Pressure Sensitivity Analysis of Azeotropes - American Chemical Society

Hyprotech (A Subsidiary of Aspen Technology, Inc.), 800, 707 8th Avenue SW,. Calgary, Alberta T2P 1H5, Canada. A new algorithm for pressure sensitivit...
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Ind. Eng. Chem. Res. 2003, 42, 207-213

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Pressure Sensitivity Analysis of Azeotropes Stanislaw K. Wasylkiewicz,* Leo C. Kobylka, and Francisco J. L. Castillo Hyprotech (A Subsidiary of Aspen Technology, Inc.), 800, 707 8th Avenue SW, Calgary, Alberta T2P 1H5, Canada

A new algorithm for pressure sensitivity analysis of azeotropes has been developed. The algorithm applies bifurcation theory together with an arc length continuation. This allows not only the compositions of azeotropes to be tracked as they change with pressure, but also all new azeotropes that appear within specified pressure range to be found. The algorithm is very efficient, and the computational time required for the pressure sensitivity analysis is on the order of the time necessary to calculate all azeotropes at constant pressure by using a homotopy method. Introduction Azeotropes and their types determine distillation boundaries. Therefore, knowledge of the temperatures and compositions of all azeotropes in a multicomponent mixture at specified pressure, as well as their pressure dependence, is crucial for the design of distillation separation systems. Strong nonlinearities of vaporliquid-liquid equilibrium and the presence of multiple solutions, both real and spurious, complicate the problem of computing temperatures and compositions of all azeotropes in heterogeneous mixtures at a constant pressure. One can try to find all solutions by starting a nonlinear solver from several starting points. This approach, however, does not guarantee that all azeotropes will be found, even with extreme calculation effort. Several techniques have been applied to solve this problem, e.g., the Levenberg-Marquardt algorithm,1 the interval Newton method,2 and the global optimization method.3 The most reliable and robust method for the calculation of all azeotropes in a homogeneous mixture has been proposed by Fidkowski et al.4 Recently, the method was generalized by Wasylkiewicz et al.5 to include heterogeneous liquids. The method, together with an arc length continuation and a rigorous stability analysis,6 gives an efficient and robust scheme for finding all homogeneous as well as heterogeneous azeotropes predicted by a thermodynamic model at a specified pressure. The sensitivity of azeotropes to changes in pressure was discovered long ago.7 Since then, many theoretical and experimental studies have been performed on this effect.8-14 In some mixtures, azeotropes appear or disappear as the pressure changes. The magnitude of the pressure effects depends on the mixture. Sometimes, the composition of an azeotrope changes significantly; sometimes, the composition changes very little. An example of a pressure-insensitive azeotrope is ethanolwater (see Figure 1). Even a large pressure increase does not significantly change the composition of the azeotrope. Consequently, a pressure-swing distillation scheme is impractical for this mixture. However, there are azeotropes for which the composition changes rapidly with pressure. An example is the acetonemethanol azeotrope shown in Figure 2. In this case, a * Corresponding author. Tel.: +1-403-303-1000. Fax: +1403-303-0927. E-mail: [email protected]. Internet: http: www.aspentech.com.

Figure 1. x-y diagram for the mixture of ethanol and water at pressures of 100, 500, and 2000 kPa. The binary azeotrope ethanol-water is at an ethanol mole fraction of 0.9188 for 100 kPa, 0.8986 for 500 kPa, and 0.8792 for 2 MPa. NRTL model.

