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Approximate Calculation of Distillation Boundaries for Ternary Azeotropic Systems J. A. Reyes-Labarta,* M. D. Serrano, R. Velasco, M. M. Olaya, and A. Marcilla Departamento Ingeniería Química, Universidad de Alicante, Apartado 99, Alicante 03080, Spain
bS Supporting Information ABSTRACT: As it is very well known, especially when dealing with the simulation and design of distillation processes and columns, it is essential to know the limitations defined by the vaporliquid equilibrium (VLE) of the system under consideration. This paper uses a new straightforward algorithm to calculate and completely define the distillation boundaries in different ternary azeotropic systems. The method allows, using cubic splines, the calculation of different distillation trajectories and the selection of those corresponding to the searched distillation boundaries. The algorithm has been applied to eight ternary liquidvapor systems to test its validity. To simplify the optimization process, an empirical but very accurate equation has been used to calculate the VLE.
1. INTRODUCTION Distillation is the most important separation technique, even though it is an expensive operation in terms of capital and operating costs. Additionally, there is a considerable industrial interest in the design and optimization of homogeneous and heterogeneous azeotropic distillation columns and sequences of distillation columns, in order to find the most clean and economical option. In theses cases, it is essential to bear in mind the critical convergence problems and possible infeasibilities, due mainly to the equilibrium restrictions, that are present in these complex systems. Residue curve maps are commonly used in developing separation flow schemes by checking the existence of distillation regions. A distillation region is a composition subspace that contains the different residue curves with a common origin and end. Obviously, the existence, location, and curvature of distillation region boundaries (that initially cannot be crossed by simple distillation) are very important in the synthesis of azeotropic distillation sequences. In this sense, the curvature of the boundary has a significant impact on whether or not it is possible to cross it. In the literature it is remarked that boundary crossing is only possible if the feed is located on the concave side of the distillation boundary.14 In this sense, many strategies have been searched in the past to define the distillation boundaries present in a multicomponent system with azeotropic compositions. Thus, basic theory and several topological and approximated methods have been developed to predict the general location of these boundaries,2,513 bearing in mind their importance when dealing with azeotropic distillation processes. Nevertheless, no direct procedure has been developed to date. In the present work, we calculate and completely define the trajectory of the distillation boundaries present in different ternary azeotropic systems, applying the concept that defines such boundaries as distillation trajectories passing through specific singular points, previously classified. In this sense, it is important to remark that explicit functions for representing the trajectory of the different distillation boundaries are obtained r 2011 American Chemical Society
after the optimization calculations and that the calculation of residue curve maps is unnecessary. To simplify the optimization process, an empirical but very accurate equation to reproduce the vaporliquid equilibrium (VLE) has been used.
2. MATHEMATICAL TREATMENT 2.1. Explicit Equation for the VLE Calculations. In many cases, especially when dealing with complex procedures of optimization and convergence problems, it would be very interesting to have the possibility of using explicit equations to relate the phase equilibrium compositions. In this sense, a set of explicit equations to calculate the composition of the vapor in equilibrium with a given liquid mixture at fixed pressure, and its bubble point temperature, has been used:
½xi = equil
yi
¼
3
3
∑ ai , j xj j¼1 3
∑ ½xq = j∑¼ 1 aq, jxj
ð1Þ
q¼1
where xi and yi are the molar fractions of the component i in the conjugated liquid and vapor phases in equilibrium, respectively. The subscripts j and q refer to the different components of the mixture. ai, j are the nine correlation parameters. Such parameters must be obtained by correlation of experimental data. The equations have been applied satisfactorily to different types of ternary liquidvapor equilibria.14 The explicit equations allow the direct calculation of the vapor composition, thus avoiding the need for iterative calculations and the characteristic convergence difficulties associated with the solution of the typical set of nonlinear equations involved in such problems. Received: September 9, 2010 Accepted: April 26, 2011 Revised: February 8, 2011 Published: May 11, 2011 7462
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Additionally, the proposed equations allow calculating the composition of the azeotropes solving the corresponding system of equations. In the case of binary azeotropes, it is possible to obtain explicit expressions to calculate their compositions: equil
yi
A 0 x i 2 þ B 0 x i þ C0 ¼ 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B0 ( ðB0 Þ2 4A0 C0
¼ xi ¼
w
2A0
w
A0 B0
xi
C0
ð2Þ
12 a1,2 a1, 1 a2, 2 þ a2,1 a1,1 a2, 1 2a1, 2 þ 2a2,2 a1,2 a2,2
binary azeotrope 13 a1,3 a1, 1 a3, 3 þ a3,1 a1,1 a3, 1 2a1,3 þ 2a3,3 a1,3 a3,3
23 a2,3 a2, 2 - a3, 3 þ a3,2 a2,2 a3, 2 2a2, 3 þ 2a3,3 a2,3 a3,3
2.2. Topological Condition Used To Calculate Distillation Boundaries. As it is very well known,5 distillate and residue curve
Figure 1. Relationship between distillation region boundaries and liquidvapor tie lines.
