Application of Plate and Component Efficiency Correlations in

To do that, first of all, a computational program was developed, in Fortran language, to simulate distillation columns using the equilibrium stage mod...
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Ind. Eng. Chem. Res. 2006, 45, 5755-5760


SEPARATIONS Application of Plate and Component Efficiency Correlations in Homogeneous Azeotropic Distillation Processes Miria H. M. Reis,† Anto´ nio A. C. Barros,‡ Antonio J. A. Meirelles,§ Rubens Maciel Filho,† and Maria Regina Wolf-Maciel*,† Chemical Processes Department, Chemical Engineering School, State UniVersity of Campinas, P.O. Box 6066, 13081-970, Campinas SP, Brazil, Chemical Engineering Department, Center of Technological Science, UniVersity of Blumenau, Campus II, Rua Antoˆ nio da Veiga, 140, P.O. Box 1507, 89010-971, Blumenau SC, Brazil, and Food Engineering Department, Food Engineering School, State UniVersity of Campinas, P.O. Box 6121, 13083-862, Campinas SP, Brazil

In this work, plate and component efficiency correlations have been applied and their performances have been investigated to predict the behavior of extractive distillation columns (homogeneous azeotropic distillation processes). These correlations were developed based on physical and thermal characteristics of the system, as well as on mass and heat transfer mechanisms. The simulation results obtained with the efficiency correlations have been compared with those calculated using the nonequilibrium stage model and also with the experimental data of a pilot-scale extractive distillation column. Very good agreement was obtained. Therefore, it can be concluded that the efficiency correlations are reliable and applicable to extractive distillation processes. The use of these correlations allows the calculations of distillation behavior to be conducted more precisely and with a lower computational burden. 1. Introduction Nowadays, there is an increasing interest in proposing feasible separation schemes, mainly to separate nonideal mixtures. For such separations, the most applied technique is distillation. Although it is a well-known process, it is still necessary to optimize the involved variables, because of the high energy consumption and, sometimes, the sharp specifications to be met in this process. Pinto et al.1 reported that the extractive distillation for ethanol and water separation consumes ∼50%-80% of the total energy used in a typical fermentation ethanol manufacturing process. Extractive distillation (homogeneous azeotropic distillation) processes are widely used in the chemical industry to separate nonideal mixtures. In this process, a solvent, which is the heaviest component, is added to the system to be separated, which leads to an increase in the relative volatility of the key components. New azeotropes must not be formed, and the solvent must be completely miscible in the mixture (in contrast to the heterogeneous azeotropic distillation process). The main advantage of the extractive distillation process, in comparison with the heterogeneous azeotropic distillation process, is its operational simplicity and the absence of two liquid phases in some of the internal plates of the column. To study the feasible separation schemes, computer-aided tools have been constantly developed. The currently available * To whom correspondence should be addressed. Tel.: +55 19 37883957. Fax: +55 19 37883965. E-mail: [email protected] † Chemical Processes Department, Chemical Engineering School, State University of Campinas. ‡ Chemical Engineering Department, Center of Technological Science, University of Blumenau. § Food Engineering Department, Food Engineering School, State University of Campinas.

tools to simulate distillation columns are generally based on the concept of equilibrium stages. The equilibrium stage model assumes that the liquid and vapor phases that leave each stage are in thermodynamic equilibrium. Taylor et al.2 published a review about the growing necessity to use more-realistic models to study distillation processes. Baur et al.3 and Nava and Krishna4 analyzed the influence of efficiency values on composition trajectories, to better characterize multicomponent mixtures. The authors conclude that the equilibrium stage model can lead to some erroneous conclusions. Relatively new computing procedures use nonequilibrium stage models, i.e., the rate-based approach, which seems to be a more realistic approach, especially for multicomponent systems. This method uses information on flow rates, masstransfer coefficients, interfacial area, and liquid holdup, which can conveniently be combined into one variable: the number of transfer units (NTU). However, the rate-based approach is computationally intensive and also requires much more input information. Alternatively, one usual way to correct the weakness in the equilibrium stage model is the application of efficiency values. The overall efficiency for distillation columns was defined by Lewis5 as a relationship between the theoretical and real plates required to perform a specified separation. In 1925, Murphree6 defined the plate efficiency, relating the behavior of a real plate with that of an ideal plate, through the degree of contact between the vapor and liquid phases. In this definition, the efficiencies for the vapor and liquid phases are generally different for a same plate. Two empirical correlations, which have found wide use, are the correlation derived by Drickamer and Bradford7 and a modification of it that was derived by O’Connell.8 AIChE9 gives

