APPLICATION OF THE MATHEMATICAL MODELS OF DISTILLATION EQUILIBRIA FOR DESIGNING

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1 AA Ars Separatoria Acta 9-10 ( ) APPLICATION OF THE MATHEMATICAL MODELS OF DISTILLATION EQUILIBRIA FOR DESIGNING THE DIFFERENTIAL DISTILLATION PROCESS Edward SOBCZAK 1, Tomasz RINGEL 1*, Robert ŁATAŚ 2 1 Department of Technology and Apparatus of Chemical and Food Industry Faculty of Technology and Chemical Engineering University of Technology and Life Sciences Seminaryjna 3, Bydgoszcz, Poland 2 Faculty of Civil Engineering Cracow University of Technology Warszawska 24, Cracow, Poland ABSTRACT This paper presents the development of mathematical models for positive and negative azeotropes in the range of the more volatile concentrations component: x [0; a z ] [a z ; 1]. The coefficients of these models, which, on their basis, were determined the relative volatility of α and the azeotrope concentration a z, defined with the high accuracy (R 2 [0,926; 0,999]) and presented them to the table for all subjects of distillation systems. Based on these relationships, the integration of the mass balance equations of the distillation differential process d(sx)=yds was carried out and determined the dependence of the distillate obtained from 1 mole of pig iron D/S 0 from the α, a z coefficients and x w, x s concentrations. Based on these dependencies, charts D/S 0 were prepared from predefined by the authors parameter t=[0,05; 0,25; 0,5; 0,75] which is the change in the concentration of pig iron in the process of concentration. Dependencies, presented for distillation balances of positive and negative azeotrope, in the tested ranges, for similar values of α, have the same mileage. The calculations were made for the 126 distillation systems. Keywords: colorants, asymmetric metal complexes, heteroleptic complexes, resistance to UV radiation, acetylacetone * Corresponding author: Tomasz Ringel; tomek@utp.edu.pl

2 Edward Sobczak, et al. Ars Separatoria Acta 9-10 ( ) INTRODUCTION Two-component distillation balances can be divided into perfect ones, which can be described by the mole fraction dependence of component A in pair (y A ) on the mole fraction of that component in the liquid (x A ): y = () (1) and imperfect ones, having either a positive azeotrope (minimum boiling point) or a negative azeotrope (maximum boiling point). Relative volatility α depends on the activity coefficients (γa, γb) and the partial pressure of components A and B (P A, P B ): α= (2) where activity coefficients are dependent on the temperature (T) of distillation processes and the composition of the liquid, according to the authors Serwiński (1982), Ciborowski (1976) and Troniewski (2006) due to: =exp (3) For distillation systems with the positive azeotrope, a linear relationship of the vapor composition was developed from the boiling liquid composition (Sobczak, 2010). According to Yu and Coull (Bandrowski J. T., 1980) for the positive and negative azeotropes, distillation equilibrium can be described by the formula: =a (4) where: a and b are constants for certain distillation equilibria. However, the equation does not satisfy the azeotropes boundary conditions: y A = x A = a z. These conditions did not meet the model developed for the positive azeotropes (Anderson and Doherty, 1984): () y = (5) where y =y A /a z i x =x A /a z Also, the coefficient of determination α is analytically impossible to determine the condition of minimizing the error σ 2 y : = y () (6) (() ) =2 (() ) =0 (7) 104

3 Ars Separatoria Acta 9-10 ( ) Application of the mathematical models... Furthermore Anderson and Doherty (1984) did not use the selection of optimal concentrations of the azeotrope (a z ) for minimizing the error R (6), which made their method unsuitable for determination of the dependencies of deformed equilibria (such as ethanol-water). DETERMINATION OF MATHEMATICAL MODELS For the distillation system with the positive azeotrope denoting the composition of the vapor and liquid x A = x and y A = y (for y>x x [0;a z ]) the authors have developed a linear composition of the vapor from the boiling liquid composition: where: = +1 =ax+b (8) α= a = (10) whereas for distillation systems with the negative azeotrope (y < x x [0;a z ]): (9) where: = +1 =ay+b (11) α = = (12) a = (13) For distillation systems with the positive azeotrope (y < x x [a z ;1]), the authors have developed a linear composition of the vapor from the boiling liquid composition: where: = (x a ) =ax+b (14) a = (15) α = = whereas for distillation systems with the negative azeotrope (y > x x [a z ;1]: where: = (16) (y a ) =ay+b (17) 105

