Microsoft Word - 476hernandez.docx CHEMICAL ENGINEERINGTRANSACTIONS VOL. 43, 2015 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Chief Editors:SauroPierucci, JiříJ. Klemeš Copyright © 2015, AIDIC ServiziS.r.l., ISBN 978-88-95608-34-1; ISSN 2283-9216 Modeling of Solid-Liquid Equilibria Using a Group Based NRTL Equation Youcef Moudjari, Wahida Louaer, Abdeslam-Hassen Meniai* Laboratoire de l’Ingénierie des Procédés de l’Environnement (LIPE), Faculté du Génie des Procédés Pharmaceutiques Université Constantine 3, BP B60 Centre Djefal Amar Khroub 25100 Algeria meniai@yahoo.fr The NRTL equation has shown great capabilities in predicting phase equilibria data. However its major drawback is the non-availability of the required molecular interaction parameters. Consequently in the present study a new approach is proposed based on the introduction of the group contribution concept into the original NRTL equation to lead to the proposed Group Contribution NRTL model (GC-NRTL). Similarly to UNIFAC the molecular activity coefficient is made of two parts: the first one is the combinatorial contribution which deals with differences in group sizes and shapes, and the second one is the residual contribution and is concerned with the different functional group interactions and is estimated by the proposed GC-NRTL model. Group interaction parameters for the NRTL equation are calculated by minimizing an objective function defined in terms of the sum of the squared differences between the calculated values and the experimental ones reported in the literature. As a first assessment, the GC-NRTL model was tested with a number of hydrocarbon binary solid - liquid systems mainly involving current functional groups like CH2, CH3, (CH2)cyclic, C, CH, ACH, ACCH3, ACCH2, AC and CH=CH. The agreement between the predicted results and the experimental values was very encouraging and much better than the case when using the UNIFAC model. However the group interaction parameters matrix should be completed further including a greater number of functional groups. 1. Introduction Thermodynamic models such as UNIQUAC and NRTL (Non Random Two liquids) have shown great capabilities in predicting the activity coefficient which is an important parameter required in any phase equilibria calculation. However their major drawback is the fact that they involve molecular interaction parameters which are not always available, hence excluding their use for a large number of chemical systems. The adopted approach to solve this problem in the case of UNIQUAC (Universal Quasichemical Activity Coefficient) (Abrams and Prausnitz, 1975) was to introduce the group contribution concept to give the well- known UNIFAC model (UNIQUAC Functional Activity Coefficient) (Fredenslund et al., 1975). Therefore in the present work the same group contribution approach is again applied and introduced into the initial NRTL equation leading to the GC-NRTL (Group Contribution NRTL) model, contrarily to the previous approach presented by the same authors where a completely different way for property additivity was proposed (Bouneb and Meniai, 2013 ). For GC-NRTL, similarly to UNIFAC, the molecular activity coefficient is made of two parts: the first one concerns the contribution due to differences in molecule sizes and shapes and is estimated by using the athermal Staverman- Guggenheim equation Staverman (1950) and Guggenheim (1952) whereas the second one deals with the contribution due to molecular interactions. DOI: 10.3303/CET1543312 Please cite this article as: Moudjari Y., Louaer W., Meniai A.H., 2015, Modeling of solid-liquid equilibria using a group contribution based nrtl equation, Chemical Engineering Transactions, 43, 1867-1872 DOI: 10.3303/CET1543312 1867 In fact the fundamental idea of solution of a group model is to use existing phase equilibrium data for predicting phase equilibria for systems for which no experimental data are available. The calculation of thermodynamic properties using molecular models requires information about all binary combinations in the