CHEMICAL ENGINEERING TRANSACTIONS VOL. 57, 2017 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Sauro Pierucci, Jiří Jaromír Klemeš, Laura Piazza, Serafim Bakalis Copyright © 2017, AIDIC Servizi S.r.l. ISBN 978-88-95608- 48-8; ISSN 2283-9216 Modeling Vapor Liquid Equilibrium of Binary and Ternary Systems of CO2 + Hydrocarbons at High-Pressure Conditions Dennys C. Silvaa, Reginaldo Guirardellob, Antonio C. D. Freitas*b aChemical Engineering Coordination, Exact Sciences and Technology Center, Federal University of Maranhão (UFMA), Av. dos Portugueses, 1966, Bacanga, 65080-805, São Luís-MA, Brazil. bSchool of Chemical Engineering, University of Campinas (UNICAMP), Av. Albert Einstein 500, 13083-852, Campinas-SP, Brazil acd.freitas@ufma.br In this work, binary and ternary systems composed by hydrocarbons and CO2 in liquid-vapor equilibrium conditions (LV) were thermodynamically modeled using Peng-Robinson (PR) and Patel-Teja (PT) equations of state (EoS) in combination with van der Waals mixing rule with two adjustable parameters (vdW -2, kij and lij). The model was formulated as a minimization of the Mean Absolute Deviation (%AAD) between the predicted and experimental values for liquid and vapor phases using the simplex algorithm, through the software Phase- Equilibrium 2000 (PE-2000). Low deviations, %AAD = 2.33% for PR EoS and %AAD = 3.06% for PT EoS were observed for binary systems in the evaluation of 160 experimental points (EP). Ternary systems were modeled with low deviations too, %AAD = 1.12% for EoS PR and %AAD = 1.18% for EoS PT were observed in the evaluation of 69 EP. Both tested EoS proved to be useful to represent LV equilibrium in this kind of system. 1. Introduction The use of CO2 in the chemical industry has increased in recent years due to its properties as a solvent, especially at supercritical conditions. Accurate data for liquid-vapor equilibrium for systems composed by carbon dioxide, hydrocarbons, ethanol and water are important for designing separation processes and synthesis reactions from syngas, such as Fischer-Tropsch synthesis (Freitas and Guirardello, 2015) and are useful for supercritical fluids technology, applications can be found in chemical, petrochemical and food industry (Gallegos et al., 2006). The prediction and description of thermodynamic properties and phase equilibria of multicomponent mixtures remains a major challenge in the scientific community at the same time becomes a necessity in the industrial environment. Despite the wide range of applications, the use equations of EoS to model systems composed by CO2 and hydrocarbons are very rare in the literature. Studies that determine the parameters to represent liquid vapor equilibrium for this kind of systems using cubic EoS are scarce, however, these kind of study are essential for some applications, including the use of these equations in modeling and simulation of chemical processes with greater confidence. In this context, the main objective of this paper was the use of Peng-Robinson (PR) and Patel-Teja (PT) equations of state in combination with van der Waals mixing rule with 2 adjustable parameters (vdW -2, kij and lij) to determine binary interaction parameters for systems composed by CO2 and hydrocarbons in binary and ternary systems. For this, the software Phase Equilibrium 2000 (PE-2000) was used in combination with simplex Nelder-Mead algorithm to minimize an objective function based on average deviations observed between molar fractions of liquid and vapor phases. 