when they simulated the operation of a continuous column for the separation of a mixture of acetone, benzene, and chloroform. However, column composition profiles for a MVC when applied to an azeotropic mixture have never been reported. It should be noted that these column profiles for a MVC are different from those in a continuous column because the middle vessel composition lies on the column profile curve, unlike the feed to a continuous column. Further, the top and bottom products are not constrained to be collinear with the feed composition as is the case in a continuous column. To illustrate this difference, column composition profiles, for two cases, (1) low reflux case and (2) total reflux case, were calculated (using the 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.] process modeling environment). An example of such a column composition profile for the MVC at low reflux and reboil ratios, with the corresponding residue curve passing through the feed tray/middle vessel composition, is shown in Figure 9 for the mixture of acetone, benzene, and chloroform, with 100 trays and a reflux/reboil ratio of 5. Note that the middle vessel composition and the top and bottom product compositions are not collinear. A further example for the MVC is provided in which the curvature of the column composition profile causes it to cross the separatrix of the residue curve map at finite reflux rates. The system is acetone, benzene, and


Figure 9. Middle vessel column composition profiles compared to the corresponding residue curve at finite reflux. The solid line is a separatrix.

Figure 10. Middle vessel column composition profiles cross a separatrix at finite reflux.

chloroform as before, but the residue curve selected now lies close to the stable separatrix of the residue curve map. The column composition profile depicted in Figure 10 is for a column with 100 trays, operated at Rd ) Rb ) 5, and because of the extreme curvature of the column composition profile, it unambiguously crosses this separatrix into the neighboring simple distillation region. Widagdo and Seider also point out that as the number of trays approaches infinity, the limiting compositions in the top and bottom trays of the continuous column approach the fixed points of the residue curve map. It was calculated and found that this behavior also extended to the MVC. In particular, the limiting composition of the top and bottom tray in the MVC will asymptotically approach the R and ω limit sets, respectively, for the residue curves of the basic distillation region in which the feed tray composition resides. As defined by Ahmad and Barton,18 the R limit set is given by R(xM) ) limξf-∞ φ(ξ,xM), and the ω limit set is given by ω(xM) ) limξf∞ φ(ξ,xM), where φ are simple distillation residue curves through the specified composition of xM. As such, for the purposes of a limiting analysis (ND, NB, Rd, Rb f ∞), it is appropriate to use the R and ω limit sets of the residue curves to estimate the composition of the top and bottom trays of a tray

Figure 11. Column composition profiles for the first feed composition compared to the corresponding residue curve at total reflux.

Figure 12. Column composition profiles for the second feed composition compared to the corresponding residue curve at total reflux.

column, or to use the R and ω limit sets of the residue curve to estimate the composition at the top and bottom of packed bed columns. This is further confirmed by calculating the expected total reflux column profile in a tray MVC and comparing this profile to the residue curve that passes through the feed tray composition. As shown in Figures 11 and 12 for each of the initial still pot compositions illustrated in Figures 9 and 10, there is very little difference between the residue curve and the discrete points that make up the tray composition profile at total reflux. More importantly, the limits of the distillation line sequence (i.e., the top and bottom composition of the column as the number of trays approaches infinity), unambiguously approach the fixed points of the residue curves (i.e., R and ω limit sets of φ(xM)). In summary, our results show that the conclusions of Widadgo and Seider regarding the continuous distillation column can be extended to the MVC: (1) separatrices may be crossed by the MVC column profile at low reflux/reboil ratios, and (2) at high (or total) reflux, the MVC column profiles approach close to the residue curves and the top and bottom compositions approach


the R and ω limit sets of the corresponding residue curve through the feed tray composition. However, despite the similarities in the limits of the distillation line sequences (as ND f ∞ and NB f ∞) and the fixed points of the residue curve (as ξ f (∞), it should be categorically recognized that they are inherently not equivalent to each other. This applies both to distillation lines for packed and tray columns. An extension of Widagdo and Seider’s analysis reveals that a residue curve is composed of an uncountable infinity of points, and as such is a different mathematical entity when compared to the tray column composition profile/ distillation line which exists as a countable infinity of points as ND f ∞ and NB f ∞. A total reflux column profile for a tray column with Rd, Rb f ∞ and ND, NB f ∞ is composed from a countably infinite number of linear line segments which connect each of the tray compositions in the tray column, where the direction of each line segment is equal to the tangent to the residue curve through the composition point defining the beginning of the line segment (i.e., the liquid composition in the lower tray). As such, it is an entirely different mathematical object from the residue curve which is a continuous curve. The countable sequence of points thus defined does, however, converge to the R limit set of the residue curves in the rectifying section of the column and the ω limit set of the residue curves in the stripping section of the column. The reason for this behavior is that every line segment on the distillation curve is tangent to a residue curve, and this tangent line segment points into the fixed point of the residue curves if the tray composition is close enough to the fixed point on that residue curve. It should also be noted that despite the fact that the composition profile for a packed column is a continuous line (because of the continuous nature of the packed distillation column), it is still not equivalent to the residue curve. However, as Rd, Rb f ∞ and the height of the packed distillation column approaches infinity, the packed column distillation lines will approach the residue curves asymptotically. Infinite Reflux, Infinite Trays in the MVC. On the basis of the above analysis of distillation profiles in an infinite column, the approximation that the limiting composition profile (Rd f ∞) in the column is given by the simple distillation residue curve between xD and xM can be used to analyze the behavior of the still pot composition in a batch rectifier. The value of N (number of theoretical stages in the column) determines the position of the distillate composition on the simple distillation residue curve. If N was low, the distillate composition xD would be relatively near the still pot composition xM, whereas if N was high, the distillate composition would be further from the still pot composition but remain on the same simple distillation residue curve. Finally, in the limit, as N f ∞, the distillate composition will be given by the R limit set of a given basic distillation region. As highlighted by Van Dongen and Doherty,19 the still pot composition is governed by the following equation:

ND f 0, xD f xM ND f ∞, xD f R(xM)

(15)

It also follows that the column composition profile in the stripping section of the MVC is approximated by points on the simple distillation residue curve running through xM, between xM and the ω limit set of xM (which is either a stable node or a saddle point). As with the distillate, the bottoms composition is located close to this residue curve φ(ξ,xM) somewhere between xM and ω(xM). The exact location of the bottoms composition is then defined by the number of trays in the stripping section of the middle vessel column, with the limits given by

NB f 0, xB f xM NB f ∞, xB - ω(xM)

M

dx ) xM - xD dξr

composition moves vectorially away from the composition of the product. A tie line can thus be drawn between the composition of the product withdrawn from the column and the new still pot composition, with the old still pot composition being the lever point. The length of the two sections of the tie line are determined by the amount of product drawn from the column versus the holdup originally in the column. On the basis of the work of Van Dongen and Doherty, Bernot et al.20 extended the theoretical analysis to batch strippers. They pointed out that if, at large values of Rd, the distillate composition lies on the same simple distillation residue curve as the still pot composition, it follows that, at large values of Rb, the bottoms composition xB in a batch stripper also lies on the same simple distillation residue curve as the still pot composition xM. Equivalently, the column profile of the batch stripper is approximated by the simple distillation residue curve between the compositions xM and xB. At large values of N, the bottoms composition is given by the ω limit set of a given basic distillation region. As before, the still pot composition moves vectorially away from the composition of the bottoms product withdrawn. Following the arguments and analysis presented above, the MVC at a high (or infinite) reflux ratio and with an infinite number of trays should exhibit a similar behavior. The points which mark the composition of individual trays can be approximately traced by the residue curves under these limiting conditions. It then follows that the column composition profile in the rectifying section will be approximated by points on the simple distillation residue curve running through the still pot composition xM, between xM and the R limit set of xM. The R limit set of the residue curves (as given by the functionality φ) is a fixed point of the current basic distillation region (either an unstable node or a saddle point). The distillate composition is located close to the curve φ(ξ,xM) somewhere in the curve segment bounded by xM and R(xM). The exact location of the distillate composition depends on the number of trays in the rectifying section of the MVC, but with the limits given by

(16)

(14)

where xM gives the still pot composition, xD gives the distillate composition, and ξr is the dimensionless warped time for the batch rectifier. Thus, the still pot

From eqs 15 and 16, it follows that, for our limiting analysis of a MVC with an infinite number of trays in both sections of the column operated at an infinite reflux/reboil ratio, the distillate composition is given by the R limit set of the basic distillation region in which


Figure 13. Invariance of product composition with infinite trays and infinite reflux and reboil ratios.