mixture. Because of the large number of compounds which are relevant in the chemical industry, some binary parameters are often unknown. Group contribution methods such as ASOG (Derr and Deal, 1969), UNIFAC (Fredenslund et al, 1975) or modified UNIFAC (Larsen et al) offer a considerable simplification because the number of relevant groups can be restricted to a much more reduced size. The advantage of any group contribution method for the calculation of fluid phase equilibria key parameters is mainly due its ability to predict mixture properties for which no experimental data exist at all. The group contribution concept is based on the assumption that a chemical compound mixture can be treated, accurately enough as a mixture of functional groups making up these compounds. A priori GC-NRTL group interaction parameters were estimated through the minimization of objective functions expressed in terms of the activity coefficients or/and the molar fractions of the considered system components. As usual the objective function is expressed as the sum of the squared deviations between the calculated and the experimental values, generally reported in the literature. The minimization method was based on the Nelder-Mead Simplex algorithm (Nelder and Mead, 1965). The GC-NRTL model was tested considering a great number of binary solute-solvent liquid systems mainly involving current functional groups like CH CH2, CH3, (CH2)cyclic, C, CH, ACH, ACCH3, ACCH2, AC and CH=CH. The agreement between the predicted results by means of the GC-NRTL and the experimental phase equilibrium data was encouraging and the interaction parameters table should be completed to include a greater number of functional groups. 2. Theoretical aspects GC-NRTL model like UNIFAC is mainly based on the assumption that the contribution to the activity coefficient of compound can be separated into two parts, namely, a combinatorial, entropic contribution due to differences in molecular size and shape and a residual, enthalpy contribution due primarily to differences in intermolecular forces. In GC-NRTL, the Staverman- Guggenheim equation was used for the combinatorial part of the activity coefficient and the NRTL equation was used for the determination of group residual activity coefficients. The activity coefficient of compound i can then be calculated using the following equation: = + (1) with C and R denoting combinatorial and residual, respectively. The combinatorial part of the activity coefficient depends only on the number size groups, in the various molecules that make the mixture. The Staverman- Guggenheim equation was used for the combinatorial part as: = Φ + Φ + − Φ ∑ (2a) = ( − ) − ( − 1) (2b) = ∑ (2c) Φ = ∑ (2d) In these equations, xi is the mole fraction of compound i , Ө is the area fraction, and ɸ is the segment fraction, wich is similar to the volume fraction. Pure component parameters ri and qi are measures of molecular van der Waals volumes and molecular surface areas. Parameters ri and qi are calculated as the sum of group volume and area parameters and . ri = ∑ ( ) (3a) qi = ∑ ( ) (3b) Where ( ) , always an integer, is the number of groups of type k in molecule i. group parameters Rk and Qk are obtained from the van der Waals group volume and surface area and given by Bondi [16] = , (4a) 1868 = , . (4b) The residual part of the activity coefficients are calculated from the basic NRTL equation and the molecular activity coefficients term is expressed as: lnγi = ∑ ( ) [ln Γ – ln Γ ( ) ] (5) With γi the molecular activity coefficient, ( ) the number of group (k) in the molecule (i), Γ the activity coefficient of group (k) in the mixture, Γ ( ) the activity coefficient of group (k) in reference solution containing only molecules of type (i). The term ln Γ ( ) is necessary to attain the normalization that molecular activity coefficient γi becomes unity as xi 1. The activity coefficient of group (k) is calculated as follows: lnΓ = ∑∑ + ∑ ∑ − ∑∑ (6a) With xk is the mole fraction of group (k) in the mixture is calculated as follows: XK= ∑ ( )∗∑ ∑ ( )∗ (6b) xi is the mole fraction of molecule (i). Glk = (− ) (6c) is the interaction parameters between goup (l) and (k) is expressed as follows: = = (6d) with g the interaction energies between the corresponding groups, a the group interaction parameters between the corresponding groups. Note that a has units in Kelvins a ≠ a . 