10.1021/ie051182e CCC: $33.50 © 2006 American Chemical Society Published on Web 07/07/2006


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the best-established theoretical method for predicting overall column efficiency. The definition of point efficiency was developed by West et al.,10 who presented the most accurate study about plate efficiencies. Many other definitions for efficiencies were presented, such as those of Medina et al.,11 Holland,12 and Hausen,13 among others. They presented modifications to improve the Murphree definition. In fact, following the Murphree efficiency definitions, several methods for efficiency estimation have been developed. For example, Prado and Fair14 proposed a method where bubbling and jetting are taken into account. Chen and Chuang15 developed a more mechanistic model for predicting point efficiency; however, it requires evaluation against a commercial-scale distillation databank. However, there are still some difficulties in using the available efficiency correlations. Most of the empirical and semiempirical correlations have restricted applications, because the main variables involved in the distillation process are not considered. Moreover, the efficiency concepts developed for conventional distillation are usually extended to extractive distillation. Gerster16 and Weiss and Arlt17 observed that the efficiencies in extractive distillation columns are generally smaller than the efficiencies observed in conventional distillation processes. Gerster16 justified the smallest efficiency values as being due to the increase in the liquid flow rate and the decrease in the contact time between the liquid and vapor phases. According to Weiss and Arlt,17 the efficiencies are reduced because of the decrease in interfacial area that is caused by the extractive agent. In this work, robust plate and component efficiency correlations were applied in a homogeneous azeotropic distillation to predict the behavior of ternary mixtures in this type of process. For the evaluation of these correlations, the simulated data were compared with experimental results reported by Meirelles et al.18 and also with profiles calculated using the nonequilibrium stage model of Krishnamurthy and Taylor.19 2. The Efficiency Correlations The efficiency correlations consider the effects of heat- and mass-transfer mechanisms.20-22 Basically, the procedure used in the development of the efficiency correlations was based on the perturbation of the efficiency values along the column, using simulation techniques to observe all the variations of its operational parameters. To do that, first of all, a computational program was developed, in Fortran language, to simulate distillation columns using the equilibrium stage model. Basically, this program solves the MESH equations (M - mass balance, E - equilibrium equations, S - summation equations, H - energy balance) through application of the DASSL subroutine,23 which also allows the program to simulate distillation columns in the dynamic regime. The DASSL subroutine uses the backward differentiation formulas to solve a system of algebraic and differential equations simultaneously. Process variable specifications were imposed in such a way to cover all possible operating ranges for each column simulation. With this procedure, the physical and thermal parameters that have a greater influence on the column profile, for different specifications, were analyzed. The selected parameters were as follows: thermal conductivity, density, heat capacity, viscosity, molecular weight, and diffusivity. These parameters were calculated according to equations, methods, or correlations that have been proposed in the literature.24,25 Table 1 presents the

applied correlations in the calculation of pure component properties, which are used in the component efficiency correlation. The thermophysical and transport parameters were combined in an empirical equation to represent the component efficiency. To adjust the numerical coefficients of the efficiency correlations, the profiles for the distillation column were calculated, and then they were compared with different profiles, using fixed global efficiency values. These efficiency values ranged from 20% to 100%, with increment of 5%. To evaluate the performance of the efficiency correlations, a conventional extractive distillation column for the separation of ethanol and water, using ethylene glycol as a solvent, was considered. The column has 25 stages. The solvent and the ethanol-water streams are fed at plates 22 and 7, respectively. The plates are counted from the bottom to the top of the column. The solvent is fed, in pure form, at 393.15 K. The ethanol-water stream is at equimolar composition at 353.15 K. The column pressure is 101.32 kPa, and the pressure drop was not considered. The solvent-to-feed stream ratio is equal to 1.75. The adjustment of the proposed correlations was performed through the application of factorial design techniques. Based on the quasi-Newton methodology, the most reliable mathematical equation was determined to represent the efficiency values. A maximal deviation of 10-4 was established, and convergence was obtained after 46 interactions. The obtained efficiency correlation for extractive distillation columns is described in eq 1, as a function of the following mixture properties: thermal conductivity (ki,m), density (Fi,m), heat capacity (Cpi,m), viscosity (µi,m), molecular weight (Mi,m), and diffusivity (Di,m). To determine the component efficiencies, instead of using mixture properties, the pure component properties were considered (see eq 2):

