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1: Van der Waals- and Peng-Robinson-type Equation of State for Hydrocarbons, Based on Simple Molecular Properties Kamal I. Al-Malah Head of the Department of Chemical Engineering, University of Hail, Hail, Saudi Arabia Mobile: +966-556334640; +962-795204340; E-mail: [email protected] Abstract Four hundred and twenty nine (429) hydrocarbons were utilized to fit their critical pressure and temperature as a function of molecular weight and carbon atomic fraction. The critical parameters and appearing in van der Waals, Peng-Robinson, and Peng-Robinson-Stryjek- Vera equation of state (EoS), were replaced by two curve-fitted equations. The afore-mentioned original equations were tested and contrasted versus their counterparts as far as predicting the molar volume of both liquid and vapor at a given pair of pressure and temperature for a given hydrocarbon. Using either an experimental or reference datum for the estimated molar volume, it was found that the substituted equation did predict the volumetric properties with accuracy as good as the original equation. The replacement will ease the calculation of volumetric properties of liquid and vapor with a reasonable accuracy. It was found that the accuracy of vdW and vdWtype models for predicting vapor molar volumes stayed reasonable up to (Pr/Tr) < 0.4. Moreover, vdW and vdW-type models were the least accurate among the examined models in terms of predicting the liquid molar volume. In addition, except for a high molecular weight hydrocarbon (i.e., above C14), P-R-type and P-R-S-V-type models were found to predict well the liquid molar volume with a reasonable accuracy (Percent Relative Error, PRE <10%). Finally, at the critical condition (i.e., Tr=1.0 and Pr=1.0), all models, except vdW-type (with PRE< 5%), failed to predict the critical molar volume (Vc) with a reasonable accuracy (PRE < 10 %). Keywords van der Waals; Peng-Robinson; Peng-Robinson-Stryjek-Vera; Hydrocarbons 2: Introduction An equation of state (EoS) represents the heart of thermodynamic models. EoS can be used to represent phase equilibria as a function of temperature and pressure for a pure substance, in addition to estimation of thermal and volumetric properties. The van der Waals equation is an equation of state for a fluid composed of particles that have a non-zero size and a pairwise attractive inter-particle force (such as the van der Waals force). It was derived by Johannes Diderik van der Waals in 1873 [1], based on a modification of the ideal gas law, who received the Nobel Prize in 1910 for "his work on the equation of state for gases and liquids". The equation approximates the behavior of real fluids, taking into account the nonzero size of molecules and the attraction between them. The Peng–Robinson equation was developed in 1976 [2] in order to satisfy the following goals: 1. The parameters should be expressible in terms of the critical properties and the acentric factor; 2. The model should provide reasonable accuracy near the critical point, particularly for calculations of the compressibility factor and liquid density; 3. The mixing rules should not employ more than a single binary interaction parameter, which should be independent of temperature pressure and composition; and 4. The equation should be applicable to all calculations of all fluid properties in natural gas processes. A modification to the attraction term appearing in the Peng-Robinson equation of state was published by Stryjek and Vera in 1986 [3], which had significantly improved the model's accuracy by introducing an adjustable pure component parameter and by modifying the polynomial fit of the acentric factor. Ahlers and Gmehling [4] modified the Peng–Robinson (P-R) equation of state (EoS) to obtain a better description of saturated liquid densities for the pure compounds. The liquid densities of most compounds were described better by the P-R EoS. Only for a few small spherical