the middle vessel pot composition (xM) lies. Similarly, the bottoms composition is given by the corresponding ω limit set. A major difference between a column with an infinite number of trays and a finite column is that in an infinite column the distillate and bottoms product compositions are unchanging for long periods of time as compared to those of the finite column where they are continually changing with time. This is because from the definition of a basic distillation region, all residue curves in the same basic distillation region lead to the same R limit and ω limit sets.9,10,18 It should be noted that this slightly modified definition of a basic distillation region takes into account pot composition boundaries, which are in themselves basic distillation regions without any volume in composition space. Hence, while the still pot composition of a MVC operating under limiting conditions remains in the same basic distillation region, the same “R limit set composition” and “ω limit set composition” are always drawn as the distillate and bottoms product, respectively, irrespective of the instantaneous MVC still pot composition (Figure 13). The distillate and bottoms composition only change when the still pot composition encounters a separatrix or an edge of the composition simplex in the ternary system, thus entering the one-dimensional line or edge. To simplify this analysis, we assume straightline boundaries, but keep in mind that naturally occurring separatrices tend to be curved,22 further complicating the results. Because of the assumption of straightline boundaries, the still pot compositions cannot cross the separatrices that separate the basic distillation regions,19 it can only enter the separatrices. Thus, when the MVC still pot composition enters a linear separatrix, the mixture effectively becomes a lower dimensional system, with a new pair of R limit and ω limit sets. It is these new R and ω limit sets which form the new distillate and bottom products from the MVC, respectively. Separatrices in the ternary system thus serve as pot composition boundaries for the MVC, a concept first defined by Ahmad and Barton18 and illustrated in Figure 14. In Figure 14a, an unstable separatrix is encountered and a change in the ω limit set occurs, while in Figure 14b, a stable separatrix is encountered and a change in the R limit set occurs.

Figure 14. Change in the R limit set and ω limit set as a linear separatrix is encountered.

It is also possible to analyze the dynamics of the still pot composition relatively easily given that the product compositions (both distillate and bottoms) are not continually changing with time, but instead undergo discrete changes only when the still pot composition encounters a separatrix or composition edge. This change in product compositions can be conveniently expressed as a function of the still pot composition xM:

xM ∈ [basic distillation region A] xD ) R limit set(region A)

(17)

xB ) ω limit set(region A) xM ∈ [edge/separatrix of region A] xD ) R limit set (edge/separatrix of A)

(18)

B

x ) ω limit set (edge/separatrix of A) Hence, the product compositions and consequently the MVC still pot dynamics is completely defined by the location of the current MVC still pot composition with respect to the basic distillation region and separatrices. A schematic representation of the vector cone which now restricts the motion of the still pot composition is shown in Figure 15, for the following two cases: (a) a generic ternary ideal system and (b) for the acetonechloroform-methanol system (which exhibits some curvature in its boundaries). As shown, the direction of motion for the MVC still pot composition will lie within the vector cone and will be a function of the operating parameter λ which we defined for the MVC. λ may or may not be a function of time, depending on the objective function of the operating procedure. For example, if the objective is efficiency in terms of CAP (capacity factor first defined by Luyben25 as a measure


with the solution of the equation given by

(

ln

P xM i (ξ) - xi

P xM i (0) - xi

)

)ξ

∀ i ∈ [1 ... NC]

(24)

or alternatively

xM(ξ) ) xP(1 - exp(ξ)) + xM(0) exp(ξ), xP ) xP(λ) (25)

Figure 15. Vector cone of possible motion under limiting conditions in a MVC.

of the amount of product drawn per unit time), then the best operating procedure for separating a binary ideal mixture will be a constant A such that net product xP is equal to the still pot composition xM.6 As mentioned, we will, however, not explore the topic of the optimal operating profile for λ(ξ) in this paper. Revisiting the MVC model, eqs 6 and 7 can now be written as

xD(ξ) ) R(xM(ξ))

(19)

xB(ξ) ) ω(xM(ξ))

(20)

and

Equations 19 and 20 can then be substituted into eq 4 to obtain the following differential equation:

dxM ) xM - xP(ξ) dξ

(21)

where xP is defined by

xP ) λxD + (1 - λ)xB

(22)

and xP represents the net product drawn from the column, dependent only on warped time ξ through the middle column parameter λ(ξ). Equation 21 is easily separable and solved; thus, the analytical solution of the equation will depend primarily on the time dependency of λ. For example, if λ is independent of time, then eq 21 can be rearranged for each of the components in the mixture, ∀ i ∈ [1 ... NC]: M xiM(ξ) dxi xM i (0) M xi - xPi

∫

)

∫0ξdξ,

∀ i ∈ [1 ... NC]

(23)

Thus, for a given value of λ which is invariant, eqs 19 and 20, which define the bottoms and distillate products composition, can be used in conjunction with eq 22, which defines the net product drawn from the column, to obtain the value of xP. Equation 25 then defines the dynamics of the still pot composition as a function of xP, which remains a function of λ. Note, however, that the solution as given in eq 25 will only apply as long as the R limit set and ω limit set of the system remain unchanged (i.e., xM remains in the same basic distillation region). As mentioned, the R and ω limit set of the system will change when the still pot composition enters a separatrix or pot composition boundary of the current basic distillation region and results in a discrete change in the net product xP withdrawn, and a new solution to xM(ξ). For nontrivial formulations of λ as a function of time or warped time ξ, eqs 21 and 5 with eqs 19 and 20 will characterize completely the behavior of the middle vessel still pot composition as a function of the parameter λ(ξ). Given that xP remains only as a function of ξ and not of xM, eq 21 can be solved for xM(ξ) with the simple use of integrating factors. Next, we will explore the graphical implications of this limiting version of the MVC model for a ternary system. As seen in Figure 15, the distillate and bottoms product are invariant in time until the still pot composition encounters a pot composition boundary. Supposing that the parameter λ was kept constant throughout the operation of the column, the following behavior will be expected: (1) The still pot composition will move away in the opposite direction from the net product composition, as given by xP and defined by eq 22. This will continue until the still pot composition encounters a pot composition boundary of the basic distillation region. (2) Once the still pot composition encounters the pot composition boundary, it is now restricted in motion by the line which defines the pot composition boundary. A pot composition boundary of a ternary system has dimension one, which means that the motion of the still pot composition is now more restricted than before. (3) Once it enters the hyperplane that contains the pot composition boundary, the still pot composition obtains new R limit and/or new ω limit sets. This results in a new value for xP, which defines the new net product drawn from the column and a new vector cone of possible motion by the still pot composition. The new vector cone must necessarily lie on the line which defines the pot composition boundary (i.e., a onedimensional vector cone, or equivalently just a regular vector). (4) The still pot composition then moves along the pot composition boundary until it enters either the R limit set or the ω limit set of the pot composition boundary. Once the still pot composition enters the R limit set or


Figure 17. Variation of product composition in the presence of curved separatrices.

Figure 16. Distillate and bottoms composition in a MVC.

the ω limit set, no additional composition change is possible, and the still pot composition remains constant until the still pot runs dry (ξ f ∞). The above sequence of events for the operation of the MVC is illustrated in Figure 16 as before, for (a) a generic ternary ideal system and for (b) the ternary system of acetone-chloroform-methanol (with curvature in its boundaries). As an aside, it should be noted that, in the presence of curved separatrices, the still pot composition may cross the separatrices (i.e., pot composition boundaries and separatrices are no longer equivalent). As such, the distillate and bottoms product may not be the R limit set nor the ω limit set of the system. The still pot composition may be forced to follow the curvature of the separatrix, and the distillate and bottoms product composition are formed accordingly. These product compositions are indicative of the mass balance which must occur as the still pot composition is forced to trace out the curvature of the separatrix, and thus, the distillate or bottoms composition may actually sweep out a line of varying compositions as the still pot composition moves along the curvature of the separatrix. This behavior as illustrated in Figure 17 for the acetone-benzene-chloroform system in a MVC was first observed for rectifiers by Van Dongen and Doherty19 and later substantiated by Bernot et al.20 Next, it is appropriate to define the batch distillation region for a MVC. Similar to the definition of a batch distillation region for a batch stripper or a batch rectifier, the batch distillation region of a MVC is given by the set of composition points which result in exactly the same sequence of products in the rectifying and the stripping sections of the column over time. Suppose that the n-sequence of cuts from a MVC were given as ([D1,B1], [D2,B2], ... [Dn,Bn]) for a given initial MVC still pot composition R (where the first term in the square bracket represents the distillate product, the second

