Multicomponent homogeneous azeotropic distillation 3. Column sequence synthesis

Chemical Engineering Science 56 (2001) 4417– 4432

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Multicomponent homogeneous azeotropic distillation 3. Column sequence synthesis Dennis Y.-C. Thong ∗ , Megan Jobson Department of Process Integration, UMIST, PO Box 88, Manchester M60 1QD, UK Received 27 April 2000; received in revised form 2 February 2001; accepted 19 February 2001

Abstract An algorithmic distillation column sequence synthesis procedure for multicomponent azeotropic mixtures is proposed. The synthesis procedure is divided into several stages, and makes use of the feasibility test for classes of splits, as de6ned by Thong and Jobson (Chem. Eng. Sci. (2001a) submitted), to systematically generate a range of feasible and potentially feasible column sequences for a speci6ed separation requirement. A set of rules is applied to generate a recycle superstructure for every identi6ed sequence. Stream compositions and recycles are then 6nalised. Column design methods presented in Thong and Jobson (Chem. Eng. Sci. (2001b) submitted) are then applied to every column in the sequence. ? 2001 Elsevier Science Ltd. All rights reserved. Keywords: Distillation; Azeotropic; Multicomponent; Sequence synthesis; Recycles; Boundary crossing

1. Introduction All distillation sequencing techniques for nonazeotropic mixtures are sequential in nature, i.e., all splits are identi6ed for a given feed composition and quality. All further splits are then identi6ed for every product stream from all the possible 6rst splits. The process continues until no further splits are possible. This sequential approach is a comprehensive way of identifying all feasible column sequences that do not have internal recycles. The presence of recycle streams in the column sequence breaks down the ordered procession of splits and changes the nature of the problem from sequential to iterative. Ternary and quaternary diagrams allow the mixing and splitting material balances to be represented graphically, thereby facilitating an intuitive solution to complex sequencing problems with three or four component mixtures. While being a useful tool for three and four component mixtures, graphical representation of higher dimensional composition spaces is not possible. The lack of visual aids therefore complicates column sequencing in mixtures with more than four components. ∗

Corresponding author. Tel.: +44-161-200-4381; fax: +44-161236-7439.

A variety of column sequencing methods (Wahnscha=t, Le Rudulier, & Westerberg, 1993; Bauer & Stichlmair, 1998; Rooks, Julka, Doherty, & Malone, 1998) have been proposed for the separation of multicomponent azeotropic mixtures. None of these methods require visual aids and are, in theory, applicable to mixtures with any number of components. All these methods are sequential, with stream compositions determined at each successive step. This approach limits the number of sequences that can be generated as there are restrictions on feasible products from a column with a completely speci6ed feed stream (Wahnscha=t, Koehler, Blass, & Westerberg, 1992; Fidkowski, Doherty, & Malone, 1993). In this work a column sequence synthesis procedure is developed for application to multicomponent azeotropic mixtures. The synthesis procedure uses feasible and potentially feasible classes of splits, de6ned in Thong and Jobson (2001a) as splits producing product regions, or sets of product compositions. Synthesis is carried out sequentially, with each product region being the feed to the next split. Because precise stream compositions do not have to be speci6ed, the feed regions (product regions from the previous split) only serve to constrain the split to lie in certain zones in the composition space, in which many di=erent splits are possible. This approach

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also makes full use of the potential bene6ts from recycling for the manipulation of feed compositions. 2. Background Wahnscha=t et al. (1993) present a general synthesis methodology for the separation of multicomponent mixtures. The methodology is sequential in nature, and makes use of process simulations to determine product composition ranges for a speci6ed feed composition. Recovery and purity speci6cations are represented by pseudobinary ‘separation tasks’, one for every binary pair in the mixture. The methodology starts by sequentially identifying, by repeated process simulations, all the possible column sequences for a given feed composition. Some splits are then combined, and recycles are assigned. Because recycles alter the feed compositions and Fow rates to process units, the relevant units will normally have to be simulated again, until the Fowsheet simulation converges. The synthesis methodology is applicable to processes involving any type of unit operation, and is, in principle, applicable to the separation of n-component mixtures, although its implementation su=ers when there are more than three components in the mixture as obtaining the appropriate recycle compositions becomes diGcult. The methodology can miss interesting sequences because of the use of the pseudobinary representation of stream compositions and splits which, in azeotropic mixtures, does not convey the full extent of the limits imposed by non-ideal VLE on the unit operations. This limitation is compounded by the use of process simulations to determine product compositions—some feasible splits will not be identi6ed as the (distillation) simulations are carried out at large reFux ratios. Bauer and Stichlmair (1998) develop a superstructure of columns, each of which performs a ‘preferred separation’, which is a thermodynamically ideal separation. A ternary example is illustrated in Fig. 1. A preferred separation is performed in an adiabatic column, which is not the same as a reversible separation. Starting with a speci6ed feed composition, a series of preferred separations is identi6ed. This series forms the superstructure, which is then optimised using mixed integer non-linear programming routines, the objective being the minimisation of the total annualised cost. Recycle streams are determined prior to the optimisation step (Fig. 1). This necessarily requires the user to specify mixing operations and therefore necessitates some understanding of the limits imposed by the non-ideal VLE. Ternary and quaternary residue curve maps are used to aid the selection of suitable recycling options for three and four-component mixture. The application of the method to mixtures containing more than four components is therefore dependent on the existence of an appropriate recycling scheme, which they do not attempt to develop. The application