3. Interaction parameters estimation In this work, the required group interaction parameters were determined considering, a priori, only hydrocarbons. Experimental solid-liquid equilibrium data reported in the literature for different systems at different temperatures were used for this step. The binary group interaction parameters required for the GC-NRTL model were retrieved in the present work using, as mentioned above, the Nelder-Mead method for the minimization of the following objective function (Fobj) defined as the sum of the squared deviations between the experimental and calculated mole fractions: F = ∑ ( ) ( ) (7) with N denoting the number of data points. The obtained binary group and molecular interaction parameters for the GC-NRTL models are presented in the following table. Table 1: Binary group interaction parameters for GC-NRTL model CH3 CH2 CH C ACH AC ACCH3 ACCH2 (CH2)cycl CH=CH CH3 0.00 217.95 757.72 -153.57 -10.17 -62.40 -0.33 -1180.85 153.78 1017.48 CH2 119.29 0 .00 107.86 4159.72 -1873.86 51.28 8908.20 -890.08 164.42 633.13 CH -226.74 -102.80 0.00 -2551.55 660.72 98.88 1581.83 1611.91 -142.95 -837.60 C 1072.13 -189.55 -111.53 0.00 -61.91 -702.54 432.14 -779.35 1113.39 -789.66 ACH 128.87 265.08 1435.34 1087.03 0 .00 -824.88 -4377.34 4334.31 -293.55 1833.66 AC 441.46 -92.84 -93.94 1458.46 -348.89 0 .00 301.81 -230.20 320.64 -1430.07 ACCH3 1812.02 1241.34 499.62 -519.67 3943.68 -567.70 0 .00 336.00 15.81 1729.90 ACCH2 -917.34 -67.01 231.77 2578.85 771.80 294.60 332.84 0 .00 -1676.07 -397.32 (CH2)cycl -165.32 -321.69 -102.91 -515.64 -242.38 -675.75 -27.07 132.94 0 .00 -1579.73 CH=CH -345.37 -363.33 -660.67 -168.81 -1416.33 446.44 -2105.64 5483.26 -135.43 0.00 1869 Table 2 shows the solubility data results obtained by means of GC-NRTL and UNIFAC equations as well as the experimental data reported by Acree Jr (2013) for the considered systems presented in this work. Table 2: Comparison of experimental and calculated solubilities SOLUTE SOLVENT T (K) Xexp XGC -NRTL XUNIFAC Cyclooctane 0.2194 0,2206 0.22530 Biphenyl 2,2,4 Trimethylpentane 0.1094 0,1095 0.21330 Nonane 298.15 0.1551 0,1519 0.17170 Decane 0.1636 0,1656 0.17430 Hexadecane 0.2151 0,2278 0.19810 Biphenyl Octadecane 298.57 0.266 0,2540 0.21500 Hexane 0.05192 0,0485 0.07896 Cyclohexane 0.07043 0,0859 0.08235 Acenaphtane Octane 0.06826 0,0697 0.08227 Nonane 298.15 0.0721 0,0762 0.08418 Heptane 0.06075 0,0606 0.08034 Hexadecane 0.1065 0,1043 0.10430 Cyclooctane 0.09739 0,0835 0.08655 Hexane 0.03189 0,0277 0.07018 Heptane 0.03888 0,0373 0.07072 Octane 0.04443 0,0449 0.07177 Phenantrene Nonane 0.04785 0,0508 0.07297 Decane 0.05531 0,0569 0.07531 Undecane 0.0598 0,0614 0.07730 Dodecane 298.15 0.06348 0,0653 0.07944 Hexadecane 0.07972 0,0784 0.08918 Cyclohexane 0.03648 0,0441 0.07138 Methylcyclohexane 0.04572 0,0460 0.14290 Cyclooctane 0.06002 0,0473 0.07507 2,2,4 Trimethylpentane 0.02486 0,0247 0.11880 298.15 0.1901 0,1928 0.27420 Phenantrene methylbenzene 308.15 0.2486 0,2434 0.33930 318.15 0.3200 0,3157 0.4145 328.15 0.4000 0,4086 0.5000 Nonane 0.002085 0,0015 0.00253 Decane 0.002345 0,0017 0.0026 Undecane 0.002585 0,0018 0.00268 Antharcene Dodecane 298.15 0.0028 0,0020 0.00276 1,2 Dimethylbenzene 0.008458 0,0093 0.02369 1,3dimethylbenzene 0.007056 0,0093 0.02380 298.20 0.001187 0.0009 0.00448 Antharcene 2,2,4 Trimethylpentane 308.20 0.001515 0,0013 0.00669 318.20 0.001768 0,0019 0.00975 290.25 0.05948 0,0583 0.09454 294.30 0.07054 0,0686 0.1082 Naphtalene Heptane 297.85 0.07982 0,0785 0.1214 301.37 0.08893 0,0895 0.1357 303.04 0.09771 0,0964 0.1439 309.63 0.12340 0,1245 0.1780 312.60 0.14120 0,1413 0.1971 291.35 0.23950 0,2442 0.2347 301.45 0.31030 0,3148 0.3062 Naphtalene Methylbenzene 308.53 0.37390 0,3727 0.3653 318.55 