A Simple Mixing Rule for the Deiters Equation of State: Prediction of Interaction Second Virial Coefficients and PVTx Properties of Binary Mixtures

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1 Journal of Chemical Engineering of Japan, Vol. 4, No., pp., 7 Research Paper A Simple Mixing Rule for the Deiters Equation of State: Prediction of Interaction Second Virial Coefficients and PVTx Properties of Binary Mixtures Amir Hossein JALILI, Mohammad Ali KHODAGHOLI and Behrouz NONAHAL Gas Division Department, Research Institute of Petroleum Industry (RIPI), National Iranian Oil Company (NIOC), Qom Road, P.O. Box , Tehran, Iran Keywords: Second Virial Coefficient, Binary Interaction Parameter, Mixing Rule, Equation of State, Volumetric Properties The interaction second virial coefficients, B, and PVTx behavior for the binary systems consisting of N, CO, CH 4, C, C, and normal C 4 H are estimated by means of the Deiters equation of state (EoS). In this way a new combining rule based on the modified Lorentz Berthelot combining rule is proposed for this EoS. The binary interaction parameters for these combining rules are calculated by fitting the expression for the interaction second virial coefficient derived from the Deiters EoS to the experimental cross virial coefficient data. The calculated B values from the optimized binary interaction parameters and Deiters EoS are in good agreement with experimental data. Also a van der Waals type mixing rule with the two binary interaction parameters k ij and l ij obtained from the experimental B values is presented for the Deiters equation of state, which is able to predict the volumetric properties of the above mixtures to a good degree of accuracy in the specified temperature and pressure range. Also it has been shown that the Deiters EoS has higher accuracy relative to the Peng Robinson EoS for the prediction of the volumetric properties of the mixtures studied in this work. Introduction The thermophysical properties of mixtures containing carbon dioxide, nitrogen, and hydrocarbons are important for the natural gas production as well as enhanced oil recovery processes. The behavior of binary mixtures containing the components of interest provides basic information about the effects of interactions between unlike substances for describing the behavior of multicomponent mixtures. The experimental interaction second virial coefficient data may be used for the evaluation of parameters that describe the interactions between unlike molecules. Interaction (or cross) second virial coefficients, B, are of both theoretical and practical significance. On one side they are necessary for thermodynamic calculations at low pressures by means of the truncated virial equation, and on the other hand because of a direct statistical mechanical basis it reflects interactions between unlike molecules and provides insight for theoretical mixture models. One way to estimate the second virial coefficients of pure substances and mixtures is by means of equations of state (EoS). Received on April 4, 6; accepted on August, 6. Correspondence concerning this article should be addressed to A. H. Jalili ( address: jaliliah@ripi.ir). Equations of state are important and most widely used tools for the prediction and correlation of thermodynamic properties of pure substances and mixtures. Generally there are three types of EoSs: empirical, semi-empirical, and theoretical. Empirical EoSs usually contain a large number of parameters specific to the substance under consideration, with little or no physical meaning. These parameters are obtained by fitting to experimental data. These EoSs are very accurate for the fluid of interest within the range that fitting to experimental data is performed, and may be unreliable outside this range, and for other fluids. Theoretical EoSs are based on statistical thermodynamic concepts with fewer substance-dependent parameters, which have physical meaning. Due to the complexity and limitations of theory, such EoSs tend to be less accurate and have less applicability. Semi-empirical (or semi-theoretical) EoSs, like cubic EoSs, combine features of theoretical and empirical equations. These are the most widely used types of EoSs for prediction and correlation of phase equilibrium, PVT behavior of fluids, and other properties by means of a very few number of adjustable parameters. One of the semi-empirical equations of state, which has a statistical mechanics basis, is the Deiters EoS (Deiters, 98a). This EoS has been obtained by employing a square well model of the intermolecular pair potential in the pressure Copyright 7 The Society of Chemical Engineers, Japan

2 equation of statistical mechanics. The equation contains three adjustable parameters, which have been estimated from the critical constants by Deiters (98b), and also by Baonza et al. (99) for a number of pure substances. Mixing rules for the parameters of this EoS have been devised by Deiters (98) from a quasichemical lattice model to predict the vapor liquid, liquid liquid, and gas gas equilibria (Chokappa et al., 98; Dabrowska, ; Guedes et al., ) as well as the critical properties of binary mixtures (Mainwaring et al., 988). The binary interaction parameters for mixing rules may be estimated from vapor liquid equilibrium data and related properties (Gunning and Rowlinson, 97), critical and azeotropic states (Teja and Rowlinson, 97), saturated liquid densities (Mollerup and Rowlinson, 974), and interaction second virial coefficients (Teja, 978). It has also been shown (Teja, 978; Poling et al., ) that values of binary interaction parameters calculated from one property can be used to predict another thermodynamic property with confidence. In this work, by means of the relations existing between the three parameters of the Deiters EoS, a, b, and c and the square well potential parameters ε and σ, an alternative easy to use mixing rule with two binary interaction parameters is developed. The binary interaction parameters k ij and l ij, which appear in these mixing rules, have been estimated by fitting the relation for the second virial coefficient resulting from the Deiters EoS to the experimental B data, for binary systems. By means of the estimated parameters k ij and l ij, and using the mixing rules presented here, the density of the binary systems consisting of the main constituents of the natural gas (N, CO, CH 4, C, C, and nc 4 H ) are calculated from the Deiters and Peng Robinson EoSs, and are compared with the experimental data and with each other at the specified temperature and pressure ranges.. Theory A complete derivation and description of the Deiters EoS can be found in the original paper (Deiters, 98a) and here we include briefly the definitions for the convenience of the reader. The explicit form of the Deiters EoS, expressing the dependence of pressure, P on the molar volume V m and the absolute temperature T, is as follows: RT 4η η P = + cc Vm ( η) abrt eff exp( T ) I eff V ρ m [ ] ( ) () The three adjustable parameters a, b, and c stands for the characteristic temperature, the covolume, and shape parameter respectively. The parameter c corrects for the nonsphericity of the molecules being c = for spherical molecules, and c =.6887 is a universal constant which accounts for the deviation of the real pair potential from the rigid core model. Other definitions are as follows: η = π σ NA V 6 m ( ) b ρ = () V m T T ct = ( 4) a eff = T + λρ y ( ) where η is the packing fraction, σ is the distance parameter of the square-well potential, λ =.69c is a parameter which accounts for the influence of threebody interactions, I ( ρ ) and y are complex functions of ρ and c derived from statistical mechanics: γ I ( ρ)= c where i= i i ( i + ) h i γ ρ ( 6) h = , h =.84, h =.69, h = 4.668, h 4 = 8.468, h = and γ =.69786( c ).. y( ρ)= f c f( f )+ 6 f 7 c ( ) ( ) being f = exp[cc (η 4η)/( η) ] the relative free volume. The parameters a and b are related to square well potential parameters by ε a = () 8 k B N b = Aσ ( 9) 4 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

3 where ε and σ are the energy and distance parameters of the potential, N A and k B being Avogadro s and Boltzmann s constants respectively. The modified Lorentz Berthelot combining rules for potential parameters ε and σ are ε ( )( ) ( ) / = k ε ε ( )( + ) ( ) σ = l σ σ where k and l are binary interaction parameters for intermolecular potential of unlike molecules, and account for deviation to the geometric and arithmetic mean rules, respectively. Applying Eqs. () and () to Eqs. (8) and (9), the combining rules for the Deiters EoS binary parameters a and b can easily be derived as follows: ( )( ) ( ) / a = k a a [( )( + ) ] ( ) b = l b / b / Because of the complex relation between the repulsive part of the Deiters EoS and the shape parameter c, deriving a theoretically based combining rule for c is very difficult. Because of the direct relation between parameters a and c (Eq. (4)), the following combining rule was arbitrarily chosen for c parameter, which is similar to that of a : ( )( ) ( ) / c = k c c 4 Now there remains the estimation of the binary interaction parameters k and l from experimental data, which enables us to compute a, b, and c for mixture calculations. In this study the experimental cross second virial coefficient data were employed for this purpose. By analogy with Baonza et al. (99) the interaction second virial coefficient expression for binary systems obeying the Deiters EoS is as follows: [ ( )] b exp a / ct B( T)= 4. bc + c ( ) ( ). 867c. 78c. 84 This equation may be fitted to experimental B data at various temperatures, by means of the nonlinear least squares procedure and the binary parameters a, b, and c can be obtained in this way. In order to obtain a unique value for the parameters a, b, and c the following constraint must be incorporated into the fitting procedure: a = c a a c c / ( 6) The incorporation of this constraint can be rationalized by the fact that the k values obtained from either Eqs. () or (4) must be the same. In this manner, unique values are obtained for the quantities k and l. According to these combining rules the binary interaction parameters are both constant and temperature and density independent. The mixing rules for the mixture parameters a m, b m, and c m proposed in this work are as follows: a b c = x x a m i j ij i j = x x b / i j m i j ij = x x c ( 7) m i j ij i j The mixing rules for a m and c m are very much like the van der Waals mixing rules for the cubic EoSs (Poling et al., ) and the mixing rule for the b m parameter has a different form. These mixing rules are examined in the prediction of volumetric properties of the binary mixtures of natural gas, the results of which are presented in the next section.. Results and Discussion The experimental interaction second virial coefficients B were divided into two classes: primary data (pd) and secondary data (sd), according to Boushehri et al. (987) and Bzowski et al. (99). The division into two classes was based on (a) the author s statement of precision and accuracy, and (b) a direct intercomparison of the results obtained from different laboratories and different methods. The primary data were exclusively used in the correlations to determine the optimum values of parameters. Thus by fitting Eq. () to primary experimental data the binary parameters a, b, and c were evaluated. By employing Eqs. () and () and the known values of pure substance parameters a ii, b ii, and c ii the binary interaction parameters k and l were calculated for the systems of interest, which are tabulated in Table. The a, b, and c parameters for pure substances (Table ), which are calculated from critical data were chosen from Baonza et al. (99). The deviations of the primary and secondary experimental B data from calculated values are obtained from the relation VOL. 4 NO. 7

4 Table The calculated binary interaction parameters for the systems studied in this work System k l N CO.9. N CH N C.86.8 N C.4.89 N nc 4 H CO CH CO C..4 CO C..6 CO nc 4 H CH 4 C.4. CH 4 C CH 4 nc 4 H..44 C C.44.9 C nc 4 H C nc 4 H * *k and l were estimated from the experimental critical temperature and pressure of the mixture (Kreglewski and Kay, 969) Gorski and Miller, 9 (pd) Dymond and Smith, 98 (pd) Dymond and Smith, 98 (sd) Martin et al., 98 (pd) Esper et al., 989 (pd) Dymond and Smith, 98 (sd) Table Pure substance parameters for the Deiters EoS (Baonza et al., 99) Fig. of N CO Substance a [K] b [dm /mol] c N CO.4.. CH C C n-c 4 H Deviation = B,exp B,cal Fig. - 4 Dymond and Smith, 98 (pd) Martin and Trengove, 98 (pd) Dymond and Smith, 98 (sd ) of N CH 4 where B,exp and B,cal are the experimental and calculated values of B respectively. The deviation plots for the interaction second virial coefficients of the binary systems investigated in this work are summarized in Figures 4. There is no experimental B values for the binary system C nc 4 H in the open literature available to the authors and thus the binary interaction parameters was estimated from experimental critical temperature and pressure of this mixture by the method described by Baonza et al. (99). It can be seen that the deviations of the primary B data from calculated values by means of the Deiters EoS does not exceed ± cm mol for the pairs consisting of N, CO, CH 4, and C and increases to at most ±6 cm mol for the other heavier pairs consisting of propane and normal butane. The same trend exists for pure alkanes, as we proceed from methane to heavier alkanes the deviation calculated by means of the Deiters EoS and experimental B values increases. Comparing the deviation plots for the unlike second virial coefficients of this study with that of Bzowski et al. (99) reveals 6 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

5 Achtermann et al., 99 (pd) Calvin and Reed, 97 (sd) Dymond and Smith, 98 (sd) Mason and Eakin, 96 (sd) Dymond and Smith, 98 (pd) Dymond and Smith, 98 (sd) Dymond and Smith, 98 (sd) Fig. of N C Fig. Dymond and Smith, 98 (pd) Wormald et al., 996 (pd) of N nc 4 H Mason and Eakin, 96 (pd) Wormald et al., 996 (pd) Fig. 4 of N C Fig. 6 of CO CH 4 that the B values calculated by means of the Deiters EoS is of both higher accuracy and simpler to use than the extended corresponding states principle employed by Bzowski and his coworkers. The densities of the mixtures investigated in this work are calculated by means of the Deiters EoS and mixing rules of Eq. (7). The Deiters EoS is a pressure explicit equation and in order to solve this equation for density, we confront with a very difficult task. Thus a simple algorithm was selected in which after the k and l for the system under study estimated from experimental B values then the a mix, b mix, and c mix were calculated for the specified composition of VOL. 4 NO. 7 7

6 Mason and Eakin, 96 (pd) Dymond and Smith, 98 (pd) Holste et al., 98 (pd) Mason and Eakin, 96 (sd) McElroy et al., 99 (sd) Fig. 9 of CO nc 4 H Fig. 7 of CO C Fig Dymond and Smith, 98 (pd) Dymond and Smith, 98 (sd) Dymond and Smith, 98 (pd) McElroy et al., 99 (pd) of CO C the mixture. The density of the mixture was scanned at. g/cm steps until the difference between the calculated and experimental pressures, P = P cal Wormald et al., 979 (sd) Mason, and Eakin, 96 (sd) Van Nhu, Deiters, 996 (pd) Dantzler et al., 968 (sd) Fig. of CH 4 C P exp was minimized at the specified temperature of the mixture. The density corresponding to minimum P was reported as the density of the mixture. In Table deviations of calculated from the experimental densities are presented. Deviations are computed by the 8 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

7 Dymond and Smith, 98 (sd) Dymond and Smith, 98 (pd) Dymond and Smith, 98 (sd) Dymond and Smith, 98 (pd) Dymond and Smith, 98 (sd) Dymond and Smith, 98 (pd) Fig. of CH 4 C Fig. of C C Dymond and Smith, 98 (pd) Dymond and Smith, 98 (sd) Dantzler et al., 968 (sd) Dymond and Smith, 98 (sd) Mason and Eakin, 96 (pd) Fig. of CH 4 nc 4 H Dymond and Smith, 98 (pd) Dymond and Smith, 98 (sd) Dymond and Smith, 98 (pd) Fig. 4 of C nc 4 H percentage absolute average deviations (AAD%), which is defined by the following relation: AAD% = N N i= ρ exp ρ exp ρ cal where N is the number of experimental data points and VOL. 4 NO. 7 9

8 Table Percentage absolute average deviations (AAD%) in calculated densities of binary mixtures System References Number of points Temperature range [K] N CO Seitz et al. (996), Esper et al. (989) Maximum pressure [MPa] Composition AAD% b AAD% c AAD% Maximum % range, x Deviation a 9(g) + (g) N CH 4 Seitz et al. (996), Haynes and McCarty (98), Achtermann et al. (986) 9(g) + 8(g) + (g) N C Achtermann et al. (99) 477(g) CO CH 4 Seitz et al. (996), Esper et al. (989), Hwang et al. (997) 94(g) + 9(g) + (l, g) CO C Lau et al. (997), Magee (99) 88(l, g) + 6(l, g) CO C Galicia-Luna et al. (994) 7(l) CO nc 4 H Hsu et al. (98), Niesen (989) 4(l, g) + 4(l, g) CH 4 C Haynes et al. (98) 44(l, g) CH 4 C Arai and Kobayashi (98) 4(l, g) C C Parrish (984) (l) C nc 4 H Parrish (986) (l) Overall a percentage absolute maximum deviation of the Deiters EoS b calculated by setting k = l = in the Deiters EoS c calculated from the Peng Robinson EoS l: liquid g: gas JOURNAL OF CHEMICAL ENGINEERING OF JAPAN

9 ρ exp and ρ cal are the experimental and calculated densities respectively. It can be seen that the AAD% does not exceed.% (C nc 4 H mixture) and overall the average deviation is.8%. The percentage absolute maximum deviation of each binary system is shown in the next column to promote a better understanding of the accuracy of the Deiters EoS and method described here. Also the AAD%s for the case of k = l = are represented, which show that deviations does not change significantly by neglecting the binary interaction parameters except in the cases of CO C, CO C, and CO nc 4 H, which implies that these two parameters for CO alkane mixtures play an important role. Calculation of the density of the abovementioned mixtures by means of the k and l parameters obtained from secondary B data gave rise to poor results, which indicates that proper selection of experimental B data is of central importance in the method employed in this work. In the last column of Table the AAD%s for the density of the mixtures calculated by the Peng Robinson EoS (which is proofed to be a good EoS for pure and mixture of nonpolar molecular systems) employing the one-fluid van der Waals (- vdw) mixing rule are presented. The binary interaction parameters were taken from Knapp et al. (98), which are estimated from experimental binary vapor liquid equilibria. In all cases except CO nc 4 H and C C the Deiters EoS can predict the volumetric behavior of the binary systems investigated in this work with higher accuracy than the Peng Robinson EoS. Even in the case where k = l = the overall AAD%s resulting from the two EoSs in calculating the density of the mixtures are comparable to each other. Conclusions An alternative simple mixing rule with two binary interaction parameters for a cubic EoS with a statistical mechanical basis namely the Deiters equation is presented. The parameters of this mixing rule are estimated by the experimental interaction second virial coefficient data and are used to predict the volumetric behavior of binary mixtures consisting of the main constituents of the natural gas, i.e., N, CO, CH 4, C, C, and nc 4 H. It is shown that the proper selection of experimental B values is essential for the accurate prediction of the PVTx properties of binary mixtures. The method and mixing rule explained here can be used to calculate the volumetric properties of other binary mixtures as well as multi-component mixtures provided there exists accurate and reliable experimental B data for the systems of interest. Acknowledgments We wish to thank Dr. John H. Dymond who provided helpful second virial coefficient data. Literature Cited Achtermann, H. J.,T. K. Bose, H. Rogener and J. M. St-Arnaud; Precise Determination of the Compressibility Factor of Methane, Nitrogen, and Their Mixtures from Refractive Index Measurements, Int. J. 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