ηi ) 19.37272



ki,mDi,mFi,mMi,m 2

ηi,j ) 19.37272




ki,jDiFi,jMi,j Cpi,jµi,j2





where ηi is the tray efficiency and ηi,j is the efficiency of component j on tray i. These correlations are called the Barros and Wolf plate correlation (eq 1) and the component efficiency correlation (eq 2) for the extractive distillation columns,20-22 as stated in previous published works. To calculate the mixture properties, the following equations were applied, for plate i, according to recommendations that have been presented in the literature24,25 (molar partial properties were neglected): c

ki,m )

ki,jMj2/3xj ∑ j)1 c

Mj2/3xj ∑ j)1



Cpi,m )

Cpi,jxj ∑ j)1


Fi,jxj ∑ j)1



Fi,m )

Ind. Eng. Chem. Res., Vol. 45, No. 16, 2006 5757 c

µi,m )


Di,m )

µi,jMj1/3xj ∑ j)1 c

Mj1/3xj ∑ j)1



∑ ∑Di,j,kxi,jxi,k j)1 k)1 c

for k * j





j ) 1k ) 1

With regard to the diffusivity calculation, Taylor and Krishna26 have noted that the binary diffusion coefficients in ternary mixtures are similar to those in the respective binaries. We will limit our investigations to ternary mixtures. 3. Homogeneous Azeotropic Distillation Columns: Operational Conditions To apply the obtained correlations, a computational program was developed, using the equilibrium stage model, to simulate extractive distillation columns. In this software, a subroutine to calculate the efficiency values was included, using the presented correlations. In this way, the efficiency values, which were calculated using the Barros and Wolf correlations, were considered in each plate and for each component. It is also possible to consider only plate efficiencies. For equilibrium calculations, the UNIQUAC model and the Virial equation were used to calculate the liquid and vapor nonidealities, i.e., the activity and the fugacity coefficients, respectively. The vapor pressures were calculated by the Antoine equation. The liquid and vapor enthalpies were calculated according to methods recommended by Prausnitz et al.27 Two case studies were selected: the separation of the azeotropic ethanol-water and acetone-methanol binary mixtures, using ethylene glycol and water as solvents, respectively. Figures 1 and 2 present the input data and the column specifications (denoted in boldface type in the figures) for the ethanol-water-ethylene glycol and methanol-acetone-water mixtures, respectively. These values were obtained after a parametric optimization, considering a global efficiency of 100%. For the ethanol-water separation, the column has 25 stages, counted from the bottom to the top of the column, including the reboiler and the condenser (Figure 1). Pure solvent, at a temperature of 393.15 K, is fed at stage 22. The ethanol-water mixture is fed at stage 7, near the azeotropic composition (80 mol % water and 20 mol % ethanol), and at a temperature of 313.15 K. Two specifications were imposed: the reflux ratio was 1.5 and the distillate flow rate was 78.8 mol/h. Table 1. Equations, Correlations, or Methods Used in the Calculation of the Pure Compound Physical and Transport Properties24,25 property thermal conductivity, kj density, Fj heat capacity, Cpj for alcohols for other compounds viscosity, µj diffusivity, Dj,k

Figure 1. Input data and specifications for the ethanol (1)-water (2)ethylene glycol (3) mixture.

Figure 2. Input data and specifications for the acetone (1)-methanol (2)water (3) mixture.

