Using a New Proposed Influence Parameter of Gradient Theory for CH 4 / n-alkane Binary Systems : What Advances Can Be Achieved ?

The main aim of this modeling investigation is to improve the performance of the gradient theory for binary systems of methane and n-alkane. To achieve this aim, the gradient theory (GT) is combined with the volume-translated PengRobinson equation of state (VTPR EOS) for accurate description of phase equilibrium and surface tensions of methane/ hydrocarbon systems. To improve the phase equilibrium calculations, the binary interaction parameters of VTPR EOS for methane/hydrocarbon systems have been determined based on the mole fraction of methane in liquid phase. A new correlation of the influence parameter based on the densities of bulk phases has been suggested. To make the present model predictive, the binary interaction parameters of the mixture influence parameter have been set equal to zero. The results of the model show that the predictions of the present model agree well with experimental surface tensions, especially for very low values of surface tensions (near-critical interfaces).


Introduction
The accurate determination of the physical properties for (oil+gas) systems plays a dominant role in the petroleum industry.Among these physical properties, surface tension is a significant property in the exploration, production and processing of petroleum fluids.It is also an important property for crude oil recovery by using miscible gas injection technique in which very low surface tensions exist.On the other hand, it is known that methane is major component in reservoir fluids so that the surface tensions of methane/hydrocarbon binary systems must be accurately known [1,2].
Since the experimental measurement of the surface tension and properties is quite difficult, developing a suitable and a reliable model for predictions of surface tension and properties has been an intriguing subject of modeling investigations.According to the previous investigations, the preferential adsorption in the vapor-liquid interface depends on the size and interactions of the molecules; therefore, the models combining density (or volume) with the surface tension are more successful than the models utilizing only surface tension.The gradient theory of fluid interfaces [3][4][5] is one of the prediction models combining density (or volume) with the surface tension.The gradient theory has two inputs containing the Helmholtz free energy density and the influence parameter for the homogeneous and the inhomogeneous fluids, respectively.The Helmholtz free energy density is calculated based on the thermodynamic model selected.On the other hand, the influence parameter is a molecular theoretical parameter and it is difficult for researchers to use it in practice so that the semi-empirical expression is usually applied for this input of the model.
In recent years, three groups of empirical expressions have been used to determine the influence parameter of the gradient theory: (1) constant influence parameter [6][7][8]; (2) temperature dependent influence parameter [9,10]; (3) the influence parameter based on the densities of the vapor and liquid phases [11][12][13][14][15][16].The first group makes the theory easer.On the other hand, it was concluded that the second group lead to a significant improvement, but it has some limitations, for example, the temperature dependent influence parameter used by Zuo and Stenby [9,17,18] is not possible to be applied under supercritical conditions so that the reduced temperature should be set to a single value.In recent years, the third group of influence parameter of the gradient theory [11][12][13][14][15][16] has been successfully applied to refrigerant mixtures and water/gas systems.The third group of influence parameter has shown the potential to overcome the limitations of the first and second groups.
Since the knowledge of surface tension is necessary for methane/hydrocarbon binary systems to be used in the petroleum industry, this investigation has been dedicated to modeling the surface tension of methane/hydrocarbon binary systems.The gradient theory (GT) in combination with the volume-translated Peng-Robinson equation of state (VTPR EOS) [19] has been applied to describe the surface tension and density profiles of the interface for CH 4 /n-alkane binary systems.Furthermore, a new correlation based on the densities of the vapor and liquid phases have been proposed.The performance of new suggested influence parameter has been also compared with the previous proposed influence parameter in [11][12][13][14][15][16].Furthermore, the performance of suggested influence parameter has been discussed.

Model description 2.1 Gradient theory of fluid interface
The details of gradient theory exist in [2,10,20], therefore; the important equations of gradient theory has been presented here.Miqueu et al. [2,10] formulated the gradient theory of fluid interfaces for the mixtures as follows: In Eq. (1), Ω B = −P; P shows the pressure at equilibrium condition; ρ is the local mole density through the interfacial phase; σ denotes the surface tension.ρ L and ρ V are mole densities relating to the liquid and vapor phases, respectively.Subscript ref relates to the reference component.dρ i /dρ ref is mole density profile of component i and κ ij is the influence parameter.The grand thermodynamic potential, Ω(ρ), is computed by using the following equation: is the Helmholtz free energy density of the homogeneous fluid.µ is denotes the chemical potential of component i relating to the bulk phases.
The density profile that is perpendicular to the interface can be computed by using Eq.(3):