substances like N2, CH4 or Ar, the S-R-K EoS provided better results. A simple improved volume translation together with a temperature dependent volume correction delivered an accurate representation of this property near and far from the critical point for polar and nonpolar substances. The fitted parameters were generalized as a function of the critical compressibility factor. Farrokh-Niae et al. [5] proposed a new cubic three-parameter equation of state for PVT and VLE calculations for simple, high polar and associating fluids. The parameters were temperature dependent in the sub-critical region, but temperature independent in the super-critical region. The results for 42 simple and 14 associative pure compounds indicated that the calculated saturation properties and volumetric properties over the whole temperature range, up to high pressures, by the proposed equation of state (EOS), were in better agreement with the experimental data, 3: compared with those obtained by the five well-known EOSs (P–R, P–T, Adachi et al., Yu–Lu, and M4). Their results indicated the high capability of the proposed EOS for calculating the thermodynamic properties of pure and fluid mixtures. Hinojosa-Gómez et al. [6] proposed two modifications to the Peng–Robinson-fitted equation of state where pure component parameters were regressed to vapor pressure and saturated liquid density data. Their first modification was a method that enhanced the equation of state pure component property predictions through simple temperature dependent pure component parameters. In their second modification, they proposed a temperature dependency for covolume b in the repulsive parameter of the EoS, and revised the temperature function in the attractive term. The agreement with experimental data for 72 pure substances, including highly polar compounds, was found to be good with a maximum average absolute deviation in saturated liquid density to be less than 1% for all examined substances. Kontogeorgis and Economou [7] used a methodological approach based on the excess Gibbs energy and activity coefficient expressions derived from cubic equations of state (EoS) for analyzing and understanding the capabilities and limitations of those classical models. They have showed that cubic EoS of vdW-type have a functional form similar to well-known polymer models and provide quantitatively a correct representation of size/free-volume (combinatorial) effects, dominant in mixtures of alkanes of different size and other asymmetric systems; hence, cubic EoS could be extended to include fluid polymers. Model Development Four hundred and twenty nine (429) hydrocarbon compounds were used in the non-linear regression process for finding the best fit for their critical properties. The database of hydrocarbon compounds includes the following categories: 1. Normal paraffin: Example: n-alkane 2. Non-normal paraffin: Example: iso-alkane, methyl-alkane, ethyl-alkane, & methyl-ethylalkane 3. Naphthene: The major structure is saturated ring; example: cyclo-alkane 4. Olefin: Contains a single C=C double bond; example: alkene, methyl-alkene, ethylalkene, & di-methyl-alkene 5. Diolefin: Contains two C=C double bonds; example: alkadiene, methyl-alkadiene, and ethyl-alkadiene 6. Cyclic olefin: Contains a single C=C double bond within the otherwise saturated ring; example: cyclo-alkene, methyl-cyclo-alkene, & ethyl-cyclo-alkene 7. Alkyne: Contains a triple bond between carbons; example: acetylene, methyl acetylene, pentyne, and hexyne 4: 8. Aromatic: Contains a single ring; example: benzene, toluene, & xylene 9. Aromatic with attached olefin side chain: Example: Styrene, ethenyl-benzene, and propenyl-benzene 10. Aromatic with multiple rings directly connected by C-C bonds between the rings: Example: bi-phenyl and 1-methyl-2-phenylbenzene 11. Aromatic with multiple rings connected through other saturated carbon species: Example: Di-phenyl-methane and 1,1-di-phenyl-dodecane 12. Aromatic with multiple rings