2. Methodology 2.1. Peng-Robinson EoS (PR) The expression for Peng-Robinson (PR) EoS (Peng and Robinson, 1976) is represented by: DOI: 10.3303/CET1757247 Please cite this article as: Silva D.C., Guirardello R., Freitas A.C.D., 2017, Modeling vapor liquid equilibrium of binary and ternary systems of co2 + hydrocarbons at high-pressure conditions, Chemical Engineering Transactions, 57, 1477-1482 DOI: 10.3303/CET1757247 1477 𝑃 = 𝑅𝑇 𝑉 − 𝑏 − 𝛼(𝑇) 𝑉(𝑉 + 𝑏) + 𝑏(𝑉 − 𝑏) (1) Here, P is the pressure, V is the molar volume, T is the temperature, α(T) is the temperature-dependent attractive parameter, b is the co-volume and R is the ideal gas constant. The co-volume parameter is temperature independent and is calculated using the critical properties of the pure component as follows: 𝑏 = 0,0778 𝑅𝑇𝐶 𝑃𝐶 (2) The attractive parameter, α(T), is calculated by: 𝛼 = 0,45724 (𝑅𝑇𝐶 )² 𝑃𝐶 [1 + (0,37464 + 1,5422𝜔 − 0,26992𝜔2)(1 − 𝑇𝑟 0,5 )] 2 (3) 2.2. Patel-Teja EoS (PT) The expression for Patel and Teja (PT) EoS (Patel and Teja, 1982) is represented by: 𝑃 = 𝑅𝑇 𝑉 − 𝑏 − 𝑎 𝑉(𝑉 + 𝑏) + 𝑐(𝑉 − 𝑏) (4) The variables are the same explained above for PR EoS. The parameters a, b and c are: 𝑎 = 0,66121 − 0,761057𝑍𝐶 𝑅2𝑇𝐶 2 𝑃𝐶 2 [1 + (0,46283 + 3,58230𝜔𝑍𝐶 + 8,19417(𝜔𝑍𝐶 ) 2)(1 − √𝑇𝑟 )] 2 (5) 𝑏 = 0,02207 − 0,20868𝑍𝐶 𝑅𝑇𝐶 𝑃𝐶 (6) 𝑐 = 0,57765 − 1,87080𝑍𝐶 𝑅𝑇𝐶 𝑃𝐶 (7) 2.3 van der Waals mixing rule PR and PT EoS were applied in combination with the van der Waals mixing rule with two adjustable parameters (kij and lij) (vdW-2). The 𝑎𝑚𝑖𝑥 and 𝑏𝑚𝑖𝑥 parameters are presented in equations (8) and (9) respectively. 𝑎𝑚𝑖𝑥 = ∑ ∑ 𝑥𝑖 𝑥𝑗 (𝛼𝑖 𝛼𝑗 ) 0,5 (1 − 𝑘𝑖𝑗 ) (8) 𝑗𝑖 𝑏𝑚𝑖𝑥 = ∑ ∑ 𝑥𝑖 𝑥𝑗 1 2 (𝑏𝑖 + 𝑏𝑗 )(1 − 𝑙𝑖𝑗 ) (9) 𝑗𝑖 with kij = kji and lij = lji. 2.4. PE 2000 software The software Phase Equilibrium 2000 (PE-2000), developed by Brunner and coworkers (Pfohl and Petkov, 2000), uses the Simplex modified algorithm to regression of interaction parameters, minimizing the objective function of the absolute average deviation (%AAD) to molar fractions of liquid and vapor phase, as shown in: %𝐴𝐴𝐷 = 100 𝑁𝑃 (∑|𝑥𝑖 𝑒𝑥𝑝 − 𝑥𝑖 𝑝𝑟𝑒𝑑 | + ∑|𝑦𝑖 𝑒𝑥𝑝 − 𝑦𝑖 𝑝𝑟𝑒𝑑 | 𝑁 𝑖=1 𝑁 𝑖=1 ) (10) Here, xi exp is the experimental liquid mole fraction data; xi pred is the predicted value; yi exp is the experimental vapor mole fraction data; yi pred is the predicted value and NP is the number of experimental data points used in the regression procedure. Several initial estimates were used to avoid the local minima in the regression in order to guarantee that the obtained values ate the global minimum to evaluated parameters (kij and lij). The software PE 2000 has already been used in other studies with excellent results to correlate and predict phase behavior for different systems under conditions of liquid-liquid-vapor and liquid-vapor equilibrium (Freitas et al., 2013). 