4 Edward Sobczak, et al. Ars Separatoria Acta 9-10 ( ) a = (18) α= (19) The results of calculations α and α and a z are shown in Tables 1-4 and in the article (Sobczak et al., 2010) whereas accuracy is denoted by the coefficient R 2 : where: is linear is the mean value: = =1 ( ) ( ) Table 1. List of parameters α and a z for the positive azeotropes for y > x x [0; a z ] No. System Press. [kpa] R 2 a z α 1. Acetone Carbon tetrachloride Acetone Methanol Acetone Carbon tetrachloride Acetone Ethanol Acetone Benzene Acetone Water Ethanol Water Ethanol Water Ethanol Water Carbon disulphide Carbon tetrachloride Chloroform - Methanol Methanol - Toluene Carbon tetrachloride - Ethanol Water n-butanol Cyclohexane Toluene Cyclohexane Aniline n-hexane n-heptane n-hexane Benzene Methanol Water Ethanol - Benzene Carbon tetrachloride Toluene Carbon tetrachloride Cyclohexane Carbon tetrachloride Benzene Benzene Acetic acid Benzene n-heptane Benzene Toluene Benzene Cyclohexane Ethanol Amyl alcohol Ethanol n-butanol

5 Ars Separatoria Acta 9-10 ( ) Application of the mathematical models... Table 1 continued 30. Methanol Benzene Methanol Ethanol Cyclohexane n-heptane Benzene Aniline n-heptane Toluene Water Acetic acid Acetone Methanol Acetone Water Acetone Water Methanol n-butanol Methanol Amyl alcohol Methanol Water Methanol Water Methanol Water Ethanol Water Ethanol Water Benzene Aniline Water Acetic acid Water Acetic acid Water Acetic acid Water Acetic acid Water Acetic acid Water Acetic acid Methanol Benzene Methyl palmitate Ethyl stearate Lauric acid Myristic acid Methyl laurate Lauric acid TAME tert-amyl-ether 1-Butanol Acetone Ethyl acetate Acetone Hexane Hexane Ethyl acetate Diethyl ketone 3-Pentanol Pentanone 3-Pentanol Acetone Dimethyl carbonate Dimethyl carbonate 2-Pentanone Butanone - Dimethyl carbonate Acetone Methanol Dibutyl ether o-xylene Ethanol EMC DMC EMC EMC DEC Methanol EMC Methyl acetate Methanol n-heptane n-butyl alcohol n-heptane n-butyl alcohol

6 Edward Sobczak, et al. Ars Separatoria Acta 9-10 ( ) Table 1 continued 75. n-heptane n-butyl alcohol n-heptane n-butyl alcohol Ether n-heptane Ether n-heptane Ether n-heptane Benzene 2,2,2-Trifluoroethanol Toluene 2,2,2-Trifluoroethanol Acetone Isopropenyl acetate Butanone Isopropenyl acetate Isopropenyl acetate Acetylacetone Tetrahydrofuran (1-Methylethyl) benzene Tetrahydropyran (1-Methylethyl) benzene N, N-dimethylformamide 1-Chloro-2- ethylhexane N, N-dimethylformamide - 2-Ethyl-1-ol Toluene 1-Chloro-2-ethylhexane Toluene 2-Ethyl-1-ol Cumene N-ethylacetamide Cumene N, N-dimethylacetamide MTBE 1-Propanol DIPE 1-Propanol TAME 1-Propanol Acrylonitrile Hexane Acrylonitrile Cyclohexane Acrylonitrile Toluene Acrylonitrile Benzene Tetrachloroethylene 1-Pentanol Tetrachloroethylene 3-Methyl-1-butanol Tetrachloroethylene 2-Methyl-1-butanol Tetrachloroethylene 2-Methyl-1-butanol IBA Isooctane IBA Toluene IBA Methylcyclohexane Propanol Water METHF Cumene Acetate - Isopropylbenzene HFC-4310mee 2-Methylfuran