of computationally intensive MINLP routines to complex superstructures separating n-component mixtures, where the size of the superstructure is potentially overwhelming, is a further limitation of the method. To overcome this limitation, the authors simplify the superstructures by combining some splits in the complex superstructure. Although this simpli6cation reduces the size of the problem, there is no guarantee of the solution quality as clear guidelines for combining splits do not exist. The ‘common saddle test’ (Rooks et al., 1998) for column feasibility requires the composition pro6les from both speci6ed product compositions to approach the same saddle at total reFux. This behaviour is inferred from the reachability matrix (Knight & Doherty, 1990). Repeated application of the common saddle test to a speci6ed feed composition yields a variety of feasible column sequences within the distillation region that contains the feed composition. Recycling is not accommodated, although an algorithm to determine feed compositions that allow complete recovery of a component is presented in Rooks et al. (1998). The sequencing procedure does not incorporate any information on optimality since column design is not performed. The common saddle criterion is a suGcient condition for split feasibility. Thus, the criterion precludes other feasible splits (Thong & Jobson, 2001a), resulting in the procedure missing some potentially promising feasible column sequences. 3. Distillation column sequence synthesis procedure 3.1. Problem speci5cation The 6rst step in any sequencing procedure is the speci6cation of the problem. After the feed composition has been identi6ed, the adjacency and reachability matrices can be calculated using methods outlined in Knight and Doherty (1990). The distillation regions and boundaries can then be identi6ed (Rooks et al., 1998). The desired products are then speci6ed. A simple check will reveal if the feed and products are in di=erent distillation regions. Section 3.5 describes a variation of the sequence synthesis procedure that includes columns that straddle boundaries. Other boundary crossing techniques, e.g., phase splitting, liquid–liquid extraction, and boundary shifting by changing the operating pressure or mass transfer regime (Castillo & Towler, 1998) are not accounted for here, although they can be incorporated in the general synthesis framework. The procedure continues at the next stage in the absence of a distillation boundary between the feed and product compositions. The problem speci6cation is summarised in the following algorithm: 1. Identify the mixture and feed composition. 2. Calculate all azeotropes in the mixture (Fidkowski et al., 1993).

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Fig. 1. A complex superstructure of preferred splits, as represented on a ternary diagram. A preferred split is formed by extending the vector joining a liquid composition x, with the vapour composition in equilibrium with it, y∗ (Bauer and Stichlmair, 1998).

3. Calculate the adjacency and reachability matrices (Knight & Doherty, 1990). 4. Identify all distillation regions and boundaries (Rooks et al., 1998). 5. Identify all distillation compartments (Thong & Jobson, 2001a). 6. Determine which distillation region the feed lies in by calculating a residue curve from the feed composition. Assuming linear compartmental boundaries, determine geometrically the compartment which contains the feed composition (Thong, 2000). 7. Specify the desired products or components to be recovered. 8. See Section 3.5 if a distillation boundary separates the feed from one or more of the desired products. If not, proceed to Section 3.2. An equimolar mixture of methyl acetate, methanol, ethanol and isopropanol is to be separated into its constituent components. Fig. 2 shows the quaternary diagram, adjacency and reachability matrices, and lists the distillation regions, boundaries and compartments. One binary minimum-boiling azeotrope exists. All four components are to be recovered, and because no distillation boundary exists in the mixture, no boundary crossing is required. The feed composition lies in the (1-3-4-5) compartment. 3.2. Preliminary screening of column sequences The sequence synthesis procedure exploits the fact that internal recycle streams allow the manipulation of the feed compositions to any column in the sequence. Previous approaches to the synthesis problem (Wahnscha=t et al., 1993; Bauer & Stichlmair, 1998; Rooks et al., 1998) only considered exact feed compositions to the

various splits. Because the feed composition constrains the range of feasible product compositions (Wahnscha=t et al., 1992; Fidkowski et al., 1993), many feasible splits will be overlooked. Instead of specifying splits sequentially from 6xed feed compositions, the synthesis procedure generates sequences of columns, each performing a class of split (Thong & Jobson, 2001a). Precise compositions are not speci6ed, and the feeds to the various splits only serve to constrain the product regions from these classes of splits to certain distillation regions or compartments, and not to limited ranges of product compositions, as do precise feed compositions. Once all the feasible and potentially feasible classes of splits for an n-component mixture have been identi6ed, all possible column sequences are generated by recursive searching, starting with (C-1)-dimensional splits and ending with binary splits. The column sequences that recover the desired components are accepted while the others are rejected. A promising column sequence can then be selected for further study. All stream compositions in all column sequences, except the overall feed composition, do not have to be speci6ed at this point. No recycling exists at this stage. The full algorithm for screening feasible column sequences is as follows: 1. Identify all (C-1)-dimensional and lower-dimensional classes of splits using the algorithms in Thong and Jobson (2001a). 2. Select either a (C-1)-dimensional split in the compartment which contains the feed, or a cross-compartmental split involving the feed compartment and an adjacent compartment. 3. For the selected (C-1)-dimensional split identify the sub-region (compartment) that the product regions lie in.

