760
Ind. Eng. Chem. Res. 1991,30, 760-772
= coefficient in eqs 33-36 describing molecular interaction A and j effects in cages containing- i molecules of type .. molecules oftype B Biji = coefficient in eq 38 (for ternary system) describing interaction effect in cage containing i molecules of type A, j molecules of type B, and k molecules of type C p i = chemical potential of component i 4 = surface potential (see Larionov and Myers (1971) for precise definition and relationship to surface tension) yi = activity coefficient of component i defined according to definition of Larionov and Myers y[ = activity coefficient defined according to eq 13 (for a system in which n is independent of composition, y i = y[) yi' = activity coefficient defined according to Henry's law (yp = Yi/KiXi) Bi = fractional coverage of component i in Langmuir formulation Bij
Registry No. OX,95-47-6;MX,108-38-3;PX,106-42-3;EB,
100-41-4.
Literature Cited Goddard, M.; Ruthven, D. M. Adsorption of CsAromatics on KY Faujasite. Sirth International Zeolite Conference,Reno, 1983, Proceedings; Olson, D., Bisio, A., Eds.; Butterworths: Guildford, UK, 1984;pp 268-275. Hill, T. L. Introduction to Statistical Thermodynamics; Addison Wesley: Reading, MA,1960;p 124. Kemball, C.; Rideal, E. K.; Guggenheim, E. A. Thermodynamica of Monolayers. Trans. Faraday SOC.1948,44,94&954. Larionov, 0.G.; Myers, A. L. Thermodynamics of Adsorption for Non-Ideal Soltuions of Non-Electrolytes. Chem. Eng. Sci. 1971, 26,1025-1030. Myers, A. L.; Prausnitz, J. M. Thermodynamics of Mixed Gas Adsorption. AZChE J. 1965, fl, 121-129. Minka, C.; Myers, A. L. Adsorption from Ternary Liquid Mixtures. AZChE J. 1973,19,453-459.
Received for reoiew April 19, 1990 Revised manuscript receioed September 26, 1990 Accepted October 19,1990
Superscripts O , *, 0 = quantities at standard state (e.g., p i o ,p i o , etc.)
Automatic Screening of Entrainers for Homogeneous Azeotropic Distillation Etienne
R.Foucher, Michael F. Doherty,* a n d Michael F.Malone
Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003
Extended rules and an automatic procedure for the determination of the structure of simple distillation residue curve maps for ternary mixtures are described. The procedure makes use of results from directed graph theory and is based on an adjacency matrix representation. Simple distillation boundaries are determined as thermodynamically and topologically consistent connections between the singular points (i.e., azeotropes and pure components) and correspond to the nonzero entries of the adjacency matrix. The procedure requires only a knowledge of boiling temperatures and approximate compositions of any azeotropes in the mixture. This is sufficient to uniquely determine the structure of the map in most cases and enables the rapid identification of feasible entrainers. The incorporation of topological and thermodynamic constraints also makes it possible to detect certain inconsistencies in the boiling temperature data. We show three classes of maps of feasible entrainers for breaking minimum boiling azeotropes, with a corresponding column arrangement in each case; one of these is the classic extractive distillation. The method is applied to screen numerous entrainers for the separation of ethanol/water and butanol/butyl acetate mixtures. Introduction The synthesis of a distillation-based separation system is a challenging task involving two major subproblems. The fmt is to decide if the addition of another component as an entrainer is desirable and to choose one or more canidate entrainers and the corresponding column configurations. In a second stage, the optimal design, energy integration, and operability can be examined when sufficiently detailed vapor-liquid equilibrium and other data are available. It is critical in many cases to find good solutions to the first problem rapidly and with minimal data in order to limit the scope and expense of more detailed studies. In this paper, we focus exclusively on the first problem. For multicomponent, homogeneous, azeotropic mixtures where it is ineffective to break azeotropes by pressure shifting, the addition of an extraneous component may facilitate the separation. Usually, there are only a few extraneous components which render the separation feasible or economical, and it is these entrainers that we seek to identify. Van Dongen and Doherty (1985) showed that, in azeotropic mixtures, there can be boundaries that con-
fine the composition profiles in a continuous column to lie in a limited portion of the composition space. These boundaries can be found by tracking the 'pinches" (fixed points in the distillation model equations) as a function of the column parameters, e.g., reflux or reboil ratio, feed quality, or entrainer to feed ratio. This approach is exact but quite demanding computationally and requires a detailed vapor-liquid equilibrium description. An efficient method for doing this is described by Fidkowski et al. (1990).