0.47170 0,4685 0.4630 327.55 0.58590 0,5695 0.5660 299.42 0.29180 0,2946 0.2929 1870 Table 2: Comparison of experimental and calculated solubilities (cont.d) SOLUTE SOLVENT T (K) Xexp XGC -NRTL XUNIFAC 309.84 0.37150 0,3800 0.3789 Naphtalene Dimethymbenzene (Mix) 319.25 0.47910 0,4719 0.4712 330.46 0.60860 0,6027 0.6026 337.97 0.71960 0,7045 0.7046 Hexane 0.00960 0,0092 0.09516 Cyclohexane 0.01374 0,0141 0.09624 Trans- Heptane 0.01085 0,0110 0.09551 Stilbene Octane 298.15 0.01241 0,0129 0.09717 Ethyl Benzene 0.05331 0,0533 0.2653 Benzene 0.06232 0,0624 0.3340 The results are assessed by calculating the average absolute relative deviation (AAD) defined as follows: AAD = ∑ ( ) ( ) (8) Where n is the number of experimental points; ( ) is the solubility calculated using GC-NRTL and UNIFAC equation; ( )is the experimental solubility reported in literature Acree Jr (2013). 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 x e xp x calc Comparison of experimental solubility values and calculated results by GC-NRTL (a) 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 x e xp x calc Comparison of experimental solubility values and calculated results by UNIFAC (b) Figure 1: Assessment of the models a) GC-NRTL; b) UNIFAC The obtained AAD values were 0.001432787 and 0.06656557 for GC-NRTL and UNIFAC models, respectively. This comparison is also shown by Figures 1a and b where clearly the predicted results using the proposed model i.e. GC-NRTL are in a very good agreement with the experimental values, contrarily to UNIFAC where important deviations are shown. 0,06 0,08 0,10 0,12 0,14 290 295 300 305 310 315 T (K ) Naphtalene solubility Naphtalene-n-Heptane Calculated using GC-NRTL Experimental 0,03 0,06 0,09 0,12 0,15 0,18 0,21 290 295 300 305 310 315 T (K ) Naphtalene solubility Naphtalene-n-Heptane Calculated using UNIFAC Experimental (a) (b) Figure 2: Solid-liquid equilibrium for Naphthalene- n-Heptane system by a) GC-NRTL and b) UNIFAC models 1871 For a better representation of solid-liquid phase equilibrium, the corresponding diagrams are shown for a chosen system, namely Naphthalene in n-Heptane, for which experimental data are reported in the literature (Cui et al, 2009). Figures 2a & b show the comparisons of the obtained results by means of GC-NRTL and UNIFAC with the experimental values. Once more the GC-NRTL results were in a very good agreement with the experimental values, contrarily to UNIFAC. 4. Conclusion The GC-NRTL model was tested for different binary solid - liquid systems mainly involving current hydrocarbon functional groups likeCH2, CH3, (CH2)cycl , C, ACH, ACCH3, ACCH2, AC and CH=CH. The agreement between the predicted results by means of the GC-NRTL and the experimental phase equilibrium data is encouraging and the interaction parameters table should be completed to include a greater number of different functional groups. Generally the models using molecular interaction parameters lead to more accurate results compared to models using group interaction parameters like the GC-NRTL or UNIFAC. This is perhaps due to the fact that any group contribution approach is assumed approximately to be additive. However the non-availability of the required interaction parameters is also a serious limitation, justifying the introduction of the group contribution concept into the initial equation using molecular interaction parameters. Also great efforts should be made in improving the minimization techniques very often local minima are reached rather than the global minimum, inducing errors. References Abrams D. S., Prausnitz J. M., 1975, Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy of Partly or Completely Miscible Systems, AIChE Journal, Vol. 21, No. 1, 1975, p 114-128 Acree Jr W. E., 2013, IUPAC-NIST Solubility Data Series. 98, Solubility of Polycyclic Aromatic Hydrocarbons in Pure and Organic Solvent Mixtures—Revised and Updated. Part 3. Neat Organic Solvents, Journal of Physical and Chemical Reference Data 42, 013105; DOI: 10.1063/1.4775402. 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