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Fig. 2. A quaternary diagram representing the composition space of a four-component azeotropic mixture. Also shown are the adjacency and reachability matrices. The mixture displays one distillation region and two compartments.

4. For both product streams from the 6rst split, identify all possible splits in the compartments in which these products lie, as well as any cross-compartmental splits involving adjacent compartments. 5. Repeat Step 4 for all further product streams. Stop when no further splits are possible. 6. Combine any identical splits in di=erent parts of the same sequence, i.e., combine feed streams to columns performing the same class of split. 7. An intermediate stream that does not contain any desired components can either be fed to any column or left unseparated. 8. Repeat steps 2–5 for all (C-1)-dimensional splits in the feed compartment and cross-compartmental splits involving adjacent compartments. This generates many column sequences. 9. For each column sequence, determine if all desired components are recovered. Accept the sequence if they are. Reject it otherwise. Table 1 lists the feasible and potentially feasible classes of splits identi6ed for the quaternary azeotropic mixture presented in Section 3.1. All quaternary, ternary and binary splits are listed and labelled according to split type. Type D splits are omitted in this example. To illustrate the screening algorithm, a 1=3-4-5 split is arbitrarily selected (Step 2). The distillate composition is that of the azeotrope, so no further splits are possible with this stream. The bottoms product lies in the (3-4-5) compartment, in which three Type A splits are possible (Table 1). Fig. 3 shows the separation tree corresponding to the 1=3-4-5 split as the 6rst split. None of the options in Fig. 3 lead to the recovery of methyl acetate (2), which is a desired product. As a result, this sequence is rejected. Instead of the 1=3-4-5 split, any of the other quaternary splits in the feed com-

Fig. 3. Separation tree for a 1=3-4-5 6rst split. In all cases the methyl acetate-methanol azeotrope (1), methanol (3), ethanol (4) and isopropanol (5) are recovered. Superscripts denote the split type.

partment, as well as the Type C splits between compartments can be selected to be the 6rst split. Enumeration of all these splits leads to a total of 46 di=erent potentially feasible sequences, i.e., sequences where all four components are recovered. Fig. 4 lists these potentially feasible sequences. The asterisks indicate identical splits that can be combined. All the sequences listed in Fig. 4 are potentially feasible as they all involve Type C splits— if the particular Type C split is infeasible, the sequence becomes infeasible too. 3.3. Generating recycle options for a sequence of columns Once all potentially feasible sequences have been identi6ed, suitable recycling options can be identi6ed using a simple procedure. First, a superstructure of recycling options is constructed. Every end product from a column (component, azeotrope or a stream with no desired

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Table 1 Feasible and potentially feasible classes of splits for the quaternary mixture in Fig. 2. Superscripts denote the split type 4-distillation regions

1-2-3-4-5

Compartments

C#1 (1-2-4-5)

C#2 (1-3-4-5)

1=4-5A 1=2-4-5A 1-2=4-5A 1-2=2-4-5A 1-2-4=5A 1-2-4=4-5A 1-2-4=2-4-5B

1=4-5A 1=3-4-5A 1-3=4-5A 1-3=3-4-5A 1-3-4=5A 1-3-4=4-5A 1-3-4=3-4-5B

Quaternary splits (Types A & B)

C#1xC#2 1-2=3-4-5 1-3=2-4-5 1-2-4=3-4-5 1-3-4=2-4-5

Quaternary splits (Type C)

3-distillation regions

1-2-3-4

1-2-3-5

2-4-5

3-4-5

Compartments

C#1 (1-2-4)

C#2 (1-3-4)

C#3 (1-2-5)

C#4 (1-3-5)

C#5 (2-4-5)

C#6 (3-4-5)

Ternary splits

1=4A 1=2-4A 1-2=4A 1-2=2-4A

1=4A 1=3-4A 1-3=4A 1-3=3-4A

1=5A 1=2-5A 1-2=5A 1-2=2-5A

1=5A 1=3-5A 1-3=5A 1-3=3-5A

2=4-5A 2-4=4-5A 2-4=5A

3=4-5A 3-4=4-5A 3-4=5A

(Types A & B) Ternary splits (Type C)

C#1xC#2 1-2=3-4 1-3=2-4

Binary splits

1=2A

C#3xC#4 1-2=3-5 1-3=2-5 1=3A

components) is a potential recycle stream. Every feed to a column is a potential destination for every recycle stream. The size of this superstructure is then reduced using a set of rules, which are as follows: 1. Azeotropes can either be recovered, partially recovered, or recycled completely. 2. Never recycle a stream to the column that produces it. 3. Never mix a recycle stream with a feed to a column performing a split where the recycle stream contains one or more components that are not present in either product stream. 4. Never mix streams with compositions in di=erent compartments. The exception to this are recycle streams to columns performing Type C splits, which can lie in either compartment that the split traverses. The 6rst rule accounts for di=erent requirements, e.g., the azeotrope might be recycled to the reactor, or might be recycled internally in the separation scheme. The second rule is rigorous for one-feed, two-product columns as