Figure 2. x-y diagram for a mixture of acetone and methanol at pressures of 100, 500, and 2000 kPa. The binary azeotrope acetone-methanol is at an acetone mole fraction of 0.7914 for 100 kPa, 0.5042 for 500 kPa, and 0.2197 for 2 MPa. NRTL model.

pressure-swing distillation scheme can be an economically attractive solution for the separation of this mixture. Knapp and Doherty15 list more pressuresensitive binary azeotropes that are amenable to pressure-swing distillation. The sensitivity analysis of an azeotrope with pressure provides the dependence of its composition and temperature on pressure. When upper and lower pressure limits are provided, the analysis can start from one

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pressure limit, and then the azeotrope composition can be calculated for gradually increasing or decreasing pressure. A simple parameter continuation procedure can efficiently follow any azeotrope that is already known. However, such an approach cannot find new azeotropes that appear as the pressure changes. Another way to find the pressure sensitivity of the system is to apply a rigorous azeotrope calculation procedure at each pressure interval. However, such a method would be extremely time-consuming. To overcome these difficulties, we have developed a new pressure sensitivity analysis method that applies bifurcation theory together with an arc length continuation. Within the specified pressure interval, the method finds all bifurcation pressures at which an azeotrope appears or disappears. The computation time required for the whole pressure analysis is similar to the time necessary for the homotopy method to find all azeotropes at one specified pressure. In this paper, we include a few examples of pressure sensitivity analyses of azeotropes in three- and four-component mixtures. Azeotrope Search Using the Homotopy Continuation Method In a c-component mixture, for a given vapor-liquid equilibrium (VLE) model and selected pressure P, heterogeneous as well as homogeneous azeotropes can be calculated by solving the following set of equations5

f(z,T,P) ≡ y(z,T,P) - z ) 0 c

∑ i)1

c

∑ i)1

xki ) 1,

p

yi ) 1, zi )

∑

k)1

(1) p

φkxki ,

∑ φk ) 1

(2)

k)1

xki g 0, yi g 0, φk g 0, i ) 1, ..., c; k ) 1, ..., p (3) yi )

γik(T,xk) Psi (T) k xi , i ) 1, ..., c; k ) 1, ..., p (4) φi(T,y)P

where z is a composition vector of the overall liquid composed of p liquid phases. Each liquid phase k, defined by its composition vector (xk) and liquid-phase fraction (φk), is in equilibrium with the vapor phase defined by its composition vector y. Fugacities (φi) and activity coefficients (γik) are highly nonlinear functions of the temperature and compositions of the vapor or liquid phases. Pure-component saturation pressures (P si ) rapidly increase with temperature. This is a highly nonlinear, constrained problem with multiple solutions. Conventional root-finders cannot be used robustly to find all of the solutions for a multicomponent mixture. To solve this problem for homogeneous mixtures, Fidkowski et al.4 introduced an artificial equilibrium relationship

[

]

γi Psi y˜ i ) (1 - t) + t x , i ) 1, ..., c φi P i

(5)

which represents Raoult’s law when the homotopy parameter t ) 0 and the nonideal equilibrium relationship (eq 4) when t ) 1. By changing the homotopy parameter from 0 to 1, the equilibrium surface can be gradually “deformed” from the ideal one described by Raoult’s law to the nonideal equilibrium one with all azeotropes. Then, Fidkowski et al.4 applied a homotopy

continuation method to do this gradual deformation in a systematic way. They also showed that higher-order solution branches are connected to lower-order solution branches (fewer nonzero components) and eventually to branches, which start from pure components. This makes the method very robust and guarantees that all of the homogeneous azeotropes predicted by a model will be found. Recently, Wasylkiewicz et al.5 generalized the method to include heterogeneous liquids. Bifurcation Pressure Knapp14 applied a bifurcation theory to predict the effect of pressure on the composition of homogeneous azeotropes. Contrary to Fidkowski’s4 approach, he did not introduced any artificial equilibrium relationship but used pressure as the continuation parameter. He introduced the concept of a bifurcation pressure, where an azeotrope appears or disappears, and showed necessary conditions for a homogeneous binary azeotrope to bifurcate. There is an azeotrope on one side of a bifurcation pressure and a tangent pinch on the other side. As the pressure increases or decreases away from the bifurcation pressure, the severity of the tangent pinch decreases. Therefore, a distillation column should not operate near a bifurcation pressure.14 Pressure Sensitivity Analysis of Azeotropes In our pressure sensitivity analysis, we utilize an arc length continuation method to follow efficiently individual azeotrope branches as the pressure changes. We also make use of bifurcation theory to find all bifurcation pressures at which azeotropes appear or disappear within a specified pressure range. This allows us not only to trace the composition change of already-known azeotropes but also to find any new ones that appear. All solutions of the set of eqs 1-4 are tracked as the pressure is changed smoothly from the initial pressure, P1, to the final pressure, P2, by changing homotopy parameter t from 1 to 2 in the homotopy function

h(z,t) ) y(z,P) - z

(6)

where the pressure P depends linearly on the homotopy parameter

P ) P1 + (P2 - P1)(t - 1)

(7)

To start, there are as many branches of solutions as there are pure components and azeotropes at the initial pressure P1. We find all of them by applying a homotopy continuation search for all azeotropes at constant pressure.5 Then, we follow each branch and determine whether there is any bifurcation point where a new branch appears or the branch we are tracking disappears. It can be shown that all solution branches are connected, i.e., an n-component branch can only emerge from or collapse on an (n - 1)-component branch.14 This makes the method very robust and guarantees that all of the azeotropes predicted by a model will be found. The algorithm is very efficient, and the computational time required for the pressure sensitivity analysis is of the same order as that required by the homotopy method for finding all azeotropes at one pressure.5 As an example, consider a mixture of acetone, ethanol, and water between 100 and 5000 kPa. Branches of solutions are shown in a composition triangle (Figure 3) and in a composition bifurcation diagram (Figure 4).

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Figure 3. Homotopy branches for pressure analysis of azeotropes for a mixture of acetone (A), ethanol (E), and water (W) between 100 and 5000 kPa. B1, B2, and B3 are bifurcation points. NRTL model.

Figure 5. Distillation region diagram for a mixture of acetone (A), ethanol (E), and water (W) at pressures of 100, 200, 500, and 2000 kPa. NRTL model.

Figure 4. Composition bifurcation diagram for a mixture of acetone (A), ethanol (E), and water (W) between 100 and 5000 kPa. B1, B2, and B3 are bifurcation points. NRTL model.

In this pressure range, three bifurcation pressures have been found: (B1) 117 kPa, at which the binary acetonewater (AW) branch bifurcates from the acetone vertex and the acetone vertex type changes from an unstable node to a saddle; (B2) 375 kPa, at which the binary acetone-ethanol (AE) branch bifurcates from the acetone vertex and the acetone vertex type changes from a saddle to a stable node; and (B3) 857 kPa, at which the ternary branch (AEW) bifurcates from the binary acetone-water (AW) branch, and the acetone-water azeotrope type changes from an unstable node to a saddle. The most pressure sensitive azeotrope in the system is the binary acetone-ethanol azeotrope (see AE branch in Figure 3; pressure range 375-5000 kPa). The least sensitive azeotrope in the system is the binary ethanolwater azeotrope (branch EW in Figure 3; pressure range 100-5000 kPa). Pressure and RCM Topology Pressure sensitivity analysis of azeotropes provides the dependence of the composition (see, for example, Figure 4) and boiling temperature of the azeotropes and pure components on pressure. This is very useful information for the design of azeotropic distillation sequences. Even more important is information about bifurcation pressures, that is, pressures at which the residue curve map (RCM) topology changes dramati-