2.5. CO2+hydrocarbons systems The 7 binary and 2 ternary systems studied in this work and the source of experimental data are presented in Table 1. The characteristics of the studied systems, including pressure, temperature and composition range of the experimental data are presented in this table as well. The experimental data were taken from the literature as presented in Table 1 for all studied binary and ternary systems. The critical properties of all compounds studied in this work are obtained in literature as well (Poling et al., 2001). 1478 Table 1: Identification of binary and ternary systems studied in this work. System EP Range of data References T (K) P (bar) XCO2 Binary CO2 + CH4 24 240.35-261.25 18.3-82.4 0.0193-0.4206 Nasir et al. (2015) Systems CO2 + C3H8 20 252.95-273.15 3.3-33.9 0.013-0.953 Nagahama et al. (1969) CO2 + C4H10 13 273.15 2.4-31.5 0.03-0.911 Nagahama et al. (1969) CO2 + C8H18 28 322.39-372.53 20.13-137.72 0.1422-0.8892 Gallegos et al. (2006) CO2 + C10H22 29 319.11-372.94 32.41-160.6 0.2146-0.9731 Gallegos et al. (2006) CO2 + C5H10 27 323.15-344.65 6.3-96.2 0.0442-0.855 Sima et al. (2016) CO2 + C6H12 19 323.15-353.15 10-110.1 0.108-0.7754 Sima et al. (2016) Ternary Systems CO2 + CH4 + C3H8 54 230-270 8-80 0.0021-0.7377 Webster and Kidnay (2001) CO2 + CH4 + C2H6 15 250 21-30 0.0812-0.8992 Davalos et al. (1976) 3. Results and discussion 3.1 Binary systems The determined adjustable parameters, kij and lij and the %AAD as function of systems temperature are shown in Table 2 for all 7 binary systems analyzed here (17 isotherms with a total of 160 EP). Figures 1-2 shows the comparison between calculated and experimental data for the mole fraction of liquid and vapor phase for all seven systems at all temperatures and deviations in calculated pressure for both EoS analyzed. Figure 1 shows the results for PR-vdW-2 EoS determinations and Figure 2 shows the results for PT-vdW-2 EoS determinations. Analyzing Figures 1, 2, and Table 2, it can be verified that a good correlation between experimental and calculated data was obtained for most part of systems analyzed, with low computational time (less than 5 minutes) for all cases. Larger deviations were observed for PT EoS. The phase behavior of all analyzed systems showed an increase in the solubility of CO2 in hydrocarbons with increasing in system pressure. This behavior was observed for all compounds evaluated in this work, and the increase in temperature resulted in a decrease in CO2 composition in the vapor phase, this trend was observed for all studied systems too. In general was observed that liquid phase predictions from EoS are worse than that observed for vapor phase. The most dissonant points for the upper curve, in both cases, refer to the system C5H10 + CO2, which may indicate some deviations in experimental data obtained by Sima et al. (2016). In addition, there is a higher concentration of the experimental data in the region of higher to vapor phase compositions. This behavior can be explained by the lower solubility of CO2 in hydrocarbons with more than 4 carbons in chain. Similar trends were described in Freitas et al. (2013) for systems composed by ionic liquids and CO2 at high pressure conditions. Table 2: Binary interaction parameters and deviation at different temperatures for PR and PR EoS combined with vdW-2 mixing rule. PR PT System T (K) kij lij %AAD T (K) kij lij %AAD CO2 + CH4 240.35 0.0567 -0.0615 3.87 240.35 0.0595 -0.0575 4.01 250.00 0.0691 -0.0514 2.70 250.00 0.0723 -0.0465 2.89 261.25 0.0555 -0.0571 1.70 261.25 0.0587 -0.0536 1.90 CO2 + C3H8 252.95 0.0786 -0.0384 2.90 252.95 -0.0150 -0.2256 3.90 273.15 -0.0226 -0.2066 3.81 273.15 0.0769 -0.0502 2.94 CO2 + C4H10 273.15 0.0795 -0.0365 1.28 273.15 0.0825 -0.0393 3.40 CO2 + C8H18 322.39 0.0931 -0.0099 1.16 322.39 0.1006 -5.36E-5 1.72 348.25 0.1117 -0.0033 1.71 348.25 0.0825 -0.0096 0.08 372.53 0.1149 -0.0082 2.31 372.53 0.1036 -0.0048 2.26 CO2 + C10H22 319.11 0.0885 -0.0268 1.06 319.11 0.0489 -0.0356 4.15 344.74 0.0883 0.0122 3.42 344.74 0.0687 0.0248 3.64 372.94 0.0985 0.0001 1.64 372.94 0.0790 0.0115 1.72 CO2 + C5H10 323.15 0.1159 0.1131 6.17 323.15 0.1131 0.1209 6.11 333.15 0.1234 0.1630 8.19 333.15 0.1179 0.1752 8.16 344.65 0.1638 0.1999 8.16 344.65 0.1669 0.2281 8.16 CO2 + C6H12 323.15 0.1006 -0.0287 1.54 323.15 0.1031 -0.0328 1.63 353.15 0.1101 0.0301 2.36 353.15 0.1099 0.0296 3.06 Mean deviation - - - 2.36 - - - 3.06 1479 (a) (b) Figure 1. Comparison of calculated and experimental data for the mole fraction of liquid phase (a) and vapor phase (b) using PR-vdW-2 (binary systems). (a) (b) Figure 2. Comparison of calculated and experimental data for the mole fraction of liquid phase (a) and vapor phase (b) using PT-vdW-2 (binary systems). Overall, and as expected for this type of thermodynamic model, the ability of the PR and PT EoS when combined with the vdW-2-mixing rule, for representation of vapor phase behavior was greater than that observed for the liquid phase. The results obtained by Shariati et al. (1998) for multicomponent systems formed by hydrocarbons and carbon dioxide, using PR-EoS showed similar behavior to that observed in this work. 3.2 Ternary systems The adjustable parameters, kij and lij and the %AAD as function of pressure and temperature are presented in Table 3 (results for kij) and Table 4 (results for lij) for the 2 ternary systems analyzed in this work, totalizing a total 69 experimental data points. Table 3: Interaction parameters kij (kij = kji) for the two ternary systems studied at different temperatures (K) and pressures (bar) for PR and PT EoS (kij, i=j = 0,000). PR PT System T (K) P (bar) kij i = 1; j = 2 kij i = 1; j = 3 kij i = 2; j = 3 AAD (%) kij i = 1; j = 2 kij i = 1; j = 3 kij i = 2; j = 3 AAD (%) CO2 + CH4 + C3H8 230 8 1.273 0.119 0.498 5.32 0.285 0.122 -0.148 5.04 230 40 0.098 0.069 0.011 0.79 0.103 0.106 0.006 0.42 230 70 0.187 0.095 -0.001 0.48 0.195 0.097 -0.005 0.21 270 28 -0.200 0.216 0.236 6.81 -0.164 -0.374 0.141 6.84 270 55 0.113 0.191 0.025 0.26 0.107 0.189 0.024 0.26 270 80 0.143 0.093 0.007 0.21 0.150 0.094 -0.0006 0.10 CO2 + CH4 + C2H6 250 21 -0.017 0.004 -2.332 3.73 0.004 0.524 -2.761 3.22 250 25 0.129 0.041 -0.038 1.12 0.151 -0.107 -0.053 1.18 250 30 0.080 0.284 -0.161 1.56 0.062 0.451 -0.115 1.54 Mean deviation - - - - - 1.12 - - - 1.18 Figures 3 and 4 shows the comparison between calculated and experimental data for the mole fraction of liquid and vapor phase for both systems at all conditions analyzed. Figure 3 presents the results for PR-vdW-2 EoS determinations and Figure 4 presents the results for PT-vdW-2 EoS determinations. 