7 Ars Separatoria Acta 9-10 ( ) Application of the mathematical models... Table 2. List of parameters α=1/α' and a z for the negative azeotropes for y<x x [0;a z ] No. System Press. [kpa] R 2 a z α 1. Acetone Chloroform Methanol Diethylamine methylfuran THF Acetone Chloroform Water Formic acid Water Formic acid Water Formic acid Table 3. List of parameters á=1/á' and a z for the positive azeotropes for y<x x [a z ;1] No. System Press. [kpa] R 2 a z α 1. Methanol Benzene Acetone Hexane Hexane Ethyl acetate Acetone Methanol Methyl acetate Methanol n-heptane n-butyl alcohol n-heptane n-butyl alcohol n-heptane n-butyl alcohol n-heptane n-butyl alcohol N, N-Dimethylformamide 1-Chloro-2- ethylhexane Cumene N-Methylacetamide Cumene N, N-Dimethylacetamide TAME 1-Propanol Acrylonitrile Hexane Acrylonitrile Cyclohexane Acrylonitrile Benzene Tetrachloroethylene 1-Pentanol Tetrachloroethylene 3-Methyl-1-butanol Tetrachloroethylene 2-Methyl-1-butanol Tetrachloroethylene 2-Methyl-1-butanol IBA Isooctane IBA Toluene IBA Methylcyclohexane HFC-4310mee - 2-Methylfuran Ethanol MTBE Ethanol MTBE Ethanol MTBE

8 Edward Sobczak, et al. Ars Separatoria Acta 9-10 ( ) Table 4. List of parameters α and a z for the negative azeotropes for y>x x [a z ;1] No. System Press. [kpa] R 2 a z α 1. Acetone Chloroform Methanol Diethylamine Acetone Chloroform Phenol Benzyl alcohol Cyclohexanol Phenol Water Formic acid Water Formic acid Water Formic acid THE USE OF MATHEMATICAL MODELS TO CALCULATE THE DIFFERENTIAL DISTILLATION To calculate a simple distillation process, integration of the mass balance equation was carried out d(sx) = yds for x [x w ;x s ]: ln = (20) Using Equations (8, 11, 14 and 17) to the integral (20), the amount of the distillate D = S 0 -S in relation to the amount of the solution used S 0 was calculated. The following was obtained for the positive azeotrope for y > x x [0; a z ]: =1 (21) For the negative azeotrope for y <x x [0; a z ], the following was obtained: =1 (22) For the positive azeotrope for y <x x [a z ; 1], the following was obtained: =1 (23) For the negative azeotrope for y> x x [a z ; 1], the following was obtained: =1 (24) The calculations were carried out for the following parameter distillation value t = [0.05, 0.25, 0.5, 0.75], as proposed by the authors, which defines the driving force of the distillation process: 110

9 Ars Separatoria Acta 9-10 ( ) Application of the mathematical models... t= (25) where: x L is the boundary concentration level achieved by the liquid exhausted in the distillation process, where D/S 0 1 to x w x L (x w = x L ) is the zero place in the dependence from (21) to (24). Moreover, the authors defined the distribution parameter k [0; 1], defining the composition of the solution. For x [0;a z ] the concentration of the solution x s equals to: x =ka dla y > x xl = 0 (26) x = (1 k)a dla y < x xl = az (27) However, for x [a z ;1] the concentration of x s equals to: x =a + (1 k)(1 a ) dla y<x xl = 1 (28) x =a +k(1 a ) dla y>x xl = az (29) After carrying out the calculation, the value D/S 0 for the adopted range t=[0.05, 0.25, 0.5, 0.75] and k = 0.5, obtained according to D/S 0 from the parameter t; the results are shown in the graph of (1) to (5). DISCUSSION OF THE RESULTS Because the relationships D/S 0 from (21) to (24) for the approximated values α and α' are convergent (Fig. 5) the authors, using Equation (26), replaced x w by t in Equations (21) to (24): x =x +t(x x ) (30) and then, using the dependence from (26) to (29), expressed the concentration x s by k. In effect, the following was obtained: =1 t (31) for y > x α z =α, and for y < x α z =α = α -1 The above dependences are presented for k = 0.5 shown in Figures 1, 2, 3 and 4. Convergence of the dependence, shown in Figure 5 and obtained for k = 0.5, has been proved by the authors (Equations (31) and (32)). The relationship D/S 0 depends only on the parameters α and t for the concentration of the solution x s defined by the parameter distribution k, for y > x: k= dla x < a z (32) k= ( ) ( ) dla x > a z (33) 111