cally. As an example, consider three distillation region diagrams (DRDs) for a mixture of acetone, ethanol, and water shown in Figure 5. Because three bifurcation pressures have been found in the pressure range 1005000 kPa (117, 375 and 857 kPa), the range can be divided into four topologically different types of DRDs: (1) 100 < P < 117, one binary azeotrope (EW), two distillation regions. (2) 117 < P < 375, two binary azeotropes (EW, AW), two distillation regions. (3) 375 < P < 857, three binary azeotropes (EW, AW, AE), three distillation regions. (4) 857 < P < 5000, one ternary (AEW) and three binary azeotropes (EW, AW, AE), three distillation regions. By increasing or decreasing the operating pressure of individual columns, we can move distillation boundaries in the composition space or even make azeotropes appear or disappear. This can have a tremendous effect on the topology of the residue curve map and the feasibility of distillation sequences. By varying the pressure, we can provide opportunities for pressureswing distillation and heat integration between columns in the sequence.16 Branch Tracking in Pressure Sensitivity Analysis In our implementation of pressure sensitivity analysis, it is crucial to track efficiently all homotopy branches of eq 6. This includes both the known azeotropes at the initial pressure P1 as well as the new ones that appear during the calculations. At the starting point (t ) 1), we have as many branches as there are pure components and azeotropes at the initial pressure P1. We can track the branches by gradually increasing the value of the homotopy parameter t and, at the same time, the pressure according to eq 7. We use the most recently calculated point as an initial guess for the next point on the branch. Unfortunately, this type of parameter continuation is prone to fail at turning points (or other singularities) where the homotopy parameter must decrease to follow the branch. To overcome this difficulty, an arc length continuation was applied. One

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additional equation defining the arc length g(z,t,s) was added to the system of eqs 6 to form the following augmented system

G(z,t,s) )

[

]

h(z,t) )0 g(z,t,s)

(8)

There are c unknowns (z1, z2, ..., zc-1, t) and one parameter s (arc length) in the system of eqs 8. The current branch can be effectively tracked by gradually increasing the arc length, s, and solving the system of eqs 8. Because the homotopy parameter, t, is one of the unknowns, it can decrease; however, the arc length always increases. Instead of the exact nonlinear arc length relation,5 we use the pseudo-arc-length equation as defined by Keller17 c-1

g(z, t, s) ) Θ

∑ i)1

[zi - zi(sk)]

dzi(sk)

+

ds

(1 - Θ)[t - t(sk)]

dt(sk) ds

- (s - sk) (9)

The tuning factor Θ ) 0.666 has been selected to place more emphasis on composition variables than on the homotopy parameter. z(sk) is the composition, and t(sk) is the homotopy parameter known from the previously calculated point for the arc length sk. [dzT(sk)/ds, dt(sk)/ ds] is the vector tangent to the homotopy path at point k. It is approximated by the secant vector connecting points k - 1 and k and is used in a predictor step during branch tracking. In the corrector step, we solve the system of eqs 8 by a constrained Newton-Raphson method with an optimization of the Newton step length.4 The arc step size (s - sk) is set initially to 0.01 and then reduced if the branch is strongly nonlinear or increased if it is close to a linear function and far from any bifurcation point. If the determinant of the Jacobian matrix of the system of eqs 8 changes sign between two consecutive points on the branch, we know that two continuation branches intersect at a bifurcation point. The point is then calculated using a secant or bisection method and stored, as is the initial direction of the new branch [dznewT/ds, dtnew/ds]. This direction is computed from the following approximation of the algebraic bifurcation equation4,17

[ ][ ] [ ] [ ][ ] ∂hi ∂hi ∂zj ∂t dzTold dtold ds ds

dznew 0 ds dtnew ) 0 ds

(10)

where the subscript old indicates derivatives at the bifurcation point for the old branch and the subscript new is for the new branch. At the beginning, we know initial conditions only for branches corresponding to pure components and azeotropes that exist at the initial pressure P1 (t ) 1). Then during branch tracking, each new bifurcation point provides one additional initial condition (starting point and initial direction) that is stored for later exploration. Initial directions for pure-component branches at t ) 1 are set to 1.0 for the homotopy parameter and to 0.0 for all components (composition at all pure-component

branches is constant; we track pure-component branches only to find bifurcations). Initial directions for azeotrope branches at t ) 1 are estimated from the Newton step of eq 1 calculated at the known initial composition (azeotrope at P1) and a slightly different pressure P ) P1 + δ(P2 - P1), where δ is usually set equal to 0.01. The azeotrope pressure analysis procedure is finished when all initial conditions have been explored for branch tracking up to the final pressure P2 (t ) 2) or to a composition border, where it merges with another lowerorder branch. Consistency Test From the eigenvalues of the Jacobian matrix of eq 1, we can determine the stabilities (types) of all azeotropes or pure components at a particular pressure. If all eigenvalues are positive, the stationary point is an unstable node,18 i.e., a minimum-boiling azeotrope/ component. Residue curves move away from this singular point. If all eigenvalues are negative, the stationary point is a stable node, i.e., a maximum-boiling azeotrope/component. Residue curves move toward this singular point. If some eigenvalues are negative and some positive, the stationary point is a saddle, i.e., an intermediate-boiling azeotrope/component. Residue curves move toward and then away from this singular point. If some eigenvalues are zero, the stationary point is a tangential node. Given the azeotropes’ types, the topological consistency of the residue curve map (RCM) can be checked via the topological constraint13 c

∑ 2k(N+k + S+k - N-k - S-k ) ) (-1)c-1 + 1 k)1

(11)

+ where N+ k and Sk are the numbers of k-component nodes and saddles, respectively, with index +1, whereas Nk and Sk are the corresponding singular points with index -1. If the number of negative eigenvalues of the Jacobian matrix is even, then the index is equal to +1. If the number is odd, then the index is equal to -1. For a more comprehensive explanation of the indices of stationary points in a vector field, see Appendix B in Wasylkiewicz et al.6 The topological constraint (eq 11) is a necessary condition for topological consistency. If the constraint is not fulfilled, one or more azeotropes are missing, or their properties have been estimated incorrectly. However, fulfillment of the topological constraint does not guarantee that all azeotropes have been found and the system properties have been determined correctly. The topological constraint is not a sufficient condition for consistency in azeotrope calculations.5

Pressure Sensitivity Analysis for Quaternary Mixture A quaternary mixture of acetone (A), chloroform (C), benzene (B), and methanol (M) was analyzed between the pressures 100 kPa and 10 MPa. Our starting point was the solution of the search for all of the azeotropes in the system at 100 kPa. We found the solution by applying our homotopy continuation azeotrope calculation method.5 The corresponding homotopy branches are shown in Figure 6, and the results of the azeotrope calculations together with the results of the consistency

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Figure 6. Homotopy branches for azeotrope calculations for a mixture of acetone (A), chloroform (C), benzene (B), and methanol (M) at 100 kPa. NRTL model.

Figure 7. Homotopy branches for pressure analysis of azeotropes for a mixture of acetone (A), chloroform (C), benzene (B), and methanol (M) between 100 and 10 000 kPa. NRTL model.

Table 1. Results of Azeotrope Calculations for the Mixture of Acetone (A), Chloroform (C), Benzene (B), and Methanol (M)a,b components and azeotropes acetone (A) chloroform (C) benzene (B) methanol (M) AC AM CM BM ACM

azeotrope composition (mole fraction)

0.339 804 0.791 421 0.654 137 0.377 886 0.343 508

0.660 196 0.208 579 0.345 863 0.622 114 0.225 519

boiling point (°C) typec index

55.68 60.70 79.70 64.15 64.05 54.98 53.23 57.91 0.430 973 56.85

SA SA SN SN SA UN UN SA SA

-1 -1 -1 -1 +1 +1 +1 +1 -1

a Pressure 100 kPa. NRTL model. b N- ) 2, S- ) 2, N+ ) 2, 1 1 2 + + 2 3 4-1 + S+ 2 ) 2, S3 ) 1, - 2(N1 + S1 ) + 2 (N2 + S2 ) - 2 S3 ) (-1) 1 ) 0. c SN ) stable node, UN ) unstable node, SA ) saddle point.

test are presented in Table 1. There are four binary homotopy branches, leading to four binary azeotropes. The ternary ACM branch starts at the bifurcation point of the binary branch CM and ends at the ternary azeotrope ACM. For the pressure sensitivity analysis, we start a branch for each pure component and azeotrope found at 100 kPa. Each branch is followed and continually checked for new bifurcation points from 100 kPa to 10 MPa or until the branch disappears. The corresponding continuation branches are shown in Figure 7, and the results of the bifurcation pressure search in Table 2. Some azeotropes are not particularly pressure-sensitive and are present in the whole range of investigated pressures (see, e.g., the AC and BM binary azeotrope branches in Figure 7). On the other hand, the AM and CM azeotropes are very pressure-sensitive. They contain more and more methanol as the pressure increases and eventually disappear at the methanol vertex. There are three bifurcation points where new branches appear: (1) quaternary branch ACMB starts from branch ACM at bifurcation point B1, (2) ternary branch ABM starts from branch BM at bifurcation point B2, and (3) ternary branch BCM starts from branch BM at bifurcation point B5. There are five bifurcation points where branches disappear: (1) quaternary branch ACMB collapses on branch ABM at bifurcation point B3, (2) ternary branch ABM collapses on branch AM at bifurcation point B4, (3) ternary branch ACM collapses on branch AM at bifurcation point B6, (4) ternary branch BCM collapses

Figure 8. Composition bifurcation diagram for four selected branches (AM ], ACM 3, ABM 4, ACBM 0) of pressure analysis of azeotropes for a mixture of acetone (A), chloroform (C), benzene (B), and methanol (M) between 100 and 2575 kPa. NRTL model.

Figure 9. Bifurcation diagram for determinant of the Jacobian matrix for four selected branches of pressure analysis of azeotropes for a mixture of acetone (A), chloroform (C), benzene (B), and methanol (M) between 100 and 2575 kPa. NRTL model.

on branch CM at bifurcation point B7, and (5) binary branch AM collapses on vertex M at bifurcation point B8. Four selected branches (AM, ABM, ACM, and ACMB) are also shown in a composition bifurcation diagram (Figure 8) and a bifurcation diagram for the determinant of the Jacobian matrix (Figure 9) between 100 and 2575 kPa. Bifurcation points, where branches merge (B3, B4, B6) or emerge (B1, B2), occur at the points where the determinant of the Jacobian matrix changes sign. This criterion is used to detect and exactly calculate the bifurcation points.

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Table 2. Results of Bifurcation Pressure Search for the Mixture of Acetone (A), Chloroform (C), Benzene (B), and Methanol (M) between 100 and 10 000 kPaa bifurcation composition (mole fraction)

bifurcation

homotopy

pressure (kPa)

A

C

B

M

temperature (°C)

B1 B2 B3 B4 B5 B6 B7 B8

1.005 71 1.008 56 1.025 79 1.063 24 1.102 18 1.199 59 1.338 51 1.539 92

156.53 184.78 355.29 726.12 1111.55 2075.92 3451.24 5445.18

0.331 739 0 0.153 707 0.430 706 0 0.211 627 0 0

0.196 613 0 0 0 0 0 0.223 007 0

0 0.348 106 0.221 810 0 0.242 729 0 0 0

0.471 648 0.651 894 0. 624 483 0.569 294 0.757 271 0.788 373 0.776 993 1.000 000

69.47 74.78 95.12 120.99 137.07 166.41 190.40 216.35

a

NRTL model.

Because eight bifurcation pressures (bifurcation points B1-B8 in Figure 7) were found in the pressure range 100-6000 kPa, the range can be divided into nine topologically different types of DRDs. Each type of DRD can be divided into a distinct set of distillation regions giving different opportunities and constraints for separation. A particular mixture can be separated into numerous products depending on the operating pressure selected. Knowledge about the behavior of the mixture throughout a range of practical pressures can help in selection of the optimal operating pressure for a distillation column to achieve a desired separation. Conclusions A new pressure sensitivity analysis method has been developed in which bifurcation theory is applied, together with an arc length continuation method, to find all bifurcation pressures in a specified pressure limit. This method allows not only individual azeotropes to be followed efficiently, but also any azeotrope that appears in the pressure range to be found. The sensitivity analysis of azeotropes to changes in operating pressure provides new opportunities in the design of azeotropic distillation sequences. Increasing or decreasing operating pressures in individual columns can cause the appearance or disappearance of azeotropes and distillation boundaries. This can have an enormous effect on the topology of the residue curve map and the feasibility of distillation sequences. The pressure sensitivity analysis of azeotropes shows opportunities for pressure-swing distillation and heat integration between columns in the sequence. Pressure-swing distillation can often be considered as an attractive alternative for breaking homogeneous azeotropes and sometimes can considerably simplify complex separation systems. The pressure sensitivity analysis method can also be used to verify the accuracy of the VLE model throughout a pressure range. If the system is known to have a particular set of azeotropes at various pressures, the algorithm will be able to confirm whether the VLE prediction is as expected or not. In addition, if the bifurcation pressure is known from laboratory data, it can be compared to the VLE prediction. In real distillation columns, the pressure changes from stage to stage. In some cases, this can lead to a switch in topology of distillation regions and cause serious problems in the convergence of calculations in steady-state or dynamic simulators. Pressure sensitivity analysis is an indispensable tool in troubleshooting in such cases for both simulators and real operations. The implementation of our method is available in the commercial program Distil.19

Appendix Binary interaction parameters for a mixture of acetone, chloroform, benzene, and methanol are given in Table 3. Binary interaction parameters for a mixture of acetone, ethanol, methanol, and water are given in Table 4. In both cases, the NRTL model was used as described, e.g., in the DECHEMA Chemistry Data Series:20 Table 3. Binary Interaction Parameters for a Mixture of Acetone (A), Chloroform (C), Benzene (B), and Methanol (M)a aAC ) 301.8388b aCA ) -651.1909 aBA ) -253.6262 aMA ) 118.0802 RAC ) 0.3054 RCB ) 0.3034

aAB ) 701.8184 aCB ) -144.3553 aBC ) 57.414 39 aMC ) 1364.651 RAB ) 0.3050 RCM ) 0.2932

aAM ) 296.2431 aCM ) -134.3576 aBM ) 758.2957 aMB ) 1072.38 RAM ) 0.3003 RBM ) 0.4837

a NRTL model. b Binary interaction parameters a are in cal/ ij mol.

Table 4. Binary Interaction Parameters for a Mixture of Acetone (A), Ethanol (E), Methanol (M), and Water (W)a aAE ) 375.3497b aEA ) 45.3706 aMA ) 118.0802 aWA ) 750.3181 RAE ) 0.3006 REM ) 0.3029

aAM ) 296.2431 aEM ) -157.3179 aME ) 145.875 aWE ) -109.6339 RAM ) 0.3003 REW ) 0.3031

aAW ) 1299.395 aEW ) 1332.312 aMW ) 610.403 aWM ) -48.6725 RAW ) 0.5856 RMW ) 0.3001

a NRTL model. b Binary interaction parameters a are in cal/ ij mol.

Nomenclature c ) number of components g ) pseudo-arc length h ) vector homotopy function P ) pressure Ps ) saturation pressure p ) number of liquid phases s ) arc length T ) temperature t ) homotopy parameter x ) vector of mole fractions in the liquid phase xi ) mole fraction of component i in the liquid phase y ) vector of mole fractions in the vapor phase yi ) mole fraction of component i in the vapor phase y˜ i ) pseudo-mole fraction of component i in the vapor phase z ) vector of overall liquid mole fractions zi ) overall liquid mole fraction of component i γ ) activity coefficient φ ) fugacity coefficient φ ) liquid-phase fraction Θ ) tuning factor in arc length equation