For entrainer selection and column sequencing, it is necessary to acknowledge the effects of these boundaries. However, detailed parametric calculations can be prohibitively time-consuming and often require more extensive data than are available. An alternative is to examine the simple distillation residue curves (the phase plane for a simple open evaportion) whose boundaries are an approximation of those in a continuous column. While it is known that these simple distillation boundaries can be crossed by the profiles in a continuous column, it is rarely the case that this effect can be exploited for engineering purposes. Furthermore, we will presume a sufficient
0888-5885/91/2630-0760$02.50/00 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 761 condition for the feasibility of a distillation is that the desired products are not divided into different regions by distillation boundaries. With this background we can easily select entrainers based on the structure of the residue curve map. It is straightforward to compute the residue curve map given a vapor-liquid equilibrium model or even to determine it experimentally. For instance, exhaustive search algorithms have been developed for ternary mixtures containing only binary azeotropes (Petlyuk et al., 1975, 1977). It is also possible to focus on the boundary structure of the residue curve map but with much less information than the calculations demand. The topology of general ternary maps has also been extensively studied (Doherty and Perkins, 1978, 1979; Reshetov et al., 1983; Doherty, 1985) and a topological constraint formulated that is useful for analysis. Knight (1986) developed a graph-theoretic representation of the boundary structure for general ternary mixtures. In this paper we describe an improved and more complete analysis than those mentioned, which accounts for all possible configurations within the fairly general assumptions stated below and which also incorporates data consistency checks. In particular, a combined thermodynamic and topological analysis of the mixture allows us to determine the basic structure of the map, with minimal initial information. As a result, we show how to identify feasible separations based only on boiling temperatures of the pure components and azeotropes without the need for a detailed vapor-liquid equilibrium model.
Residue Curve Maps A simple distillation residue curve map is a graph of liquid composition paths, which are solutions of the following ordinary differential equation (ODE) for various initial conditions:
where is the warped time, x designates the liquid mole fraction vector, and y is the vapor mole fraction in equilibrium with x at a given pressure P. The evolution of x from any initial condition can be found by integrating (1). A relation of y to x is required, and this can be accomplished for a large number of mixtures using activity coefficient models, e.g., Gmehling and Onken (1982). The singular points of these equations are the pure components and azeotropes (Le., satisfying x = y). It has been shown (Doherty and Perkins, 1978) that these singular points are elementary (saddles and nodes) and isolated in the composition space. Another important property is that the temperature always increases along a residue curve. As a consequence, it is possible to put an orientation on any path joining two singular points (i.e., simple distillation boundaries or portions of the edges of the composition space) by marking it with arrows in the direction of increasing temperature. Although thermodynamics does not rule out multiple azeotropes in a binary mixture, experimental examples are extremely rare (e.g., Wade and Taylor, 1973). For multiple ternary azeotropes, we only know of one example which was computed with a model (Tamir and Wisniak, 1978) and for which there is no experimental evidence. For these reasons, we limit this study to ternary systems containing at most one binary azeotrope per edge (with a total number of binary azeotropes in the mixture B I3), and at most one ternary azeotrope. We choose to label the three pure components in order of increasing boiling point at the
pressure of interest L, I, and H (for light, intermediate, and heavy, respectively). Also, the number of pure component nodes is denoted by N1and pure component saddles is denoted by Sl;N2 and S2are the number of binary nodes and saddles, respectively; N3 and S3are the number of ternary nodes and saddles. With this notation, the restrictions that we imposed on the complexity of the system can be written as N 1 + Si = 3 N2 + Sz = B 5 3 N3 + S3 = 1 or 0 (2) In most cases, a knowledge of the boiling temperatures of all the species, along with their position in the triangle, allows us to determine the nature of each vertex and to sketch the boundaries of the simple distillation residue curve map. We have developed and encoded an algorithm that determines the connections between the vertices of the residue curve map. This gives a first approximation to the actual residue curve map, which is quantitatively accurate when the distillation boundaries are straight. The procedure also makes it possible to eliminate sets of data for which the temperature distribution violates the topological and thermodynamic constraints of the residue curve maps. In a few instances, detailed in Appendix B, it is shown that an indeterminacy occurs because two or more different maps are consistent with one set of temperatures and compositions. More precisely, global or local indeterminacy can arise because there might be several possible connections to a saddle, or because a binary saddle and a binary node can switch roles without violating the topology. Robust tests are derived to detect these indeterminate cases, based on the values of the topological parameters, but at this point, a thermodynamic model and integration of (1)are necessary to distinguish between the possible maps. When there is no indeterminacy, the automatic procedure determines the existence and approximate location of the boundaries, based on a set of simple properties that are summarized in Appendix A.
Structure of the Algorithm The main body of the reasoning relies on thermodynamics and makes use of directed graph theory. Residue curve maps are represented as directed graphs with all species (Le., pure components and azeotropes) as the vertices of the graph and distillation boundaries connecting the species as the arcs of the graph. Determining the structure of the map can be translated into building an adjacency matrix A that represents the connections between the different vertices. For a system of N species, A is a N X N matrix whose entry Aij is equal to 1if there is a connection going from component i toward component j and equal to 0 otherwise (Knight, 1986). The diagonal terms Aii are all equal to 0 because a residue curve cannot return to its point of origin. The vertices are indexed in order of increasing boiling temperature, and therefore A is upper triangular because temperature increases along residue curves. All upper triangular entries the initialized to -1 and correspond to potential connections. Once the existence of a connection is confirmed or ruled out, the value of the corresponding entry in the adjacency matrix is changed to +1 or 0 accordingly. The main steps of the algorithm are described below and illustrated afterward with an example: Step 1. Fill in the Edges of the Triangle. Knowing the boiling temperatures of the pure components and
762 Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 90°C
Figure 1. Example demonstrating rule 2 on a hypothetical mixture.
binary azeotropes, the edges of the triangle are given an orientation in the direction of increasing temperature. Step 2. Determine the Nature of the Pure Component Vertices. Because the singular points are elementary, the nature of each pure component vertex (node or saddle) is determined, and some infeasible connections can be ruled out according to the following: Rule 1. A pure component vertex is a saddle if one of the adjacent pure components or binary azeotropes has a lower boiling temperature and the other one has a higher value. There can be no other connections except along the edges of the triangle. A pure component vertex is a stable (unstable) node if the two adjacent pure components or binary azeotropes each boil at a lower (higher) temperature. There can be no connections to any higher (lower) boiling species. Step 3. Determine the Nature of the Ternary Azeotrope. A necessary condition for a ternary azeotrope to be a saddle is the existence of two higher and two lower boiling species, to and from which it could connect, respectively. A pure component saddle is not connected to any species other than the two adjacent species along the edges, and it can never be connected to a ternary saddle. Therefore, the two lower and higher boiling species cannot be pure component saddles. With this in mind, it is possible to determine cases where the ternary azeotrope cannot be a saddle: Rule 2. A ternary azeotrope is a node if (i) N 1 + B C 4 (there are less than four possible connections) and/or (ii) excluding the pure component saddles, the ternary azeotrope is the highest, second highest, lowest, or second lowest boiling species. Otherwise, the data are inconsistent. These conditions are sufficient to detect a ternary node; from the many examples studied, we believe that they are also necessary, though a proof is currently lacking. An example demonstrating this rule is given in Figure 1,where the ternary node (which boils at 100 "C)is the third lowest boiling species, but only the second lowest when the pure component saddles are excluded. As a convention, all the maps are drawn with the heavy pure component as the lower right vertex and the light pure component as the upper left vertex. Step 4. Make Connections with the Ternary Saddle. In part A of Appendix B, the case of a system containing a ternary saddle is studied and it is shown that there is an indeterminacy if N 1 + B = 6. It is also shown that when there is no indeterminacy, the ternary saddle is connected to all the binary azeotropes and pure component nodes. The remaining steps deal only with systems containing no ternary saddle azeotrope. Step 5. Determine the Number of Binary Nodes and Saddles When There Is No Ternary Saddle. When there is no temary saddle in the system, the number of binary nodes and saddles is computed from the topological constraint (Doherty and Perkins, 1979): N2= ( 2 + B - N 1 - 2N3 + 2S3)/2 (3)
70°C
Figure 2. Hypothetical example of data inconsistency. Methanol
64.7"C
100°C
92.7%
131.4"C
Figure 3. Boiling temperature and composition for the mixture methanol, water, and methyl chloracetate at 1-atm pressure, reported by Yamakita, Shiozaki, and Matsuyama (1983).
with S2 = B - N , and S3 = 0. This set of equations also provides a data consistency check. An example of such a test is presented in step 6. Step 6. Check the Consistency of the Data. We first count the number of binary azeotropes that are neither the highest or the lowest boiling species of the system. Thermodynamics requires that at least these binary azeotropes be nodes, whereas the topology demands that the number of binary nodes be exactly N2. Therefore, if N 2 is smaller than the number of extrdmum boiling binaries, the data are necessarily inconsistent. An example is given in Figure 2. Excluding the pure component saddle (which boils at 70 "C), the ternary azeotrope is the second lowest boiling species and therefore cannot be a saddle (step 3). If the ternary azeotrope is then assumed to be a node, the topological constraint in (3) gives N2 = 0, whereas the number of extremum boiling binary azeotropes is equal to 1 (namely, the binary azeotrope that boils at 65 "C is the lowest boiling species of the system). Therefore, the data are inconsistent. Yamakita et al. (1983) describe a test to detect that the boiling temperature of the ternary azeotrope is erroneous in the mixture methanol/water/methyl chloroacetate (Figure 3). Their idea is to compare the node/saddle type of the ternary azeotrope, as given by (i) a rule similar to rule 2 and (ii) an expression of the topological constraint. Their approach requires that experiments be carried out to determine the number of binary nodes and saddles and data inconsistency is identified whenever the two results disagree. With our method, rule 2 indicates that the ternary azeotrope is a node. Substituting for N 3 in the topological constraint yields N2 C 0, which is impossible. This method does not require any experiment, but cannot ensure that the faulty datum is that of the ternary azeotrope. However, Yamakita, Shiozaki, and Matsuyama's rule to determine the nature of the ternary azeotrope is incomplete and will yield spurious results in some instances (e.g., the ternary azeotrope in Figure 1would be diagnosed as a saddle). A second consistency test follows from property 6, Appendix A, which requires that the number of binary saddles
Ind. Eng. Chem. Res., Vol. 30, No. 4, 1991 763
n
diOXnne
input compositions and temperatures
101.35OC
119.5O C (77%wL scctic acid) =tic
acid
118.5'C
Determine pure component with pure components
1\
l-b"o,3-mCthylbutMe
108.65o c
120.65 OC
(38%w(. =tic acid)
101.35
saddle to all binary
101.35
Calculate B (number of intermediate
component nodes
saddles, when passible
118.5
108.65
120.65
118.5
108.65
120.65
Figure 4. Example of indeterminate system. Both maps are consistent with the boiling temperature data and the topological constraint.
is no greater than the number of intermediate boiling binary azeotropes, S2IBib. Step 7. Indeterminacy. As described in Appendix B, a local indeterminacy occurs when the number of binary saddles and the number of intermediate boiling binary azeotropes are different. In such cases, the actual residue curve map has to be computed in order to determine the exact structure. Although it is possible to know precisely the different possible structures that fit the temperature distribution, this information is not sufficient for entrainer screening purposes since one feasible and one infeasible map is always obtained among the indeterminate maps. Figure 4 illustrates an indeterminacy for the mixture dioxanef acetic acid/ 1-bromo-3-methylbutane at 1 atm (Horsley, 1973). The information on the binary edges gives rise to two possible maps. More information is needed about the ternary mixture to solve the indeterminacy. Note that even if a ternary azeotrope is present, there would still be an indeterminacy, unless its boiling point is bounded between 108.65and 119.5 "C (this would make it a ternary saddle). Step 8. Make Final Connections When There Is No Indeterminacy. As shown in Appendix B, when there is no indeterminacy and if a ternary node exists, it is conneded to every binary saddle whose stability is compatible. Then the infeasible connections of the remaining binary saddles are ruled out (for instance, the connections of a maximum boiling saddle to any lower boiling species). Finally, the remaining feasible connections are established. A flowchart of the procedure is given in Figure 5. Illustration of the Procedure We consider the example shown in Figure 1 and work through the steps of the algorithm. The systematic determination of the structure for the residue curve map is represented in Figure 6. Step 0 Initialization. The temperature-composition information is provided by the user. Step 1: Fill in the edges of the triangle. Step 2: Determine the nature of the pure component vertices. The pure component boiling at 120 "C is a stable node. The other two pure components are saddles, and therefore any connection with them is ruled out. To rule out these connections, all the -1 entries of the adjacency
Rule out infeasible connections indeterminacy
L
-
Make connections for the binary saddles (step 8)
Compute actual residue curve map
Figure 5. Flowchart of the algorithm.