maps present a set of singular or critical points related to the distillation trajectories that can be classified (by checking the boiling temperatures or the signs of the derivatives of the function (yi xi) in the singular points), as follows: stable node, critical point with all paths approaching; unstable node, critical point with all paths departing; saddle, singular point with finitely many paths both approaching and departing. As commented before, the applied algorithm is based on the concept that distillation boundaries are distillation trajectories that pass through specific singular points such as points that represent azeotropic mixtures (binary or ternary) or pure components. Additionally, the concept of the tie lines being chords of the distillation curves is also used (Figure 1). Thus, the topological condition used to calculate the distillation boundaries is the following: in a composition diagram, a distillation boundary is a distillation trajectory from a stable node to an unstable node that contains not only the liquid
Figure 2. General scheme of the method applied to calculate ternary distillation boundaries. 7463
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Table 1. Parameters (ai, j) of the Empirical Equation Used To Calculate the VLE a ai, j system A
B
C1
C2
C3
component 1
component 2
component 3
1
0.0980
0.0687
0.2616
2
0.0985
0.1370
0.2537
3 1
0.4304 0.0760
0.3822 0.0849
1.0000 0.1112
2
0.0451
0.3960
0.0745
3
0.0000
0.0761
1.0000
1
0.7064
0.1700
0.1675
2
1.0000
0.3204
0.1594
3
0.0042
0.0013
0.4933
1
1.0000
0.3486
0.2472
2 3
0.9029 0.0071
0.3978 0.0021
0.0635 0.6125
1
0.9321
0.3540
0.2022
2
1.0000
0.4753
0.1293
3
0.0070
0.0028
0.5989
1
0.6872
0.0624
0.0000
2
0.1922
0.6811
0.2874
3
0.0000
0.1657
1.0000
E
1 2
1.0000 0.2206
0.2407 0.4940
0.4747 0.1316
3
0.1274
0.0449
0.7106
F
1
0.1473
0.0363
0.0155
2
0.0872
0.1718
0.0000
3
0.2597
0.0000
1.0000
1
0.4918
0.3780
0.1701
2
0.3343
0.7220
0.0183
3 1
0.0945 1.0000
0.0211 0.5959
1.0000 0.4307
2
0.1929
0.3101
0.5712
3
0.0194
0.4748
0.3260
D
G
H
a
component
See text for descriptions of systems AH.
compositions but also the vapor compositions in equilibrium. In this sense, the distillation boundary presents a trajectory where the corresponding liquidvapor tie lines have a minimum length. 2.3. Calculation Procedure. Figure 2 shows the general scheme of the calculation algorithm applied to find the trajectory of the distillation boundary searched by minimizing the corresponding objective function, eq 3: nipt
equil 2 ðy1, k ycs ∑ 1, k Þ ¼ 0 k¼1
ð3Þ
Taking advantage of using an explicit equation to directly calculate the VLE, the calculation algorithm has been easily implemented in the spreadsheet Excel for Windows, using its “Solver” tool in the optimization process. To generate the probe distillation trajectories, that after the optimization process will be the searched distillation boundary, cubic splines are used (although any other adequate equation could be applied). The procedure consists of the following main steps: (1) Initial classification of the different singular points of the composition diagram, i.e., the azeotropic compositions (binary
and ternary) and pure components present in the system, as stable, unstable, or saddle points. (2) Definition of the origin and the end of the boundary trajectory searched, the number of intermediate points to be calculated homogeneously that are distributed along the distillation trajectories (nipt), and the number of intermediate points (nodes) to be used for the cubic spline function (nipcs). (3) Definition of the composition of the component that will be the independent variable (e.g., x2; the composition of the component with the largest variation along the boundary distillation searched usually is a good selection). (4) Selection of the independent variable values for each intermediate point homogeneously distributed along the trajectory searched, i.e., x2,k (k = 1, 2, ..., nipt) and x2,k0 (k0 = 1, 2, ..., nipcs). (5) Necessary guesses, to start the optimization process, of the initial values for the variable to be optimized, i.e., the composition x1,k0 of the cubic spline function nodes. cs (6) Calculation of xcs 1,k = f (x2,k,x1,k0 ,x2,k0 ), using the corresponding cubic spline function (cs) and x2,k. (7) Calculation of yequil i,k , using the empirical eq 1 at each liquid point xi,k. cs equil (8) Calculation of ycs 1,k = f (y2,k ,x1,k0 ,x2,k0 ), using the cubic equil spline function and y2,k . (9) Calculation of x1,k0 , using an optimization solver over the cubic spline function and the objective function given by eq 3 (these values will define the distillation boundary trajectory searched (passing through the selected singular points of origin and end)).