For the acetone-methanol separation, the column has 22 stages (Figure 2). Pure solvent is fed at stage 10 and at a temperature of 333.15 K. The acetone-methanol mixture is fed at stage 5, approximately at the azeotropic composition (55.78 mol % acetone and 44.22 mol % methanol), and at a temperature of 313.15 K. The specifications were as follows: the bottom flow rate was 221.0 mol/h and the reboiler duty was 0.7 × 107 kJ/h. For both cases, pressure drops were not considered, and the column pressure was fixed at 101.32 kPa. The condenser and the reboiler were considered to be stages with an efficiency equal to 100%.

equation, method, or correlation Sato and Reidel equation Goyal correlation Missenard group contribution method Rowlinson and Bondi method Orrick and Erbar method Wilke and Chang method for diluted solutions, corrected by the Vignes method for concentrated liquid mixtures

4. Efficiency Profiles Figures 3a and 3b present the plate and component efficiency profiles for both systems studied. These profiles are dependent on the system specifications; however, some general conclusions can be observed. Disturbances in the profiles at feed and solvent positions can be observed (mainly for the system represented in Figure 3b); because the solvent is the least-volatile compo-


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Figure 3. Plate and component efficiency profiles: (a) ethanol-water-ethylene glycol mixture and (b) acetone-methanol-water mixture.

Figure 4. Comparison of (a) vapor-phase and (b) liquid-phase profiles calculated using the nonequilibrium stage model and the corrected equilibrium stage model (Barros and Wolf correlation) for the ethanol-water-ethylene glycol mixture.

nent, its influence is higher. Moreover, it can be noted that the efficiency values for each component are quite different, ranging from 35% up to 90%. This range agrees with the profile calculated by O’Connell.8 O’Connell8 compared the calculated efficiency profiles with data from commercial and laboratory columns, obtaining good agreement. Jordan,28 near the end of the 1960s, already reported the necessity to consider different values for each component in multicomponent mixtures. Figures 3a and 3b also show that the plate efficiency profile is located between the component efficiency profiles, and that plate and component efficiency profiles have the same qualitative behavior along the column. Chan and Fair29 also calculated point and component efficiencies, and, although the authors did not make this comparison, it is possible to observe this behavior from their results. 4.1. Comparison of the Obtained Profiles with the Nonequilibrium Stage Model Calculations. Nowadays, the most reliable modeling to represent an actual distillation process is, certainly, the nonequilibrium stage model developed by Krishnamurthy and Taylor.19 In a nonequilibrium stage model, separate balance equations are written for each phase, including terms to represent the mass and heat transfers from one phase to another. Equilibrium is assumed only at the interface between the two phases. This modeling generates a larger number of

equations, and more physical properties are necessary. Thus, the nonequilibrium stage model, although being more realistic, has a greater computational burden. Details about the nonequilibrium stage model simulator used in this work can be obtained in Pescarini et al.30 The calculated nonequilibrium profiles were used to evaluate the Barros and Wolf efficiency correlations. For the nonequilibrium calculations, the diameter of the column for the ethanol-water-ethylene glycol mixture was considered to be equal to 0.2 m. The column design and the specifications are those presented in Figure 1. For the acetonemethanol-water case study, the diameter of the column was considered to be equal to 0.4 m. Figure 2 presents the column design and the specifications for this last case study. Figure 4 presents a comparison between the liquid and vapor mole fractions calculated using the nonequilibrium and the equilibrium stage models, taking into account the efficiency values that have been calculated by the Barros and Wolf correlations, for the ethanol-water-ethylene glycol mixture. These profiles present excellent agreement. For the acetone-methanol-water mixture, the profiles obtained with the nonequilibrium stage model were also compared with those obtained by the equilibrium model corrected using the Barros and Wolf correlations (Figure 5).

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Figure 5. Comparison of (a) vapor-phase and (b) liquid-phase profiles calculated using the nonequilibrium stage model and the corrected equilibrium stage model (Barros and Wolf correlation) for the acetone-methanol-water mixture.

Figure 6. Comparison of liquid flow rate profiles calculated using the nonequilibrium stage model for different column diameter (d) values, the equilibrium stage model, and the corrected equilibrium stage model (Barros and Wolf correlation) for the acetone-methanol-water mixture.

Also, for this nonideal system, the results calculated by both models are practically the same. The influence of the column diameter on the agreement of the calculated liquid flow rate profiles was also investigated. Figure 6 shows that, even for different column diameters, the profile using efficiency values, calculated by Barros and Wolf correlations, are closer to the nonequilibrium stage model profile than the equilibrium stage model (for a global efficiency of 100%). 4.2. Comparison of the Obtained Profiles with Experimental Data. To evaluate the simulated results, the obtained profiles were compared with the experimental data reported by Meirelles et al.18 The authors conducted some experiments in a pilot-scale plant that consisted of two columns with a diameter of 0.08 m. The main extractive column contained 60 stages (without counting the condenser and the reboiler). The ethanolwater mixture was fed in the main extractive column at plate 29 and the solvent was fed at plate 58 (counted from bottom to top). Table 2 reports some information about the experimental data. Other information and the numerical data needed for the aforementioned pilot test were taken from Meirelles,31 with regard to this experiment. However, the data were not sufficient to perform the simulation, for example, the feed and solvent