10 Edward Sobczak, et al. Ars Separatoria Acta 9-10 ( ) whereas for y < x: 1 k= dla x < a z (34) 1 k= ( ) dla x > a ( ) z (35) Dependencies of the equilibria (14) and (17) using the selection of parameters α and a z guarantee the highest precision of the relationships. In addition, the authors showed that using the Trouton s rule: =85 (36) relative volatility, α: α= exp ( ) (37) decreases with increasing temperatures. 1,2 1,0 0, D/S0 0,6 0,4 0,2 0,0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 t Fig. 1. The dependence D/S 0 on the parameter t for the positive azeotrope and x [0;a z ], calculated from Equation (21): 1 2-butanone-Isopropenyl acetate (α = 1.62), 2 Isopropenyl acetate-acetylacetone (α = 4.16), 3 1-Propanol-Water (α = 34.25) 1,2 1,0 0, D/S 0 0,6 0,4 0,2 0,0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 t Fig. 2. The dependence D/S 0 on the parameter t for the negative azeotrope and x [0;a z ], calculated from Equation (22): 1 Acetone Chloroform (α'= 1.51), 2 Methanol Diethylamine (α' = 2.2), 3 Water Formic acid (α'= 3.28) 112

11 Ars Separatoria Acta 9-10 ( ) Application of the mathematical models... 1,2 1,0 0, D/S0 0,6 0,4 0,2 0,0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 t Fig. 3. The dependence D/S 0 on the parameter t for the positive azeotrope and x [a z ;1], calculated from Equation (23): 1 Acetone Methanol (α'= 1.31), 2 n-heptane n-butyl alcohol (α'= 4.33), 3 Acrylonitrile Hexane (α' = 13.21) 1,2 1,0 0, D/S 0 0,6 0,4 0,2 0,0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 t Fig. 4. The dependence D/S 0 on the parameter t for the negative azeotrope and x [a z ;1], calculated from Equation (24): 1 Acetone Chloroform (α = 1.84), 2 Cyclohexanol Phenol (α = 4.49), 3 Phenol Benzyl alcohol (α = 5.87) 1,2 1,0 0, D/S0 0,6 0,4 0,2 0,0 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 t Fig. 5. The dependence D/S 0 on the differential distillation parameter t for the positive and negative azeotropes, and x [0;a z ] [a z ;1] and similar values: 1 Acetone Ethyl acetate α = 2.22; calculated from the Equation (21), 2 Methanol Diethylamine α'= 2.2; calculated according to (22), 3 N, N-Dimethylformamide 1-Chloro-2-ethylhexane α' = 2.25; calculated from the formula (23), 4 Water Formic acid α = 2.281; calculated from Equation (24) 113

12 Edward Sobczak, et al. Ars Separatoria Acta 9-10 ( ) CONCLUSION Mathematical models, obtained for positive and negative azeotropes, were obtained with high accuracies R 2 = Relative volatility α for y > x and its inverse α = 1/α for y < x, which are present in these models, both for the positive and negative azeotropes, are consistent with experimental data, as evidenced by identical dependences D/S 0 on the parameters t defining the relative distillation driving force and, for a given k, defining the composition of the solution. The dependences presented for the distillation balance with the positive and negative azeotrope in the ranges x [0;a z ] and x [a z ;1] for the approximated values of α for y > x and α = 1/α for y < x are identical Figure 5 and Equation (31) indicate that the coefficients α for y > x and α y < x are the corresponding parameters. The authors showed that using the Trouton s rule, the relative volatility, α, decreases with increasing temperatures. REFERENCES [1] Orchilles A. V., Miguel P. J., Vercher E., Martínez-Andreu A., Ionic Liquids as Entrainers in Extractive Distillation: Isobaric Vapor-Liquid Equilibria for Acetone + Methanol + 1-Ethyl-3-methylimidazolium Trifluoromethanesulfonate. J. Chem. Eng. Data 52, [2] Arce A., Rodil E., Soto A., Experimental Determination of the Vapor- Liquid Equilibrium at kpa of the Ternary System 1-Butanol + Methanol + TAME. J. Chem. Eng. Data 45, [3] Pereiro A.B., Rodríguez A., Canosa J., Tojo J., Measurement of the Isobaric Vapor-Liquid Equilibria of Dimethyl Carbonate with Acetone, 2-Butanone, and 2- Pentanone at kpa and Density and Speed of Sound at K. J. Chem. Eng. Data 50, [4] Gomez-Marigliano A. C., Arce A., Rodil E., Soto A., Isobaric Vapor- Liquid Equilibria at kpa and Densities, Speeds of Sound, and Refractive Indices at K for MTBE or DIPE or TAME + 1-Propanol Binary Systems. J. Chem. Eng. Data 55, [5] Dejoz A., González-Alfaro V., Llopis F.J., Miguel P.J., Vázquez M.I., Vapor-Liquid Equilibrium of Binary Mixtures of Tetrachloroethylene with 1-Pentanol, 3-Methyl-1-butanol, and 2-Methyl-1-butanol. J. Chem. Eng. Data 44, [6] Mejia A., Segura H., Cartes M., Vapor-Liquid Equilibria and Interfacial Tensions of the System Ethanol + 2-Methoxy-2-methylpropane. J. Chem. Eng. Data 55, [7] Aucejo A., Gabaldón C., Loras S., Marzal P., Sanchotello M., Phase Equilibria in the Binary and Ternary Systems Composed of Diethyl Ketone, 2-Pentanone, and 3-Pentanol at kpa. J. Chem. Eng. Data 48,

13 Ars Separatoria Acta 9-10 ( ) Application of the mathematical models... [8] Aucejo A., Loras S., Muñoz R., Wisniak J., Segura H., Phase Equilibria and Multiple Azeotropy in the Associating System Methanol + Diethylamine. J. Chem. Eng. Data 42, [9] Atik Z., Experimental Isobaric Vapor-Liquid Equilibria of Binary Mixtures of 2,2,2-Trifluoroethanol with Benzene or Toluene. J. Chem. Eng. Data 52, [10] Gill B. K., Rattan V. K., Kapoor S., Experimental Isobaric Vapor-Liquid Equilibrium Data for Binary Mixtures of Cyclic Ethers with (1-Methylethyl)benzene. J. Chem. Eng. Data 53, [11] Gill B. K., Rattan V. K., Kapoor S., Vapor-Liquid Equilibrium Data for N-Methylacetamide and N,N-Dimethylacetamide with Cumene at 97.3 kpa. J. Chem. Eng. Data 54, [12] Bandrowski, J., Palica, M Materiały pomocnicze co ćwiczeń i projektów z inżynierii chemicznej. Gliwice: Wydawnictwo Politechniki Śląskiej. [13] Ciborowski, J Inżynieria chemiczna. Inżynieria procesowa. WNT Warszawa. [14] Graczova E., Steltenpohl P., Vapor-Liquid Equilibria of Binary Systems Comprising 1-Chloro-2-ethylhexane and 2-Ethyl-1-hexanol. J. Chem. Eng. Data 53, [15] Vercher E., Rojo F.J., Martínez-Andreu A., Isobaric Vapor-Liquid Equilibria for 1-Propanol + Water + Calcium Nitrate. J. Chem. Eng. Data 44, [16] Haan, A. B., Bölts R., Gmehling J., Vapor-Liquid Equilibria and Excess Enthalpies for Binary Mixtures of Acrylonitrile with Hexane, Cyclohexane, Benzene, Toluene, 2-Butanone, and Acetonitrile. J. Chem. Eng. Data 41, [17] Monick J.A., Allen H.D., Marlies C.J., Vapor-Liquid Equilibrium Data for Fatty Acids and Fatty Methyl Esters at Low Pressures. Journal of the American Oil Chemists' Society 23(6), [18] Acosta J., Arce A., Martínez-Ageitos J., Rodil E., Soto A., Vapor-Liquid Equilibrium of the Ternary System Ethyl Acetate + Hexane + Acetone at kpa. J. Chem. Eng. Data 47, [19] Cai J., Cui X., Zhang Y., Li R., Feng T., Vapor-Liquid Equilibrium and Liquid-Liquid Equilibrium of Methyl Acetate + Methanol + 1-Ethyl-3- methylimidazolium Acetate. J. Chem. Eng. Data 56(2), [20] Korea Thermophysical Properties Data Bank. Pobrano Styczeń 12, 2011 z lokalizacji Chemical Engineering Research Information Center: research/kdb/hcvle/hcvle.php [21] Vijayaraghavan S.V., Deshpande P.K., Kuloor N.R., Vapor-liquid Equilibrium Data for the Systems Diisopropyl Ether-n-Heptane and Diisopropyl Ether-Carbon Tetrachloride at Medium Pressures 12(1), [22] Thiede S., Horstmann S., Gmehling J., Vapor-Liquid Equilibrium Data, Excess Enthalpy Data, and Azeotropic Data for the Binary System Dibutyl Ether + o-xylene. J. Chem. Eng. Data 55, [23] Kapoor S., Gill B.K., Rattan V.K., Isobaric Vapor-Liquid Equilibrium of Binary Mixture of Methyl Acetate with Isopropylbenzene at 97.3 kpa. World Academy of Science, Engineering and Technology 47, [24] Serwiński M., Tabele liczbowe z Inżynierii Chemicznej. Politechnika Łódzka. 115

14 Edward Sobczak, et al. Ars Separatoria Acta 9-10 ( ) [25] Serwiński M., Zasady inżynierii chemicznej i procesowej. WNT Warszawa. [26] Sobczak E.R., Matematyczny model równowagi destylacyjnej i jego zastosowanie do projektowania procesów destylacji i rektyfikacji. Inżynieria i aparatura chemiczna 49(41), [27] Loras S., Aucejo A., Montón J.B., Wisniak J., Segura H., (2002). Phase Equilibria for 1,1,1,2,3,4,4,5,5,5-Decafluoropentane + 2-Methylfuran, 2-Methylfuran + Oxolane, and 1,1,1,2,3,4,4,5,5,5-Decafluoropentane + 2-Methylfuran + Oxolane at 35 kpa. J. Chem. Eng. Data 47, [28] Troniewski L. (red.), (2006). Tablice do obliczeń procesowych. Politechnika Opolska. [29] Rattan V.K., Gill B.K., Kapoor S., Isobaric Vapor-Liquid Equilibrium Data for Binary Mixture of 2-Methyltetrahydrofuran and Cumene. World Academy of Science, Engineering and Technology 47, [30] Martınez-Soria V., Peña M. P., Montón J.B., Vapor-Liquid Equilibria for the Binary Systems Isobutyl Alcohol + Toluene, + Isooctane, and + Methylcyclohexane at kpa. J. Chem. Eng. Data 44, [31] Vijayaraghavan V., Deshpande P.K., Kuloor N.R., Vapor-liquid Equilibrium Data for the System n-heptane-n-butyl Alcohol at Medium Pressures. J. Chem. Eng. Data 12(1), [32] Cui W., Zhu J., Liu W., Wu B., Chen K., (2008). Isobaric Vapor Liquid Equilibria for Binary Systems of Acetone + Isopropenyl Acetate, 2-Butanone + Isopropenyl Acetate, and Isopropenyl Acetate + Acetylacetone at kpa. J. Chem. Eng. Data 53, [33] Zhang X., Zuo J., Jian Ch., Experimental Isobaric Vapor-Liquid Equilibrium for Binary Systems of Ethyl Methyl Carbonate + Methanol, + Ethanol, + Dimethyl Carbonate, or + Diethyl Carbonate at kpa. J. Chem. Eng. Data 55,

Introduction (1) where ij denotes the interaction energy due to attractive force between i and j molecules and given by; (2)

Introduction (1) where ij denotes the interaction energy due to attractive force between i and j molecules and given by; (2) (7)7 Prediction of Vapor-Liquid Equilibria of Binary Systems Consisting of Homogeneous Components by Using Wilson Equation with Parameters Estimated from Pure-Component Properties Shigetoshi KOBUCHI, Kei