2=4A

2=5A

3=4A

3=5A

4=5A

the recycling of a product stream to the parent column will result in the material balance over the column being violated. The third rule is a consequence of the synthesis procedure, which generates splits in a backward direction, i.e., products from a column are selected 6rst, then the required feed to the column is calculated. If a feed stream to a column performing a pre-speci6ed split has more components than both product streams from the column, the split will be infeasible. The fourth rule is based on the observation that crossing boundaries by recycling always leads to infeasible sequences (Doherty & Caldarola, 1985). This rule is conservative, as there are cases where mixing across the boundary is possible. An example is presented in Fig. 5. Here, streams b1 and d 1 are mixed although they exist in di=erent distillation regions. The column sequence in Fig. 5 can be seen to be feasible since all the internal material balances are satis6ed. This sequence is feasible because of the curvature of the distillation boundary, which allows the material balance over the second and

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third columns to be satis6ed, i.e., composition b1 lies in the triangle formed by b2, b3 and d 3. Furthermore, even straight boundaries can be crossed by mixing as the position of the boundary can be manipulated by changing the operating pressure or mass transfer regime. The recycling options are generated for one of the potentially feasible sequences in Fig. 4. Fig. 6 shows a four-column sequence with potential recycles indicated by dashed lines. In Fig. 6 the type of splits that the columns perform are indicated by letter. As the azeotrope (1) is recycled internally in this example, the only possible location for this stream, identi6ed by elimination using the recycling rules, is the feed stream to the 6rst column. The feed stream to the 6rst column can accept all recycle streams as there are no distillation boundaries in the mixture, and the column performs a Type C split that traverses both compartments. Similar recycling superstructures can be generated for the other 45 sequences listed in Fig. 4. 3.4. Determining stream compositions and 5nalising recycle streams in a column sequence Only product regions have been determined thus far in the synthesis procedure. The recycle streams enable a split to be performed, but the appropriate recycle options can only be 6nalised after the stream compositions have been determined. Determining stream compositions is an iterative procedure and starts with specifying product compositions for a particular split. An appropriate test for feasibility (Thong & Jobson, 2001a; Bausa, von Watzdorf, & Marquardt, 1998) is applied to test for product feasibility. If the selected product compositions are infeasible, another pair of compositions can be speci6ed. The process of searching for feasible product compositions continues until a pair of feasible compositions is found. This process is carried out for every column in the sequence as follows:

Fig. 4. Potentially feasible sequences for the recovery of all constituent components in the quaternary mixture in Fig. 2.

1. For all columns performing a Type B, C or D split, identify feasible product compositions by arbitrarily specifying product compositions and checking for feasibility using an appropriate feasibility test. Feasible product compositions are identi6ed independently for each column performing a Type B, C or D split. 2. For all columns performing Type A splits that do not have a speci6ed feed composition (a product stream from a column performing a Type B, C or D split), arbitrarily specify a pair of product compositions. There is no need for a feasibility test as any product compositions in the product regions will be feasible (Thong & Jobson, 2001a). 3. If a column performing a Type A split has a speci6ed feed composition, specify either a distillate or bottoms composition and calculate the co-product composition using a material balance. If the co-product

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Fig. 5. A feasible three-column sequence to recover acetone and chloroform using toluene as the entrainer. The sequence includes a boundary-crossing operation using mixing. Solid lines indicate separations and dashed lines indicate mixing operations.

Fig. 6. The recycle superstructure for a potentially feasible four-column sequence to recover all constituent components in a quaternary mixture (Fig. 2). Solid lines represent de6nite recycles. Dashed lines indicate potential recycles.

composition is in the appropriate product region, the product compositions will be feasible. If not, specify arbitrarily both product compositions. This process is illustrated using the column sequence in Fig. 7, which is identical to the sequence in Fig. 6. Feasible product compositions are identi6ed for the 6rst column, which performs a Type C split. The bottoms product stream from the 6rst column is fed to a column which performs a direct (Type A) split. As methanol (3) is to be recovered as the distillate from this column,

a material balance determines the bottoms composition, which is on the (4-5) composition boundary. These product compositions are therefore feasible. The columns performing binary splits do not have to be checked for feasibility as they will always be feasible. It was stated in Thong and Jobson (2001a) that any product composition in the appropriate product regions will be feasible if the split is a Type A split. This does not mean that any feed composition in the appropriate compartment will enable a Type A split to be realised. Consider the quaternary mixture in Fig. 8. A feasible Type A split is the 1=5-6 class of split. Geometrically, this split requires a feed composition to lie in a plane formed by the points 1, 5 and 6 in Fig. 8. This feed composition to this column will be either the overall feed to the column sequence or a product from a previous split, and is unlikely to lie in the (1-5-6) plane. As a consequence, the column performing the 1=5-6 split is likely to require one or more recycle streams to enable the split to be executed. Some Type A splits therefore require recycles, an occurrence which is accounted for in Step 3 of the algorithm to determine stream compositions. Once all stream compositions have been identi6ed, appropriate recycle options are determined using a material balance across every column. This material balance can be formulated as a geometrical problem, as shown in a hypothetical four-component mixture in Fig. 9. There are always only two products from a column, d and b, which results in the line db. The overall feed to this column must lie somewhere on the line db. In Fig. 9 there are three possible recycle streams with compositions ji , ki and li . The stream fi has to be fed to this column as it is the product stream from a preceding column. For the

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Fig. 7. Stream compositions for the four-column sequence presented in Fig. 6.