Figure 7. Comparison of experimental and calculated liquid-phase mole fractions versus the stage number for the ethanol-water-ethylene glycol system. Table 2. Experimental Data of Meirelles et al.18 property


ethanol feed composition (mole fraction) water feed composition (mole fraction) solvent-to-feed ratio reflux ratio ethanol top composition (mole fraction) solvent bottom composition (mole fraction) water bottom composition (mole fraction)

0.850 0.150 0.72 1.29 0.995 0.807 0.043

temperatures were not specified. It was assumed here that the solvent and the azeotropic stream temperatures are equal to 393.15 and 313.15 K, respectively. Meirelles31 just mentioned that the solvent was previously warmed. The reflux ratio and the distillate composition were specified as given in Table 2. Figure 7 presents the calculated and experimental profiles of the liquid-phase mole fractions versus the tray number for the ethanol-water-ethylene glycol system. The liquid-phase mole fraction calculated by Barros and Wolf correlations clearly represents the experimental results of Meirelles et al.18 quite well. 5. Concluding Remarks The great advantage of the plate and component efficiency correlations used in this work is their relation to the most


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important parameters in the extractive distillation process, taking into account heat- and mass-transfer mechanisms. The obtained efficiency profiles show that (i) there are disturbances in feed and solvent entrance positions and (ii) component efficiency profiles are different from each other, and generlly are not coincident with the plate efficiency profile. The comparison of the composition profiles calculated using the Barros and Wolf correlations with experimental data and with the nonequilibrium stage model results dictates the robustness of the developed correlations. Thus, in this work, reliable efficiency correlations that are applicable to extractive distillation processes have been shown to predict the distillation behavior more precisely, and with an inexpensive computational burden. Nomenclature c ) component number Cp ) heat capacity (kg m-1 s-1) D ) diffusivity (m/s2) k ) thermal conductivity (W m-1 K-1) M ) molecular weight (kg/kg-mol) x ) liquid molar fraction Greek Letters η ) efficiency F ) density (kg/m3) µ ) viscosity (kg m-1 s-1) Subscripts i ) tray index j ) component index m ) mixture Acknowledgment The authors thank Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPQ- 571683/1997-5 + 141893/ 2002-8 + 521011/1995-7) for financial support. Literature Cited (1) Pinto, R. T. P.; Wolf-Maciel, M. R.; Lintomen, L. Saline Extractive Distillation Processes for Ethanol Purification. Comput. Chem. Eng. 2000, 24, 1689. (2) Taylor, R.; Krishna, R.; Kooijman, H. Real-world Modeling of Distillation. Chem. Eng. Prog. 2003, 99, 28. (3) Baur, R.; Taylor, R.; Krishna, R.; Copati, J. A. Influence of Mass Transfer in Distillation of Mixtures with a Distillation Boundary. Chem. Eng. Res. Des. 1999, 77, 561. (4) Nava, J. A. O.; Krishna, R. Influence of Unequal Component Efficiencies on Trajectories during Distillation of a Homogeneous Azeotropic Mixture. Chem. Eng. Process. 2004, 43, 305. (5) Lewis, W. K. The Efficiency and Design of Rectifying Columns for Binary Mixtures. J. Ind. Eng. Chem. 1922, 4, 492. (6) Murphree, E. V. Rectifying Column Calculation. Ind. Eng. Chem. 1925, 17, 747. (7) Drickamer, H. G.; Bradford, J. R. Overall Plate Efficiency of Commercial Hydrocarbon Fractionating Columns as a Function of Viscosity. Trans. Am. Inst. Chem. Eng. 1943, 39, 319.