matrix corresponding to connections with these saddles are changed to 0 (symbolized by a citcle in Figure 6). Step 3: Determine the nature of the ternary azeotrope. Application of rule 2 indicates that the ternary azeotrope is a node; N 3 = 1, S3 = 0. Therefore the program is directed to step 5 of the algorithm. Step 5: The topological constraint indicates that there are two binary saddles and one binary node; Sz= 2, Nz= 1. Step 6: Check data consistency. One of the binary azeotropes is the lowest temperature in the map. This is consistent with step 5, N z = 1. There are two intermediate boiling binary azeotropes (namely, the Intermediate-Heavy azeotrope which boils at 105 "C and the Light-Intermediate azeotrope which boils at 115 "C).Thus S25 Bib Therefore, there is no data inconsistency in the system. Step 7: Indeterminacy. We have Sz= Bib and consequently the system does not exhibit any indeterminacy. Moreover, it follows that the saddles in this example occur at the binary azeotropes that boil at 105 and 115 "C. The binary azeotrope boiling at 80 "C is an unstable node. Step 8: Establishing the inner connections. We know from property 5 of Appendix A that there are exactly as many boundaries as there are binary saddles. A logical way to establish the connections would therefore consist of looking at each binary saddle and considering the possible connections. However, a binary saddle could be connected to either a binary node or a ternary node, but it would not be possible at this point to decide which connection is the correct one. Because there is no indeterminacy (cf. step 7)) any consistent map is the unique and correct solution. We first establish the connections with the ternary node and then study the remaining binary saddles. The only binary saddle that can be connected to the ternary node is the one that boils at 105 "C because a connection from the ternary node to the maximum boiling binary azeotrope
764 Ind. Eng. Chem. Res., Vol. 30, No. 4,1991 90
::a LB Remarks
Residue curve map
I
Sequence
I
CASE 1
7he enuainer i s intermediate boiling and does no! i n d u c e a new amuopc.
110
105
Step 0
120
E
Step I
CASE 11
A
i\
90
Saddle
115
Saddle 110-
90
B 1
7
5
E
i1 Node \
Node
105
120
110
step 2
105
120
A
bi
Step 3
90
90
E
B
A
115
-4--
80
1 1 5 n
105
120
I10
Step 8 (i)
1 5 h 105
120
A
kB
toiling ternary areotvpc lean in A.
Step 8 (ii)
90
110
105
The entrainer IS intenediate boiling and forms a maximum boiling
m u o p e with the lighter of the two pure components (,.e. A). I t may or may not form a mrnimum boiling muop with B. with or without a minimum
...E..................... 110
Classical emactwe distillation with a heavy entrainer i n d u c i n g no new azcouopc. (In some cases, B can come off the lop of the first column)
ksk
9 "C
E
........................
B
jBfl A
DO
....................
E
u A/E azeo
115°C
120
CASE Ill
Same column configuration. but the ~
A
........................ 110°C
1050~
120°C
Step 8 (iii)
Figure 6. Step by step determination of the structure of the residue curve map of a hypothetical mixture.
would make that azeotrope a stable node, whereas it is known to be a saddle. This establishes the ternary azeotrope as an unstable node, and its connection to the binary azeotrope at 105 "C is confirmed since by property 4 of Appendix A it must be connected to at least one binary saddle (see step 8(i) in Figure 6). Step 8(ii) consists of ruling out infeasible connections with the binary saddles (for example, the dotted connection in Figure 6 is ruled out by property 2 of Appendix A). Step 8(iii) establishes the last possible connection between the remaining binary saddle and the pure component stable node (using property 2 of Appendix A). The resulting map is globally consistent and we have found the correct and unique structure.
Application to Screening Entrainers Within the set of potential residue curve maps for homogeneous azeotropic distillation (Doherty and Caldarola, 19851, the seven most favorable maps for breaking mini-
B
E
entrainer i s lower tailing. (Note that !he thi4 map also bclongs to this category by
L
A
inletchanging A and E).
B
Figure 7. Favorable residue curve maps for homogeneous azeotropic distillation.
mum boiling binary azeotropes are illustrated in Figure 7, where A is the light pure component, B is the heavy pure component, and E is the entrainer. Three of the simplest corresponding column configurations are also shown (I, 11, and 111). In cases I and 111, a "direct split", where A is removed as the distillate from the first column and B as the bottoms in the second column, is also possible. However, this might require the first column to have multiple feeds for optimal operation. It is interesting to notice that in the last five maps (corresponding to case 111), the column sequence must operate in the portion of the composition space limited by the vertices corresponding to pure components A, B, and the binary azeotrope A/E. Furthermore, in this subtriangle, the temperature distribution is: T(azeo A/B)