Fig. 8. A quaternary azeotropic mixture at 1 atm. The shading indicates the distillation boundary.

Fig. 9. A geometrical representation of a quaternary split and recycle options. ji and ki are feasible recycle options as the plane ji ki fi intersects the line db.

split to work, the geometrical entity formed by mixing any of ji , ki and li with fi , must intersect the line db. As the composition space is three dimensional, a minimum of three streams, which form a 2-D plane, will be necessary to ensure a mathematical chance of feasibility (Thong, 2000). In Fig. 9 the mixing of streams ji , ki and fi result in a 2-D plane that intersects the line db. The mixture of streams ji , ki and fi therefore constitutes a feasible recycle option. There are, in general, multiple recycle options when there are more than (C-2) potential recycle streams in a (C-1)-dimensional split. When the recycling rules are applied to the sequence in Fig. 7, with the recycle superstructure shown in Fig. 6, one feasible recycle option is shown in Fig. 10. In this example, the azeotrope (1), methyl acetate (2) and isopropanol (5) are recycled to the 6rst column. When

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Fig. 10. One of several recycle options for the sequence and stream compositions in Fig. 7. The material balance over the two-column subsystem (dashed line) is not satis6ed—this requires a composition change in the stream (3-4-5).

internal recycles are present, however, material balances overcolumnsubsystemswithintheoverallsequencehaveto be satis6ed. The control volume over the 6rst two columns, represented by the polygon in Fig. 10, shows one material balance that is not satis6ed for this particular choice of stream compositions. The recycle structure is in itself feasible as the manifold formed by joining points F and 5, and the manifold formed by joining points 2 and points 3, 4 and 5 (representing product region (3-4-5), intersect. This intersection implies that there should be stream compositions that enable this particular material balance to be satis6ed. The columns in the material balance essentially perform a direct split to recover methyl acetate (2). Geometrically, the material balance requires the intersection of the line (F-5), which indicates the inputs, with the line [2-(3-4-5)], which indicates the outputs. The material balance in Fig. 10 can also be interpreted in terms of component Fow rates. There is no problem with the methyl acetate and isopropanol balances as methyl acetate is completely removed in stream (2). The relative amount of the isopropanol recycle (5) can be varied, allowing any mole fraction of methyl acetate in the (3-4-5) stream. Because the feed is equimolar in all components, however, the mole fractions of methanol and ethanol in stream (3-4-5) have to be identical, as stream (3-4-5) is the only outlet for methanol and ethanol. The current composition of stream (3-4-5) does not allow the material balance to be satis6ed. Varying the amount of the isopropanol recycle (5) does not help. Because the compositions of streams F, 2 and 5 are invariant, the composition of stream (3-4-5) has to be changed.

Fig. 11. Geometrical representation of the material balance in Fig. 10. The lines (F-5) and [2-(3-4-5)1 ] do not intersect but the lines (F-5) and [2-(3-4-5)2 ] intersect.

All the information discussed in the preceding paragraph can be represented succinctly in geometrical form, which is shown in Fig. 11. The line (F-5) does not intersect the line [2-(3-4-5)]. Because the compositions of all other streams are 6xed, the composition of stream (3-4-5) has to be modi6ed. This is achieved by selecting a point on the line (F-5), point X in Fig. 11, and extending the line from this point to the point (2) till a composition in the (3-4-5) product region is reached. This is equivalent to selecting a recycle-to-feed ratio for the isopropanol (5) recycle and forcing the closure of the material balance. There are two conditions that have to be satis6ed concurrently—the material balance and the split feasibility. A new composition for the stream (3-4-5), indicated in Fig. 10 and calculated using the method described here,

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Fig. 12. A four-column sequence with modi6ed stream compositions and new recycles. Feasible column parameters for all non-binary splits are listed. The feed is equimolar in all components.