3. RESULTS AND DISCUSSION The following systems have been studied (Table 1 shows the corresponding ai, j equilibrium correlation parameters): system A, benzene (1) þ cyclohexane (2) þ toluene (3), at 760 mmHg with one binary azeotrope;15 system B, diethyl ether (1) þ ethanol (2) þ water (3), at 2156.3 mmHg with one binary azeotrope;16 systems C1C3, 2-butanol (1) þ 2-butanone (2) þ water (3), at 200, 600, and 760 mmHg, respectively, with two binary azeotropes;15 system D, water (1) þ ethanol (2) þ toluene (3), at 760 mmHg with three binary azeotropes and one ternary azeotrope;15 system E, 2-propanol (1) þ benzene (2) þ water (3), at 760 mmHg with three binary azeotropes and one ternary azeotrope;15 system F, ethanol (1) þ benzene (2) þ water (3), 760 mmHg with three binary azeotropes and one ternary azeotrope;15 system G, acetone (1) þ methanol (2) þ cyclohexane (3), at 760 mmHg with three binary azeotropes and one ternary azeotrope.15 system H, methanol (1) þ acetone (2) þ chloroform (3), at 760 mmHg with three binary azeotropes and one ternary azeotrope;16 Figure 3 shows the results for the studied ternary systems with one and two binary azeotropes (systems AC) or with three binary azeotropes and one ternary azeotrope (systems DH). As it can be observed, different type of distillation boundaries, for instance, between two binary azeotropes, between one binary and one ternary azeotrope, or between a pure component and a binary or ternary azeotrope, have been obtained. In some cases, such as the distillation boundary starting from the two to three binary subsystem in system H, the distillation boundary calculated presents a complex or singular curvature. As commented before, the knowledge of these curvatures is very important in the analysis of possible separation flow schemes.14 7464
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Figure 3. Calculated distillation boundaries for the studied systems AH (9, cubic spline nodes; , liquid trajectory; 2, vapor compositions).
The nature of the criterion used to define the distillation boundaries, which represents a necessary condition for the VLE
in order to limit the different distillation regions present in a system, guarantees that the distillation boundaries calculated are 7465
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Industrial & Engineering Chemistry Research totally consistent with the corresponding residue curve map of the studied system. In this sense, Figure S1 of the Supporting Informationshows the VLE tie lines map with the corresponding predicted distillation boundaries for system D. Table S1 of the Supporting Information shows, for all the studied systems, the numerical data (cubic spline nodes) corresponding to the different distillation boundary trajectories calculated. As an example, Table S2 of the Supporting Information shows, with more detail, the results for the different calculation steps in the case of system E (2-propanol þ benzene þ water), and for the calculation of the distillation boundary from the ternary azeotrope composition to the one to three binary equil azeotropes, that presents a singular curvature:x2,k, xcs 1,k, y1,k , y2, equil cs , y1,k. The independent variable in this case is x2 because its k variation (Δx2 = 0.5113 0.0000 = 0.5113) is larger than the change of x1 (Δx1 = 0.5261 0.0947 = 0.4314).
4. CONCLUSIONS The algorithm applied to directly calculate distillation boundaries and obtain their corresponding explicit functions is very easy to implement and produces very good results in the different azeotropic ternary systems studied, avoiding the calculation of the residue curve map in the whole range of compositions. If the number of intermediate points for the cubic spline (nipcs) along the distillation trajectory were increased, the results would be improved. On the other hand, the use of adequate explicit equations, as good estimators, to calculate the equilibrium = f(xi,ai, j), introduces important advantages compositions, yequil i in the procedure, simplifying its programming and especially avoiding typical convergence problems. Additionally, the proposed algorithm can be easily extended to multicomponent systems. ’ ASSOCIATED CONTENT
bS
Supporting Information. Figure S1 showing a VLE tie lines map and predicted distillation boundaries for one of the studied systems (system D, waterethanoltoluene) at 760 mmHg, Table S1 lisitng the distillation boundary results for the studied systems with optimized variables (i.e., the dependent composition of the cubic spline function nodes), and Table S2 detailing intermediate points for the distillation trajectory (nipt = 30) corresponding to the distillation boundary from the ternary azeotrope to the binary azeotrope 13 (system E). This information is available free of charge via the Internet at http://pubs.acs.org/.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We thank the Vice Presidency of Research (University of Alicante) and Generalitat Valenciana (Project GV/2007/125) for financial support. ’ LIST OF SYMBOLS ai, j = binary correlation parameters of the explicit equation for the VLE az = azeotrope OF = objective function
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nincs = number of intermediate nodes to be used in a cubic spline nipt = number of intermediate points to be calculated along the trajectory tested xi = molar fraction of component i in the liquid phase yi = molar fraction of component i in the vapor phase Subscripts
i, j, q = different components k = index for nipt k0 = index for nincs Superscripts
equil = equilibrium value from the corresponding model used cs = cubic spline
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