The free energy density and VTPR EOS
The Helmholtz free energy density of a homogeneous fluid can be computed by utilizing the following equation [20]: In this modeling investigation, the volume-translated Peng-Robinson (VTPR) equation of state [19] has been employed both for computing the thermodynamic properties of the vapor and liquid phases and the Helmholtz free energy density of the homogeneous fluid.The VTPR EOS is defined as: 2 in which a denotes the energy parameter and b is the volume parameter.The parameters of the PR EOS can be expressed as a function of critical properties.The critical properties of pure fluids [21] are shown in Table 1.
in which P is the pressure, T is the temperature, v is the molar volume and R is the universal gas constant.The subscripts c and r show the critical and reduced properties, respectively.The other parameters of PR EOS can be illustrated as follows:  The temperature-dependent volume translation parameter, c, is determined as follows: (1) (2) In Eqs.( 10)-( 14), Z c is the critical compressibility factor.
In this work, the classical van der Waals mixing rule is applied to the vapor-liquid phase equilibrium: in which x i denotes the mole fraction of component in phases.
The phase equilibrium calculations has been performed based on the applied flash calculations in [13,14].

The proposed influence parameter
As it is mentioned in the introduction section, the influence parameter is the second required input of the gradient theory.In previous investigations [11][12][13][14][15][16], the following form of influence parameter was used as a function of liquid and vapor densities.
In this work, this form of influence parameter has been proposed for gradient theory.
in which v c is the critical molar volume.
In order to apply gradient theory (GT) to mixtures, the influence parameter for each component in the mixture (κ i ) must be described in terms of the mole density of each component i in the liquid and vapor phases.Also, it is assumed that the binary interaction coefficient of the influence parameter (l ij ) is zero.
In Eqs. ( 20) and ( 21), x and y are the mole fractions of component i in the liquid and vapor phases, respectively.

The gradients of molar density
The gradients of molar densities for each component through the interfacial phase has been calculated by solving an algebraic equation on the basis of using the geometric combing rule for the influence parameters (l ij = 0).Equation (22) has been derived according to the minimization of Helmholtz free energy [2,10,20]: 3 Results and discussion

Pure fluids
Prior to modeling the surface tension and interfacial properties of methane/hydrocarbon systems, the influence parameters for each pure hydrocarbon containing methane (C 1 ), propane (C 3 ) n-butane (nC 4 ), n-hexane (nC 6 ), n-heptane (nC 7 ) and n-decane (nC 10 ) should be determined.The surface tensions of pure fluids have been taken from [22].The coefficients of influence parameter of the gradient theory have been presented in Table 2.The average absolute deviation of surface tension in liquid phase has been defined as follows:

Binary systems
As it is mentioned in the introduction section, the gradient theory has been used for computing the surface tension of (C 1 /hydrocarbon) binary systems.To achieve this aim, first, the phase equilibrium of the applied binary systems has to be determined.Based on the mole fraction of methane in liquid phase, the temperature dependent binary interaction coefficient is determined.The average absolute deviation of mole fraction in liquid phase has been defined as follows: Then, the surface tension of (C 1 /hydrocarbon) binary systems can be computed by using the gradient theory of fluid interfaces.In this section, first, the C 1 /C 3 binary system has been considered.The experimental phase equilibrium data are taken from [23].The temperature and pressure of experimental data range from 227 K to 344 K and 21 bar to 85 bar, respectively.The experimental surface tensions are taken from [24].
The following temperature dependent binary interaction coefficient has been determined for C 1 /C 3 binary system. .
The AAD% of 2.76 for phase equilibrium shows a good description of phase equilibrium for C 1 /C 3 binary system by using VTPR EOS.The experimental and calculated surface tensions are presented in Table 3 and Fig. 1.The results of model prove that the results of gradient theory in combination with applied influence parameters (Eqs.( 20) and ( 21)) are in a quite good agreement with experimental surface tensions for C 1 /C 3 binary system.Furthermore, one can see that the gradient theory in combination with the influence parameter obtained from Eq. ( 21) allows a very good description of surface tension for near critical condition so that a new correlation of the influence parameter can be appropriate for describing nearcritical interfaces.The density profiles of C 1 and C 3 through the interface have been shown in Fig. 2. Figure 2 indicates that the density profile of C 1 is not a monotonic function of z so that C 3 should be considered as reference fluid to produce these surface tensions.Also, the density profile of C 1 has a peak which shows that C 1 adsorbs at the interface.
Fig. 1 The experimental [23] and calculated surface tension vs. pressure for C 1 /C 3 binary system at T = 303.15K  The second system considered here is C 1 /nC 4 binary system.Although the gradient was applied by Nilssen [25] to this system, this calculation repeated in this investigation to recognize the advantages of new influence parameter.The experimental phase equilibrium data and surface tensions are from [26] and [27], respectively.To illustrate the effects of temperature and pressure, the experimental phase equilibrium calculations are done at temperatures ranging (294-344) K and pressures ranging (82-130) bar.The calculations concerning C 1 /nC 4 binary system are similar to the ones performed for C 1 /C 3 binary system.Based on the equilibrium compositions of C 1 in liquid phase, the following temperature dependent binary interaction coefficient has been regressed for C 1 /nC 4 binary system (AAD% = 3.54).
Table 4 compares the results of the present model with the experimental data and the model used by Nilssen [25].Also, nC 4 has been considered as a reference fluid for calculation of surface tension.One can see that the coupling of gradient theory (GT) and Eq. ( 21) as an influence parameter has the best performance, especially for very low surface tensions (nearcritical interfaces).In comparison with work of Nilssen [25], the present model is more accurate for description of near critical interfaces so the present model shows an advance for description of near critical interfaces.
Subsequently, the present model has been done for two other binary systems containing C 1 /nC 5 and C 1 /nC 6 at 310.93 K and 298.15 K, respectively.The range of pressures considered here is (10-168) bar.Similar to previous systems, binary interaction coefficient has been determined for these two binary mixtures.The experimental phase equilibrium data have been taken from [28,29].The optimum values of binary interaction coefficient are −0.0261 and 0.0307 for C 1 /nC 5 and C 1 /nC 6 , respectively.The AADs% of phase equilibrium are 3.31 and 0.77 for C 1 /nC 5 and C 1 /nC 6 systems, respectively.
The experimental [30,31] and calculated surface tensions of these two binary systems have been compared in Figs. 3  and 4. nC 5 and nC 6 are considered as reference fluids.For C 1 /nC 5 binary system, the AAD σ % of surface tension is 13.85 when gradient theory is used in combination with Eq. ( 20).This value is 5.82 when gradient theory is used in combination with Eq. ( 21); therefore, the new proposed influence is more accurate than the one in Eq. ( 20) suggested in previous investigations [11][12][13][14][15][16].For C 1 /nC 6 binary system, the performance of these two influence parameters is approximately the same.The AAD σ % are 6.15 and 6.12 by using Eq. ( 20) and Eq. ( 21) in combination with gradient theory, respectively.The present model has been extended to two additional binary systems including C 1 /nC 7 and C 1 /nC 10 .To our knowledge, the gradient theory was used by Miqueu et al. [2] to these systems.The gradient theory has been reapplied to recognize the advances when new proposed influence parameters are used.Similar to previous systems, the binary interaction coefficient are determined.The ranges of temperature and pressure are presented in Table 5.The binary interaction coefficient of C 1 /nC 7 is obtained as follows: Fig. 3 The experimental [30] and calculated surface tension vs. pressure for C 1 /nC 5 binary system at T = 310.93K Fig. 4 The experimental [31] and calculated surface tension vs. pressure for C 1 /nC 6 binary system at T = 298.15K Table 6 shows the range of thermodynamic conditions in which the surface tension modeling has been performed.The references of experimental surface tension are also reported in Table 6.As it is shown in Table 6, the performance of Eq. ( 20) is better than Eq. ( 21) for C 1 /nC 7 binary system and the performance of these two equations are approximately the same for C 1 /nC 10 binary system.Miqueu et al. [2] applied gradient theory for these two systems only at 310.93 K.In this study, by using the proposed influence parameter and suggested influence parameter in previous investigations [11][12][13][14][15][16], gradient theory is applicable for wider ranges of temperature, so an advance can be seen in comparison with the work of Miqueu et al. [2].Figures 5 and 6 compare the experimental and calculated surface tensions for C 1 /nC 7 and C 1 /nC 10 binary system.One can see that the present model is accurate for modeling the surface tension of both systems, especially for near critical interface.For C 1 /nC 7 binary system, the density profiles of C 1 and nC 7 has been shown in Fig. 7. Similar to C 1 /C 3 binary system, C 1 adsorbs at the interface.
Fig. 5 The experimental [30] and calculated surface tension vs. pressure for C 1 /nC 7 binary system at T = 338.7 K Fig. 6 The experimental [30] and calculated surface tension vs. pressure for C 1 /nC 10 binary system at T = 366.48K

Fig. 2
Fig. 2 The density profiles of C 1 and C 3 through the interface at T = 258.15K and P = 40.81bar

For C 1 /
nC 10 , the binary interaction parameter is expresses as follows:

Fig. 7
Fig. 7 The density profiles of C 1 and nC 7 through the interface at T = 338.87K and P = 108.84bar

Table 1
Critical properties of fluids

Table 2
The coefficients of influence parameter of pure components

Table 3
The experimental and calculated surface tensions of C 1 /C 3 binary system

Table 4
The experimental and calculated surface tension of C 1 /nC 4 binary system

Table 5
The range of thermodynamics conditions and AAD% of phase equilibrium calculations for C 1 /nC 7 and C 1 /nC 10 binary systems

Table 6
The range of thermodynamics conditions and AAD σ % of surface tension calculations for C 1 /nC 7 and C 1 /nC 10 binary systems