Geometrically Locating Azeotropes in Ternary Systems - American

Ind. Eng. Chem. Res. 2005, 44, 3709-3719

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Geometrically Locating Azeotropes in Ternary Systems Zhiwen Qi† and Kai Sundmacher*,†,‡ Max-Planck-Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, D-39106 Magdeburg, Germany, and Otto-von-Guericke-University Magdeburg, Process Systems Engineering, Universita¨ tsplatz 2, D-39106 Magdeburg, Germany

A geometrical method is provided to locate all kinds of azeotropes, i.e., nonreactive, kinetic, and reactive azeotropes, in homo- and heterogeneous ternary systems. The mass balance conditions of azeotropes are geometrically demonstrated as reaction kinetic surface (RKS), which can be further generalized as the potential singular point surface (PSPS). The azeotropes can be directly located by two sets of RKS’s based on different components. However, it is more convenient to use PSPS with one set of RKS due to its important property; i.e., the PSPS consists of all kinds of azeotropes and all pure components such that the PSPS can be predicted starting from pure components. A continuation method is used to determine the PSPS and the RKS. Several nonreactive and reactive systems with homogeneous and heterogeneous liquid mixtures are illustrated. Introduction Since the knowledge of azeotropes is important to identify the limitation in separation operations, the determination of the existence and the composition of azeotropes are very important for process design. For the past 2 decades, several researchers, most notably Doherty and co-workers, have focused on the properties of azeotropes to produce insights concerning the operation of azeotropic distillation columns, as well as to develop design methodologies.1 Residue curve maps have been successfully applied for predicting process scheme and potential products of traditional (or nonreactive) distillation column.2 It has also been extended to reactive cases in homogeneous3-9 and heterogeneous systems.10-12 To construct maps of residue curves and distillation lines, all of the azeotropes in the mixture must be known and their stabilities should be characterized, which is very expensive if done strictly from experiment alone. Therefore, a computation method is strongly needed for locating all azeotropes, which has attracted significant attention in the last half century (e.g., Aristovich and Stepanova;13 Teja and Rowlinson;14 Wang and Whiting;15 and Chapman and Goodwin16). However, the highly nonlinear form of the thermodynamic equations describing the phase equilibria makes the computation of azeotropes a particularly difficult problem, especially when the kinetic effect of a chemical reaction is taken into account. An excellent review on nonideal distillation, including a discussion on the computation of azeotropes, was published by Widagdo and Seider.17 To be more useful, a computational method for locating azeotropes should be fully reliable, capable of finding all azeotropes when one or more exists, and capable of verifying when none exist, which was the considerable interest of the recently developed technolo* To whom correspondence should be addressed. Phone: +49-391-6110351. Fax: +49-391-6110353. E-mail: [email protected]. † Max-Planck-Institute for Dynamics of Complex Technical Systems. ‡ Otto-von-Guericke-University Magdeburg.

gies. Fidkowski et al.18 developed a continuation method that efficiently locates all of the homogeneous nonreactive azeotropic compositions by finding all of the roots of the homotopy equation. Okasinski and Doherty,8,9 Wasylkiewicz et al.,19 and Eckert and Kunicek20 applied this method for reactive azeotropes and heterogeneous nonreactive azeotropes, respectively. Although these methods have been demonstrated to be very promising, perfecting the methods of identifying the real bifurcation points and for initiating the new branches still remains. Hardings et al.21,22 have used a powerful optimization procedure to find homogeneous, heterogeneous, and nonreactive azeotropes in multicomponent mixtures. Their approach is based on developing convex underestimators which are coupled with a branch and bound framework. Maier et al.23,24 provided a new approach for reliably finding homogeneous and reactive azeotropes of a multicomponent mixture, and for verifying when none exists. The technique is based on interval analysis, in particular the use of an interval-Newton method with a generalized bisection algorithm. This method is mathematically and computationally guaranteed and does not require initial starting points or the construction of model specific convex underestimating functions. It can be applied to any activity coefficient model. Recently, Qi et al.25 found that the conditions of all kinds of azeotropes, i.e., nonreactive, kinetic, and reactive azeotropes, in mulicomponent systems can be generalized into a simple relationship in the form of transformed variables presented by Barbosa and Doherty3 and Ung and Doherty.26,27 These conditions are illustrated as the potential singular point surface (PSPS), where all azeotropes are located. For the ternary system, the PSPS is one curve and therefore can be applied for geometrically locating all kinds of azeotropes in homogeneous and heterogeneous mixtures, which is the purpose of this contribution. In the next, we will first demonstrate the PSPS and the reaction kinetic surface (RKS), and then will apply them to locating azeotropes in homogeneous and heterogeneous mixtures at different reaction conditions.