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Literature Cited (1) Chapman, R. G.; Goodwin, S. P. A General Algorithm for the Calculation of Azeotropes in Fluid Mixtures. Fluid Phase Equilib. 1993, 85, 55. (2) Maier, R. W.; Brennecke, J. F.; Stadtherr, M. A. Reliable Computation of Homogeneous Azeotropes. AIChE J. 1998, 44, 1745. (3) Harding, S. T.; Maranas, C. D.; McDonald, C. M.; Floudas, C. A. Locating All Homogeneous Azeotropes in Multicomponent Mixtures. Ind. Eng. Chem. Res. 1997, 36, 160. (4) Fidkowski, Z. T.; Malone, M. F.; Doherty, M. F. Computing Azeotropes in Multicomponent Mixtures. Comput. Chem. Eng. 1993, 17, 1141. (5) Wasylkiewicz, S. K.; Malone, M. F.; Doherty, M. F. Computing All Homogeneous and Heterogeneous Azeotropes in Multicomponent Mixtures. Ind. Eng. Chem. Res. 1999, 38, 4901. (6) Wasylkiewicz, S. K.; Sridhar, L. N.; Doherty, M. F.; Malone, M. F. Global Stability Analysis and Calculation of Liquid-Liquid Equilibrium in Multicomponent Mixtures. Ind. Eng. Chem. Res. 1996, 35, 1395. (7) Roscoe, H. E.; Ditmar, W. On the absorption of hydrochloric acid and ammonia in water. J. Chem. Soc. 1859, 12, 128. (8) Roscoe, H. E. On the composition of the aqueous acids of constant boiling point. J. Chem. Soc. 1860, 13, 146. (9) Roscoe, H. E. On the composition of the aqueous acids of constant boiling pointsSecond comunication. J. Chem. Soc. 1862, 15, 270. (10) Horsley, L. H. Graphical method for predicting azeotropism and effect of pressure on azeotropic constants. Anal. Chem. 1947, 19, 603. (11) Zawisza, A. C. Change of azeotropic parameters with pressure. Bull. Acad. Polon. Sci. Ser. Sci. Chem. 1961, 9, 147.

(12) Malesinski, W. Azeotropy and Other Theoretical Problems of Vapor-Liquid Equilibrium; Interscience: New York, 1965. (13) Zharov, W. T.; Serafimov, L. A. Physicochemical Fundamentals of Distillation and Rectification; Khimiya: Leningrad, U.S.S.R., 1975. (14) Knapp, J. P. Exploiting pressure effects in the distillation of homogeneous azeotropic mixtures. Ph.D. Dissertation, University of Massachusetts, Amherst, MA, 1991. (15) Knapp, J. P.; Doherty, M. F. A New Pressure-SwingDistillation Process for Separating Homogeneous Azeotropic Mixtures. Ind. Eng. Chem. Res. 1992, 31, 346. (16) Wasylkiewicz, S. K.; Kobylka, L. C.; Castillo, F. J. L. Pressure Sensitivity Analysis of Azeotropes in Synthesis of Distillation Column Sequences. Hung. J. Ind. Chem. 2000, 28, 41. (17) Keller, H. B. Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems. In Application of Bifurcation Theory; Rabinowitz, P. H., Ed.; Academic Press: New York, 1977. (18) Wasylkiewicz, S. K.; Kobylka, L. C.; Satyro, M. A. Designing Azeotropic Distillation Columns. Chem. Eng. 1999, 106 (8), 80. (19) Distill, version 5.0; Hyprotech Ltd.: Calgary, Alberta, Canada, 2001 (http://www.hyprotech.com). (20) Gmehling, J.; Onken, U.; Arlt, W. Vapor-Liquid Equilibrium Data Collection; DECHEMA: Frankfurt am Main, Germany, 1981.

Received for review January 25, 2002 Revised manuscript received October 17, 2002 Accepted October 18, 2002 IE020079B