is feasible, i.e., the Type C split performed by the 6rst column is feasible. When a stream composition changes, downstream columns will naturally be a=ected. New downstream compositions can be determined using the algorithm described earlier in this Section. As stream compositions change, new recycle options must be identi6ed. This is illustrated in Fig. 12, which displays the new stream compositions and recycles after the stream composition changes in Fig. 10. All the material balances are satis6ed in this sequence, and the methyl acetate (2) recycle is no longer required. Fig. 12 also shows feasible column parameters for every non-binary split, calculated using the column design method presented in Thong and Jobson (2001b). The recycle-to-feed (R=F) ratios in Fig. 12 are determined from the positions of the intersections between the various geometrical entities. 3.5. Boundary crossing by exploiting boundary curvature Synthesising column sequences that include a boundary crossing operation is identical to the algorithms presented previously, except for the column that straddles the boundary. The type of boundary crossing implied here is the split with a feed in one region and products in an adjacent region. The other type of boundary crossing which involves open leaves are not accounted for here. Thong and Jobson (2001a) introduce a simple test to estimate the direction of curvature of a distillation boundary. Bound-

ary crossing is possible if the feed composition is in the concave segment of the distillation boundary. As indicated in Section 3.1, boundary crossing is required if a distillation boundary separates the feed from one or more desired products. The sequence synthesis procedure is carried out as described in Sections 3.1 and 3.2, until a compartment on a distillation boundary is produced as a product region. At this point, a simple check is applied to determine if the product region contains any of the components that are to be recovered. This check is necessary since not only boundaries but their lower-dimensional constituents can be crossed. This check is illustrated using the quaternary mixture in Fig. 8. If the feed composition is in the (1-2-3-5-6) region and the boundary curves towards the other region, a boundary crossing operation is feasible. If, say, pure methanol (4) is a desired product, the boundaries of interest are then (1-3) and (1-2-3-5). Product regions on the (2-5) and (3-5) boundaries do not contain methanol (4) and are therefore not worth crossing as there is no possibility of recovering methanol in subsequent columns. At the boundary, splits in the adjacent region are then carried out and the synthesis procedure is applied normally. Splits with one of the product regions on the distillation boundary are not considered. A ternary example is shown in Fig. 13. The product from the 6rst split lies on the distillation boundary (3-4) and is in the (1-3-4) region. This product is, therefore, a candidate feed composition for a column that straddles the boundary. Three splits are identi6ed for the (2-3-4) region, two of which

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Fig. 13. An example of a boundary crossing operation in a ternary mixture. The 1=3-4 split exists in region II while the 2-3=4 split is e=ectively in region I.

Fig. 14. A quaternary diagram representing the composition space of a four-component azeotropic mixture. Also shown are the adjacency and reachability matrices. The mixture displays two distillation regions.

have product regions on the distillation boundary. These two splits are therefore eliminated, leaving the 2-3=4 split as the remaining possible class of split for the column that crosses the boundary. The 2-3=4 split is, in this case, a feasible split for the boundary crossing operation (Fig. 13). It can be observed from Fig. 13 that a 2-3=3-4 split is a feasible boundary crossing operation although the bottom product region is the distillation boundary. The range of feasible product compositions for this class of split is, however, limited. Although conservative, the rule for eliminating these types of splits allows the identi6cation of splits that maximise the use of boundary curvature, e.g., the 2-3=4 split in Fig. 13. Note that the splits identi6ed

here are equivalent to sharp splits in non-azeotropic mixtures as only product regions on the composition or distillation boundaries are considered. This procedure does not identify products in the interior of the composition space, which might be feasible products from a column that straddles a distillation boundary. Fig. 13 shows the product compositions for the various splits only to illustrate the procedure. At the preliminary screening stage (Section 3.2), only product regions will be known. At the next stage (determining stream compositions), appropriate product compositions have to be identi6ed for the column that straddles the boundary. This is accomplished by trial and error.

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Table 2 Feasible classes of splits for the quaternary mixture in Fig. 14 4-distillation regions

1-3-4-5

2-3-4-5

Compartments

C#1 (1-3-4-5)

C#2 (2-3-4-5)

1=3-4-5A 1-3=4-5A 1-3=3-4-5A 1-3-4=5A 1-3-4=4-5A 1-3-4=3-4-5B

2=3-4-5A 2-3=4-5A 2-3=3-4-5A 2-3-4=5A 2-3-4=4-5A 2-3-4=3-4-5B

Quaternary splits (Types A & B)

3-distillation regions

1-3-4

1-3-5

1-4-5

2-3-4

2-3-5

2-4-5

Compartments

C#1 1-3-4

C#2 1-3-5

C#3 1-4-5

C#4 2-3-4

C#5 2-3-5

C#6 2-4-5

Ternary splits (Types A & B)

1=3-4A 1-3=3-4A 1-3=4A

1=3-5A 1-3=3-5A 1-3=5A

1=4-5A 1-4=4-5A 1-4=5A

2=3-4A 2-3=3-4A 2-3=4A

2=3-5A 2-3=3-5A 2-3=5A

2=4-5A 2-4=4-5A 2-4=5A

Binary splits

1=3A

1=4A

1=5A

Table 3 Feasible classes of splits on the distillation boundary (3-4-5) of the quaternary mixture in Fig. 14 4-distillation boundaries

3-4-5

Compartments

C#1 (3-4-5)

Quaternary splits (Type A)

3-4=5 3-4=4-5

3-distillation boundary

3-4

3-5

Compartments

C#1 (3-4)

C#1 (3-5)

Ternary splits (Type A)