10.1021/ie049031j CCC: $30.25 © 2005 American Chemical Society Published on Web 04/19/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005

Figure 1. Reactive kinetics surface for a nonreactive mixture of acetone (A), chloroform (C), and methanol (M) at 1 atm.

constant was applied and the Damko¨hler number is defined as Da ) Hcatkf,ref/V0. The overall reaction term Θ is given by

Θ)

k′f k′′f βR ′(x′) + (1 - β)R ′′(x′′) kf,ref kf,ref

(2)

which represents the overall reaction driving force in the liquid phases, β is the molar ratio of the holdup of the extract phase to the overall mixture holdup, and superscripts ′ and ′′ refer to the two corresponding liquid phases. In eq 1, the vapor phase mole fractions yi and the liquid-phase mole fractions xi have to fulfill the vaporliquid-liquid equilibrium (VLLE) in the case of heterogeneous liquid mixtures and the vapor-liquid equilibrium (VLE) in homogeneous mixtures. For the latter, the reaction term Θ in eq 1 is simplified to Figure 2. Locating nonreactive azeotropes by two sets of RKS for mixture of acetone (A), chloroform (C), and methanol (M) at 1 atm.

Conditions for Azeotropes To formulate the conditions for all kinds of azeotropes, we consider a general case, i.e., with chemical reaction and heterogeneous liquid mixture. The single reversible reaction is

ν1A1 + ν2A2 + ... + νNCANC S 0 where νi is the stoichiometric coefficient of component i. The summation of νi yields νT. The reaction rate is expressed as

R phase(xphase) rphase ) kphase f with kf as the rate constant of the forward reaction at boiling temperature and R as the normalized reaction rate. The mass balances in the vessel are obtained as10

dxi ) (xi - yi) + (νi - νTxi)DaΘ i ) 1, ..., NC (1) dζ where dζ ) (V/HL)dt and kf,ref is the rate constant at reference conditions. The heating policy V ) V0 )

Θ)

kf R (x) kf,ref

(3)

At azeotropic compositions, dxi/dζ ) 0, which leads eq 1 to

0 ) (xi - yi) + (νi - νTxi)DaΘ i ) 1, ..., NC (4) Equation 4 is the general condition for azeotropes in the kinetically controlled mixture and is defined as the reaction kinetic surface. It can be further reduced for nonreactive azeotrope in a nonreactive mixture (Da ) 0) and reactive azeotrope in a chemical equilibrium controlled mixture (Da f ∞):

For a nonreactive azeotrope: 0 ) (xi - yi)

(5)

For a reactive azeotrope: 0 ) (νi - νTxi)Θ

(6)

Applying eq 4 for a reference component k and eliminating the common term DaΘ yield

Xi ) Yi i ) 1, ..., NC - 2

(7)

with Xi and Yi as the transformed liquid and vapor mole

Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3711

fractions:

Xi ≡

νkxi - νixk νkyi - νiyk and Yi ≡ νk - νTxk νk - νTyk i ) 1, ..., NC - 2 (8)

Equation 7 can be geometrically demonstrated as the potential singular point surface. The PSPS has several important properties: (i) Containing all pure components and all kinds of azeotropes; (ii) dependent only on reaction stoichiometry and physical thermodynamics, i.e., VLE or VLLE; (iii) independent of the reaction rate expression; (iv) valid for all reaction conditions. Solution Method The PSPS and the RKS can be predicted using the continuation method in the computer-aided modeling and simulation environment DIVA (Mangold et al.28). For the RKS, each condition from eqs 4-6 can be represented by a (NC - 1) dimensional hypersurface. Since the PSPS passes through all pure components, the composition of pure components can be used as the starting point of branch continuation, which guarantees to find all azeotropes. For a given one-reaction system, one has only a one-dimensional surface for the PSPS. Principally, azeotropes in multicomponent mixtures can be directly located as the nonpure component intersection points of the (NC - 1) sets of RKS. If no such points are enclosed in the composition triangle, we say there exists no azeotrope in the system. Since the azeotropes at two extreme cases (i.e., nonreactive and chemical equilibrium cases) indicate the qualitative branch regions of all kinds of azeotropes, it is more convenient to use PSPS with one set of RKS, especially for reactive mixtures, as will be demonstrated later. However, one may encounter a visual problem for systems with more than four components, which restrict the application of this geometrical method to ternary systems as discussed in this paper. In ternary systems, the PSPS and the RKS will be curves in the twodimensional composition triangle. For nonreactive mixtures, one can suppose different reaction stoichiometric coefficients νi such that the PSPS can be predicted. The obtained PSPS’s are then applied to locate the nonreactive azeotropes with other PSPS’s based on different sets of νi. For chemical equilibrium controlled systems, one can directly plot the chemical equilibrium surface. A reactive azeotrope is the nonpure component intersection point of the PSPS with the chemical equilibrium curve. In the following sections, we will first intensively illustrate how to locate azeotropes by RKS and PSPS through a nonreactive homogeneous mixture acetonechloroform-methanol. Then we apply the PSPS for a heterogeneous mixture and reaction mixtures. In this paper, we also plot the behavior outside the physical composition triangle since they follow the same properties as those inside the triangle. In addition, one may be note that generally it takes seconds to generate one curve of the PSPS or RKS. Nonreactive Homogeneous System: Acetone (A)-Chloroform (C)-Methanol (M) The first example is a homogeneous nonreactive mixture acetone (A)-chloroform (C)-methanol (M),

Figure 3. Potential singular point surfaces for mixture of acetone (A), chloroform (C), and methanol (M) at 1 atm under assumed reactions (a) A + C S M, (b) A + C S 2M, and (c) A S C + M.

which has been studied by Fidkowski et al.18 (Wilson model) and Maier et al.18 (Wilson and NRTL models). Here we use the Wilson model to describe the liquid activity coefficient. The thermodynamic parameters are listed in Table 5. For this nonreactive mixture, the RKS at 1 atm is predicted from eq 5 based on chloroform (i.e., xC - yC ) 0, Figure 1a). Note that one part of the RKS is the line xC ) 0 and only one of the other two parts passes

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Figure 4. Locating nonreactive azeotropes by the PSPSs for mixture of acetone (A), chloroform (C), and methanol (M) at 1 atm.

Figure 5. Locating nonreactive azeotropes by RKS and PSPSs mixture of acetone (A), chloroform (C), and methanol (M) at 1 atm. Table 1. Comparison of Calculation Results for the Mixture of Acetone (A), Chloroform (C), and Methanol (M) with Literature Data Maier et al.23

this work azeotrope

mole fraction (ACM)

AC AM CM ACM

(0.3259, 0.6741, 0.0000) (0.2740, 0.0000, 0.7260) (0.0000, 0.3918, 0.6082) no ternary azeotrope

AC AM CM ACM

(0.3661, 0.6339, 0.0000) (0.7976, 0.0000, 0.2024) (0.0000, 0.6629, 0.3371) (0.3444, 0.2155, 0.0401)

T (°C)

mole fraction (ACM)

At 15.8 atm, Wilson Model 181.46 (0.324, 0.676, 0.0000) 155.32 (0.275, 0.000, 0.725) 151.14 (0.000, 0.392, 0.608) no ternary azeotrope

Fidkowski et al.18 T (°C)

mole fraction (ACM)

T (°C)

181.46 155.32 151.44


185.92 154.40 151.45

(0.3437, 0.6563, 0.0000) (0.7944, 0.0000, 0.2056) (0.0000, 0.6482, 0.3518) (0.3234, 0.2236, 0.4530)