1480 The same behavior observed for binary systems was observed here for the ternary systems too. The largest deviations were observed in the description of liquid phase, this trend is common for this type of model when used to perform the calculations carried out by this work using phi-phi (𝜙 − 𝜙) formulation. The most dissonant points for the upper curve, in both cases, refer to lower pressure (8 and 28 bar to system 1 and 21 bar to the system 2). Table 4: Interaction parameters lij (lij = lji) for the two ternary systems studied at different temperatures (K) and pressures (bar) for PR and PT EoS (lij, i=j = 0,000). PR PT System T (K) P (bar) lij i = 1; j = 2 lij, i = 1; j = 3 lij i = 2; j = 3 %AAD lij i = 1; j = 2 lij i = 1; j = 3 lij i = 2; j = 3 %AAD 230 8 1.092 -0.036 0.353 5.32 0.215 -0.040 -0.127 5.04 230 40 -0.039 -0.055 0.008 0.79 -0.052 -0.002 0.009 0.42 230 70 0.089 -0.028 -0.029 0.48 0.113 -0.028 -0.032 0.21 270 28 -0.392 -0.094 0.186 6.81 -0.370 -0.688 0.126 6.84 270 55 -0.020 0.078 0.022 0.26 -0.042 0.082 0.033 0.26 270 80 0.019 -0.024 -0.007 0.21 0.024 -0.026 -0.012 0.10 250 21 -0.187 -0.280 -2.551 3.73 -0.173 0.247 -3.192 3.22 250 25 0.012 -0.072 -0.041 1.12 0.045 -0.228 -0.050 1.18 250 30 -0.050 0.133 -0.187 1.56 -0.095 0.335 -0.133 1.54 Mean deviation - - - - - 1.12 - - - 1.18 (a) (b) Figure 3. Comparison of calculated and experimental data for the mole fraction of liquid phase (a) and vapor phase (b) using PR-vdW-2 (ternary systems). (a) (b) Figure 4. Comparison of calculated and experimental data for the mole fraction of liquid phase (a) and vapor phase (b) using PT-vdW-2 (ternary systems). It is important to emphasize that the results obtained in this work are easily reproducible, since it were obtained by the use of a free software. Low deviations were observed, PR and PT EoS showed similar ability to represent this kind of experimental data in the ranges of pressure, temperature and composition tested in this work. Both EoS can be applied to describe the LV phase behavior of the systems studied with good results in further modelling and simulation of chemical processes. 1481 4. Conclusions In this study, binary interactions parameter for PR and PT EoS combined with vdW-2 mixing rule (kij and lij), have been optimized for 7 binary mixtures and 2 ternary mixtures composed by hydrocarbons and carbon dioxide by a minimization of %AAD function using the Nelder-Mead Simplex-algorithm implemented in free software PE2000. Experimental data were obtained from the literature, a total of 160 data points, distributed in 17 isotherms, for the binary system and 69 experimental points for the ternary systems were studied. Good results and low computational times (less than 5 minutes) were observed for all systems examined. For binary systems, regions of higher deviations were associated with conditions of low temperatures and high pressures, especially in the liquid phase, %AAD to 2.33% for PR EoS and 3.06% for PT EoS was determined for the evaluated systems. PR EoS presented lower deviations for these systems. In ternary systems the larger deviations regions remains in liquid phase %AAD of 1.12% for PR EoS and 1.18% for PT EoS were detected in low pressure regions. One of the reasons that could have led to this difference would be the small amount of experimental data obtained in the literature for the representation of these systems and the condition of low temperature in which these data was measured. 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