3=4

3=5

As an example, after the 2-3=4 split in Fig. 13 has been identi6ed as a potential boundary crossing operation, a distillate composition can be speci6ed on the (2-3) edge and a straight line plotted from the distillate to the bottoms composition (benzene, 4). Residue curves are then calculated from initial points on this straight line to determine if any point on the line lies in the (1-3-4) region. If so, these point(s) are potential feed compositions to the column that straddles the boundary. This also determines one of the product compositions for the preceding column. If none of the points lie in the (1-3-4) region, new product compositions must be found and the residue curves recalculated. The process continues until a pair

2=3A

2=4A

2=5A

4=5A

of appropriate product compositions is found. The other stream compositions are determined as described in Section 3.4. The full algorithm for synthesising column sequences that include boundary crossing operations is as follows: 1. Specify the problem as in Section 3.1. 2a. Apply the methods described in Section 3.2 to generate column sequences. When a product region is a compartment on a distillation boundary and the product region contains the desired component(s), perform all splits in the adjacent region (compartment) that do not have the boundary as a product region. 2b. A split on the boundary is possible only if the feed to the split is on the concave side of the boundary. Boundary curvature can be estimated using the method presented in Thong and Jobson (2001a). 3. Generate recycle options as in Section 3.3. The feed stream to the column that straddles the boundary cannot be mixed with any recycle streams as the only recycles that are likely to be of use will be in an adjacent distillation region, and these recycles are prohibited by rule 4 in Section 3.3. 4. Determine stream compositions and 6nalise recycle options as in Section 3.4. Always determine product compositions for the column that straddles the boundary 6rst. Step 2b accounts for the fact that in some cases a split on the boundary is not possible. As an example, a 3-4 split is possible if the (3-4) product region is a result of

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Fig. 15. Potentially feasible sequences for the recovery of all constituent components in the quaternary mixture in Fig. 6:12. Boundary-crossing operations are highlighted in bold. The asterisks and plus signs indicate splits that can be combined.

a 1=3-4 split (Fig. 13). In this case the feed to the (3-4) split will not be on the boundary, but inside the (1-3-4) region. If the (3-4) product region is a result of a 2=3-4 split, a 3-4 split is not possible. Application of the algorithm is demonstrated on a quaternary mixture of acetone, chloroform, benzene and toluene, which forms a binary azeotrope between acetone and chloroform. The quaternary diagram, adjacency and reachability matrices are shown in Fig. 14. The boundary in Fig. 14, represented by the shaded region, is not the actual boundary but a representation of the fact that the boundary curves towards the (2-3-4-5) region. The feed to the column sequence is equimolar in all components.

Table 2 lists the feasible splits for the mixture in question. Type D splits are omitted and there are no Type C splits, since there is no further division of either distillation region into compartments. Table 3 lists the feasible splits on the boundary and its lower-dimensional constituents. Fig. 15 lists the sequences that enable the recovery of all the components. In six of the options (3A, 3B, 4A, 4B, 5A, 5C), the boundary can be crossed in the toluene-free ternary composition space. In several options (5A, 5B, 5F-L) the boundary can be crossed twice, once in the quaternary composition space and again in the ternary composition space. Although possible, this sequence is

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Fig. 16. The recycle superstructure for a potentially feasible four-column sequence to recover all constituent components in a quaternary mixture (Fig. 14). The second column straddles the distillation boundary.

not preferable, as boundary crossing should generally be kept to a minimum. Several sequences in Fig. 15 have the (3-4) product region as an end product. A further separation between 3 and 4 cannot be performed as the (3-4) product region is the product of a split on the convex side of the boundary. One of the sequences in Fig. 15 is selected for further study. Fig. 16 shows this sequence and its associated recycle superstructure. The sequencing algorithm does not allow the recycling of material to the feed of a column that straddles the boundary. As a result, streams 2, 3, 4 and 5 cannot be recycled to the feed of the second column in the sequence. The binary azeotrope (3) has to be mixed with the feed to the 6rst column as there is no other suitable recycling option for this stream. Fig. 17 shows the 6nal sequence, which satis6es all the internal material balances. Only the azeotrope is recycled for this particular choice of stream compositions. All end product streams are 99.9% pure. 4. Comments and future work Although mixing across distillation boundaries is not accounted for in the sequencing procedure, column sequences that exploit this occurrence, e.g., the sequence in Fig. 5, can be synthesised using the tools that have been developed. In some cases the mixing of streams might not be so much a viable option but a necessary operation to enable a split in a distillation region that does not contain the feed composition. As the feasible classes of splits will be the same regardless of mixing strategies, the sequence-screening step will have to be modi6ed to

allow splits in adjacent distillation regions. One or more recycling rules will also have to be added to the present list to accommodate mixing across the boundary. When determining appropriate recycle streams, a situation might arise where both product streams from a column are recycled to the same point. Unless both products are desired components, the degree of separation can be reduced and one recycle stream eliminated. An example of this is shown in Fig. 18, where the bottoms stream from a 1=3-4 split provides the feed to a 3=4 split on the boundary. The desired products of the three-component feed are pure benzene, pure acetone and the acetone-chloroform azeotrope. Th azeotrope (3) could, for example, be recycled to a reactor or fed to another separation process, e.g., a membrane separation unit. A recycle of acetone (1) is necessary to enable a 3=4 split to take place. In this case both product streams from the 1=3-4 split are fed to the next split. A reduction in the degree of separation simply translates as producing a bottoms product from the 1=3-4 split that lies on the straight line joining 3 and 4, and not on the distillation boundary. This eliminates the need for an acetone (1) recycle. This particular example is possible because the (3-4) stream is not a desired end product and does not therefore have any composition restrictions placed on it. This step eliminates unnecessary recycle streams and can, as in Fig. 18, reduce the degree of separation in a column, reducing the separation cost. The problem speci6cation, sequence-screening and recycle generation steps are easily automated. The determination of the stream compositions and 6nalisation of recycle streams are carried out manually in this work. These last two steps can, in principle, be automated as feasible compositions can be found using an appropriate feasibility test for product compositions, e.g., Bausa et al. (1998). Determining appropriate recycle options can be automated, as the geometric test described in Section 3.3 is numerical and can be formulated mathematically (Thong, 2000). Any algorithm for automating this step will have to modify stream compositions to simultaneously address the internal material balances as well as split feasibility. Although quaternary mixtures have been examined, the feasibility and column design methods are general and the sequence synthesis procedure applies to mixtures with more than four components. A limitation of the procedure is the use of the recycles, which can become cumbersome in mixtures with many components. Consider a split in a quaternary mixture with a feed and both product compositions speci6ed. Because the product compositions are speci6ed independently from the feed composition, the material balance around the column will not necessarily be satis6ed. A minimum of two recycle streams are required to satisfy the material balance, as three points are the minimum required to form a 2-D plane. A similar split in a six-component mixture will require a minimum of four recycle streams. Although feasible, this might not

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Fig. 17. A four-column sequence with stream compositions and recycles. Feasible column parameters for all non-binary splits are listed.

Fig. 18. An example of eliminating a recycle stream by reducing the degree of separation in a split. The dashed line indicates a mixing operation between streams B1 and acetone (1).

be a practical option. Selection of appropriate product compositions, as in the example in Fig. 17, can facilitate a reduction in the number of required recycle streams. This can be achieved by automating the search for feasible product compositions to allow the e=ect of di=erent product compositions on the recycle structure to be explored. A potential drawback of the column synthesis procedure in its current form is the generation of a large number of column sequences. The algorithm is formulated

with the purpose of generating as many column sequences as possible, so as not to miss promising Fowsheets. As in non-azeotropic mixtures, the number of potential sequences will increase dramatically with multicomponent mixtures. Shah (1999) has developed a distillation column synthesis and optimisation method that mathematically guarantees the global optimum in the separation of non-azeotropic mixtures. Such an approach is not directly applicable in the separation of azeotropic mixtures due to its sequential nature.

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An interesting topic for future research would be the optimisation of the sequences generated here. The column sequencing procedure can be applied within a larger separation framework that includes phase splitting, extraction, pressure-swing and mass transfer e=ects. In the case of phase splitting and extraction, all that is required is an accurate liquid–liquid equilibrium model to detect instances of two-phase behaviour. This model should be able to accurately predict the e=ect of temperature on the size of the liquid–liquid region as temperature is an important design variable in decanting and extraction. Pressure and mass transfer e=ects can be employed in tandem to shift distillation boundaries. When applying these techniques, the position of the boundary would have to be determined at di=erent pressures and mass transfer regimes. Once this is done, the sequencing procedure can be applied normally as the feasible classes of splits will remain the same under all conditions. It is, however, diGcult to calculate component stage eGciencies, which are required in the calculation of mass transfer rates, in mixtures with more than three components (Castillo & Towler, 1998). The development of accurate mass transfer models for n-component mixtures will facilitate the development of novel column sequences using the synthesis procedure presented here. 5. Conclusions An algorithmic synthesis procedure for column sequences separating n-component azeotropic mixtures has been developed. Although the synthesis procedure generates the splits sequentially, precise stream compositions are not required. The procedure therefore avoids the restriction on feasible product compositions from a completely speci6ed feed stream. This approach allows a more extensive use of recycles to address the issue of material balances around the various columns and column subsystems in the sequence. The procedure is divided into four distinct stages— problem speci6cation, preliminary sequence screening, generation of recycle options and 6nally, the determination of stream compositions and 6nalisation of the recycle structure. The problem speci6cation stage involves the speci6cation of the product requirements as well as the description of the composition space of the mixture to be separated. The feasibility test for classes of splits is used to generate feasible and potentially feasible sequences of columns in the preliminary sequencing stage. A set of rules is applied to generate a recycle superstructure for every one of the sequences identi6ed in the previous stage. Thus, many di=erent column sequences can be generated with minimal computational e=ort as design calculations are not required. In the 6nal stage, stream compositions and recycle structures are determined. This stage requires some trial and error and there exist many possible solu-

tions corresponding to di=erent stream compositions and consequently, di=erent recycle structures. Column design can then be carried out using the column design methods presented earlier in this work. The sequence synthesis procedure is developed to identify feasible sequences within a distillation region as well as sequences that exploit boundary crossing. Examples of both situations are presented.

Acknowledgements The authors would like to acknowledge the support of the UMIST Process Integration Research Consortium.

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