65.47 55.34 53.91 57.58

At 1 atm, Wilson Model 65.24 55.65 53.63 57.73

through C-vertex. In the same manner, the RKS based on acetone is given in Figure 1b. If the obtained two sets of RKS are overlapped in one diagram, as shown in Figure 2, several intersection points are located, which are pure components (A, C, and M) and azeotropes (binary ones AC, AM, and CM, and ternary one ACM) of this mixture. The stabilities of these points are marked in the diagram (stable node b, unstable node O, saddle point 0). The corresponding compositions and boiling temperatures are listed in

Table 1 and are compared with results of Fidkowski et al.18 Though ternary azeotropes are found from both methods, the compositions, also of binary azeotropes, are quite different. For this nonreactive system, the PSPS’s are predicted based on three sets of assumed reaction stiochiometric coefficients, as plotted in Figure 3. It clearly shows that, independent of reaction scheme, (1) all azeotropes and all pure components are located on one set of PSPS; (2) all PSPS branches start from (or pass through) pure


Figure 6. Locating nonreactive azeotropes by the PSPS’s for mixture of acetone (A), chloroform (C), and methanol (M) at 15.8 atm. Dash-dot: reaction A + C S M. Solid: reaction A S C + M.

trope in a real ternary system. The most complex feasible topological structure of the phase diagram is similar to the ACM system (Figure 2). Since the PSPS from one set of reaction schemes can locate all binary nonreactive azeotropes, the question that is left is how can we verify the existence of the ternary azeotrope. The simple way is to overlap the PSPS from different reaction schemes, as illustrated in Figure 4. In Figure 4a, the PSPS’s based on reactions A + C S M (dashdot) and A S C + M (solid) are overlapped, while the PSPS’s from reactions A + C S 2M (dash-dot) and A S C + M (solid) are superposed in Figure 4b. As we can see, independent of reaction scheme, we located the same azeotropes as in Figure 2. Another option is that we can arbitrarily superpose one set of PSPS with one set of RKS, as shown in Figure 5. In Figure 5a, the PSPS (dash-dot) is predicted from the assumed reaction A S C + M and the RKS (solid) from xA - yA ) 0, i.e., Figure 1b. In Figure 5b, the assumed reaction is A + C S M and the RKS comes from xC - yC ) 0, i.e., Figure 1a. Again, all azeotropes and all pure components are enclosed as in Figures 2 and 4. This approach is not only valid for nonreactive mixtures but also for reactive systems, as demonstrated later. Figure 6 gives azeotropes at 15.8 atm located by two sets of PSPS’s from different assumed reaction schemes. Comparing to behaviors at 1 atm, first we can find that no ternary azeotrope exists now. Moreover, the PSPS based on reaction A S C + M (solid) is similar but the one from reaction A + C S M (dash-dot) is completely different. The PSPS branches now hold different shapes and pass through different binary azeotropes. The compositions of all binary azeotropes are given in Table 1 and compared with the literature. They are close to the results in Maier et al.,23 but have big differences from Fidkowski et al.,18 which may be due to different thermodynamic properties and parameters used. We are confident about our results since it is guaranteed by the properties of the PSPS rather than by the mathematical method.

Figure 7. Liquid-liquid envelope for mixture of n-propanol (P), water (W), and toulene (T) at 1 atm.

Nonreactive Heterogeneous System: n-Propanol (P)-Water (W)-Toulene (T)

components; (3) all intersection points of the PSPS branches with the edges are binary azeotropes. The last two phenomena are especially important compared with the RKS in Figure 1, which implies the convenience of using the PSPS rather than the RKS alone. According to Serafimov’s work,29 there are a maximum of three binary azeotropes and one ternary azeo-

In the mixture of n-propanol, water, and toluene, liquid-phase splitting will take place. Wasylkiewicz et al.19 have computed azeotropes of this system based on homogeneous and heterogeneous models to illustrate their method. Here, we use PSPS to locate all azeotropes. The thermodynamic properties are listed in Table 6.

Table 2. Comparison of Calculation Results for the Mixture of n-Propanol (P), Water (W), and Toluene (T) with Literature Data Wasylkiewicz et al.19

this work azeotrope

a

mole fraction (PWT)

PW PT WT PWT

(0.3989, 0.6011, 0.0000) (0.6525, 0.0000, 0.3475) (0.0000, 0.5547,a 0.4453) no ternary azeotrope

PW PT WTb PWTb

(0.3989, 0.6011, 0.0000) (0.6525, 0.0000, 0.3475) (0.0000, 0.5581, 0.4419) (0.1736, 0.4819, 0.3445)

T (°C) Homogeneous Model 90.54 95.77 72.09 Heterogeneous Model 90.54 95.77 84.55 82.20

mole fraction (PWT)

T (°C)


90.19 95.29 72.05

(0.4096, 0.5904, 0.0000) (0.6671, 0.0000, 0.3329) (0.0000, 0.5586, 0.4414) (0.1807, 0.4785, 0.3408)

90.19 95.29 84.52 81.99

Underlines mark the slight difference between the homo- and heterogeneous models. b Heterogeneous azeotropes.

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Figure 8. Comparison of the PSPS’s from the homogeneous and heterogeneous models for system n-propanol (P), water (W), and toulene (T) at 1 atm under assumed reactions (a) P S W + T and (b) P + W S T.

Figure 9. Locating nonreactive azeotropes by the PSPSs for system n-propanol (P), water (W), and toulene (T) at 1 atm. (a) Homogeneous model; (b) heterogeneous model. Table 3. Azeotropes in MTBE System at 10 atm

Figure 10. Potential singular point surface (dash-dot) for the MTBE system at 10 atm. Solid line ) chemical equilibrium surface.

First, let us predict the liquid-liquid envelope for this mixture, as shown in Figure 7 with liquid-liquid tie

mole fraction (IB/Me/M)

T (°C)

Da ) 0 IB-Me: (0.9223, 0.0777, 0.0000) Me-M: (0.0000, 0.5637, 0.4363)

69.58 129.65

Da ) 0.05 KA1: (0.0198, 0.3951, 0.5850) KA2: (0.0477, 0.0890, 0.8633) KA3: (0.9301, 0.1207, -0.0508)

126.57 130.99 67.83

lines. As we can see, there are two critical points. One is inside the physical composition triangle (filled). Another is outside this triangle, which has no physical meaning. Figure 8a is the comparison of the PSPS’s from the assumed reaction P S W + T based on the homo- and heterogeneous models. Outside the liquid-liquid envelope, the PSPS branches (1 and 4) are the same for the two liquid models. However, inside the liquid-liquid envelope, the PSPS branches are different (branch 2 for homogeneous model and branch 3 for heterogeneous model), which bifurcate at two branch points that are located on the liquid-liquid envelope. It is worth noting that branches 2 and 3 intersect the W-T edge at two


Figure 11. Locating nonreactive (a, Da ) 0) and kinetic (b, Da ) 0.05) azeotropes in MTBE system by the RKS and the PSPS at 10 atm. Table 4. Azeotropes in the Hypothetical System with Liquid Phase Splitting and Reaction A + B S C at 1 atm homogeneous model mole fraction (ABC)

heterogeneous model T (°C)

mole fraction (ABC)

T (°C)

AB: (0.7527, 0.2473, 0.0000) BC: (0.0000, 0.6760, 0.3240) ACa: (0.7966, 0.0000, 0.2034) ABC1a: (0.6875, 0.1320, 0.1805)

74.95 92.55 72.75 70.05

KA1: (0.0192, 0.4578, 0.5230) KA2: (0.0085, 0.0735, 0.9180) KA3a: (0.6961, 0.1126, 0.1913)

88.35 95.88 72.04

RA1a: (0.6933, 0.1184, 1883)

72.04

Da ) 0 AB: (0.7527, 0.2473, 0.0000) BC: (0.0000, 0.6760, 0.3240) AC: (0.7231, 0.0000, 0.2769)

74.95 92.55 71.05

KA1: (0.0192, 0.4578, 0.5230) KA2: (0.0085, 0.0735, 0.9180) KA3: (0.6713, 0.1979, 0.1308)

88.35 95.88 72.00

RA1: (0.6701, 0.1778, 0.1521) a Heterogeneous azeotropes.

72.00

Da ) 1

Da f ∞

From Figure 8, all binary azeotropes are located, which are the intersection of the PSPS branches with the triangle edges (also see Figure 9). It is clear that, outside the liquid-liquid envelope, the binary azeotropes (PW and PT) are the same for the homo- and heterogeneous models. But the binary azeotropes inside the liquid-liquid envelope (WT) are different. In Figure 9a, two sets of PSPS from the homogeneous model are superposed in order to determine the existence of the ternary azeotropes. The same is true in Figure 9b for the heterogeneous system. As a result, no ternary azeotropes can be found in the homogeneous mixture. However, two ternary azeotropes are located in the heterogeneous mixture. One (PWT1) is inside, and another (PWT2) is outside the triangle. The compositions and the corresponding boiling temperatures of all azeotropes (except PWT2 since it has no physical meaning) are listed in Table 2, which shows a slight difference from Wasylkiewicz et al.19 Figure 12. Potential singular point surfaces for the hypothetical reaction system with liquid splitting.

points, i.e., two binary azeotropes WT between water and toluene, whose compositions are slightly different (see Table 2). As an interesting feature, branches 3 and 4 form an isola shape PSPS from the heterogeneous model, which moves through the W-vertex and the two binary azeotropes (PW and WT). In Figure 8b, the PSPS branches based on the assumed reaction scheme P + W S T are compared for the homo- and heterogeneous models. They show the feature similar to that in Figure 8a.

Reactive Homogeneous System: MTBE Synthesis MTBE synthesis is a reversible reaction:

isobutene (IB) + methanol (Me) S MTBE (M) (9) Vanimadhavan et al.5,6 studied the kinetic effects on the residue curve maps through this system. The thermodynamic properties are given in Table 7. The reaction rate is

(

r ) kf aIBaMe -

1 a K M

)

(10)

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Figure 13. Locating nonreactive, kinetic, and reactive azeotropes in the hypothetical reaction system by the RKS and the PSPS. (a)-(c) Homogeneous model; (d)-(f) heterogeneous model. Dash-dot ) PSPS; solid ) RKS.

with kf ) 74.40 exp(-3187.0/T) and K ) exp(6820.0/T - 16.33). The PSPS for this reaction system is predicted in Figure 10 with the chemical equilibrium surface at 10 atm. It is easy to conclude that there is no reactive azeotrope in this system simply because there exists no

nonpure component intersection point between the PSPS and the chemical equilibrium surface. In addition, two nonreactive azeotropes are found since the PSPS passes through edges IB-Me and Me-M of the triangle. Their compositions are listed in Table 3 and stabilities are given in Figure 11a for the nonreactive case (Da )

Ind. Eng. Chem. Res., Vol. 44, No. 10, 2005 3717 Table 5. Thermodynamic Properties for the Mixture of Acetone (A), Chloroform (C), and Methanol (M)23 1. Antoine Equation Parameters component

A

B

C

1566.690 1488.990 1521.230

273.419 264.915 233.970

ωi

V′i (cm3/mol)

Tc (K)

0.304 0.218 0.556

73.52 80.17 40.51

508.1 536.4 512.6

acetone (1) 7.63130 chloroform (2) 7.44777 methanol (3) 7.97010 Antoine equation: sat log(psat i ) ) A - B/(C + T) with pi in mmHg, T in °C 2. Racket Equation Parameters component acetone (1) chloroform (2) methanol (3) Zcr,i ) 0.29056-0.08775$i -(1-Tr,i′)2/7 Vscr,i ) V′iZcr,i with T′r,i ) T/293 (1-Tr,i)2/7 Vi ) Vscr,iZcr,i with Tr,i ) T/Tc A11 ) 0 A21 ) -484.3856 A31 ) 583.1054 Wilson equation:

3. Binary Interaction Parameters for Wilson Model (cal/mol) A12 ) 28.8819 A22 ) 0 A32 ) 1760.6714

A13 ) -161.8813 A23 ) -351.1964 A33 ) 0

with

0). From Figure 11a, one can see that no ternary nonreactive azeotrope exists in this system. At kinetically controlled conditions, e.g., Da ) 0.05, the two branches of the RKS in the nonreactive case move upward and intersect with the PSPS at three kinetic azeotropes. One of them (i.e., KA3) is outside the triangle, which has no physical meaning. At higher Damko¨hler number, these two RKS branches will merge and only one ternary kinetic azeotrope (KA3) will remain, which is not given here. Reactive Heterogeneous System: Hypothetical System The hypothetical reaction system was studied in Qi et al.10 and Qi and Sundmacher11 to analyze the effect of the kinetics on the residue curve maps. The thermodynamic parameters, i.e., in Table 8, are chosen such that binary as well as ternary azeotropes occur along with pronounced liquid-phase splitting. A rate expression in terms of liquid-phase activities is used to describe the reaction kinetics:

(

rphase ) kphase aphase aphase f A B

)

aphase C K

(11)

assuming identical rate constants in both liquid phases, i.e., k′f ) k′′f, and a constant value of the equilibrium constant, K ) 3.5. The PSPS of this system is depicted in Figure 12 together with the liquid splitting information (the liquid-liquid envelope, the liquid-liquid tie lines, and the critical point) and compared with those from the homogeneous model. Like the PWT system, the PSPS branches outside the liquid splitting region, i.e., branches 1 and 4, are identical for the homo- and heterogeneous models. However, inside this envelope, the PSPS branches (branch 2 for the heterogeneous model and

branch 3 for the homogeneous model) are different, which bifurcate on two branch points on the envelope. Now let us locate the azeotropes for the homo- and heterogeneous mixtures at different reaction conditions, as illustrated in Figure 13. Generally, the RKS and the azeotropes are identical outside and different inside the liquid-liquid envelope. For all azeotropes with physical meaning, their compositions and the corresponding boiling temperatures are listed in Table 4. In the nonreactive case (Figures 13a and 13d), the RKS consists of two branches. These RKS’s locates three binary nonreactive azeotropes for both mixtures. Two of them (AB, BC) are identical but the AC azeotropes hold different compositions and stabilities. In addition, one heterogeneous ternary azeotrope (ABC1) can be found for the heterogeneous mixture in the composition triangle. For both mixtures, trivial ternary azeotropes outside the triangle (ABC in Figure 13a and ABC2 in Figure 13d) are detected, which have no physical meaning. Note that they always exist at any kind of reaction conditions, as shown in the rest of the figures. With increasing the Damko¨hler number (Da ) 1.0, Figures 13b and 13e), the nonreactive RKS from xA ) 0 in Figures 13a and 13d turn into the triangle due to the chemical reaction, which locate two kinetic azeotropes (KA1 and KA2) outside the liquid-liquid envelope. The nonreactive isola RKS branches change their shapes in both mixtures. The one in the homogeneous mixture even breaks off. They locate two additional kinetic azeotropes (KA3 and KA4). By further increasing the Damko¨hler number, the two branches of the RKS join to one curve and they reach the chemical equilibrium surface at Da f ∞ (Figures 13c and 13f). In addition, there exists a straight RKS, which comes from the stoichiometric condition, i.e., νi - xiνT ) 0. These RKS’s locate two reactive azeotropes for both mixtures, only one of them (RA1) is of interest.

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Table 6. Thermodynamic Properties for the Mixture of n-Propanol (P), Water (W), and Toluene (T)30 1. Antoine Equation Parameters component

A

n-propanol (1) 8.37895 water (2) 8.07131 toulene (3) 6.95087 Antoine equation: sat log(psat i ) ) A - B/(C + T) with pi in mmHg, T in °C. A11 ) 0 A21 ) 1177.90 A31 ) 622.14 Rij ) Rji ) 0.2, Rjj ) 0 NRTL equation:

B

C

1788.02 1730.63 1342.31

227.438 233.426 219.187

2. Binary Interaction Parameters for NRTL Model (K) A12 ) -334.16 A22 ) 0 A32 ) 936.24

A13 ) -155.55 A23 ) 1583.5 A33 ) 0

with τij ) Aij/T and Gij ) exp(-Rijτij) Table 7. Thermodynamic Properties for the Mixture of Isobutene (IB), Methanol (Me), and MTBE (M)5 1. Antoine Equation Parameters component

B

C

Vi (cm3/mol)

-2125.74886 -3643.31362 -2571.58460

-33.160 -33.434 -48.406

99.33 44.44 118.8

A

isobutene (1) 20.64556 methanol (2) 23.49989 MTBE (3) 20.71616 Antoine equation: sat ln(psat i ) ) A + B/(C + T) with pi in Pa, T in K

2. Binary Interaction Parameters for Wilson Model (cal/mol) A11 ) 0 A12 ) 169.9953 A21 ) 2576.8532 A22 ) 0 A31 ) 271.5669 A32 ) -406.3902 Wilson equation, see Table 5

A13 ) -60.1085 A23 ) 1483.2478 A33 ) 0

Table 8. Thermodynamic Properties for the Hypothetical Mixture with Liquid Phase Splitting10 1. Antoine Equation Parameters component

A1

A (1) 13.31187 B (2) 10.14589 C (3) 9.81339 Antoine equation: sat sat log(pi ) ) A1 - A2/(A3 + T) with pi in Pa, T in K A11 ) 0 A21 ) 478.60 A31 ) 1070.484 Margules equation:

A2

A3

4068.457 1936.010 1669.898

139.572 5.291 -41.350

2. Binary Interaction Parameters for Margules Model (K) A12 ) 478.60 A22 ) 0 A32 ) 626.90

Conclusions A method is presented for locating all kinds of azeotropes in homo- and heterogeneous ternary systems at different reaction conditions by intersecting the potential singular point surface with the reaction kinetic surface. The PSPS is dependent only on the reaction stoichiometric coefficients and independent of the reaction kinetics. It connects all azeotropes and all pure components and therefore it is convenient to predict the PSPS starting from the pure components. For the nonreactive case, arbitrary reaction stiochiometric coefficients can be chosen for obtaining the PSPS. The azeotropes can be located by superposing PSPS and/or RKS. For the

A13 ) 1070.484 A23 ) 626.90 A33 ) 0

kinetic controlled case, the RKS can be predicted from the vertexes of reactants. We have applied the method for many homogeneous and heterogeneous ternary systems, and we did not encounter any problems. This is simply due to the property of the potential singular point surface; i.e., all azeotropes should be consistent. In other words, the method is guaranteed by the physical properties of the system rather than by the mathematical method. One more advantage is that it is not restricted to the liquid activity coefficient model. Though the PSPS is suitable for all azeotropes in multicomponent systems, the drawback of this method


limits its application for systems of more than four components due to the visualization problem. Appendix In the Appendix, we list all thermodynamic properties used in this paper. See Tables 5-8. Nomenclature ai ) activity of component i in the liquid phase Da ) Damko¨hler number HL ) molar holdup of the liquid phase, mol Hcat ) molar holdup of the catalyst, mol kf ) forward reaction rate constant, 1/s kf,ref ) forward reaction rate constant at reference temperature, 1/s NC ) number of components P ) system pressure, Pa psat ) saturated vapor pressure for component i, Pa i r ) reaction rate, mol/(mol‚s) R ) dimensionless rate in pseudo-homogeneous liquid phase, eq 6 T ) boiling temperature, K V0, V ) initial and instantaneous molar vapor flow rates, kmol/s xi, yi ) mole fraction of component i in liquid and vapor phases, respectively Xi, Yi ) transformed composition variable for component i in liquid and vapor phases, respectively Greek Letters Θ ) overall reaction driving force in the liquid mixture β ) molar ratio of the holdup of the extract phase to the overall mixture holdup ζ ) dimensionless time γi ) activity coefficient of component i in the liquid phase νi ) stoichiometric coefficient of component i νT ) total mole change of reaction

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Received for review October 6, 2004 Revised manuscript received December 13, 2004 Accepted January 3, 2005 IE049031J

Geometrically Locating Azeotropes in Ternary Systems - American

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