term in the square bracket represents the bottoms product of that given cut, and each set of terms in square brackets represents different cuts obtained from the MVC). If the sequence of cuts for another initial composition β was given by ([d1,b1], [d2,b2], ... [dn,bn]), then R and β are in the same middle vessel column batch distillation region if and only if {di ) Di ∀ i ) 1...n} and {bi ) Bi ∀ i ) 1...n}. It should be noted that these middle vessel batch distillation regions will vary with λ and are only specified for a given value of λ. To illustrate this point, the batch distillation regions for a stripper correspond to the middle vessel batch distillation regions when λ ) 0, and the batch distillation regions for a rectifier correspond to the middle vessel batch distillation regions when λ ) 1. These rectifier and stripper batch distillation regions are not necessarily equivalent.18 Similarly, batch distillation regions for the middle vessel at different values of λ between 0 and 1 are not necessarily equivalent either. Finally, the concept of steering the middle vessel composition first introduced in Figures 4 and 7 will also be explored here. As mentioned, variation of λ with time will result in an ability to steer the still pot composition within the basic distillation region. Since the bottoms and distillate products are much more predictable under limiting conditions of operation (they are unchanging within a given basic distillation region), the results of steering (or varying A) are also much more predictable as a result. A simple illustration is provided in Figure 18. As we see in Figure 18, suppose that there are certain compositions that we wish to avoid (denoted by regions marked by A), we would be able to steer the still pot composition in such a way that the light (L) and the heavy (H) components in their pure forms continue to be drawn, but at varying rates over time. Calculating the exact values of t1, t2, t3, and so forth, will only be a matter of conducting an overall material balance to see how long it would take to draw the required amount of M M M material before the target points xM 1 , x2 , x3 , and x4 were reached. Theoretically, it would thus be possible to control the path of the still pot composition by varying λ with time (or warped time ξ) in an open-loop fashion, without a need for an elaborate feedback control loop. Higher Dimensionality Systems. The same concepts used in the earlier section for the analysis of ternary systems can also be applied to systems of higher dimensionality. To illustrate this, we will attempt to generalize the above description of the behavior of the still pot composition to a generic n-component system. Finally, a graphical analysis will also be presented for the generic quaternary system to show the applicability of the analysis presented.


Figure 18. Steering the still pot composition for the limiting case of infinite reflux and infinite trays.

As before, the distillate and bottoms compositions are unchanging when the still pot composition remains within the same basic distillation region. The distillate and bottoms composition only change when the still pot composition encounters a pot composition boundary in the n-component system and enters the (n-2)-dimension hyperplane containing the pot composition boundary. This is because upon entering the (n-2)-dimension hyperplane, the mixture effectively becomes an “(n-1)component system”, with new R limit and ω limit sets. It is these R and ω limit sets which form the new distillate and bottom product compositions, respectively. The hyperplane within which the pot composition boundary lies can also be thought of as a separate distillation region of lower dimension. It is thus possible to analyze the dynamics of the still pot composition relatively easily even for a fourcomponent system, given that the product compositions (both distillate and bottoms) are not continually changing with time, but instead undergo discrete changes only when the still pot composition encounters a pot compo-

sition boundary and enters the hyperplane containing the pot composition boundary. It should be noted that this change in the product compositions can be expressed as a function of time, but is more conveniently expressed as a function of the still pot composition xM, as was the case for the ternary system. A sample formulation is given by

xM ∈ [basic distillation region A] xD ) R limit set (region A)

(26)

B

x ) ω limit set (region A) xM ∈ [basic distillation region B] xD ) R limit set (region B)

(27)

xB ) ω limit set (region B) where basic distillation region B ≡ pot composition boundary of region A. Hence, the product compositions


Figure 19. Distillate and bottoms composition in a MVC for a four-component system.

and consequently the still pot dynamics are still completely defined by the current still pot composition and its location with respect to the basic distillation regions of various dimensionalities. The geometrical implications of this limiting version of the MVC model for an n-component system are also explored. As explained, the distillate and bottom products are unchanging in time until the still pot composition encounters a pot composition boundary. Supposing that λ was kept constant throughout the operation of the column, the following behavior will be expected: (1) The still pot composition will move away in the opposite direction from the net product composition, as given by xP and defined by eq 22. This will continue until the still pot composition encounters a pot composition boundary of the basic distillation region. (2) Once the still pot composition enters a pot composition boundary, it is restricted in motion by the hyperplane within which the pot composition boundary lies. This is because the new R and ω limit sets also lie in this hyperplane, so any vector in the vector cone of possible motion must also lie in the same hyperplane. Typically, a pot composition boundary of an n-component system (which was restricted within the (n-1)n xi ) 1) has dimension hyperplane as defined by ∑i)1 dimension n - 2, which means that the motion of the still pot composition is now more restricted than before. (3) Once it enters the pot composition boundary hyperplane, the still pot composition now has a new ω limit set or a new R limit set or both. This results in a new value for xP, which defines a new net product drawn from the column and a new vector cone of possible motion by the still pot composition, which must necessarily lie within the same hyperplane. (4) The process repeats itself 1-4, until the still pot composition finally enters an R limit set or an ω limit set and no further composition change of the still pot composition is possible. The above list of events for the operation of the MVC is illustrated in Figure 19 for a generic quaternary system. The vector cone of possible motion for the still pot composition remains on a plane, despite being in a three-dimensional composition simplex. This is due to

the fact that only two products are drawn from the column at any given time; hence, the products, the initial still pot composition, and the resulting still pot composition must all lie in the same plane. It is evident that the above principles extend to systems with an arbitrary number of components. As before, in the presence of curved boundaries or with a finite reflux/ reboil ratio and/or number of trays, the still pot composition may be able to cross the separatrices or pot composition boundaries, because of the topology of the residue curve map. However, more often than not, when a curved boundary is encountered, the still pot is forced to trace a path along the curved boundary. As such, the distillate and bottoms product may no longer be the fixed points of the residue curve map. As the still pot composition is forced to trace out the curvature of the separatrix, the distillate and bottoms product composition are formed accordingly.19,20 A Bifurcation Analysis of the MVC Batch Distillation Regions. The MVC is a generalization of traditional batch distillation columns (i.e., rectifier and stripper) since it encompasses both the rectifier (λ ) 1) and the stripper (λ ) 0). Given that the MVC represents a range of behavior between the two limiting cases of a batch stripper and rectifier (0 e λ e 1), it would be appropriate to consider the behavior of the MVC as a hybrid between the two configurations. In particular, for mixtures where the batch distillation regions for the stripper and the rectifier are not the same, some sort of bifurcation would be expected in the behavior of the column and the batch distillation regions of the column from that of a stripper to that of a rectifier as λ varies from 0 to 1. To illustrate our analysis in this section, we consider one of the ternary systems enumerated by Doherty and Caldarola26 and Matsuyama and Nishimura,27 designated as the 001 system. It is one of the simplest ternary systems with only one binary azeotrope, but it will suffice for our analysis since the 001 system contains batch-rectifying regions which are not equivalent to the batch-stripping regions. Using the tools developed by Van Dongen and Doherty and Bernot et al. for the analysis of the batch rectifier and batch strippers, batch distillation regions for the rectifier and stripper were found and labeled accordingly in Figure 20. As before, limiting operating conditions (infinite trays and infinite reflux/reboil ratios) were assumed. From Figure 20, there are two batch distillation regions (R and β) with non-zero volume for the batch rectifier, but only one batch distillation region with nonzero volume for the batch stripper (γ). Each of the A-B and B-C binary edges of the composition simplex, the line segments A-AC, AC-C and B-AC, and each fixed point of the residue curve map, are individual batch distillation regions of zero volume (one-dimensional for the edges; zero-dimensional for the fixed points), as each of these “regions” produce a unique sequence of cuts. Of greater interest, however, are the non-zero volume batch distillation regions for the batch rectifier (R and β) and the batch stripper (γ) and how these volumes transform as the value of λ varies from 0 to 1 (i.e., as the middle vessel column deforms from being a pure stripper to being a pure rectifier). For the rectifier, any interior point of the composition simplex (i.e., one that does not lie in an edge or a fixed point) would draw azeotrope AC (which is the R limit


Figure 20. Batch distillation regions for the stripper and the rectifier in the 001 system.

set of all the residue curves interior to the simplex) as the distillate product. Hence, any point with composition interior to region R would draw AC as the product until it encounters the A-B edge, where it will enter the batch distillation region of the A-B line segment, draw B, and finally A as the product. Thus, the product sequence is (AC, B, A). Correspondingly, any point with composition in region β draws AC as the first product until it encounters the B-C edge, where it enters the batch distillation region of the B-C edge, draw C and finally B as the product. The product sequence associated with β is (AC, C, B). The product sequences for R and β are not equivalent, hence, they are distinct batch distillation regions. For the stripper configuration, any point interior to the composition simplex (i.e., not lying in one of the edges or fixed points) will draw pure A as a product, as it is the ω limit set of the residue curves passing through any point interior of the composition simplex. However, from the topology of the residue curve map, any point in the simplex that draws A as a product will encounter the B-C edge, after which it will draw B as the new ω limit set, and finally C as the last product. Hence, there is only one batch-stripping region, with product sequence (A, B, C). Now, consider the operation of the MVC at a fixed value of λ where 0 < λ < 1. It would be useful to consider the dynamics of the still pot composition in terms of its movement away from a “net product” as given by xP (see Figure 3c), which is the weighted average of the two products drawn (weights given by λ and 1 - λ, respectively, as expressed in eq 22). Thus, for a given composition point interior to the composition simplex (R(xM) ) AC and ω(xM) ) A), the net product lies on line segment AC-A connecting the two products drawn from the column, namely, AC and A. Suppose a value of λ is chosen such that a net product xP(0) as shown in Figure 21a is produced. Next, draw a line from xP(0) to the fixed point of pure B, dividing the residue curve map into two regions, δ and . Further divide region δ into δ1 and δ2 by drawing a line from xP(0) to xPδ , where xPδ is given by xPδ ) λB + (1 - λ)A, the net product drawn from the

Figure 21. Batch distillation regions at a given value of λ.