connected through other carbon species with triple bond: Example: Di-phenyl-acetylene 13. Aromatic with multiple condensed rings: Example: naphthalene, pyrene, methylnaphthalene, and nonyl-naphthalene 14. Aromatic with attached saturated rings: Examples: 1,2,3,4-tetra-hydro-naphthalene and 1methyl-2,3-dihydro-indene 15. Aromatic with attached unsaturated (but not aromatic) rings: Example: indene and 1methyl-indene It should be noticed that the attractive term a and the co-volume repulsive term b appearing in equations of state, like: van der Waals (vdW), Peng-Robinson (P-R), Peng-Robinson-StryjekVera (P-R-S-V), Relich-Kwong (R-K), and Soave-Redlich-Kwong (S-R-K), do contain and , respectively. Hence, it is worth estimating such two quotients based on molecular properties of a hydrocarbon molecule. The carbon atomic fraction (X) and molecular weight (Y) were chosen as the independent variables and both quotients, defined by Z1 and Z2, respectively, represented the dependent variable from regression point of view: (1) (2) For example, given methane (CH4), then its Cfrac will be 1/(1+4)=0.20. Moreover, its MW is simply equal to . The results of non-linear regression for both equations (1) and (2), with 95% confidence interval, are: (3) The goodness of fit for Eq. (3) is given by the R-square as 0.999 and adjusted R-square as 0.999. (4) 5: The goodness of fit for Eq. (4) is given by the R-square as 0.9978 and adjusted R-square as 0.9978. To demonstrate the validity of the proposed equations, will replace and will replace in the following three models of EoS; namely, van der Waals (vdW), Peng-Robinson (P-R), & Peng-Robinson-Stryjek-Vera (P-R-S-V). Three models of EoS will be sufficient evidence that such a replacement of quotients by their equivalence via equations (3) and (4) is reasonable and justifiable. Van der Waals Equation of State (5) (6) (7) van der Waals-Type Equation of State Eq. (6) becomes: (8) (9) Eq. (7) becomes: (10) Substitute a and b in Eq. (5) to get: (11) Peng-Robinson (P-R) Equation of State (12) Where (13) 6: (14) (15) (16) Peng-Robinson (P-R)-Type Equation of State (17) Where (18) (19) (20) (21) Peng-Robinson-Stryjek-Vera (P-R-S-V) Equation of State (22) Where (23) (24) (25) (26) (27) Peng-Robinson-Stryjek-Vera (P-R-S-V)-Type Equation of State (28) Where 7: (29) (30) (31) (32) (33) Solution of the Cubic Equations of State Regarding the cubic in molar volume, equations of state (see equations (5), (11), (12), (17), (22), and (28)); each equation was put in the polynomial form where it shows the coefficients associated with each order in molar volume, . For example, van der Waals’ equation of state (Eq. (5) can be put in the form of: (34) If the absolute values of pressure and temperature (P and T) are given for a pure substance, then Eq. (34) will be a non-linear algebraic equation in which will have, in general, three roots for the molar volume at the given pressure and temperature. MATLAB® code, in the form of an mfile, was written to utilize the MATLAB® built-in root-finding algorithm for finding the roots for such an equation. Two roots were chosen such that the imaginary part is zero, which correspond to the volume of a gas and volume of liquid. Any additional parameters appearing in the main EoS, like a and b, have to be defined by some additional equations other than the main equation itself. The percent relative error, PRE is defined as: (35) Results and Discussion Hydrocarbons with similar and different molecular weight (i.e., size) and carbon atomic fraction (i.e., carbon to hydrogen ratio) were selected to examine the applicability of the suggested model(s). Low-, medium-, and high-molecular weight hydrocarbons were examined. Table 1 shows the estimated molar volume of both the liquid and vapor for a given substance at a given pressure and temperature. The following trends can be observed as far as the accuracy (manifested via the magnitude of PRE) of the model is concerned: 8: 1. In general, all models, the original and the substituted/modified equations, do predict well the vapor molar volume, with a good accuracy (i.e., a PRE value less than 10%). On the other hand, the accuracy of vdW and vdW-type models stays reasonable up to (Pr/Tr) < 0.4. 