Nanoscale-extended alpha functions for pure and mixing confined fluids

Fluid Phase Equilibria 482 (2019) 64e80

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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d

Nanoscale-extended alpha functions for pure and mixing confined fluids Kaiqiang Zhang a, Na Jia a, *, Lirong Liu b a b

Petroleum Systems Engineering, Faculty of Engineering and Applied Science, University of Regina, Regina, Saskatchewan S4S 0A2, Canada Institute for Energy, Environment and Sustainable Communities, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada

a r t i c l e i n f o

a b s t r a c t

Article history: Received 26 July 2018 Received in revised form 18 October 2018 Accepted 22 October 2018 Available online 1 November 2018

In this paper, two new nanoscale-extended attractive (alpha) functions in Soave and exponential types are developed for the first time, which are applied and evaluated for the calculations of the thermodynamic and phase properties of confined fluids coupled with a modified equation of state (EOS). Moreover, a novel method is proposed and verified to determine the nanoscale acentric factors. The behaviour of several important parameters, i.e., minimum reduced temperature, nanoscale acentric factor, alpha function and its first and second derivatives, are specifically analyzed at different temperatures and pore radii. The newly-developed alpha functions are validated to accurately calculate the thermodynamic and phase properties in bulk phase (rp ¼ 1000 nm) and nanopores. The minimum reduced temperature from the Soave alpha function occurs at the acentric factor of u ¼ 0.295211 while the exponential function is monotonically related to the temperatures without any minimum conditions. Moreover, the acentric factors and intermolecular attractivities are found to be increased with the pore radius reductions at most temperatures, wherein they remain constant or slightly increase by reducing the pore radius at rp  50 nm while become quickly increased at rp < 50 nm. It should be noted that the alpha functions are decreased with the pore radius reduction at the critical temperature (Tr ¼ 1). The intermolecular attractivities are found to be stronger for the heavier or high carbon number components. Furthermore, the first and second derivatives of the Soave and exponential alpha functions to the temperatures are continuous at T  4000 K. Overall, the two original (Soave and exponential) and two nanoscale-extended alpha functions are proven to be accurate for the thermodynamic and phase calculations in bulk phase and nanopores. © 2018 Elsevier B.V. All rights reserved.

Keywords: Alpha functions Nanoscale Thermodynamic and phase properties Confined fluids Equations of state

1. Introduction Cubic equations of state (EOS) are widely applied in academic researches and industrial applications due to their simplicity and accuracy for predicting pure and mixing fluid phase and thermodynamic properties in vapour and liquid phases [1e5]. Since the well-known van der Waals (vdW) EOS was initiated in 1873 [6], numerous cubic EOS have been proposed for the thermodynamic equilibrium calculations during the past one and half centuries, such as the Redlich‒Kwong (RK), Soave‒Redlich‒Kwong (SRK), and Peng‒Robinson (PR) EOS etc. [7e9]. The capacity of these EOS in calculating the pure and mixing phase properties largely depends on the appropriate selections of

* Corresponding author. E-mail address: [email protected] (N. Jia). https://doi.org/10.1016/j.fluid.2018.10.018 0378-3812/© 2018 Elsevier B.V. All rights reserved.

attractive functions [10e13]. The attractive functions in the cubic EOS play important roles in accurately predicting the characteristics of a real pure component deviated from its ideal behaviour [14]. In principle, the existing alpha functions can be divided into two categories: Soave- and exponential-type alpha functions. A remarkable success in the alpha function development was achieved by Soave's work [7]. The original Soave alpha function was developed with respect to the acentric factors, which makes it possible to adequately correlate the phase behaviour of pure and mixing fluids containing non-polar or slightly polar components [15,16]. However, a major limitation of the Soave alpha function comes from its abnormal extrema at the supercritical region, where the attractive function performs a concave upward parabola and does not decrease monotonically with the temperature increases. Many efforts had been made to modify the original Soave alpha function and overcome its inherent limitations [17e20]. However, some further modifications usually introduced some new deviation

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

terms while the inherent limitation was still remained. Later, the exponential-type alpha function, which demonstrates an asymptotic behaviour when the reduced temperature is approaching infinity, was proposed by Heyen [21] and further modified by Trebble and Bishnoi [22] as well as Twu et al. [23]. Recently, a series of consistency tests for the alpha functions of the cubic EOSs were conducted and presented [24]. Overall, the following three basic conditions are required for the alpha functions: 1. the alpha function must be finite and positive at all temperatures; 2. the alpha function equals to the unity at the critical point; 3. The alpha function approaches a finite value as the temperature becomes infinity [25]. The three requirements can be easily satisfied if the Soave- and exponential-type alpha functions are utilized concurrently. Although numerous alpha functions have been developed and/or modified in the past decades, all existing alpha functions are for the bulk phase case while no work has been developed for the phase properties of confined fluids in nanopores. In recent years, confined fluids in porous media, especially in the nanoscale spaces, attract more attentions due to its wide and practical applications, for example, inorganic ions pass through the cell membranes [26], industrial separation process and heterogeneous catalysis [27,28], and oil/gas production from shale reservoir [29]. The confinement effect is significantly strengthened when the pore radius reduces to the nanometer scale and comparable to the molecular size, which causes drastic changes for fluid phase properties even in qualitative perspectives [30e32]. More specifically, capillary pressures become considerably large [33], critical properties of the confined fluids shift to different extent [34], molecule‒molecule and molecule‒wall interactions are enhanced [35,36], all of which are resulted from the strong confinement effects and could cause substantial changes in terms of the thermodynamic phase properties at small pore radius. Previous study indicates that the bubble point pressure of a mixing hydrocarbon‒ CO2 system is significantly decreased, while the upper dew-point pressure increases and lower dew-point pressure decreases with an increasing confinement effect for a gas condensate system [37]. The solubility parameter and minimum miscibility pressure (MMP), which is defined as the lowest operating pressure at which the oil and gas phases can become miscible in any portions at an oil reservoir temperature [38], are significantly decreased with the reduction of the pore radius [39,40]. At the current stage that the experimental approaches are incapable of fully exploring the nanoscale phase properties, the modified EOS coupled with nanoscale-extended alpha functions are necessarily and immediately required. In this study, two new nanoscale-extended alpha functions in the Soave and exponential types are developed by considering the confinement effects, which are applied to calculate the thermodynamic and phase properties of pure and mixing confined fluids in bulk phase (rp ¼ 1000 nm) and nanopores by coupling with a modified SRK EOS. It should be noted that rp ¼ 1000 nm is assumed to be the bulk phase as a result of a series of trial tests, where rp ¼ 10,000, 100,000, and 1,000,000 nm have been also used for the calculations and their results are almost equivalent to those at rp ¼ 1000 nm. On the other hand, some previous studies have validated that the phase behaviour at rp > 100 nm are similar to the bulk phase cases [29,34,40,41]. The modified SRK EOS is developed by following the similar theoretical manner of the modified vdW EOS in the previous study [34]. The calculated phase and thermodynamic properties from the new models are compared to and validated with the measured data in bulk phase and nanopores. Moreover, a new method is proposed and verified to determine the nanoscale acentric factors. A series of important parameters, which are the minimum reduced temperature, nanoscale acentric factor, alpha functions and their first and second derivatives, are