MVC (at this given λ) when the still pot composition is on the A-B binary edge. Region is also further divided by drawing a line from xP(0) to xP , where xP is given by xP ) λC + (1 - λ)B and denotes the net product drawn from the column (at the current λ) when the still pot composition lies on the B-C binary edge. There are thus a total of four middle vessel batch distillation regions interior to the composition simplex with non-zero volume. It should be noted that each of the lines joining xP(0) to xPδ and xP are in themselves a separate middle vessel batch distillation region of zero volume. Any initial still pot composition that lies within regions δi then draws the net product xP(0) and eventually encounter the simplex edge given by line segment A-B. At this point, the R limit set for the still pot composition changes to pure B, and the products drawn from the column are now pure A (bottoms) and pure B (distillate), as illustrated in Figure 21b. The new net product drawn from the MVC is then denoted by xPδ . The middle vessel column will continue to draw this net product such that the still pot composition moves away from xPδ until it enters the fixed points A (if the initial still pot composition was in δ2) or B (if the initial still pot composition was in δ1). Once the still pot composition enters one of these fixed points, the top and bottom products will both correspond to the still pot composition because the R and ω limit set of a fixed point is itself. The cuts from the middle vessel column for an initial still pot composition in region δ1 can thus be characterized as ([AC,A], [B,A], [B,B]). A similar sequence obtained for δ2 is ([AC,A], [B,A], [A,A]). Using a similar analysis for initial still pot composition in regions i, recognizing that xP is the net product drawn from the MVC once the still pot composition encounters the B-C binary edge, the middle vessel batch distillation sequences are deduced for regions 1 and 2 and summarized in Table 1. As such, δ1, δ2, 1, and 2 are four different MVC batch distillation regions in the spirit of Ewell and Welch’s definition. The pot composition boundaries between

1520 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 Table 1. Middle Vessel Batch Distillation Sequence for Regions δ1,2 and E1,2 of Non-Zero Volume region

first cut

second cut

third cut

δ1 δ2 1 2

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

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

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

these regions are then (1) the line segment that joins the fixed point B to the initial net product xP(0), where xP(0) is given by

xP(0) ) λ(composition of AC) + (1 - λ)(composition of A) (28) and (2) the line segment that joins the initial net product xP(0) to the net product on the A-B binary edge given by xPδ , where xPδ is given by

xPδ ) λ(composition of B) + (1 - λ)(composition of A) (29) (3) The line segment that joins the initial net product xP(0) to the net product on the B-C binary edge given by xP , where xP is given by

xP

) λ(composition of C) + (1 - λ)(composition of B) (30)

As seen from our analysis, the motion of the still pot composition is more restricted at a given λ (0 < λ < 1) for the MVC, with four different distillation regions, as compared to either the batch stripper or batch rectifier. In fact, when we explore systems which contain separatrices,14,17 we note that (1) stable separatrices served as pot composition boundaries for the batch rectifier but not for the stripper, (2) unstable separatrices served as pot composition boundaries for the batch stripper but not the batch rectifier, but (3) stable and unstable separatrices are both pot composition boundaries for a MVC at a given λ, where λ * 0 or 1. Thus, the still pot composition of a MVC is actually more restricted in motion than that of either a stripper or a rectifier if λ is kept constant. However, the flexibility of the MVC lies in the fact that λ can be varied during an operation. λ can take on any value between 0 and 1 inclusive, which means that it can cross both the stable separatrices (when λ ) 0) and the unstable separatrices (when λ ) 1) in a single operation. Hence, the still pot composition in a MVC is less restricted in motion than either a stripper or a rectifier if λ varies during operation of the MVC. Thus, the pot composition boundaries enumerated in our analysis above only exist for the given value of λ that we had assumed. For a larger value of λ, the initial net product xP(0) drawn from the MVC will be nearer to the unstable node (AC), while for a smaller value of λ, the initial net product xP(0) drawn from the MVC will be nearer to the stable node (A). Correspondingly, varying λ also varies xPδ and xP accordingly. The pot composition boundaries between these four regions will, however, always be from xP(0) to the fixed point of B, from xP(0) to the net product on the A-B edge (xPδ ), and from xP(0) to the net product on the B-C edge ( xP ). Hence, pot composition boundaries also shift accordingly with the variation of λ.

Figure 22. Sweep of pot composition boundary as λ varies between 0 and 1.

In the limit, as λ f 1, xP(0) ) xD ) R(xM) ) AC, xPδ ) B, xP ) C, and the pot composition boundaries are transformed as follows:

xP(0)-B transforms into AC-B xP(0)-xPδ transforms into AC-B and

xP(0)-xP transforms into AC-C Given that the line segment AC-C is along the composition simplex edge, and hence naturally a pot composition boundary, the only remaining pot composition boundary interior to the composition simplex, as λ f 1, is thus given by the line segment B-AC, which is exactly the pot composition boundary for a batch rectifier. Similarly, as λ f 0, xP(0) ) xB ) R(xM) ) A, xPδ ) A, P x ) B, and the pot composition boundaries are transformed as follows:

xP(0)-B transforms into A-B xP(0)-xPδ transforms into A-A and

xP(0)-xP transforms into A-B The A-B line segment and A-A point are all edges of the composition simplex, and as such are naturally occurring pot composition boundaries. Hence, at λ ) 0, we would expect no pot composition boundaries interior to the composition simplex, which is exactly what we would expect for the stripper configuration. Thus, as λ sweeps out the value between zero and 1, the pot composition boundaries transform accordingly as illustrated in Figure 22. The first boundary given by xP(0)-B starts off as the line segment B-A (λ ) 0) and swivels around point B until it reaches the line segment B-AC at λ ) 1. The second boundary given by xP(0)xPδ starts off as the fixed point A (λ ) 0) and forms a line which spans the A-B edge and the A-AC edge. The triangle formed by A-B-AC and that formed by A-xP(0)-xPδ are similar triangles by construction. This boundary eventually ends up merging with the first boundary into the line segment B-AC at λ ) 1. Finally, the last boundary given by xP(0)-xP starts off as the

Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 1521 Table 2. Comparison of Middle Vessel Batch Distillation Sequences for Regions δ1,2 and E1,2 vs Expected Stripper Sequences MVC regions

middle vessel product sequence

resulting region at λ ) 0

expected stripper product sequence

δ1 δ2 1 2

([AC,A],[B,A],[B,B]) ([AC,A],[B,A],[A,A]) ([AC,A],[C,B],[B,B]) ([AC,A],[C,B],[C,C])

A-B edge pure A A-B edge simplex interior (γ)

(A,B) (A only) (A,B) (A,B,C)

Table 3. Comparison of Middle Vessel Batch Distillation Sequences for Regions δ1,2 and E1,2 vs Expected Rectifier Sequences MVC regions

middle vessel p oduct sequence

resulting region at λ ) 1

expected rectifier product sequence

δ1 δ2 1 2

([AC,A],[B,A],[B,B]) ([AC,A],[B,A],[A,A]) ([AC,A],[C,B],[B,B]) ([AC,A],[C,B],[C,C])

B-AC line segment region R region β AC-C line segment

(AC,B) (AC,B,A) (AC,C,B) (AC,C)

binary edge A-B (at λ ) 0) and forms a line which spans the A-AC edge and the B-C edge. This boundary eventually ends up merging, becoming the line segment AC-C, at λ ) 1. Correspondingly, the batch distillation region as given by region δ1 in Figure 21a starts off at λ ) 0 as a batch distillation region of zero volume along the edge of the line segment B-C. It slowly expands in volume as λ increases, but as λ increases further, this region then collapses back into a region of zero volume along the line segment B-AC. On the other hand, region δ2 starts off as the fixed point A slowly expands as λ increases, and finally becomes the region given by R in Figure 20 (λ ) 1). Similarly, the batch distillation region given by region 1 starts off at λ ) 0 as a region of zero volume on the edge B-C and expands as λ increases. It reaches a maximum in volume and then starts shrinking as λ increases further and finally forms the region given by β in Figure 20 at λ ) 1. On the other hand, region 2 starts off at λ ) 0 as the entire interior of the composition simplex and slowly shrinks as λ f 1, until at λ ) 1, it is a distillation region of zero volume given by the line segment AC-C. To illustrate the validity of this representation of the stripper and rectifier as specific cases of the MVC, consider the product sequence of the stripper and rectifier in each of the middle vessel batch distillation regions enumerated. Imagine the stripper as a MVC where no top product is drawn, in which case, it is the bottom product in the MVC sequence that is relevant to a stripper (i.e., the second term, y, in the ordered pairs [x,y] given for the MVC at each cut gives the product of the stripper). However, we stated that region δ1 transforms into the line segment A-B when λ f 0, but the product sequence for δ1 was ([AC,A], [B,A], [B,B]). The relevant product sequence for the stripper is then (A,A,B) or (A,B). This is indeed the product sequence drawn from the stripper if the initial composition lay on the line segment A-B; pure A is drawn until all the A is exhausted, followed by pure B. Considering also region δ2, with the relevant product sequences ([AC,A], [B,A], [A,A]), δ2 transforms into fixed point A at λ ) 0; hence, the only product drawn from the stripper would be pure A. Examining the bottoms product sequence enumerated for region δ2 in the MVC, we obtain (A,A,A) or (A), which is expected of the stripper. Conducting a similar analysis for the MVC batch distillation regions given by 1 and 2, the expected products in a stripper and a MVC are tabulated for each region and compared to each other in Table 2. From Table 2, the bottoms product of the MVC does indeed predict the product

Figure 23. Bifurcation behavior at a given point as λ varies between 0 and 1.

obtained from a stripper in the appropriate middle vessel batch distillation region. Next, consider the case of the batch rectifier. As an inverse of the above analysis, it would be the first term in the square brackets of the middle vessel column product sequence that would be relevant for a batch rectifier. A similar analysis is thus conducted on each of the regions δ1, δ2, 1, and 2 and the results are summarized in Table 3. As can be seen from Table 3, the distillate product of the MVC again predicts the product obtained from a rectifier in the appropriate middle vessel batch distillation region. Finally, consider the bifurcation in MVC behavior that occurs for a given initial composition for a point that lies interior to the composition simplex. As an example take the point φ in Figure 23. Let the initial value of λ be 0 (i.e., the MVC behaves like a stripper). There is only one batch distillation region for the whole interior of the composition simplex (γ in Figure 20 or 2


at λ ) 0), and φ consequently falls within this region, with the product sequence ([AC,A], [C,B], [C,C]). As the value of λ is increased, there would exist values of λ ) λbifur,i, i ) 1...3 at which each of the three pot composition boundaries (given by line segments xP(0)-xP , B-xP(0), and xP(0)-xPδ ) crosses the point φ, and φ changes its location from one middle vessel batch distillation region to another. λbifur,1 represents a switch in behavior for point φ from that of region 2 to that of 1. λbifur,2 represents a switch in behavior for φ from that of region 1 to that of δ1, while λbifur,3 represents a switch in behavior for φ from that of region δ1 to that of δ2. On the basis of the above analysis, initial still pot compositions which lie within the region R in Figure 20 will exhibit bifurcation behavior at three different values of λ as λ varies between 0 and 1, with λbifur,i values characteristic to each initial still pot composition. Initial still pot compositions which lie within the region of β in Figure 20 will, however, exhibit only one bifurcation point, switching in behavior from that of region 2 to that of region 1 as λ varies from 0 to 1. With this bifurcation behavior in MVCs, it can be seen that pot composition boundaries of the traditional strippers and rectifiers are no longer valid for the MVC if λ is allowed to vary during the operation of the column. It is only pot composition boundaries that correspond to both the stripper and rectifier configuration which remain as pot composition boundaries for the MVC if λ is allowed to vary during the operation of the MVC. Such a pot composition boundary exists at the same spatial location for all values of λ such that as λ varies from 0 to 1 the boundary exists at all values of λ inclusive of 0 and 1. To illustrate the removal of these traditional boundaries, consider the pot composition boundary for the batch rectifier (line segment AC-B) in the 001 system. Let a point F1 be within the batch rectifier distillation region R, as illustrated in Figure 24a. An initial composition such as F1 is unable to cross the rectifier pot composition boundary as given by line segment AC-B in a batch rectifier. However, in the MVC, we would be able to vary the value of λ, such that the pot composition boundary given initially by the line segment AC-B shifts, such that with an appropriate value of λ, the point F1 would lie in the middle vessel batch distillation region denoted by 1 in Figure 21. The products from the MVC would then be [AC,A] with the still pot composition moving toward the B-C edge. Once the still pot composition crosses the AC-B line segment and arrives at a point such as F2 in Figure 24b, the middle vessel can revert to its operation as a batch rectifier, and the pot composition would now be in the batch rectifier distillation region given by β. The still pot composition has thus effectively crossed over from region R into region β. The traditional pot composition boundary for a rectifier (line segment AC-B) is thus not a pot composition boundary in a MVC. Pot composition boundaries which are not common to both the stripper and rectifier are not pot composition boundaries for the MVC that is allowed to operate at all values of λ. It should be noted, however, that separatrices which form pot composition boundaries for either the stripper (unstable separatrices) or the rectifier (stable separatrices) remain as pot composition boundaries for all values of 0 < λ < 1. Only at λ ) 0 does a stable separatrix cease to be a pot composition boundary for the MVC, and only at λ ) 1 does an unstable separatrix

Figure 24. Removal of pot composition boundaries that are not common to both a stripper and rectifier in a MVC.

cease to be a pot composition boundary for the MVC. However, a MVC can be operated at all values of 0 e λ e 1; thus, the MVC is able to cross both the stable and unstable separatrices. Hence, it should be clarified that the true criterion for a pot composition boundary for a MVC capable of all values of λ (from 0 to 1) is that this particular boundary must not transform in any way as λ varies between 0 and 1. The removal of pot composition boundaries not common to both the rectifier and the stripper in a MVC capable of operating at all values of λ thus affords a greater degree of freedom to the MVC as compared to that of a traditional batch stripper or batch rectifier. It is this nonequivalency of batch distillation regions in the stripper and rectifier which allows the middle vessel configuration to traverse between batch distillation regions of traditional column configurations. However, it may be possible to move from one conventional (stripper or rectifier) region to another, but the reverse movement is not always possible. Taking our example, it was possible to move from region R into region β by operating the MVC cleverly, but it would not have been possible to move from the batch rectifier region β into region R, because of the way in which bifurcation occurs. In summary, the middle vessel batch distillation region as defined for a given λ is actually much more restricted than that of either the traditional stripper or rectifier. However, because of the variability of λ in the operation of a MVC, the batch distillation region for the MVC operated at varying λ thus becomes less restricted than either the stripper or the rectifier, because of its ability to cross pot composition boundaries which are not common to both the stripper and the rectifier. It should also be noted that, in our analysis, we have concentrated on “mass-balance” pot composition boundaries, and not separatrix-type pot composition boundaries. Mass-balance boundaries are boundaries that


Figure 25. Separatrix-type pot composition boundaries vs massbalance-type pot composition boundaries.

arise because of the fact that the pot composition must move in a straight line away from its net product, and hence, it is restricted in its possible product sequences. Separatrix-type boundaries, however, restrict the motion of the still pot composition because of a change in the R or ω limit set of the still pot composition. Separatrix-type boundaries do not transform continuously over varying values of λ, but undergo discrete step changes from being a boundary to not being a boundary. A stable separatrix is a boundary for the MVC for 0 < λ e 1, but ceases to be a boundary at λ ) 0. Similarly, an unstable separatrix is a boundary for the MVC for 0 e λ < 1, but ceases to be a boundary at λ ) 1. An example is illustrated in Figure 25, for the acetonebenzene-chloroform system, where the pot composition boundary for the rectifier is a separatrix-type boundary, but the pot composition boundary for the stripper is a “mass-balance” type boundary. When this insight is incorporated into the above analysis developed for “mass-balance” boundaries, bifurcation analyses can also be conducted on systems with separatrix-type boundaries. Finally, it should be noted that these concepts are also applicable to systems of higher dimensions, where there are pot composition boundaries made up of bundles of trajectories, some of which are separatrices (separatrixtype boundary) and “mass-balance” pot composition boundaries which are hyperplanes dividing the different regions in which product compositions would differ. A similar analysis to that described in this section can also be extended to residue curve maps of all dimensions for which the batch stripper regions differ from that of the batch rectifier regions. Bifurcation of these regions as a function of λ can thus be characterized appropriately. The consequent removal of batch distillation boundaries due to this bifurcation with λ can thus be used to traverse the pot composition boundaries of the traditional batch stripper and rectifier columns, and possibly afford a richer variety of separation possibilities not possible in traditional stripper or rectifier batch distillation columns. More on the Equivalency of the MVC vs a Stripper and a Rectifier. As ND f ∞ and Rd, Rb f ∞, the composition of the products drawn as a distillate from the middle vessel column approaches that of the

R limit set for the current basic distillation region. Similarly, as NB f ∞ and Rd, Rb f ∞, the composition of products drawn from the bottom of the MVC approaches that of the ω limit set. These product compositions remain unchanged as the MVC still pot composition changes within the same basic distillation region. Thus, the product drawn for the second stage of the operation (rectification/stripping) will be exactly that of the product drawn from the equivalent position (distillate/bottoms) in the MVC. Thus, the equivalence of the rectifier and the stripper to that of the MVC can now be extended to discrete operating steps with the stripper and rectifier. An example is shown for a generic ternary ideal mixture (with no azeotropes) in Figure 26. Figure 26a shows the required still pot composition path with the value of λ that would allow the middle vessel to achieve the desired path; Figure 26b shows the possible operating procedure using infinitesimal rectifier and stripper moves, so as to duplicate the original path achieved by the MVC; finally, in Figure 26c, the same final pot composition and product compositions are obtained by using first a rectifier and then followed by a transfer to a stripper, with only one transfer occurring at point R. The equivalent operation can also be achieved using a discrete step with a stripper followed by a transfer (at point β) to a rectifier with another discrete operating step, as shown in Figure 26d. The need for infinitesimal stripping and rectifying steps in nonlimiting columns was due to the changing product compositions which occur with changes in the still pot composition; with the limiting column with infinite equilibrium trays this is no longer relevant. With this equivalency of the rectifier and the stripper versus that of a MVC, it would seem that the MVC is indeed irrelevant. It is true that the MVC provides minor advantages over the stripper/rectifier combination during the operational stage of the distillation (i.e., when the products are being drawn). First, energy costs for the production of vapor in the distillation columns may be halved because, in the stripper/rectifier combination, there are two operational stages, each requiring an approximately equal amount of energy as that supplied to a MVC. However, such energy savings are probably irrelevant in pharmaceutical or speciality chemical industries, who are the main users of batch distillation technology. Secondly, there are also small savings in terms of campaign time as well, provided a relatively large number of batches are to be processed. In the MVC, two streams are drawn at the same time, the distillate and bottoms product, but in the combined operation of the rectifier and stripper, both columns can be operating at the same time (overlapping operation) after the initial start-up phase where only one column is operating. Hence, at any time after the first batch, there would be the nth batch in the stripper, and the (n+1)th batch in the rectifier; at any point in time, both columns are operating, and two product streams are being drawn. Thus, in the limit of a large number of batches, the MVC does not really afford significant theoretical time savings in terms of actual column operating time either. However, in the practical operation of any distillation column, there are always overheads involved in processing other than the actual operation of the column. There is usually a start-up stage at total reflux before any products are drawn from the column, as it takes time


Figure 26. Pot composition path using a MVC vs a stripper and a rectifier.

for the composition profile in the column to reach the desired quasi steady state. There is also processing time required for transfer between the columns. Finally, there is also the shutdown stage, where the contents in the column have to be cooled before they can be safely transferred to another distillation column or storage container. These overheads will actually still be the same for the contending configurations, if there is only one rectification and one stripping stage required (as in Figure 26, parts c and d). To see this equivalency, consider the Gantt charts for operating the MVC and the Gantt chart for the combined operation of the rectifier and the stripper with overlapping schedules for the columns, as shown in Figure 27. The total cycle time for each batch is actually still 7 h, despite using two columns (stripper/rectifier) instead of the MVC, provided that only one stripping operation and one rectifying operation is required. With a similar cycle time, the energy costs involved for total reflux in start-up and reboiler vaporization would also be equivalent. However, there may be some constraints on the still pot composition path, other than the initial and final composition, such as potentially dangerous mixture compositions that should be avoided (Figure 18), or any other reason that would require more than one set of stripping and rectifying operations to achieve the equivalent change in the still pot composition as achieved by a single operation of the MVC. This would result in more than one set of stripping and rectifying operations, which imply a longer processing time for the strippingrectifying combination as the required overhead of 2 h (0.5 h transfer in, start-up, shutdown, transfer out) would be incurred for each set of stripping-rectifying operations. In the extreme case where the still pot path must be strictly adhered to, infinitesimal operation of

the rectifier and the stripper would be required, which would result in infinitely many transfers, start-ups, and shutdowns, translating into huge overhead times for the entire process if the combined stripper-rectifier configuration was used rather than the MVC configuration. In summary, there is certainly a theoretical equivalency between the middle vessel column and a combined operation of a batch rectifier and a batch stripper. However, the final choice between the two different types of configuration would depend on the objective function that has to be optimized and the path constraints imposed on the operation. The ultimate objective of any separation is to move between the initial and final compositions, drawing the required products, in the minimum amount of time and/or cost, with time and cost weighted appropriately (as determined by the producers’ objective functions). If a single set of stripping-rectifying operations is able to achieve the separation required, then it would be cheaper to use a stripping-rectifying setup using existing strippers and rectifiers, rather than outfitting a new MVC, which would incur capital costs. However, if multiple stripping-rectifying steps are required to achieve the required separation and still pot composition path, then perhaps the MVC will be preferable, as it cuts down on overhead times, reduces energy consumption, and requires less shutdown/start-up cycles. It is ultimately a trade off between the capital cost of installing a MVC versus the higher operating costs of operating a combined rectifier and stripper column configuration, and a sound financial decision has to be made based on the operating scheme required, the demand for the product, total campaign time expected, and a host of other financial determinants (such as investment rates of return, project beta, credit interest rates, etc.).


Figure 27. Gantt charts for operating (a) a MVC and (b) a single set of stripping-rectifying operations.

It should be noted that if a new plant were to be designed from scratch (where no existing strippers/ rectifiers can be utilized), use of the MVC would be preferable over that of the combined operation of a stripper and rectifier. This is because the capital costs of building a stripper and a rectifier would be comparable (probably more) to the cost of building a MVC. For example, the stripper/rectifier combination would result in twice as many condensers, reboilers, and control systems, resulting in higher capital costs. Operational costs would also be higher as higher energy loads and perhaps more operating personnel would be required for the combined stripper/rectifier operation. Transfer pumps would also have to be installed between the strippers and rectifiers. It is true, however, that with the greater flexibility of the MVC it would also require a more complex control system, especially if λ is varied over time. Regardless, from the point of view of capital investment of a plant built from the ground up, the MVC would be more attractive, and it also offers greater flexibility than either the stripper or the rectifier and a lower energy consumption cost. Furthermore, the use of a single MVC reduces the transfer of chemicals between unit operations which reduces the risk of spillage and/or contamination and

reduces the amount of waste generated from residue remaining in still pots at the end of a batch operation, as only one still pot is required for the entire separation. Thus, product quality, safety, and environmental considerations also indicate that a MVC may be preferable to the combined stripping-rectifying operations. Thus, the MVC could prove to be far superior when compared to the analagous combined stripping-rectifying operations when all these factors are taken into consideration and monetized. A Theoretical Study of Multivessel Columns Having illustrated the increased flexibility of a middle vessel batch distillation column over that of a traditional stripper or a rectifier, it should be noted that this flexibility was achieved via the second product stream which is drawn from the column, and the fact that this stream is sufficiently different in composition from the first product stream drawn. This afforded a twodimensional vector cone in which the still pot is allowed to move, compared to the single direction in which a still pot composition must move when it is operated in a rectifier or stripper (which only draws a single product).


It is thus conceivable that a column which allows us to draw more than two sufficiently different product compositions would afford a greater degree of freedom in the motion of the still pot composition. Three product streams drawn mean three dimensions of possible motion, while n product streams drawn from the column mean n dimensions of possible motion. This observation is not novel. The drawing of split streams from a continuous distillation column to increase the variety of separations possible in the continuous column is a well-documented process. Split streams can also be drawn from a MVC with a corresponding increase in separation possibilities, which then leads to the conceptualization of the multivessel column, in which multiple vessels are located at intervals along the column with products drawn from each of these vessels. The “multivessel column” was proposed by Hasebe in 19951 as an extension of the MVC, with vessels (which have significant holdup) feeding material at several points distributed along the length of the column. Wittgens et al.28 considered the limiting case of total reflux operation in this column and conducted both a simulation and an experimental analysis of this batch distillation configuration. They also explored the control aspects of this configuration in a recent paper.29 However, a theoretical discussion of the qualitative dynamics of the multivessel column has not been conducted to date. While the additional possibilities of overall column composition motion is not exciting in a ternary system where motion is restricted to the composition simplex plane given by x1 + x2 + x3 ) 1, it does offer additional possibilities of separation for mixtures with a larger number of components. For example, an n-component mixture would have its motion restricted in an (n - 1)dimension composition simplex, and an (n - 1)-dimension vector cone would allow the still pot composition to reach points in the composition simplex which it otherwise would not be able to in a stripper or rectifier, or for that matter in a MVC. This thought experiment than leads us to elucidate the usefulness of multivessel columns in the separation of multicomponent mixtures. In particular, an n-component mixture could be separated in a column with (n - 1) holdup trays, with a product stream drawn from each of the trays, to afford a total of (n - 1) product streams, each stream substantially different from each other (i.e., there is no linear dependency of any of the vectors of motion given by xColumn - xPi , where Column denotes the overall weighted average composition in the entire column, P denotes product, i denotes the tray from which it is drawn, with i ) 1...n - 1). Formulating this more formally, in the spirit of the still pot composition steering equations developed for the MVC, we obtain the following equation for the motion of the overall column composition as a function of a warped time τ. First, we define τ for the multivessel column as follows:

dτ )

() M

ϑ1 )

P1 n-1

Pi ∑ i)1

ϑ2 )

P2 n-1

Pi ∑ i)1

l ϑn-2 )

(32) Pn-2 n-1

Pi ∑ i)1

ϑn-1 )

Pn-1 n-1

Pi ∑ i)1

such that by definition n-1

ϑi ) 1 ∑ i)1

(33)

From eqs 31 and 32 and the respective overall and component mole balances for the multivessel column total holdup (i.e., of all the trays), the following equation is derived for the motion of the overall composition for the total holdup in a multivessel column (given by composition xM, with total molar holdup in the column M):

dxM dτ

n-1

)

ϑi(xM - xPi ) ∑ i)1

(34)

Thus, the direction vectors of the possible motion for the total column holdup composition are given by the vectors ((xM - xPi ) ∀ i ) 1...n-1). It is this set of vectors which must be linearly independent in order for us to obtain a (n - 1)-dimension vector cone of motion within a multivessel column separating a n-component mixture. A detailed derivation of the above equations is provided in Appendix B. Further detailed analysis of the behavior of the multivessel column based on the system of equations developed for the multivessel column (eqs 31-34) should be pursued. An understanding of the behavior of a multivessel column would allow us to better elucidate its potential usefulness in separating multicomponent mixtures. Conclusion

n-1

Pi ∑ i)1

holdup in the entire column. We also define the n - 1 relevant parameters for the multivessel column as

dt

(31)

where Pi denotes the product flow rate from the ith holdup tray of the column and M denotes the total molar

A mathematical model of a middle vessel batch distillation column was developed on the basis of the simplifying assumptions of constant molar overflow, and quasi steady state (or negligible holdup) on the trays of the column. Extensive theories were also developed regarding the qualitative behavior of the MVC, when applied to the separation of azeotropic mixtures, based


on a limiting analysis of the model (as the number of trays in the column ND, NB f ∞, and reflux and reboil ratios Rd, Rb f ∞). This theory on the limiting behavior of the column is validated in the sequel to this paper30 by simulating the middle vessel column in the ABACUSS process modeling environment using the mathematical model developed. However, more work on this subject is encouraged, as there is the possibility that novel separations and more novel columns can be formulated in the spirit of Stichlmair et al.31 so as to increase the possibilities of separation which are not possible with the current array of separation equipment.

Multiplying the l.h.s. of eq 39 by eq41 which equals unity, the right-hand side (r.h.s.) is unchanged, and we obtain

M

dxM D + B dt ) (D + B)xM - (DxD + BxB) dt M dξ

(

)

(42)

from which M and dt can be cancelled to give

(D + B)

dxM ) (D + B)xM - (DxD + BxB) (43) dξ

Dividing eq 43 by (D + B), Acknowledgment This work was supported by the U. S. Department of Energy under Grant DE-FG02-94ER14447 and the MIT Undergraduate Research Opportunities Program.

(

)

(44)

and remembering the definition of the MVC parameter λ(t) as being

Appendix A. Derivation of Middle Vessel Column Model Equations. Provided in this appendix is the detailed derivation of the middle vessel column model equation (eqs 4 and 5) from the basic definition of warped time (eq 2), the component mass-balance equation obtained for the middle vessel column, and the overall mass balance for the MVC. Starting with the equation for component mass balance,

dMxM ) -(DxD + BxB) dt

(35)

differentiate the left-hand side (l.h.s) by parts to obtain

xM

dxM dM +M ) -(DxD + BxB) dt dt

(36)

We also have the overall mass-balance equation given as

dM ) -(D + B) dt

-xM(D + B) + M

dx ) -(DxD + BxB) dt

dxM ) (D + B)xM - (DxD + BxB) dt

(38)

(39)

But, from the definition of warped time,

(D M+ B) dt

(40)

which can be rearranged to obtain

1)

D(t) D(t) + B(t)

(45)

λ is then substituted into eq 44 to give,

dxM ) xM - λxD - (1 - λ)xB dξ

(46)

which is equivalent to eq 4. The derivation of the definition of warped time as given by eq 5 is also presented as follows. From the overall mass-balance equation as given by eq 37, we obtain

dM ) -(D + B) dt

(47)

Equation 47 can then be substituted into our definition of the dimensionless warped time given by eq 40 to obtain

dξ ) -

dM M

(48)

which implies that

dξ ) -d[ln(M)]

(49)

M

or

dξ )

λ(t) )

(37)

Substituting eq 37 into eq 36, the following expression is obtained:

M

B dxM D ) xM xD + xB dξ D+B D+B

(D M+ B) dξdt

which is exactly eq 5. As shown by Bernot et al.,16 eq 5 can then be manipulated to obtain an expression of the proportion of initial charge that has been drawn off as products, Π/M(0), where Π is the cumulative amount of distillate and bottoms removed from time ξ ) 0 to any given time ξ > 0, and M(0) is the initial molar holdup in the still pot. Using the initial conditions of ξ ) 0 at M ) M(0), eq 5 is solved as

Π(ξ) M(0) - M(ξ) ) ) 1 - exp(-ξ) M(0) M(0)

(50)

where Π is given by

(41)

Π(ξ) ) M(0) - M(ξ) )

∫0ξ(B + D) dξ

(51)


which can be re-expressed with a change of variables using eqs 37, 2, and 5, to obtain the following:

Π(ξ) )

∫0

ln[M(0)/M(t[ξ])]

M dξ

(53)

Component mole balances on each stage of the column (assuming CMO, envelope C in Figure 1) then yields the following operating line relationships:

-Ldxnd - Vdynd + Ldxnd+1 + Vdynd-1 ) 0, ∀ 1 e nd e ND (54) where y0 ) yM is the middle vessel vapor composition, and because of the total condenser assumption xND+1 ) yND ) xD. The vapor-liquid equilibrium (VLE) relationship on each tray and in the middle vessel can also be written as

yi ) yi(xi, T(xi, P), P),

i ∈ {nd}, {nb}, M

(55)

These equations, (53), (54), and (55), define the value of xD as expressed by eq 6 for a given value of xM. A similar set of equations can also be obtained for the stripping section of the column, as given by considering envelope G in Figure 1:

Lb ) Vb + B

(56)

and by considering the mass-balance envelope E in Figure 1:

-Lbxnb - Vbynb + Lbxnb+1 + Vbynb-1 ) 0, ∀ 1 e nb e NB (57) where xNB+1 ) xM is the still pot liquid composition, and because of the total reboiler assumption y0 ) x1 ) xB. The VLE relationships given by eq 55 also hold as before in the stripper trays. Equations 56, 57, and 55 then combine to define xB given xM as expressed by eq 7. Encapsulating all the mass-balance relationships given by eqs 53-57 into composite functions (6) and (7), eqs (4-7) then characterize completely the behavior of the still pot composition in the MVC. In the presence of finite reboil and reflux ratios, with our assumption of quasi steady state, a reflux ratio (Rd) and reboil ratio (Rb) can be defined accordingly and eqs 53-57 modified to obtain a set of mass-balance relationships based on Rd and Rb. First, the reflux ratio as defined for conventional batch rectifiers is given by

Ld Rd ) D

Rb )

(52)

The details of the formulation for the set of equations in (6) and (7) or for the set of equations in (8) and (9) are encapsulated in the algebraic mass balances which can be written for the column. On the basis of the QSS assumptions, given that these mass balances are valid instantaneously at any point in time, accumulation of the variables is omitted. Considering the rectifying section of the column, the total mole balance around the condenser (envelope F in Figure 1) gives

Vd ) Ld + D

while the reboil ratio can be defined as

Vb B

(59)

Next, substituting the definition of the reflux ratio as given by eq 58 into (54), a new operating line equation is obtained for the rectifying section of the column:

-Rdxnd - (Rd + 1)ynd + Rdxnd+1 + (Rd + 1)ynd-1 ) 0, ∀ 1 e nd e ND (60) where, as before, y0 ) yM is the middle vessel vapor composition, and because of the total condenser assumption xND+1 ) yND+1 ) xD. By a similar procedure, (59) can also be substituted into (57), to obtain a new operating line equation for the stripping section of the column:

-(Rb + 1)xnb - Rbynb + (Rb + 1)xnb+1 + Rbynb-1 ) 0, ∀ 1 e nb e NB (61) where xNB+1 ) xM is the still pot liquid composition, and as before, because of the total reboiler assumption, y0 ) x1 ) xB. The vapor-liquid equilibrium (VLE) relationship on each tray and in the middle vessel is unchanged and still given by eq 55. Equations 60 and 61 combined with (55) then define the distillate product xD and the bottoms product xB with respect to the reflux ratio Rd and the reboil ratio Rb in the formulation as given by eqs 8 and 9, respectively. B. Derivation of Model Equations for the Multivessel Column. Starting with the equation for component mass balance,

dMxM

n-1

)-

dt

{PixPi } ∑ i)1

(62)

we differentiate the l.h.s. by parts to obtain

xM

dM

+M

dt

dxM

n-1

)-

dt

{PixPi } ∑ i)1

(63)

But, the overall mass-balance equation is given as

dM

n-1

)-

dt

Pi ∑ i)1

(64)

Hence, substituting eq 64 into eq 62, the following expression is obtained: n-1

xM(-

Pi) + M ∑ i)1

dxM dt

n-1

)-

{PixPi } ∑ i)1

(65)

or equivalently,

M

dxM dt

n-1

) xM(

n-1

Pi) - ∑ {PixPi } ∑ i)1 i)1

n-1

(58)

)

Pi(xM - xPi ) ∑ i)1

(66)


But, from the definition of warped time τ for the multivessel column

() ()

dxM

n-1

dτ )

ϑ1-ϑn-1 are then substituted into eq 71 to give

Pi ∑ i)1

dτ dt

M

(67)

which can be rearranged to obtain

n-1

)

ϑi(xM - xPi ) ∑ i)1

which is eq 34. We can also obtain an alternate definition of warped time as follows. From the overall mass-balance equation as given by eq 64, we obtain

n-1

1)

n-1

Pi ∑ i)1

dt

M

dτ

dM ) -( (68)

Multiplying the l.h.s. of eq 66 by eq 68 which equals unity, the r.h.s. is unchanged, and we obtain the following equation:

()

dt

M

dτ

M

dxM

)

dt

Pi(x ∑ i)1

-

xPi )

(69)

dxM

Pi ∑ i)1


(70)

n-1 Dividing eq 70 by (∑i)1 Pi),

n-1

dx

)

(75)

dτ ) -d[ln(M)]

(76)

n-1

)

dτ

M

dM dτ ) M

which gives a definition of warped time different from that of eq 67.

from which M and dt are cancelled: n-1

(74)

which implies that

n-1

M

Pi) dt ∑ i)1

Equation 74 can then be substitued into our definition of the dimensionless warped time given by eq 67 to obtain

n-1

Pi ∑ i)1

(73)


dτ

n-1

(71)

Pi ∑ i)1

The definition of the multivessel column parameters were defined by eq 32 as

ϑ1 )

P1 n-1

Pi ∑ i)1

ϑ2 )

P2 n-1

Pi ∑ i)1

l ϑn-1 )

(72) Pn-2 n-1

Pi ∑ i)1

ϑn-1 )

Pn-1 n-1

Pi ∑ i)1

Literature Cited (1) Hasebe, S.; Noda, M.; Hashimoto, I. Optimal operation policy for multi-effect batch distillation system. Comput. Chem. Eng. 1997, 21, S1221-S1226. (2) Barolo, M.; Guarise, G. B.; Rienzi, S. A.; Trotta, A.; Macchietto, S. Running batch distillation in a column with a middle vessel. Ind. Eng. Chem. Res. 1996, 35, 4612-4618. (3) Barolo, M.; Guarise, G. B.; Ribon, N.; Rienzi, S. A.; Trotta, A. Some issues in the design and operation of a batch distillation column with a middle vessel. Comput. Chem. Eng. 1996, 20, S37S42. (4) 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. (5) 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. (6) Meski, G. A.; Morari, M. Design and operation of a batch distillation column with a middle vessel. Comput. Chem. Eng. 1995, 19, S597-S602. (7) 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. (8) Safrit, B. T.; Westerberg, A. W. Improved operational policies for batch extractive distillation columns. Ind. Eng. Chem. Res. 1997, 36, 436-443. (9) Safrit, B. T.; Westerberg, A. W. Algorithm for generating the distillation regions for azeotropic multicomponent mixtures. Ind. Eng. Chem. Res. 1997, 36, 1827-1840. (10) Safrit, B. T.; Westerberg, A. W. Synthesis of azeotropic batch distillation separation systems. Ind. Eng. Chem. Res. 1997, 36, 1841-1854. (11) Robinson, C. S.; Gilliland, E. R. Elements of Fractional Distillation, 4th ed.; McGraw-Hill: New York, 1950. (12) Devyatikh, G. G.; Churbanov, M. F. Methods of high purification. Znanie, 1976. (13) Lelkes, Z.; Lang, P.; Benadda, B.; Moszkowicz, P. Feasibility of extractive distillation in a batch rectifier. AIChE J. 1998, 44, 810-822. (14) Cheong, W. Simulation and analysis of a middle vessel batch distillation column. Bachelor’s thesis, Massachusetts Institute of Technology, May 1998.

1530 Ind. Eng. Chem. Res., Vol. 38, No. 4, 1999 (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) Bernot, C.; Doherty, M. F.; Malone, M. F. Patterns of composition change in multicomponent batch distillation. Chem. Eng. Sci. 1990, 45, 1207-1221. (17) Cheong, W.; Barton, P. I. Azeotropic distillation in a middle vessel batch column. 2. Nonlinear separation boundaries. Ind. Eng. Chem. Res. 1999, 38, 1531-1548. (18) Ahmad, B. S.; Barton, P. I. Homogeneous multicomponent azeotropic batch distillation. AIChE J. 1996, 42, 3419-3433. (19) Van Dongen, D. B.; Doherty, M. F. On the dynamics of distillation processes VI: Batch distillation. Chem. Eng. Sci. 1985, 40, 2087-2093. (20) Bernot, C.; Doherty, M. F.; Malone, M. F. Feasibility and separation sequencing in multicomponent batch distillation. Chem. Eng. Sci. 1991, 46, 1311-1326. (21) Ewell, R. H.; Welch, L. M. Rectification in ternary systems containing binary azeotropes. Ind. Eng. Chem. 1945, 37, 12241231. (22) Reinders, W.; De Minjer, C. H. Vapor liquid equilibrium in ternary systems. Recl. Trans. Chim. 1940, 59, 207-230. (23) Widagdo, S.; Seider, W. D. Journal review: Azeotropic distillation. AIChE J. 1996, 42, 96-130. (24) 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.

(25) Luyben, W. L. Multicomponent batch distillation. 1. ternary sytems with slop recycle. Ind. Eng. Chem. Res. 1988, 27, 642647. (26) 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. (27) Matsuyama, H.; Nishimura, H. Topological and thermodynamic classification of ternary vapor-liquid equilibria. J. Chem. Eng. Jpn. 1977, 10, 181-187. (28) Wittgens, B.; Litto, R.; Sorensen, E.; Skogestad, S. Total reflux operation of multivessel batch distillation. Comput. Chem. Eng. 1996, 20, S1041-S1046. (29) Skogestad, S.; Wittgens, B.; Litto, R.; Sorensen, E. Multivessel batch distillation. AIChE J. 1997, 43, 971-978. (30) Cheong, W.; Barton, P. I. Azeotropic distillation in a middle vessel batch column. 3. Model validation. Ind. Eng. Chem. Res. 1999, 38, 1549-1564. (31) Stichlmair, J.; Fair, J. R.; Bravo, J. L. Separation of azeotropic mixtures via enhanced distillation. Chem. Eng. Prog. 1989, 85, 63-69.

Received for review July 17, 1998 Revised manuscript received December 17, 1998 Accepted December 18, 1998 IE980469R

Azeotropic Distillation in a Middle Vessel Batch Column. 1. Model

Recommend Documents