2. In general, vdW and vdW-type models were the least accurate among the examined models in terms of predicting the liquid molar volume, with a PRE value higher than 10% while exceeding 100% for a high molecular weight hydrocarbon, like n-eicosane (C20H42). 3. P-R, P-R-type, P-R-S-V, and P-R-S-V-type models do predict well the liquid molar volume with a reasonable accuracy (PRE <10%), except for a high molecular weight hydrocarbon (above C14) where PRE exceeds 10%, in general. 4. At the critical condition (i.e., Tr=1.0 and Pr=1.0), all models, except vdW-type, failed to predict the critical volume (Vc) having PRE values > 10%. Conclusion First of all the replacement of and ratios, appearing in vdW, P-R, and P-R-S_V, by molecular properties; namely, the molecular weight and carbon atomic fraction, was successfully done. The replacement will ease the calculation of volumetric properties of liquid and vapor with a reasonable accuracy (i.e., a PRE value less than 10%), taking into account the following limitations: 1. The accuracy of vdW and vdW-type models for predicting vapor molar volumes stays reasonable up to (Pr/Tr) < 0.4. 2. vdW and vdW-type models were the least accurate among the examined models in terms of predicting the liquid molar volume. 3. Except for a high molecular weight hydrocarbon (i.e., above C14), P-R-type and P-R-S-Vtype models were found to predict well the liquid molar volume with a reasonable accuracy (PRE <10%). 4. At the critical condition (i.e., Tr=1.0 and Pr=1.0), all models, except vdW-type (with PRE< 5%), failed to predict the critical volume (Vc) with a reasonable accuracy (PRE < 10 %). References 1. Van der Waals JD (1873) On the Continuity of the Gas and Liquid State, Doctoral Dissertation, Leiden. 2. Peng DY, Robinson DB (1976) A New Two-Constant Equation of State. Industrial and Engineering Chemistry: Fundamentals 15: 59–64. doi:10.1021/i160057a011. 9: 3. Stryjek R, Vera JH (1986) PRSV: An improved Peng—Robinson equation of state for pure compounds and mixtures. The Canadian Journal of Chemical Engineering 64: 323–333. doi:10.1002/cjce.5450640224. 4. Ahlers J, Gmehling J (2001) Development of a universal group contribution equation of state I. Prediction of liquid densities for pure compounds with a volume translated Peng– Robinson equation of state. Fluid Phase Equilibria 191: 177–188. PII: S0378-3812(01)00626-4. 5. Farrokh-Niae AH, Moddarress H, Mohsen-Nia M (2008) A three-parameter cubic equation of state for prediction of thermodynamic properties of fluids. J. Chem. Thermodynamics 40: 84–95. doi:10.1016/j.jct.2007.05.012. 6. Hinojosa-Gómez H, Barragán-Aroche JF, Bazúa-Rueda ER (2010) A modification to the Peng–Robinson-fitted equation of state for pure substances. Fluid Phase Equilibria 298: 12–23. doi:10.1016/j.fluid.2010.06.022. 7. Kontogeorgisa GM, Economoub IG (2010) Equations of state: From the ideas of van der Waals to association theories. J. of Supercritical Fluids 55: 421–437. doi:10.1016/j.supflu.2010.10.023. 8. Mackay D, Shiu WY, Ma K-C, Lee SC (2006) Physical-Chemical Properties and Environmental Fate for Organic Chemicals, 2nd edition, Taylor & Francis, New York. 9. Poling BE, Prausnitz JM, O’Connell JP (2001) The Properties of Gases and Liquids”, 5th edition McGraw-Hill, New York. 10: Table 1: The prediction of molar volume of both liquid and vapor (evaluated at the noraml boiling point of the hydrocarbon) as given by the original and modified equation. Either an experimental or reference datum was chosen to calculate the percent relative error associated with each estimated volumetric property. VG: Volume of a gas; VL: Volume of a liquid; and Vc: Critical volume of a fluid. PRE represents the percent relative error: PRE = {Absolute (Estimated –Reference)/Reference} × 100%. Component vdW Volume (L/mol) (PRE) vdWType Volume (L/mol) (PRE) P-R Volume (L/mol) (PRE) P-RType Volume (L/mol) (PRE) P-R-S-V Volume (L/mol) (PRE) P-R-SV-Type Volume (L/mol) (PRE) Experimentalª (Reference*) Volume (L/mol) (PRE) 24.388* n-Pentane (C5H12) @ 309.2 K, 1 atm (Tr=0.65831 Pr=0.030078) n-Pentane (C5H12) @ 400 K, 11 atm (Tr=0.8516 Pr=0.33085) n-Pentane (C5H12) @ 400 K, 15 atm (Tr=0.8516 Pr=0.45117) n-Pentane (C5H12) @ 309.2 K, 1 atm (Tr=0.65831 Pr=0.030078) n-Hexane (C6H14) @ 341.9 K, 1 atm (Tr=0.67382 Pr=0.033637) n-Hexane (C6H14) @ 341.9 K, 1 atm (Tr=0.67382 Pr=0.033637) Benzene (C6H6)@ 353.2 K, 1 atm (Tr=0.62836 Pr=0.02069) Benzene (C6H6) @ 353.2 K, 1 atm (Tr=0.62836 Pr=0.02069) Tetraline (C10H12)@ 480.75 VG = 24.7612 (1.53 %) VG = 24.7741 (1.58 %) VG = 24.3868 (0.005 %) VG = 24.4106 (0.093 %) VG = 24.388 (0.00 %) VG = 24.4117 (0.097 %) VG = 2.4776 (9.9 %) VG = 2.4873 (10.3 %) VG = 2.2544 (0.00 %) VG = 2.2717 (0.77%) VG = 2.2544 (0.00 %) VG = 2.2717 (0.77 %) 2.2544* VG = 1.6311 (21.6%) VG = 1.6427 (22.4 %) VG = 1.3417 (0.0 %) VG = 1.3662 (1.83 %) VG = 1.3417 (0.00 %) VG = 1.3662 (1.83 %) 1.3417* VL= 0.19709 (66.5 %) VL= 0.18397 (55.4 %) VL= 0.11477 (3.1 %) V L= 0.1077 (9.03 %) VL= 0.11481 (3.03 %) VL= 0.10774 (9.00 %) 0.1184ª VG = 27.3349 (1.7 %) VG = 27.3542 (1.8 %) VG = 26.874 (0.004 %) VG = 26.9075 (0.12 %) VG = 26.8751 (0.0 %) VG = 26.9085 (0.12 %) 26.8751* VL= 0.24131 (71.6 %) VL= 0.22867 (62.6 %) VL= 0.13919 (1.0 %) V L= 0.13237 (5.8 %) VL= 0.13922 (0.98 %) VL= 0.1324 (5.8 %) 0.1406ª VG = 28.4555 (1.1 %) VG = 28.4542 (1.1 %) VG = 28.1338 (0.004 %) VG = 28.1361 (0.004 %) VG = 28.135 (0.0 %) VG = 28.1372 (0.008 %) 28.135* VL= 0.15839 (65.0 %) VL= 0.14629 (52.4 %) VL= 0.092874 (3.2 %) V L= 0.086421 (9.98 %) VL= 0.092911 (3.2 %) VL= 0.086452 (9.94 %) 0.096ª VG = 38.5084 VG = 38.6005 VG = 37.9135 VG = 38.0512 VG = 37.9151 VG = 38.0526 37.9151* 11: Component vdW Volume (L/mol) (PRE) vdWType Volume (L/mol) (PRE) P-R Volume (L/mol) (PRE) P-RType Volume (L/mol) (PRE) P-R-S-V Volume (L/mol) (PRE) P-R-SV-Type Volume (L/mol) (PRE) Experimentalª (Reference*) Volume (L/mol) (PRE) K, 1 atm (Tr=0.66757 Pr=0.030704) Tetraline (C10H12)@ 480.75 K, 1 atm (Tr=0.66757 Pr=0.030704) n-Eicosane (C20H42)@ 608 K, 1 atm (Tr=0.79259 Pr=0.090716) 4,4′Dimethylbiphenyl (C14H14) @ 568.15 K, 1 atm (Tr=0.71632 Pr=0.039898) (1.6 %) (1.8 %) (0.004 %) (0.36 %) (0.0 %) (0.36 %) VL= 0.31095 (75.3 %) VL= 0.31384 (76.9 %) VL= 0.17992 (1.4 %) V L= 0.17876 (0.77 %) VL= 0.17998 (1.45 %) VL= 0.17882 (0.80 %) 0.1774ª V G= 47.4539 (4.4 %) V G= 47.6315 (4.7 %) V G= 45.5252 (0.12 %) V G= 45.8619 (0.86 %) V G= 45.47 (0.0 %) V G= 45.8115 (0.75 %) 45.47* VL= 0.46639 (103.7 %) VL= 0.44485 (94.2 %) VL= 0.26138 (14.1 %) V L= 0.2476 (8.1 %) VL= 0.26127 (14.1 %) VL= 0.24749 (8.1 %) 0.2290ª VL= VL= VL= V L= VL= VL= n-Eicosane 0.4514ª 1.1314 1.012 0.59596 0.53954 0.59314 0.53708 (C20H42) @ 608 K, (150.6 %) (124.2 %) (32.0 %) (19.5 %) (31.4 %) (18.98 %) 1 atm (Tr=0.79259 Pr=0.090716) V c= V c= V c= V c= V c= V c= Ethyl-benzene 0.374¤ 0.53313 0.38759 0.44237 0.29417 0.44237 0.29417 (C8H10) @ 617.17 (42.5 %) (3.6 %) (18.3 %) (21.3 %) (18.3 %) (21.3 %) K, 35.62 atm (Tr=1 Pr=1) V c= V c= V c= V c= V c= n-Pentane (C5H12) Vc= 0.311¤ 0.43473 0.29641 0.36072 0.22209 0.36072 0.22209 @ 469.7 K, 33.25 (39.8 %) (4.7 %) (16.0 %) (28.6 %) (16.0 %) (28.6 %) atm (Tr=1 Pr=1) ªBased on the experimental value of molar volume of liquid at its normal boiling point, reported by: Mackay et al. (2006), 2nd edition “Physical-Chemical Properties and Environmental Fate for Organic Chemicals”, Taylor & Francis, New York. *P-R-S-V is taken as the reference; for it is considered the most accurate equation of state among the examined models. ¤ Based on the critical value of molar volume of ethyl benzene at its critical pressure and temperature, reported by: Poling et al. (2001), 5th edition “The Properties of Gases and Liquids”, McGraw-Hill, New York. 12: