Fluid Phase Equilibria

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1 Fluid Phase Equilibria 302 (2011) Contents lists available at ScienceDirect Fluid Phase Equilibria ournal homepage: Evaluating the phase equilibria of liquid water + natural gas mixtures using cubic equations of state with asymmetric mixing rules P. Reshadi a, Kh. Nasrifar a,, M. Moshfeghian b a Department Chemical Engineering, Shiraz University of Technology, Shiraz, Iran b School of Chemical & Petroleum Engineering, Shiraz University, Shiraz, Iran article info abstract Article history: Received 24 May 2010 Received in revised form 26 July 2010 Accepted 9 August 2010 Available online 13 August 2010 Keywords: Gas liquid equilibria Equation of state Mixing rule Water content Natural gas Based on a previously developed liquid liquid mixing rule we present a modified and robust mixing rule for accurate prediction of water content of natural gas mixtures and the natural gas solubility in liquid water phase. The non-density dependent mixing rule (NDD) and the new mixing rule are incorporated into the Peng Robinson (PR), Soave Redlich Kwong (SRK), and Nasrifar Bolland (NB) equations of state to investigate their accuracies in estimating the water content of the gas phase as well as the gas solubility in the aqueous phase. For each binary system water + hydrocarbon, water + carbon dioxide, water + hydrogen sulfide, and water + nitrogen, three binary interaction parameters are required to describe the gas liquid water equilibria. In this work, experimental data from literature were used to tune the parameters. The results are in good agreement with experimental data, demonstrating the reliability of the new mixing rule and the thermodynamic approach used in this work Elsevier B.V. All rights reserved. 1. Introduction Natural gas is generally saturated with water at reservoir conditions. Evaluating the water content of natural gas mixtures is important in a wide variety of applications in gas industry, e.g., for the design and optimization of operating conditions in gas pipelines and natural gas facilities. Additionally, presence of water in the gas phase may result in liquid water. The liquid water formed can damage equipments due to corrosion, two-phase flow and blockage during production and transportation. The water content of natural gas mixtures is reduced appreciably to avoid such problems. On the other hand, the solubility of hydrocarbons in water is important due to ecological concerns and new restrictions on the existence of organic pollutants in water streams. Therefore, it is of great interest to develop semi-empirical correlations, charts, or thermodynamic models for estimating the water content of natural gas mixtures and the gas solubility in the liquid water phase. Several predictive models are available in the literature which can be used to calculate the phase equilibrium in natural gas + liquid water system. In general, these models include: (1) empirical or semi-empirical correlations and charts, (2) thermodynamic models which are based on equating the chemical potential or fugacity of each component in different phases. Corresponding author. Tel.: ; fax: addresses: nasrifar@sutech.ac.ir, kh nasrifar@yahoo.com (Kh. Nasrifar). There are various empirical or semi-empirical equations and charts with different accuracies for estimating the water content of gases, e.g., ideal model, ideal model + Poynting correction factor, Bukacek correlation [1], Sharma and Campbell method [2], Robinson et al. chart [3], Behr correlation [4], Maddox et al. correlation [5], Wichert and Wichert correlation [6], Kazim correlation [7], McKetta Wehe chart [8], Ning et al. correlation [9], and Aqualibrium software [10]. In this paper, some of these methods are examined. These methods are simple and easy to use. Therefore, they have got a good popularity among the engineers in gas industry. The validity test of the available correlations and charts to determine the water content of gases does not give complete satisfaction and their improvement is possible. These methods were generally developed with limited number of experimental data, mostly at high temperatures. In fact, during the methods development, the experimental data needed to describe the phase equilibria between water and hydrocarbon gases, and also between water and non-hydrocarbon gases were not available at low temperatures (typically lower than K) [11]. Due to this deficiency, the calculated water content by the cited correlations and charts at low temperatures might not be accurate. Therefore, a simple and yet robust thermodynamic model which can be used for wide ranges of conditions is still necessary. Thermodynamic models use different approaches to describe different phases. For example, some of them use activity coefficient and Henry s constant approach for modeling the liquid water phase; however, other models use the equation of state approach /$ see front matter 2010 Elsevier B.V. All rights reserved. doi: /.fluid

2 180 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) It is likely to calculate the water content of hydrocarbons with reasonable accuracy using cubic equations of state (EoS) with a temperature-independent binary interaction parameter [12 14]. On the other hand, solubility of pure gases in water were investigated by many authors: Peng and Robinson [15,16], Mathias and Copeman [17], Kabadi and Danner [18], Michel et al. [19], Anderko [20], Victorov et al. [21], Yu and de Swan Arons [22], Soreide and Whitson [23], Daridon et al. [24], Apostolou et al. [25], and Nasrifar and Moshfeghian [26]. Due to the inability of EoS/mixing rules to represent the hydrophobic effect in the aqueous phase, the prediction of hydrocarbons solubility in water is poor. To overcome this deficiency, on the basis of experimental evidence, empirical mixing rules for the attractive energy parameter of cubic EoS were developed by Kabadi and Danner [18], Michel et al. [19] and Nasrifar and Moshfeghian [26]. The Michel et al. [19] mixing rule illustrates improvement over the Kabadi and Danner [18] mixing rule. The Nasrifar and Moshfeghian mixing rule [26] also illustrates improvement over Michel et al. [19], and Kabadi and Danner [18] models. Nasrifar and Moshfeghian [26] developed a mixing rule that seems to be accurate in prediction of heavy and cyclic hydrocarbon solubility in water for a wide temperature range. The aim of the present work is developing an asymmetric mixing rule based on the Nasrifar and Moshfeghian (NM) mixing rule [26] for the attractive energy parameter of cubic EoS. This mixing rule is used to predict the phase equilibria in natural gas + liquid water systems for a wide range of temperature. For this purpose the water content data of natural gas main components as well as natural gas components solubility in water-rich phase for some binary and multicomponent systems were collected from the literature. Three cubic EoS were used for calculations, i.e., the Peng Robinson (PR) EoS [27], the Soave Redlich Kwong (SRK) EoS [28] and the Nasrifar Bolland (NB) EoS [29]. Considering the simplicity and wide applications of the cubic EoS, it would be desirable to calculate the phase equilibria in natural gas main components + liquid water systems. 2. Thermodynamic model 2.1. Model For calculating the gas liquid equilibria of natural gas + liquid water systems, three cubic EOS are used. These EoS comprise the PR EoS [27], the SRK EoS [28] and the NB EoS [29]. A two-parameter cubic equation of state can generally be expressed by P = RT v b a (1) (v + ub)(v + wb) where P, T, v and R are the pressure, temperature, molar volume and universal gas constant, respectively. The u and w are equation of state dependent constants. For the PR EoS, u = , w = ; for the SRK equation, u = 0, w= 1; and for the NB EoS, u = w = 1/ 3. For these three EoS, the b-parameter is expressed by b = x i b i (2) i For polar non-polar interactions, however, the classical van der Waals mixing rule for the attractive parameter a is not satisfactory and an unconventional form of the classical mixing rule is required The new mixing rule for the attractive parameter In systems which contain water, the asymmetric interaction cannot efficiently be accounted for by the classical mixing rule. To overcome this deficiency several authors, e.g., Mathias and Copeman [17], Kabadi and Danner [18] and Michel et al. [19] presented unconventional mixing rules for the attractive energy parameter of the cubic EoS. Kabadi and Danner [18] developed a non-quadratic mixing rule, considering the hydrophobic interactions between water and hydrocarbons in water-rich phase. The mixing rule is expressed by a = x 2 1 a 11f (x 2 ) + x 2 2 a x 1 x 2 (a 11 a 22 ) 1/2 (1 k 12 ) (3) 0.8 G(1 Tr,1 f (x 2 ) = ) x 2 (4) a 11 where G is a constant and T r,1 = T/T C,1. In Eqs. (3) and (4) and those that follow the subscript 1 stands for water. Later, Michel et al. (MHP) [19] proposed f (x 2 ) = 1 + ˇ12 x 2 exp( x 2 ) (5) where = 10 and ˇ12 is a function of temperature according to ˇ12 = T n (6) Nasrifar and Moshfeghian [26] defined a parameter which corrects the water water interaction in the presence of hydrocarbons. This parameter accounts for the changes in the structure of water due to the presence of hydrocarbons. In fact, some of water molecules are separated from the water structure, producing free space or holes which can be occupied by the hydrocarbon molecules [31]. Nasrifar and Moshfeghian [26] presented the following equation for considering water water interactions between water molecules in the structural body and free water molecules = a 11 (7) a 11 where a 11 is the water water interaction in the absence of hydrocarbon and a in the presence of hydrocarbon. Nasrifar and 11 Moshfeghian [26] constructed a quasi-chemical reaction between m water molecules and a hydrocarbon molecule to produce a complex as follows mw + HC HCW m (8) with K E = X HCW m X m 1 X (9) 2 where K E is an equilibrium constant and X the true mole fraction. Taking it for granted that the complex is responsible for the changes in the water water interaction, Nasrifar and Moshfeghian [26] obtained the following relation for the parameter that seems to be adequate for correlating the liquid liquid equilibria of water hydrocarbon systems for a wide range of temperature = 1 + ı 12 x 9 1 x 2 (10) with ( ı 12 = m 1 + m ) ) 2 exp ( H (11) T RT Considering water water interaction in the presence of hydrocarbons, one can use the van der Waals mixing rule in the following form a = x 2 1 a 11 + x2 2 a x 1 x 2 (a 11 a 22 ) 1/2 (1 k 12 ) (12) Combining Eqs. (7) and (12) yields a = x 2 1 a 11 + x 2 2 a x 1 x 2 (a 11 a 22 ) 1/2 (1 k 12 ) (13) Eq. (13) together with Eqs. (10) and (11) complete our relationship, namely the NM mixing rule. Nasrifar and Moshfeghian [26] evaluation for the compositional dependence of shows that the function provides a maximum at x 2 = 0.1 and approaches unity as x 2 0, and x 2 1. For the case x 2 0.6, i.e., for the gas phase at

3 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) Table 1 Binary interaction parameters for water + natural gas components using different EoS with the NM mixing rule (m 1 = 0 for all binaries). Component NB PR SRK k 12 m 2 (K) H /R (K) k 12 m 2 (K) H /R (K) k 12 m 2 (K) H /R (K) Methane Ethane Propane Butane Carbon dioxide Hydrogen sulfide Nitrogen frequent conditions, Eq. (13) reduces to the van der Waals mixing rule [26]. We extended the proposed binary mixing rule, Eq. (13), to multicomponent systems through the following equations a = a C + a A (14) where a C is given by the classical quadratic mixing rule a C = x i x a i (15) i and the parameter a i is expressed by a i = (1 k i ) a i a (16) The term a A corrects for the asymmetric interactions which cannot be efficiently accounted for by the classical mixing a A = a 11 x 11 1 ı 1, x (17) =2 with ( ) ı 1, = (m 1, + m 2, T ) exp ( H /R) (18) T The new mixing rule (Eqs. (14) (18)) contains two parameters (k i, ı 1, ). The parameter k i is a constant and independent of temperature for water + natural gas components systems. However, the parameter ı 1, can be assumed to be a function of temperature as given by Eq. (18). In this work, m 1, was found to be zero for all binary systems considered in this work. However, in addition to k i, the parameters m 2, and ( H /R) were found and reported in Table 1. The new mixing rule does not suffer from the invariance problem so-called Michelsen and Kistenmacher syndrome [67]. In Eq. (17), knowing that subscript 1 is reserved for water the other components can be divided into a certain number of identical components with the same properties. The value of a A remains unchanged. In fact, Eq. (17) does not need the correction suggested by Mathias et al. [68] for the invariance problem. However, Eq. (17) is not suitable for two or more solvent systems. To evaluate the performance of the NM mixing rule, the same EoS with the non-density dependent (NDD) mixing rule [32] are used to calculate the phase equilibria of liquid water + natural gas systems. Detailed description of the NDD mixing rule is given in the supplementary materials. 3. Results and discussion The critical temperature (T c ), critical pressure (P c ) and acentric factor (ω) for each of the compounds used in this study were taken from Ref. [30] Binary systems The binary interaction parameters between natural gas components and liquid water were determined by adusting the model parameters through a Simplex algorithm using the obective function displayed by Eq. (19): N FOB = 1 x cal, x exp, M N + 1 y cal, y exp, x exp, M (19) y exp, where, N and M are the number of gas solubility and water content data points, respectively. The terms x cal, and y cal, refer to calculated gas solubility and water content, respectively, and the terms x exp, and y exp, represent the experimental gas solubility data and the water content of gas phase, respectively. Table 1 reports the adustable parameters determined from fits of data using the NM mixing rules with the three EoS. The corresponding parameters for the NDD mixing rule are provided in the supplementary materials. Detailed comparisons of the NM and the NDD mixing rules for calculating the mutual solubility of liquid water + natural gas main components using the NB EoS, the PR EoS and the SRK EoS are given in Tables 2 and 3. Table 2 presents water solubility in the gas phase and Table 3 natural gas components solubility in the liquid water Table 2 Calculated solubility (%AAD a ) of water in the gas phase for the systems water + natural gas components using different EoS. Component Temperature range (K) Ref NP NM mixing rule NDD mixing rule NB PR SRK NB PR SRK Methane [33,34] Ethane [35] Propane [36] Butane [37] 72 (59) Carbon dioxide [34,38] Hydrogen sulfide [39] Nitrogen [40 43] Overall a %AAD = (100/) xcal, x exp, /xexp,.

4 182 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) Table 3 Calculated solubility (%AAD a ) of natural gas components in water for the systems water + natural gas components using different EoS. Component Temperature range (K) Ref NP NM mixing rule NDD mixing rule NB PR SRK NB PR SRK Methane [34,44] Ethane [45] Propane [46] Butane [37] 72 (59) Carbon Dioxide [47] Hydrogen Sulfide [39] Nitrogen [48] Overall a %AAD = (100/) xcal, x exp, /xexp,. phase. Clearly, the PR EoS fits the experimental solubility data more accurately than the other two EoS; while the NB EoS and the SRK EoS exhibit more or less the same quality. The results for water solubility in the gas phase exhibit similar quality no matter which mixing rule is used. On the other hand, in the case of natural gas components solubility in water, for some components the NM mixing rule has superiority over the NDD mixing rule and for the other ones the NDD mixing rule is more accurate. Parts (a) and (b) of Fig. 1 show the water content of methane and the solubility of methane in liquid water, respectively. In Fig. 1a, the water content of methane is depicted as a function of pressure at different temperatures while in Fig. 1b the solubility of methane in liquid water is depicted. Clearly, both mixing rules reasonably describe the mutual solubility of the components for the mixture water + methane. In order to better visualize the low solubility region for the water content of the gas phase, the figures are presented in a logarithmic scale. Figs. 2 7 are similar to Fig. 1. Shown in part (a) of Figs. 2 7 are the water content of ethane, propane, n-butane, carbon dioxide, hydrogen sulfide, and nitrogen as a function of pressure. The solubilities of the same components in liquid water are depicted in part (b) of Figs Clearly, in predicting the water content of the gases, the NM mixing rule is somewhat better than the NDD mixing rule; however, in predicting the gas solubility in liquid water, the NDD mixing rule shows superiority. Fig. 8 shows that there are linear relationships with carbon number for the binary interaction parameters of the NM mixing rule. This suggests that the new mixing rule can be extended to heavy hydrocarbons using the linear relationships given in the figure with k i = 0.5. In order to further evaluate the performance of the model, comparisons were made between the results of the NM mixing rule with the PR EoS and some other correlations and equations, i.e., ideal model, ideal model + Poynting correction, Bukacek correlation [1], Kazim correlation [7], McKetta Wehe chart [8], Ning et al. correlation [9] and Aqualibrium software [10]. Table 4 shows the percent average absolute deviation (%AAD) in predicting the water content of methane using the aforementioned methods. As can be seen, in spite of relatively high pressure conditions and high non-ideality of the system, the results of our predictive model are in good agreement with the experimental data and show improvement over the other approaches. In some of these models the vapor pressure of pure water is required as an iut. Poor estimation of the vapor pressure will lead to poor estimation of water content of the gas phase. In this paper the vapor pressure was calculated from the relations reported by Daubert and Danner [49], and McCain [50] 3.2. Multicomponent systems Multicomponent experimental data for hydrocarbon + water systems are scares. Maority of them report gas phase water content data. Therefore, a robust thermodynamic method could be useful in the absence of any experimental data. None of the multicomponent Fig. 1. Mole fraction of coexisting liquid and gas phases for the system methane + water as a function of pressure: (a) water mole fraction in the methane-rich gas phase; (b) methane mole fraction in the aqueous liquid phase. Literature water content data [33,34]:, K;, K;, K;, K;, K;, K;, K;, K;, K; *, K;, K. Literature methane solubility data [34,44]:, K;, K;, K;, K;, K. Models: dashed lines stand for the PR EoS/NM mixing rule (model 1), and solid lines for the PR EoS/NDD mixing rule (model 2).

5 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) Fig. 2. Mole fraction of coexisting liquid and gas phases for the system ethane + water as a function of pressure: (a) water mole fraction in the ethane-rich gas phase; (b) ethane mole fraction in the aqueous liquid phase. Literature water content data [35]:, K;, K;, K;, K. Literature ethane solubility data [45]:, K;, K;, K;, K;, K;, K. Models: dashed lines stand for the NB EoS/NM mixing rule (model 1), and solid lines for the NB EoS/NDD mixing rule (model 2). Fig. 3. Mole fraction of coexisting liquid and gas phases for the system propane + water as a function of pressure: (a) water mole fraction in the propane-rich gas phase; (b) propane mole fraction in the aqueous liquid phase. Literature water content data [36]:, K;, K;, K;, K;, K;, K;, K. Literature propane solubility data [46]:, K;, K;, K;, K;, K;, K;, K; *, K. Models: dashed lines stand for the PR EoS/NM mixing rule (model 1) and solid lines for the PR EoS/NDD mixing rule (model 2). Fig. 4. Mole fraction of coexisting liquid and gas phases for the system n-butane + water as a function of pressure: (a) water mole fraction in the n-butane-rich gas phase; (b) n-butane mole fraction in the aqueous liquid phase. Literature water content data [37]:, K;, K;, K;, K;, K;, K. Literature n-butane solubility data [37]:, K;, K;, K; *, K;, K;, K. Models: dashed lines stand for the PR EoS/NM mixing rule (model 1), and solid lines for the PR EoS/NDD mixing rule (model 2).

6 184 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) Fig. 5. Mole fraction of coexisting liquid and gas phases for the system carbon dioxide + water as a function of pressure: (a) water mole fraction in the carbon dioxide-rich gas phase; (b) carbon dioxide mole fraction in the aqueous liquid phase. Literature water content data [34,38]:, K;, K;, K;, K;, K;, K;, K;, K;, K; *, K. Literature carbon dioxide solubility data [47]:, K;, K;, K;, K;, K. Models: dashed lines stand for the PR EoS/NM mixing rule (model 1) and solid lines for the PR EoS/NDD mixing rule (model 2). Fig. 6. Mole fraction of coexisting liquid and gas phases for the system hydrogen sulfide + water as a function of pressure: (a) water mole fraction in the hydrogen sulfide-rich gas phase; (b) hydrogen sulfide mole fraction in the aqueous liquid phase. Literature water content data [39]:, K;, K;, K;, K; *, K. Literature hydrogen sulfide solubility data [39]:, K;, K;, K;, K; *, K. Models: dashed lines stand for the PR EoS/NM mixing rule (model 1), and solid lines for the PR EoS/NDD mixing rule (model 2). Fig. 7. Mole fraction of coexisting liquid and gas phases for the system nitrogen + water as a function of pressure: (a) water mole fraction in the nitrogen-rich gas phase; (b) nitrogen mole fraction in the aqueous liquid phase. Literature water content data [40 43]:, K;, K;, K;, K;, K;, K;, K;, K;, K;, K; +, K;, K;, K; *, K. Literature nitrogen solubility data [48]:, K;, K;, K;, K;, K;, K;, K;, K;, K; *, K. Models: dashed lines stand for the PR EoS/NM mixing rule (model 1), and solid lines for the PR EoS/NDD mixing rule (model 2).

7 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) Fig. 8. The linearized parameters of the NM mixing rule versus carbon number of methane, ethane, propane, and butane: (a) the NB EoS, (b) the PR EoS, and (c) the SRK EoS. systems reported here was used to develop the binary interaction parameters reported in Table 1. All the predictions in this part are solely based on flash calculations. To achieve the best results, we used the binary interaction parameters given in the literature between the hydrocarbon and non-hydrocarbon components of the mixtures. Using Refs. [57 65], the binary interaction parameters were calculated for both the NB EoS, and the PR EoS. Detailed description of the formula is given in the supplementary materials. In order to further evaluate the performance of the NM mixing rule, some experimental water content data in the literature for eighteen ternary mixtures were collected. These solubility data were compared to the predictions of the model using the binary interaction parameters of water natural gas components gener- Table 4 Accuracies a of different models in predicting water content b of methane (this work uses the NM mixing rule with the PR EoS). T (K) P (MPa) Exp b This work Ideal model Ideal + Poynting factor Ning [9] Bukacek [1] Kazim [7] McKetta Wehe chart [8] Aqualibrium software [10] %AAD a a %AAD = (100/) xcal, x exp, /xexp,. b The water content [40] are multiplied by (10 3 ).

8 186 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) Table 5 Calculated water content of certain ternary mixtures using the PR EoS (experimental data from Ref. [51]). System Mixture No. Dry basis mole fraction Temperature range (K) %AAD a C 1 C 2 NDD mixing rule NM mixing rule H 2O+C 1 +C System Mixture No. Dry basis mole fraction Temperature range (K) %AAD a C 1 N 2 NDD mixing rule NM mixing rule H 2O+C 1 +N System Mixture No. Dry basis mole fraction Temperature range (K) %AAD a C 1 CO 2 NDD mixing rule NM mixing rule H 2O+C 1 +CO System Mixture No. Dry basis mole fraction Temperature range (K) %AAD a C 1 H 2S NDD mixing rule NM mixing rule H 2O+C 1 +H 2S System Mixture No. Dry basis mole fraction Temperature range (K) %AAD a C 1 C 3 NDD mixing rule NM mixing rule H 2O+C 1 +C a %AAD = (100/) xcal, x exp, /xexp,. ated in this work. Table 5 summarizes the comparisons made between the NM and NDD mixing rules for calculating the solubility of water in the gas phase using the binary interaction parameters generated in this work. The PR EoS was used for the equilibrium calculations. Clearly, the results are in good agreement with the experimental data. The results exhibit the reliability of the new mixing rule and the parameters reported in this work. Table 6 reports the summarized results obtained for calculating gas phase compositions for the sour quaternary systems of methane, carbon dioxide, hydrogen sulfide, and water. Prediction of the water content of sour natural gas mixtures is a very challenging test for thermodynamic models. The NDD and the new mixing rules with the NB EoS and PR EoS were used for equilibrium calculations over a temperature range from about 311 to 450 K and pressure from 4.8 to 17.3 MPa. This system herein reported is of particular importance, because the equilibrium is frequently occurred in natural gas systems. The NM mixing rule with two EOSs, especially the PR EoS, appears to be able to describe the phase behavior of liquid water + hydrocarbon containing acid gas systems quite accurately. The detailed analyses of the calculations are provided in the supplementary materials. The solubility data of hydrocarbon mixtures (two hydrocarbons or more) in water are very limited and often inaccurate [53]. Considering this limitation, we found it useful to investigate the experimental data of Dhima et al. [53] for the solubility of methane + ethane, and methane + n-butane mixtures in Table 6 Accuracies (%AAD a ) of the PR and NB EoS together with the NM and NDD mixing rules in predicting the gas phase mole fractions for the system CH 4 +CO 2 +H 2S+H 2O. The temperature ranges from 311 to 450 K and pressure from 4.82 to MPa. Experimental data are from Ref. [52]. Components PR NB NM mixing rule NDD mixing rule NM mixing rule NDD mixing rule CH CO H 2S H 2O a %AAD = (100/) xcal, x exp, /xexp,.

9 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) Table 7 Accuracies (%AAD a ) of the PR EoS with the NM mixing rule in predicting the solubility of methane, ethane, and their mixtures in pure water at K for the mixture C 1 +C 2 +H 2O. Pressure ranges from 20 to 100 MPa. Experimental data are from Ref. [54]. Table 8 Accuracies (%AAD a ) of the PR EoS with the NM mixing rule in predicting the solubility of methane, n-butane, and their mixtures in pure water for the mixture C 1 + n-c 4 at K. Pressure ranges from 20 to 100 MPa. Experimental data are from Ref. [53]. Components %AAD a Components %AAD a C C C 1 +C a %AAD = (100/) xcal, x exp, /xexp,. C n-c C 1 + n-c a %AAD = (100/) xcal, x exp, /xexp,. water. Table 7 presents the experimental and calculated solubility data for methane + ethane + water, and Table 8 for methane + n- butane + water ternary system. The PR EoS with the NM mixing rule was used for the equilibrium calculations. Amiriafari [54] and Amiriafari and Campbell [55] concluded that the solubility of the binary and ternary hydrocarbon mixtures is much greater than the solubility of the pure components at the same temperature and pressure. In the case of the methane + ethane and methane + n-butane mixtures this non-ideality becomes very important especially at high pressures. Therefore, as can be seen in Tables 7 and 8, the agreement with experimental data is reasonable. The detailed analyses of the calculations are provided in the supplementary materials. Table 9 presents comparisons made between experimental data for the water content of different sour gases that was investigated by Ng et al. [56] and the predictions from the NM mixing rule with the PR EoS. The iso-and n-butane are grouped into one component (n-butane). Good agreement with experimental data was obtained and the results demonstrated the accuracy of the NM mixing rule for estimating the water content of the sour gases. Table 9 Experimental [56] and predicted water content of different sour gas mixtures using the PR EoS with the NM mixing rule. T (K) P (MPa) Gas phase composition (mole fraction) Experimental Prediction %AD a Methane CO 2 H 2S b E E b E E b E E b E E c E E c E E b E E c E E c E E b E E Overall 7.96 a %AD = 100 xcal x exp /xexp. b Composition of this mixture is 90% methane + 6% ethane + 2.5% propane + 0.6% i-butane + 0.9% n-butane. c This mixture consists of methane and propane with a molar ratio equal to 95:5. Table 10 Experimental [66] and calculated water dew points for two gaseous mixtures using the PR EOS and NM mixing rule. Feed mol % Mixture Composition Pressure (MPa) T exp (K) T cal (K) T (K) a CH CO H 2S H 2O CH CO H 2S H 2O CH CO H 2S H 2O CH CO H 2S H 2O %AAD b 0.67 a T = T cal T exp. %AAD = (1/) T.

10 188 P. Reshadi et al. / Fluid Phase Equilibria 302 (2011) Fig. 9. Experimental [66] and predicted water dew point (L 2G), hydrocarbon dew point (L 1L 2G), and three phase bubble point (L 1L 2G) for the mixture 5% methane + 5% carbon dioxide + 40% hydrogen sulfide + 50% water. In Table 10 one can see the experimental and predicted water dew points for two sour gas mixtures. The PR EoS with the NM mixing rules were used for the calculations. The average absolute deviation was found to be 0.67 K. Clearly the agreement with the experimental data is good. For the mixture 2 given in Table 10, in addition to water dew points, the hydrocarbon dew point and three phase bubble point boundaries are depicted in Fig. 9. Shown in Fig. 9, the model accurately predicts the phase boundaries of the sour gas. Except near the critical point, the model posed no difficulty in calculating the three phase boundaries. 4. Conclusion The NM mixing rule developed for the a-parameter of the cubic EoS has been extended to multicomponent mixtures. This mixing rule needs at least two adustable parameters per each binary. The NM mixing rule accuracy is comparable to the NDD mixing rule in predicting the gas liquid equilibria of water + main natural gas components for a wide range of temperature. Reasonable agreement was achieved between the results of this method and the experimental data reported in the literature. In addition, the water content of the methane + water system has been evaluated by the NM mixing rule, which applied to the PR EoS, and compared with some empirical or semi-empirical approaches. The results have been encouraging and demonstrating the reliability of the new mixing rule and the thermodynamic approach used in this work. The PR EoS has shown a better agreement with the experimental data while the NB EoS and SRK EoS exhibit more or less similar quality. List of symbols a attraction energy parameter (MPa m 6 kmol 2 ) A kl group contribution parameter b molar co-volume parameter (m 3 kmol 1 ) B kl group contribution parameter f(x 2 ) correction factor for water water interaction parameter in KD and MHP mixing rules FOB obective function G constant in KD mixing rule (MPa m 6 kmol 2 ) HC hydrocarbon HCW m complex H enthalpy change of reaction (kj kmol 1 ) k 12 binary interaction parameter k i binary interaction parameter K E equilibrium constant l dimensionless constant for the binary interaction parameter of the NDD mixing rule N number of gas solubility data points N g number of group types M number of water content data points m constant m 1, m 2 constants (dimensionless, K) n constant for MHP mixing rule number of points P pressure (MPa) R universal gas constant ( MPa m 3 K 1 kmol 1 ) T temperature (K) u equation of state dependent constants v volume w equation of state dependent constants W water x apparent mole fraction X true mole fraction y gas phase mole fraction Greek letters a constant equal to 10 for MHP mixing rule ik fraction of molecule i occupied by group k ˇ12 binary interaction parameter in MHP mixing rule ı 12 binary interaction parameter in the new mixing rule correction factor for water water interaction in the new mixing rule a constant for MHP mixing rule Subscripts 1 water 2 Hydrocarbon/non-hydrocarbon C critical property cal calculated property exp experimental property i, molecular species k, l group type p polar components r reduced property 0 reference property Superscript indicates changes in a 11 parameter due to presence of hydrocarbon A asymmetric properties C classical properties 0 non-temperature dependent term in NDD mixing rules 1 temperature dependent term in NDD mixing rules Acknowledgment The financial support provided by the Parsian Gas Refinery under the contract number is gratefully acknowledged.

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