65

specifically studied to analyze their behaviour at different temperatures and pore radii. Finally, the original and nanoscaleextended alpha functions in Soave and exponential types are compared and evaluated in terms of the thermodynamic and phase property calculations in bulk phase and nanopores.

2. Material In this study, pure CO2, N2, and a series of alkanes from C1eC10 are used, whose critical properties (i.e., temperature, pressure, and volume), SRK EOS constants, and Lennard-Jones potential parameters are summarized [42e44] and listed in Table 1. The pressure‒ volume‒temperature (PVT) tests of the C8H18eCH4 system were conducted at T ¼ 311.15 K and the pore radii of rp ¼ 3.5 and 3.7 nm (i.e., silica-based mesoporous materials SBA-15 and SBA-16) [45]. Furthermore, the PVT tests for the N2‒n-C4H10 system were conducted by using a conventional PVT apparatus connected to a hightemperature and pressure container with a shale coreplug at the temperatures of T ¼ 299.15 and 324.15 K [46]. It should be noted that the shale coreplug was hydrocarbon-wetting and its dominant pore radius was around 5 nm, which is applied for the subsequent calculations in this study. The purities of N2 and n-C4H10 used in the experiments equal to 99.998% and 99.99%, respectively. In addition, the measured phase properties of a ternary hydrocarbon mixture systems of 4.53 mol.% n-C4H10 þ 15.47 mol.% i-C4H10 þ 80.00 mol.% C8H18 [47] are applied to verify the newly-developed model with the nanoscale-extended alpha functions. The detailed experimental set-up and procedures for conducting the above-mentioned PVT tests can be referred in the literature [45e48].

3. Theory 3.1. Modified equations of state The conventional SRK EOS is modified to consider the confinement-induced effects of pore radius and moleculemolecule interactions in nanopores. The SRK EOS is one of the most commonly-accepted and widely-used cubic EOS which are usually capable of accurately predicting the vapour‒liquid equilibrium (VLE) and fluid stability/metastability [6,7,49,50]. Suppose that a nanoscale pore system, as shown in Fig. 1, consists of some confined particles via the Lennard‒Jones potential. The canonical partition function from the statistical thermodynamics is shown as follows [51],

Q ðN; V; TÞ ¼

X 1 eEi ðN;VÞ=kT ¼ L3N qN int ZðN; V; TÞ N! i

(1)

where N is the number of molecules; V is the total volume; T is the temperature; E is the overall energy state; k is the Boltzmann constant; L is the de Broglie wavelength, L ¼ ð2phmkT Þ0:5 , h is the Planck's constant, m is the molecular mass; qint is the internal partition function; and Z is the configuration partition function, which is expressed as, 2

ZðN; V; TÞ ¼ ∬ eUðr1 ;r2 ;:::;rN Þ=kT dr1 dr2 :::drN

(2)

V

where U is the potential energy of entire system of N number of molecules which positions are described by ri, i ¼ 1,2, …N, and ri is the distance of separation between molecules. The detailed analytical derivations of the canonical partition function can be found in the previous study [34], so the pressure is expressed as,

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K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

Table 1 Recorded critical properties (i.e., temperature, pressure, and volume), Soave‒Redlich‒Kwong equation of state (EOS) constants, and Lennard-Jones potential parameters of CO2, N2, O2, Ar, and C1eC10 [42e44]. Tc ðKÞ

Component

Pc ðPaÞ

ac ðPa,m6 =mol2 Þ

5

7.38  10 3.39  106 5.04  106 4.87  106 4.60  106 4.88  106 4.25  106 3.65  106 3.80  106 3.37  106 3.29  106 3.14  106 2.95  106 2.73  106 2.53  106

304.2 126.2 154.6 150.8 190.6 305.4 369.8 408.1 425.2 469.6 507.5 543.2 570.5 598.5 622.1

CO2 N2 O2 Ar CH4 C2H6 C3H8 i-C4H10 n-C4H10 C5H12 C6H14 C7H16 C8H18 C9H20 C10H22

Vc ðm3 =molÞ 6

48

9.40  10 8.95  105 8.00  105 7.50  105 9.90  105 1.48  104 2.03  104 2.63  104 2.55  104 3.04  104 3.44  104 3.81  104 4.21  104 4.71  104 5.21  104

1.02  10 3.82  1049 3.86  1049 3.80  1049 6.44  1049 1.56  1048 2.62  1048 3.72  1048 3.88  1048 5.33  1048 6.38  1048 7.66  1048 8.99  1047 1.07  1047 1.24  1047

b ðm3 =molÞ 29

7.12  10 4.45  1029 3.66  1029 3.70  1029 4.96  1029 7.48  1029 1.04  1028 1.34  1028 1.34  1028 1.66  1028 1.85  1028 2.07  1028 2.31  1028 2.62  1028 2.94  1028

ε=k ðKÞ

s ðmÞ

294 36.4 50.7 120 207 155 120 140 118 145 199 206 213 220 226

2.95  1010 3.32  1010 3.05  1010 3.41  1010 3.57  1010 3.61  1010 3.43  1010 3.85  1010 3.91  1010 3.96  1010 4.52  1010 4.70  1010 4.89  1010 5.07  1010 5.23  1010

Fig. 1. Schematic diagrams of the nanopore network and its associated potential.

ðT ð

PðN; V; TÞ ¼ kTð

vln½ðV  N bÞN e

4εLJ 3 1 Uðr1 ; r2 :::rN Þ ∬ dV1 dV2 ¼ s f ðAr Þ V r12 > s kT kT LJ

Econf ðN; V; TÞ

∞

kT 2

vV vEconf ðN; V; TÞ NkT ¼  V  Nb vV

dTÞ



ÞN;T

(3)

c1 c þ 2 f ðAr Þ ¼ c0 þ pffiffiffiffiffi Ar Ar where c0 ¼  89p, c1 ¼ 3:5622, c2 ¼  0:6649, and Ar is the

where b is the excluded volume per fluid molecule and gðr; r; TÞ is the pair correlation function for molecules interacting through the

reduced contact area [34]. Accordingly, Econf ðN; V; TÞ is presented as,

potential UðrÞ. Econf ðN; V; TÞ is expressed as [34],

Econf ¼ Econf ¼

kTn2 C Uðr1 ; r2 :::rN Þ dV1 dV2 ∬ kT 2V 2 ri > s

(5)

(4)

Fluid interactions UðrÞ are assumed to be numerically represented through the Lennard-Jones potential, whose schematic diagram is shown in Fig. 1. Then, the integral part of Eq. (4) is solved semi-analytically as,

an2 C þ 2n2 CεLJ s3LJ ,ðpc1ffiffiffiffi þ Ac2r Þ Ar

nb

lnð

V Þ V þ nb

(6)

where εLJ is the moleculemolecule Lennard‒Jones energy parameter and sLJ is the moleculemolecule Lennard‒Jones size parameter. The modified vdW and SRK EOS for the confined fluids in nanopores are obtained by substituting Eq. (6) into Eq. (3) with a specific C [35], whose molar base formulation (i.e., divided by the mole number) is shown as follows,

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

PNPSRK ¼

RT aT c1 c  þ 2 Þ ½ac  2εLJ s3LJ ,ðpffiffiffiffiffi v  b vðv þ bÞ A Ar r

(7)

where aT is the so-called a function. Eq. (7) is the modified SRK EOS for calculating the phase behaviour of pure and mixing confined fluids in nanopores. 3.2. Critical properties in nanopores The critical temperature and pressure of the confined fluids in 2

vP Þ ¼ ð v P Þ ¼ 0, nanopores can be solved at the condition of ðvV T vV 2 T which are derived from the modified SRK EOS in Eq. (7) and shown below, respectively,

TcpSRK ¼

pffiffiffi 2 3ð 3 2  1Þ ,aT c1 c þ 2 Þ ½ac  2εLJ s3LJ ,ðpffiffiffiffiffi bR Ar Ar

(8a)

PcpSRK ¼

pffiffiffi ð 3 2  1Þ3 ,aT c1 c þ 2 Þ ½ac  2εLJ s3LJ ,ðpffiffiffiffiffi b2 Ar Ar

(8b)

where Tcp and Pcp are the respective critical temperature and pressure in nanopores. It is well known that the corresponding bulk fluid critical properties from the conventional SRK EOS are: p3 ffiffiffi p3 ffiffiffi 2 3 ,aT ,aT ac and PcSRK ¼ ð 21Þ ac . Hence, the shifts of TcSRK ¼ 3ð 21Þ bR b2 critical temperature and pressure in nanopores from the SRK EOS are shown as follows,

ð

Tc  Tcp c sLJ c sLJ ÞSRK ¼ 2pffiffiffi1 þ 2 2 ð Þ2 Tc pac rp pac rp ¼ 0:7197

ð

sLJ rp

sLJ

 0:0758ð

rp

Þ2

(9a)

Pc  Pcp c sLJ c sLJ ÞSRK ¼ 2pffiffiffi1 þ 2 2 ð Þ2 Pc pac rp pac rp ¼ 0:7197

sLJ rp

sLJ

 0:0758ð

rp

Þ2

The equations for calculating the critical shifts in Eqs. (9a) and (9b) are considered to be general/universal since they are equivalent to those from the modified vdW EOS in the previous study [34]. 3.3. Nanoscale acentric factors Acentric factor (u) is an empirical parameter reflecting the deviation of acentricity or non-sphericity of a compound molecule from that of a simple fluid (e.g., argon or xenon) [52], which was originally introduced by Pitzer et al. [14,42] and modified by incorporating the shifts of the critical properties in nanopores as follows,

u ¼ logðPrNP ÞTrNP ¼0:7  1 PrNP ¼

are calculated at different reduced pressures, which are plotted versus the reciprocal of the reduced temperatures in nanopores (i.e., 1/TrNP ). Fig. 2 shows a sample that the calculated logarithm reduced pressures for CO2, N2, and alkanes of C1‒10 at the pore radius of rp ¼ 1 nm are plotted with respect to the reciprocal of the reduced temperatures. Therein, the calculated logarithm reduced pressures at 1/TrNP ¼ 1.429 (i.e., TrNP ¼ 0.7) are applied to calculate the nanoscale acentric factors in Eq. (10). It can be anticipated that the acentric factors are various at different pore radii because the critical temperatures and pressures are dependent on the pore radius. The logarithm reduced pressures of the CO2 and alkanes of C110 are calculated at different pore radii of rp ¼ 1e1000 nm and plotted versus the reciprocal of the reduced temperatures, by means of which the acentric factors at different pore radii are determined. The determined acentric factors for the CO2, N2, O2, Ar, C1eC10, C12, C14, C16, C18, and C19 in bulk phase are listed and compared with the measured data in Table S1 of the Supplementary Materials. It is found that the calculated acentric factors agree well with the measured results, whose absolute average deviations are in the range of 0.83e22.07%.

3.4. Modified alpha functions in nanopores For a thermodynamic phase equilibria, the alpha function takes account of the attractivities between the molecules, which depends on the reduced temperature (Tr ) and acentric factor (u). As aforementioned that the existing alpha functions can be divided into two categories: Soave-type (aTS ) and exponential-type (aTE ) functions. Two typical alpha functions in these two categories have been validated for the polar or non-polar non-hydrocarbons and light hydrocarbons [17,53], which are extended for the nanoscale calculations by substituting the above-mentioned modified parameters and shown as,

aTSNP ¼ ½1 þ mSNP ð1  (9b)

(10)

Pv T ; T ¼ Pcp rNP Tcp

where PrNP is the reduced pressure in nanopores, TrNP is the reduced temperature in nanopores, and Pv is the vapour pressure. The acentric factor defined at the reduced temperature of Tr ¼ 0.7 has been validated to be accurate for the substances like CO2 and alkanes of C110 [42], which still used in nanopores because the reduced temperature is a dimensionless parameter. The logarithm reduced pressures of the CO2 and alkanes of C110

67

pffiffiffiffiffiffiffiffiffiffiffiffi 2 TrNP Þ

mSNP ¼ 0:480 þ 1:574uNP  0:176u2NP

(11a)

ENP aTENP ¼ exp½ð2 þ 1:0444TrNP Þ,ð1  T m rNP Þ

mENP ¼ 0:15683 þ 0:51494uNP  0:054124u2NP

(11b)

Eq. (11a) and (11b) are the new nanoscale-extended Soave and exponential alpha functions. It should be noted that the constants in Eq. (11b) are updated for the SRK EOS through the vapour pressure minimization of 36 pure components.

3.5. Vapour‒liquid equilibrium calculations The modified SRK EOS (i.e., Eq. (7)) is applied to calculate the VLE properties in bulk phase and nanopores. The shifts of critical properties (i.e., critical temperature and pressure) of the confined fluids are predicted by using the modified equations from Eq. 9a and b. In addition, the liquid and vapour phases are assumed to be the wetting phase and non-wetting phase, respectively [42]. Thus the capillary pressure (Pcap ) is,

Pcap ¼ PV  PL

(12)

where PV is the pressure of the vapour phase and PL is the pressure of the liquid phase. On the other hand, the capillary pressure can be expressed by YoungLaplace equation,

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K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

Fig. 2. Calculated logarithm reduced pressures for CO2, N2, and alkanes of C1‒10 at the pore radius of rp ¼ 1 nm with respect to the reciprocal of the reduced temperatures.

Pcap ¼

2g cos q rp

(13)

where g is the interfacial tension (IFT) and q is the contact angle of the vapourliquid interface with respect to the pore surface, which is assumed to be 30 according to the experimental results in the literature [47]. Therein, the IFT is estimated by means of the MacleodSugden equation [29],

g ¼ ðrL

r X i¼1

xi p i  r V

r X

component in the liquid and vapour bulk phases, i ¼ 1, 2, …, r; r is the component number in the mixture; and pi is the parachor of the ith component. The VLE calculations based on the modified SRK EOS require a series of iterative computations through, for example, the NewtonRaphson method. The calculated vapour pressures for the CO2, N2, O2, Ar, and C1eC10 in bulk phase are listed and compared with measured results in Table 2. 3.6. Enthalpy of vaporization and heat capacity

yi p i Þ4

(14)

i¼1

where xi and yi are the respective mole percentages of the ith

The new nanoscale-extended alpha functions in Eq. (11a) and (11b) are required to be validated by comparing with the experimental measured or literature recorded thermodynamic

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

properties. Some common thermodynamic properties in bulk phase may not be available in nanopores. Here, the enthalpy of vaporization is selected for the validation purpose because of its data availability in nanopores. The detailed derivations of the enthalpy of vaporization and heat capacity are specified in the Supplementary Materials.

4. Results and discussion 4.1. Model verifications The proposed VLE model, which is on a basis of the modified SRK EOS and coupled with two new nanoscale-extended alpha functions (i.e., M-Soave and M-exponential), is applied to calculate a series of phase and thermodynamic properties in bulk phase and nanopores. The calculated vapour pressures for the CO2, N2, O2, Ar, and C1eC10 in bulk phase from the new nanoscale-extended SRK EOS and alpha functions are listed in Table 2, which are compared with the experimentally measured data and calculated results from the Peng‒Robinson EOS (PR EOS) coupled with original Soave and exponential alpha functions at different temperatures from the literature [11,12,14,15]. It is found that both the PR EOS coupled with the original alpha functions and the modified SRK EOS coupled with the nanoscale-extended alpha functions are capable of accurately predicting the vapour pressures in bulk phase. The proposed models in this study provide more accurate vapour pressures with overall percentage average absolute deviations (AAD%) of 0.72% (MSoave) and 0.94% (M-exp) and maximum absolute deviations (MAD %) of 1.37% and 1.89% compared with the experimentally measured data, which are better than those from the previous studies with overall AAD% of 0.85% and 0.98% and MAD% of 3.84% and 5.03%. Furthermore, the enthalpies of vaporization and constant-pressure heat capacities for the CO2, N2, O2, Ar, and C1eC10 in bulk phase are calculated from the modified SRK EOS coupled with the M-Soave and M-exp alpha functions and listed in Table S2. In comparison with the calculated results from the previous studies, the newlyproposed models perform better in the constant-pressure heat capacity with overall AAD% of 7.77% and 10.71% and MAD% of 13.45% and 19.76% but become comparable in the enthalpy of

69

vaporization calculations. Overall, the nanoscale-extended SRK EOS and two alpha functions are accurate for calculating the phase and thermodynamic properties in bulk phase. The new models are better to be verified at the nanometer scale by comparing their results with the experimentally measured data. Fig. 3a and b shows the measured [54] and calculated enthalpies of vaporization and heat capacities for the N2 in bulk phase and nanopores at different temperatures and pore radii, keep in mind that the nanoscale experimental phase and thermodynamic data are extremely scarce currently. It is easily seen from these figures that the calculated enthalpies of vaporization and heat capacities from the nanoscale-extended SRK EOS coupled with the two new alpha functions agree well with the measured thermodynamic data in bulk phase and nanopores. Hence, the newly-developed nanoscale-extended SRK EOS coupled with the two new alpha functions are capable of accurately calculating the phase and thermodynamic properties in bulk phase and nanopores.

4.2. Parameter analyses 4.2.1. Minimum reduced temperatures The alpha functions are a function of the reduced temperature and acentric factor. The temperature-dependent features of the alpha functions are specifically studied here since the acentric factors are normally fixed at a constant pressure and/or pore radius. The first derivatives of the Soave and exponential alpha functions to the reduced temperature are shown as follows,

vaTS m ½1 þ mS ð1  pffiffiffiffiffi ¼ S vTr Tr

pffiffiffiffiffi Tr Þ

(15a)

vaTE 0:836 2mE E ¼ exp½ð2 þ 0:836Tr Þ,ð1  T m r Þ,½ mE  vTr Tr Tr  0:836ð1 þ mE Þ

(15b)

Suppose that Eq. (15a) and (15b) infinitely approach zero (please note that the second term is assumed to be zero since the first term right-hand side of Eq. (15b) won't be zero) in order to obtain the minimum conditions,

Table 2 Calculated vapour pressures for the CO2, N2, O2, Ar, and C1eC10 in bulk phase from the literature [14,15] and vapour‒liquid equilibrium model coupled with the new nanoscaleextended equation of state and alpha functions. Component

Tr

Pm v (kPa)

NDP

AAD (%) Soave [14,15]

exp [14,15]

M-Soave

M-exp

CO2 N2 O2 Ar C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 AAD (%) MAD (%)

0.714e1.000 0.500e1.000 0.420e1.000 0.556e1.000 0.476e1.000 0.780e1.000 0.698e1.000 0.753e1.000 0.745e1.000 0.581e1.000 0.687e1.000 0.632e1.000 0.607e1.000 0.528e1.000 e e

530.33e7386.59 12.52e3400.20 0.15e5043.00 68.95e4860.52 11.70e4596.09 912.99e4863.50 291.79e4239.31 456.65e3796.02 338.78e3369.00 17.31e3020.00 99.66e2740.00 29.04e2486.00 13.24e2305.00 0.87e2110.00 e e

65 77 32 67 48 30 34 48 25 54 49 25 20 94 e e

0.27 0.75 2.05 0.87 0.67 0.34 0.26 0.36 0.25 0.70 0.25 0.52 0.80 3.84 0.85 3.84

0.18 0.20 1.20 0.43 0.67 0.45 0.43 0.68 0.52 1.46 0.77 1.03 0.69 5.03 0.98 5.03

0.45 0.88 1.37 1.03 1.24 0.22 0.31 0.59 0.47 0.66 0.34 0.71 0.68 1.17 0.72 1.37

0.28 0.67 1.89 1.12 1.55 0.76 0.28 0.83 1.03 0.71 0.64 1.00 0.72 1.68 0.94 1.89

m experimentally measured. NDP number of data point. AAD average absolute deviation. MAD maximum absolute deviation.

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K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

Fig. 3. Measured [54] and calculated enthalpies of vaporization and heat capacities for the N2 from the modified equation of state with the two nanoscale-extended alpha functions at different temperatures in bulk phase and nanopores.

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ mS Trmin ¼ mS

(16a)

0:836 2mE  ¼ 0:836ð1 þ mE Þ E T Tm rmin rmin

(16b)

The equivalent conditions for Eq. (16b) can never be satisfied since its left-hand side terms are always larger than the right-hand side terms. Hence, the exponential alpha function is monotonically related with the reduced temperature so that no minimum conditions can be obtained. Eq. (16a) shows that the minimum

conditions for the Soave alpha function can be reached when mS is positive. The calculated m and 1 þ m in the Soave and exponential types are plotted versus the acentric factors in Fig. 4a. It is seen from the figure that the mS and 1þmS are always positive at pffiffiffiffiffiffiffiffiffiffiffiffiffiffi u > 0.295211. In Fig. 4b, the ratios of 1þmS to mS (i.e., Trmin ) are plotted with respect to the acentric factors. It is found from the figure that a vertical asymptote occurs roughly at u ¼ 0.295211, lower than which the ratios are always negative and become more negative by increasing the acentric factors. On the other hand, the ratios are always positive and decreased or even be a horizontal asymptote (approaching the unity) with the acentric factor at u > 0.295211. Thus, the minimum reduced temperature decreases

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

with the acentric factor increases. Cautious should be taken that an increasing acentric factor usually causes the critical properties to be larger so that the minimum reduced temperature reversely increases [16]. 4.2.2. Nanoscale acentric factors Only the nanoscale acentric factors from the newly-proposed method are specified here since the acentric factors in bulk phase have been extensively introduced in the previous publications. The nanoscale acentric factors of the CO2, N2, and C1‒10 at the pore radius of 1e1000 nm are determined by means of the new method

71

demonstrated in Fig. 2, which are plotted versus the pore radius in Fig. 5a and b. It is found from the figures that the acentric factors are increased with the pore radius reductions. More precisely, the acentric factors remain constant or slightly increase with the pore radius reductions at rp  50 nm while they become quickly increased at rp < 50 nm. As the acentric factors for different components vary in bulk phase, in a similar manner, their behaviour in nanopores are also different. Normally, for alkanes, the acentric factors in bulk phase and nanopore are increased with the carbon number increase. For the twelve components, the acentric factors of the CO2 and C1‒10 perform similar patterns and are sensitive to

Fig. 4. Calculated (a) m and mþ1 from the Soave and exponential type alpha functions and (b) ratios of mþ1 to m from the Soave type alpha function with respect to the acentric factors from 0.5 to 2.

72

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

the pore radius except that the acentric factors of N2 are slightly insensitive. 4.2.3. Attractive functions in bulk phase and nanopores Figs. S1a‒l in the Supplementary Materials show the calculated Soave and exponential alpha functions and dimensionless attractive term of A (i.e., A ¼

ac aT P ) ðRTÞ2

for the CO2, N2, and C1‒10 in bulk phase

at different temperatures. The alpha functions in different types perform significantly different. The Soave alpha functions are initially decreased with the temperature increase but increase once passing the minimum points, whereas the exponential alpha

functions are monotonically decreased with the temperature and asymptotically approach zero. The minimum Soave alpha functions at the supercritical conditions are various for different components that the high carbon-number alkanes reach the minimum conditions at higher temperatures in comparison with those of the low carbon-number alkanes or gaseous components like N2. Moreover, at extremely high temperatures (4000 K), the Soave alpha functions of the alkanes are almost equivalent at around 0.5, which is lower than the CO2 (1.4) and N2 (2.3) cases. On the other hand, the dimensionless attractive term A are quickly reduced with the temperature at low temperatures and become almost constant

Fig. 5. Calculated acentric factors for CO2, N2, and alkanes of C1eC10 at different pore radii of rp ¼ 1e1000 nm.

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

afterwards. It should be noted that the A in the Soave and exponential types are equivalent. However, they perform marginally different for different components. In a similar manner with the alpha functions, the A of the CO2, N2 and lighter alkanes like C1 and C2 are decreased more quickly with the temperature increase in comparison with those of the higher carbon-number alkanes. Hence, it may be concluded that the attractivities between molecules are reduced with the temperature and become asymptotically to zero when temperatures exceed some certain high values. In addition, the attractivities between the higher carbon-number alkanes are stronger than those of the lighter components. The Soave alpha functions of the CO2, N2, and C1‒10 are calculated by means of the new models at the pore radii of 1e1000 nm, which are plotted with respect to the temperatures in Figs. S2a‒l. The Soave alpha functions in nanopores are found to be much different from those in bulk phase but also presented as various concave upward parabola curves with respect to the temperatures. More specifically, the alpha functions at the pore radius of 1000 nm equal to those in bulk phase and become different by reducing the pore radius. As the pore radii become smaller, both the initial reductions and subsequent increases of the Soave alpha functions become faster, their minimum conditions occur at lower

73

temperatures, and their values at high temperatures are significantly larger. Moreover, the temperature effect on the Soave alpha functions are dependent on the pore radii. For example, in comparison with the CO2 case, the Soave alpha functions of the N2 in bulk phase have a more drastic decrease and increase versus temperatures and a lower temperature for the minimum condition while become less sensitive at rp  10 nm. In a similar manner with that in bulk phase, the CO2, N2, or lighter alkanes like C1 and C2 cases are still sensitive to the temperature increases in nanopores. Furthermore, the alpha function behaviour of the CO2 and alkane cases initiate to be substantially different at rp < 50 nm while that of the N2 case occurs at rp < 10 nm. Thus, it may be concluded that the alpha functions of some components, such as the N2, are sensitive to the temperatures but can be insensitive to the pore radii. For alkanes, their sensitivities of the alpha functions to the temperatures and pore radii are weakened with the carbon number increase. The exponential alpha functions of the CO2, N2, and C1‒10 in nanopores are much different from those in bulk phase but also decreased monotonically with the temperatures, which are similar to the Soave alpha functions in nanopores and presented in Figs. S3a‒l. Figs. S4a‒l and S5a‒l show the dimensionless attractive term A

Fig. 6. a and b Calculated alpha functions in Soave type for CO2, N2, and alkanes of C1‒10 at different pore radii of rp ¼ 1e1000 nm and reduced temperatures of (a1 and a2) Tr ¼ 0.01 and (b1 and b2) Tr ¼ 1. 6c and d Calculated alpha functions in Soave type for CO2, N2, and alkanes of C1‒10 at different pore radii of rp ¼ 1e1000 nm and reduced temperatures of (c1 and c2) Tr ¼ 3 and (d1 and d2) Tr ¼ 8.

74

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

in Soave and exponential types for the CO2, N2, and C1‒10 in nanopores at different temperatures and rp ¼ 1e1000 nm. The calculated A in Soave type from Figs. S4a‒l are initially reduced with the temperature to the minimum and then reversely increase afterwards, which are similar to the patterns of the Soave alpha functions in nanopores but different from the A vs. temperature curves in bulk phase. The corresponding temperatures for the minimum A in Figs. S4a‒l are the exactly same with those for the minimum alpha functions in Figs. S2a‒l. The values of the A in nanopores are larger than those in bulk phase at most temperatures. It is easily seen from the figures that the apparent differences in terms of the A values for the CO2 and alkanes are initiated at rp < 50 nm and for the N2 case at rp < 10 nm, which can be also observed from the Soave and exponential alpha functions in Figs. S2 and S3 as well as the calculated A in exponential type in Figs. S5a‒l. Overall, the attractive values of the heavier components, such as the intermediate alkanes, become larger than those of the light alkanes like C1‒2, CO2 and N2. From Figs. S2‒S5, the alpha functions in Soave and exponential types always become larger with the reduction of pore radius at most temperatures but may be different at some specific temperatures. Fig. 6a‒d show the calculated Soave alpha functions for the

CO2, N2, and C1‒10 at rp ¼ 1e1000 nm and Tr ¼ 0.01, 1, 3, and 8. The Soave alpha functions for all the components increase with the pore radius reductions at Tr ¼ 0.01, 3, and 8, whereas they reversely decrease by reducing the pore radius at Tr ¼ 1. The similar patterns are also presented from the exponential alpha functions at different temperatures and their specific values with respect to the pore radius at Tr ¼ 1 are shown in Fig. 7a and b. Therefore, it is concluded that in a dissimilar manner with the cases at most temperatures, the alpha functions for the CO2, N2, and C1‒10 at their critical temperatures (i.e., Tr ¼ 1) are decreased with the pore radius reductions. 4.2.4. First and second derivatives of alpha functions The first and second derivatives of the alpha functions to the temperatures (i.e., Eqs. S4 and S6 in the Supplementary Materials) are critically important for calculating the thermodynamic properties [15,16]. In this study, the mathematical behaviour of the first and second derivatives of the alpha functions in Soave and exponential types in bulk phase and nanopores are specifically investigated, whose results for the CO2, N2, and C1‒10 at different temperatures and/or pore radii are shown in Figs. S6‒S10. Both the first and second derivatives in Soave and exponential types are

Fig. 6. (continued).

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

75

Fig. 7. Calculated alpha functions in exponential type for CO2, N2, and alkanes of C1‒10 at different pore radii of rp ¼ 1e1000 nm and reduced temperatures of Tr ¼ 1.

the second derivatives in exponential type are firstly larger at low temperatures but reduce to be smaller at high temperatures. Furthermore, the first and second derivatives for all the components become more sensitive to the temperatures at smaller pore radii.

continuous with respect to the temperature at T  4000 K. More specifically, the first derivatives in the two types always increase while the second derivatives reduce with the temperature to different extent in bulk phase and nanopores. Moreover, the respective first derivatives in exponential type and the second derivatives are increased and decreased to be asymptotically approaching zero, whereas the first derivatives in Soave type increase with the temperature from negative to positive. It is obvious that the first and second derivatives for the heavier components are less sensitive to the temperature increases compared to the lighter cases. The first and second derivatives become considerably different at smaller pore radii. In comparison with those at a larger pore radius, the first derivatives in the two types are initially smaller at low temperatures and become larger at high temperatures, the second derivatives in Soave type are always larger, and

4.3. Comparisons of different alpha functions The original Soave and exponential alpha functions and their nanoscale-extended formulations are applied for all case calculations aforementioned in bulk phase and nanopores. However, it still remains unclear which one is superior for the bulk phase and nanopore calculations. Table 2 lists the calculated vapour pressures for the CO2, N2, O2, Ar, and C1eC10 in bulk phase from the four different alpha functions, where the AAD% between the calculated

Table 3 Measured [47] and calculated pressurevolumetemperature data from the modified Soave‒Redlich‒Kwong (SRK) equation of state with the nanoscale-extended Soave and exponential type alpha functions for iC4nC4C8 system in the micro-channel of 10 mm and nano-channel of 100 nm at (a) constant pressure and (b) constant temperature. Parameters

Before flash calculationa

After flash calculationa

(a) constant pressure case Temperature ( C) Pressure (Pa) Liquid (iC4nC4C8, mol.%) Vapour (iC4nC4C8, mol.%) Liquid fraction (mol.%) Vapour fraction (mol.%) IFT (mJ/m2) Pcap in micro-channel (kPa) Pcap in nano-channel (kPa)

24.9 85,260 15.47 0 100.00 0.00 e e e

71.9

(b) constant temperature case Temperature ( C) Pressure (Pa) Liquid (iC4nC4C8, mol.%) Vapour (iC4nC4C8, mol.%) Liquid fraction (mol.%) Vapour fraction (mol.%) IFT (mJ/m2) Pcap in micro-channel (kPa) Pcap in nano-channel (kPa)

71.9 839,925 61.89 18.11 0 0 100.00 0.00 e e e

a b c

4.53 0

80.00 0

20.00 0

experimentally measured data. calculated percentage absolute average deviations. average AAD% of the three values.

4.88 64.35 82.20 17.80 16.24 3.38 286.91

1.87 16.82

426,300 28.59 11.15 75.82 21.01 29.50 70.50 13.33 2.77 235.54

After flash calculation (with M-Soave type)

AADb %

After flash calculation (with M-exp type)

AADb %

93.25 18.83

5.21 64.82 83.22 16.78 14.65 2.11 196.38

1.88 18.65

92.91 16.53

2.55c 7.94c 1.24 5.73 9.79 37.57 31.55

6.53 65.98 83.85 16.15 17.12 3.79 236.24

2.98 18.27

90.49 15.75

32.04c 9.17c 2.01 9.27 5.42 12.13 17.66

60.26 3.16

21.18 72.56 25.26 74.74 12.34 1.98 189.44

8.87 20.13

69.95 7.31

20.82c 34.95c 14.37 6.01 7.43 28.52 19.57

26.01 75.58 28.23 71.77 15.01 2.67 208.32

11.35 19.86

62.64 4.56

4.92c 16.70c 4.31 1.80 12.60 3.61 11.56

76

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

and measured data from the Soave, exponential, M-Soave, and Mexp alpha functions are determined to be 0.85%, 0.98%, 0.72%, and 0.84%, respectively. It seems the M-Soave, M-exp, and original Soave alpha functions are slightly more accurate than the original exponential alpha function for calculating the vapour pressures in bulk phase. Moreover, the calculated enthalpies of vaporization for the same components from the four alpha functions are listed in Table S2, whose respective AAD% with the measured data are equal to 2.46%, 2.05%, 2.75%, and 3.21%. Here, the original exponential alpha function performs the best while the other three functions

are comparable for the enthalpy of vaporization calculations in bulk phase. On the other hand, the M-Soave alpha function with the AAD % of 7.77% is superior to the other three functions in terms of the constant-pressure heat capacity calculations in bulk phase. Given that their AAD% and MAD% are all small and close to each other, it is hard to determine which alpha function is the best for the phase and thermodynamic properties in bulk phase. The nanoscale-extended Soave and exponential alpha functions coupled with the modified SRK EOS are applied to calculate the phase behaviour of three mixtures in nanopores. Table 3 lists the

Fig. 8. Measured [45] and calculated pressure‒volume diagrams from the modified Soave‒Redlich‒Kwong (SRK) equations of state with the Soave and exponential type alpha functions for the 90.00 mol.% C8H18‒10.00 mol.% CH4 mixtures at the temperature of T ¼ 311.15 K and pore radius of (a) rp ¼ 3.5 nm and (b) rp ¼ 3.7 nm.

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

measured [47] and calculated PVT data of the iC4nC4C8 mixtures in the micro-channel of 10 mm and nano-channel of 100 nm at the constant pressure and temperature conditions. It is found from the table that the overall performances of the two alpha functions are similar for the phase calculations in nanopores. The application of the M-Soave alpha function performs slightly better in the constant pressure case while the model with the M-exponential case has more accurate results in the constant temperature case. The pressure‒volume diagrams of the C8H18eCH4 mixtures from the modified SRK EOS coupled with the two nanoscale-extended alpha functions are calculated at the temperature of T ¼ 311.15 K and pore

77

radii of rp ¼ 3.5 and 3.7 nm, which are compared with the measured data [45] in Fig. 8. In Fig. 9, the measured [46] and calculated pressure‒volume diagrams of the N2‒n-C4H10 mixtures at the pore radius of rp ¼ 5.0 nm and different temperatures of T ¼ 299.15 and 324.15 K are shown. The calculated results from the two alpha functions are almost equivalent and in reasonable agreement with the measured results for these two mixtures at different conditions in nanopores. Hence, the four alpha functions for the bulk phase calculations and the two nanoscale-extended alpha functions for the nanoscale calculations are validated to be accurate. Some of them may be slightly more accurate for some case calculations but

Fig. 9. Measured [46] and calculated pressure‒volume diagrams from the modified Soave‒Redlich‒Kwong (SRK) equations of state with the Soave and exponential type alpha functions for the 5.40 mol.% N2‒94.60 mol.% n-C4H10 mixtures at pore radius of rp ¼ 5.0 nm and the temperatures of (a) T ¼ 299.15 K and (b) T ¼ 324.15 K.

78

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

none of them can be a general alpha function which always performs the best for all case calculations in bulk phase or nanopores. Finally, it is worthwhile to mention that the constants of the exponential-type alpha function were determined from the PR EOS perspective and may cause some deviations for the SRK EOS cases. A recent study regarding the SRK-type exponential alpha functions [55] was developed, which may be compatible with the proposed models after further validations in the future. 5. Conclusions The following eight major conclusions can be drawn from this work:  Two new nanoscale-extended alpha functions, i.e., M-Soave and M-exponential, are developed analytically by considering a series of confinement effects, such as the critical shifts and molecule‒molecule interactions etc., which have been validated to accurately calculate the phase and thermodynamic properties in bulk phase (rp ¼ 1000 nm) and nanopores coupled with a modified Soave‒Redlich‒Kwong equation of state (SRK EOS).  The minimum reduced temperature from the Soave alpha function occurs at the acentric factor of u ¼ 0.295211, whereas the exponential alpha function has a monotonic relationship with the reduced temperature and no minimum conditions can be determined.  A new method is proposed to determine the nanoscale acentric factors. The acentric factors are increased with the pore radius reductions that the acentric factors remain constant or slightly increase with the pore radius reductions at rp  50 nm while they become quickly increased once the pore radius is smaller than 50 nm.  The Soave alpha functions are related to the temperatures in concave upward parabola curves while the exponential type are monotonically decreased and asymptotically approaching zero with temperature increases. The dimensionless attractive term A in the two types follow the similar patterns of the alpha functions in nanopores while they decrease monotonically and asymptotically approach zero with temperature increases in bulk phase.  The alpha functions and attractive term A in the both types for different components become more sensitive to the temperature increases with the pore radius reductions to different extent, wherein the significant differences occur at rp < 50 nm for the CO2 and alkanes and at rp < 10 nm for the N2. Moreover, the intermolecular attractivities are stronger for the heavier or high carbon number components.  Some abnormal phenomena take place with the pore radius reductions. The Soave alpha functions of the N2 are more sensitive to the temperatures in bulk phase but become more insensitive at rp ¼ 10 nm in comparison with the CO2 case. It is also found that the alpha functions are decreased with the pore radius reductions at the critical temperature (Tr ¼ 1), which are opposite to the cases at other temperatures.  The first and second derivatives of the Soave and exponential alpha functions to the temperatures are continuous at T  4000 K. The first derivatives in the two types are always increased while the second derivatives are reduced with the temperature increases to different extent in bulk phase and nanopores, all of which become more sensitive to the temperatures at smaller pore radii.  The original and nanoscale-extended alpha functions in Soave and exponential types have been validated to be accurate in bulk phase and nanopores. Some of them may be slightly better for some case calculations but none of them can be a general alpha

function which always performs the best for all case calculations.

Acknowledgements The authors would like to acknowledge the editor, Dr. Georgios Kontogeorgis, and anonymous reviewers for their insightful reviews, especially precious suggestions and detailed information provided by one reviewer. They also want to thank the Petroleum Systems Engineering at the University of Regina and the financial supports from Petroleum Technology Research Centre(PTRC) and Mitacs Canada.

Nomenclature

Notations a A Ar b c Cp CV dp E F G h H k L m mE mE-NP mS mS-NP N Q P Pc Pcap Pcp pi PL Pv PrNP qint R rp S T Tc Tcp Tr TrNP U U0 V Vr xi yi Z

equation of state constant dimensionless attractive term reduced surface area equation of state constant coefficients constant-pressure heat capacity constant-volume heat capacity pore diameter cohesive energy Helmholtz free energy Gibbs free energy Planck's constant enthalpy of vaporization Boltzmann constant length molecular mass m function in exponential type nanoscale-extended m function in exponential type m function in Soave type nanoscale-extended m function in Soave type number of molecules Canonical partition function system pressure critical pressure in bulk phase capillary pressure critical pressure in nanopores parachor of ith component pressure of the vapour phase pressure of the liquid phase reduced pressure in nanopores internal partition function universal gas constant pore radius entropy system temperature critical temperature in bulk phase critical temperature in nanopores reduced temperature reduced temperature in nanopores internal energy internal energy of the ideal gas system volume reduced volume mole percentage of ith component in the liquid phase mole percentage of ith component in the vapour phase configuration partition function

K. Zhang et al. / Fluid Phase Equilibria 482 (2019) 64e80

Greek Symbols aT alpha function aTE exponential-type alpha function aTS Soave-type alpha function aTENP nanoscale-extended exponential-type alpha function aTSNP nanoscale-extended Soave-type alpha function g interfacial tension L de Broglie wavelength ε Lennard‒Jones energy parameter s Lennard‒Jones size parameter rL density of the liquid phase rV density of the vapour phase d Hildebrand solubility parameter q contact angle of the vapour‒liquid interface with respect to the pore surface u acentric factor uNP nanoscale-extended acentric factor Acronyms AAD Exp EOS IFT LJ MAD M-exp M-Soave PVT PR PTRC RK SRK vdW VLE

[14]

[15] [16] [17]

[18] [19]

[20] [21] [22] [23]

average absolute deviation alpha function in exponential type equation of state interfacial tension Lennard‒Jone maximum absolute deviation nanoscale-extended alpha function in exponential type nanoscale-extended alpha function in Soave type pressure‒volume‒temperature PengRobinson Petroleum Technology Research Centre Redlich‒Kwong Soave‒Redlich‒Kwong van der Waals vapour‒liquid equilibrium

[24] [25]

[26]

[27]

[28] [29]

[30] [31]

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.fluid.2018.10.018. References

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