(8) O’Connell, H. E. Plate Efficiency of Fractionating Columns and Absorbers. Trans. Am. Inst. Chem. Eng. 1946, 42, 741. (9) Bubble Tray Design Manual: Prediction of Fractionation Efficiency; American Institute of Chemical Engineers (AIChE): New York, 1958. (10) West, F. B.; Gilbert, W. D.; Shimizu, T. Mechanism of Mass Transfer on Bubble Plates. Plate Efficiencies. Ind. Eng. Chem. 1952, 44, 2470. (11) Medina, A. G.; Ashton, N.; McDermott, C. Murphree and Vaporization Efficiencies in Multicomponent Distillation. Chem. Eng. Sci. 1978, 33, 331. (12) Holland, C. D. Computing Large Negative or Positive Values for the Murphree Efficiencies. Chem. Eng. Sci. 1980, 35, 2235. (13) Hausen, H. The Definition of the Degree of Exchange on Rectifying Plates for Binary and Ternary Mixtures. Chem. Ing. Technol. 1953, 25, 595. (14) Prado, M.; Fair J. R. Fundamental Model for the Prediction of Sieve Tray Efficiency. Ind. Eng. Chem. Res. 1990, 29, 1031. (15) Chen, G. X.; Chuang, K. T. Prediction of Point Efficiency for Sieve Trays in Distillation. Ind. Eng. Chem. Res. 1993, 32, 701. (16) Gerster, J. A. Azeotropic and Extractive Distillation. Chem. Eng. Prog. 1969, 65, 43. (17) Weiss, S.; Arlt, R. On the Modelling of Mass Transfer in Extractive Distillation. Chem. Eng. Process. 1987, 21, 107. (18) Meirelles, A.; Weiss, S.; Herfurth, H. Ethanol Dehydration by Extractive Distillation. J. Chem. Technol. Biotechnol. 1992, 56, 181. (19) Krishnamurthy, R.; Taylor, R. A Nonequilibrium Stage Model of Multicomponent Separation ProcesssPart I: Model Description and Method of Solution. AIChE J. 1985, 31, 449. (20) Barros, A. A. C. Ph.D. Thesis (in Portuguese), State University of Campinas, Chemical Engineer School, Campinas SP, Brazil, 1997. (21) Barros, A. A. C.; Wolf-Maciel, M. R. The New Efficiency Correlation to Evaluate the Extractive Distillation Processes. In 2nd Conference on Process Integration, Modeling and Optimization for Energy SaVing and Pollution Reduction, Budapest, Hungary, 1999; p 157. (22) Wolf-Maciel, M. R.; Soares, C.; Barros, A. A. C. Validations of the nonequilibrium stage model and of a new efficiency correlation for non ideal distillation process through simulated and experimental data. In 11th European Symposium on Computer Aided Process Engineering (ESCAPE11), 2001; p 321. (23) Petzold, L. R. A description of DASSL: A differential/algebraic system solver. In Proceedings of the IMACS World Congress, Montreal, Canada, 1982. (24) Reid, R.; Prausnitz, J.; Sherwood, T. K. The Properties of Gases and Liquids; McGraw-Hill: New York, 1988. (25) Perry, R. H.; Green, D. W. Perry’s Chemical Engineering Handbook; McGraw-Hill: New York, 1999. (26) Taylor, R.; Krishna, R. Multicomponent Mass Transfer; Wiley: New York, 1993. (27) Prausnitz, J.; Anderson, T.; Grens; E.; Eckert, C.; Hsieh, R.; O’Connell, J. Computer Calculations for Multicomponent Vapor-Liquid and Liquid-Liquid Equilibria; Prentice-Hall: Englewood Cliffs, NJ, 1980. (28) Jordan, D. G. Chemical Process DeVelopment; Wiley: New York, 1968. (29) Chan, H.; Fair, J. R. Prediction of Point Efficiencies on Sieve Trays. 2. Multicomponent Systems. Ind. Eng. Chem. Process Des. DeV. 1984, 23, 820. (30) Pescarini, M. H.; Barros, A. A. C.; Wolf-Maciel, M. R. Development of a Software for Simulation Processes Using a Nonequilibrium Stage Model. Comput. Chem. Eng. 1996, 20, S279. (31) Meirelles, A. J. A. Technologische Untersuchungen Zur Selektivdistillation Von Ethanol-Wasser-Gemischen. Ph.D. Thesis, Technische Hochschule Merseburg, Germany, 1987.

ReceiVed for reView October 24, 2005 ReVised manuscript receiVed May 25, 2006 Accepted June 8, 2006 IE051182E