Modeling the Hydrate Dissociation Pressure of Light Hydrocarbons in

Article Cite This: Ind. Eng. Chem. Res. 2019, 58, 12369−12391

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Modeling the Hydrate Dissociation Pressure of Light Hydrocarbons in the Presence of Single NaCl, KCl, and CaCl2 Aqueous Solutions Using a Modified Equation of State for Aqueous Electrolyte Solutions with Partial Ionization María A. Zúñiga-Hinojosa,† Jeremías Martínez,‡,§ Fernando García-Sań chez,∥ and Ricardo Macías-Salinas*,† Downloaded via BUFFALO STATE on July 30, 2019 at 14:35:15 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

†

Departamento de Ingeniería Química, SEPI-ESIQIE, Instituto Politécnico Nacional, Ciudad de México, 07738, Mexico Facultad de Química, Universidad Autónoma del Estado de México, Paseo Colón y Paseo Tollocan S/N, Toluca, Estado de México, 50120, Mexico § Centro Conjunto de Investigación en Química Sustentable UAEM-UNAM, Carretera Toluca-Atlacomulco, km 14.5, Toluca, Estado de México, 50200, Mexico ∥ Gerencia de Ingeniería de Recuperación Adicional, Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas Norte No. 152, Ciudad de México, 07730, Mexico ‡

ABSTRACT: In this study, we present the modeling of the hydrate dissociation pressures for methane, ethane, and propane in the presence of NaCl, KCl, and CaCl2 aqueous solutions using a modified equation of state for aqueous electrolyte solutions with partial ionization (AES-PI) coupled with the van der Waals−Platteeuw theory. The proposed thermodynamic model includes a linear correlation for describing the ionic hydration process as a function of the pressure and molality. The proposed approach is capable of representing the inhibition effect at different salt concentrations. The results showed an excellent agreement between the experimental data and the calculated values at different conditions typical of the deepwater environment.

1. INTRODUCTION Gas hydrates are crystalline structures that consist of water molecules that form a lattice by hydrogen bonds and contain gas molecules inside them, with their formation being favored at low temperatures and high pressures. The typical hydrateforming gas molecules are methane (the main component of natural gas) and ethane; however, propane, carbon dioxide, and nitrogen also form hydrates. There exist three types of hydrate structures, whose size depends on the size of gas molecules; i.e., structure I (sI) contains small molecules such as methane and ethane, whereas structure II (sII) is capable of accommodating larger molecules such as propane and isobutane. In the particular case of even larger components, those such as isopentane or neohexane can form structure H (sH) when accompanied by smaller molecules such as methane, hydrogen sulfide, or nitrogen.1 During the deepwater exploration and production stages in the oil and gas industry, gas hydrates are one of the main concerns since high pressures and low temperatures favor hydrate formation, which can result in the plugging of valves and pipelines, in delayed gas production, and in a significant increase of the operating costs. In this context, the use of inhibitors such as NaCl, KCl, and CaCl2 aqueous solutions have a crucial © 2019 American Chemical Society

role because they displace the thermodynamic stability of hydrates at higher pressures and prevent their formation in the flowlines. Consequently, thermodynamic models must be extended to the typical temperature, pressure, and molality ranges that can be found in the deepwater activities of the oil and gas industry. Diverse modeling approaches have been developed for predicting the formation and inhibition of gas hydrates. The first approach was proposed by van der Waals and Platteeuw2 (vdWP), who based their model on statistical mechanics principles. They described the hydrate phase as a theoretical metastable empty hydrate lattice. The model considers that hydrates form spherical cavities containing a guest molecule, and that the cavity is not distorted. This latter assumption has been questioned by several research groups.1,3−11 In 1972, Parrish and Prausnitz12 presented a generalized method to calculate the hydrate dissociation conditions in gas mixtures for ice−hydrate−vapor (IHV) and liquid−hydrate− vapor (LwHV). They proposed a new expression for the Received: Revised: Accepted: Published: 12369

April 5, 2019 June 3, 2019 June 5, 2019 June 5, 2019 DOI: 10.1021/acs.iecr.9b01880 Ind. Eng. Chem. Res. 2019, 58, 12369−12391

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Industrial & Engineering Chemistry Research

energy.39−52 The good thing about electrolytes models based on the Helmholtz energy is that they can be used to calculate the densities of electrolyte solutions and applied over a broader range of temperatures and pressures. Englezos and Bishnoi53 adapted the activity coefficient model proposed by Pitzer and Mayorga54 and calculated the fluid phases with the Trebble− Bishnoi equation of state55 for the gas hydrate inhibition prediction in aqueous electrolyte solutions, obtaining a maximum deviation of 6.2%. Duan and Sun56 also used the Pitzer− Mayorga approach54 in conjunction with their proposed model57 to calculate the gas hydrate formation. They evaluated the Langmuir constants from ab initio calculations. The maximum absolute average deviation (AAD) obtained was 4.5% for the inhibited gas hydrate systems with NaCl and CaCl2 aqueous solutions and their mixtures. On the other hand, Zuo and Stenby58 adapted the extended Patel−Teja EoS59 in combination with a modified Pitzer−Debye−Hückel expression. The average AAD value obtained by the authors was 4.5%; the CaCl2 aqueous solution presented maximum AAD values of 11.11 and 10.56% at 15 and 20 wt % CaCl2, respectively. Mohammadi and Tohidi60 used the Valderrama modification of the Patel−Teja EoS61 with non-density-dependent mixing rules and the adaptation of Debye−Hückel electrostatic term. These researchers considered the binary interaction parameters as a function of weight percent of the salt aqueous solution. Their results were in good agreement with the experimental data when calculating the hydrate depression temperature. Ma et al.62 also used the Patel−Teja EoS59 coupled with the Kurihara mixing rule63 to correlate the gas solubility data; accordingly, the liquid phase composition was determined. The hydrate model of Chen and Guo64 was applied to calculate the hydrate formation temperatures of the single and mixed gas hydrates with polar aqueous solutions. Their results were in good agreement with the experimental data. Haghighi et al.65 applied the CPA EoS and the Debye−Hückel electrostatic contribution term.42 They also used the binary interaction parameters between the salt and water as a function of salt concentration and temperature. Such parameters were optimized66 for calculating the methane hydrate dissociation conditions in porous media by using experimental freezing point depression data. Their results showed a reasonable agreement between model predictions and experimental data. Ji and Adidharma24−26 proposed the SAFT2 EoS, and they proved that it is reliable in representing the thermodynamic properties of aqueous electrolytes. Accordingly, Jiang and Adidharma23 used such an equation to study the hydrate dissociation conditions for light hydrocarbons in the presence of single and mixed aqueous electrolyte solutions. The SAFT2 model has four parameters for describing each ion; these parameters are functions of temperature and the ion-specific fitted coefficients. To reduce the number of parameters, the binary interaction parameter between the cation−anion interaction was assumed to be constant, whereas the other parameters were taken into account in the mixing rule for the segment energy. Their results showed a satisfactory representation of the experimental data points for the studied systems, except for the methane hydrate inhibited with NaCl aqueous solution at 24.1 wt %, which yielded an AAD value of 12.6%. In 2011 Bandyopadhyay and Klauda67 presented an update of the Klauda and Sandler8 fugacity-based model by using the predictive Soave−Redlich−Kwong (PSRK EoS)68,69 to calculate the hydrate inhibition with electrolyte solutions, e.g., NaCl and CaCl2, in a molality range less than 3.55. Their results showed an AAD value of 9%. Dufal et al.27

calculation of the Langmuir constants. One of the improvements suggested by the Holder group was to correct the Langmuir constant by using an empirical parameter as a function of the guest (gas species) size3 and performing molecular dynamic simulations.5 On the other hand, Klauda and Sandler6 developed a new approach for performing gas hydrate equilibrium calculations. They applied quantum mechanics (QM) calculations to predict the values of the model itself, removing the reference property parameters. They also observed a change in the size of the hydrate structure by using a constant lattice and different guest molecules in the quantum chemistry calculations. In a more advanced model, Klauda and Sandler7,8 applied the QM calculations by considering the guest−guest and the guest−hydrate lattice interactions. In the latter, the lattice distortion caused by the guests was taken into account. Ballard and Sloan13 included the hydrate distortion by the water activity correction in the hydrate phase instead of in the reference state of ice or liquid water phase. They also considered in their model the specific guest in the lattice. However, the crucial change was to make the cage radii as a function of PTx and to introduce a linear function of the hydrate structure parameter. Martiń and Peters9 also included a lattice distortion caused by the guest molecules in their approach and applied the cubic plus association equation of state (CPA EoS)14−16 for calculating the fluid phases. The distortion was introduced as a correlation parameter in the chemical potential. This approach is based on the fugacitybased model, but it contains less fitted parameters than the fugacity-based model. Also, Karakatsani and Kontogeorgis10 stated that it might not be valid not to include the cage distortion for larger hydrocarbon gases such as propane. Folas et al.17 and Kontogeorgis et al.18 applied the CPA EoS for predicting the hydrate dissociation conditions for single gas molecule and multicomponent mixtures with and without inhibitors, obtaining good results. On the other hand, the statistical associating fluid theory equation of state (SAFT),19 introduced by Huang and Radosz,20 has been used in different variations to model the hydrate equilibrium calculations by various research groups. Li et al.21,22 applied SAFT EoS in combination with the vdWP model for single and binary combinations of hydrocarbons and non-hydrocarbons. During their calculations, they considered inhibitors commonly used in the gas industry obtaining good results. More recently, Jiang and Adidharma23 calculated the hydrate pressure dissociation of single hydrocarbons, as well as binary and ternary mixtures, by using the ion-based statistical associating fluid theory (SAFT2)24−26 coupled with the vdWP approach.2 The results obtained were satisfactory in representing the experimental data, except at high pressures; accordingly, they suggested that Langmuir constants might be dependent on pressure as previously found by Sloan and Kho.1 Dufal et al.27 used the statistical associating fluid theory for potentials of variable range (SAFT-VR)28,29 coupled with the vdWP theory for the hydrate equilibrium calculations. Their results showed a satisfactory agreement with the experimental data. It is important to mention that the modeling of hydrate inhibition in the presence of electrolyte solutions is scarce in the scientific literature. A possible explanation is that the thermodynamic properties are more difficult to estimate for electrolytes than for nonelectrolyte solutions. The modeling of thermodynamic properties for electrolyte solutions can be split in (1) local-composition models based on the excess Gibbs energy30−38 and (2) equations of state based on the Helmholtz 12370

DOI: 10.1021/acs.iecr.9b01880 Ind. Eng. Chem. Res. 2019, 58, 12369−12391

Article

Industrial & Engineering Chemistry Research modeled the effect of different inhibitors such as electrolyte solutions on the equilibrium stability of hydrates with the SAFT-VR EoS. In the model, they included a coupling of the cell potential, related to the interaction between the guest hydrate and the water with the intermolecular pair potential used in SAFT-VR to model the fluid phases of the guest molecule. Their results showed a good representation of the experimental data. In this context, most of the reported studies of gas hydrate dissociation conditions with aqueous electrolytes solutions are based on equations of state; i.e., the electrolyte solution models are given in terms of the excess Gibbs free energy. Therefore, these approaches do not include the effect of pressure, so they cannot be used to calculate the solution density at different pressure conditions. However, the models based on the Helmholtz free energy such as the different variations of SAFT EoS11,27 and the EoS for aqueous electrolyte solutions with partial ionization (AES-PI)70 are capable of predicting the solution density at various pressure conditions. We rather favor the use of cubic EoSs such as the Peng− Robinson EoS (PR EoS)71 since they continue to be the simplest approach for modeling the phase behavior of mixtures; additionally, they are always available from any process simulator. On the other hand, to the best of our knowledge, the solvation effect has been neglected in the hydrate dissociation conditions using aqueous electrolyte solutions. The purpose of the present work is thus to develop a modeling approach that can predict the hydrate dissociation pressures of methane, ethane, and propane in the presence of single aqueous electrolyte solutions such as NaCl, KCl, and CaCl2 at the three-phase equilibria conditions of ice−hydrate− vapor (IHV) and liquid water−hydrate−vapor (LwHV). The present approach uses the modified aqueous electrolyte solutions with partial ionization equation of state (MAES-PI EoS) coupled with the van der Waals−Platteeuw theory,2 and it includes the ionic hydration process (IHP) calculation by using a linear correlation as a function of pressure and molality.

Equation 4 is the difference of the chemical potential of the water in a hypothetical empty hydrate lattice and in the hydrate phase; its calculation is given as 2 ij j ΔμwH = −RT ∑ νi lnjjj1 − jj i=1 k

θij =

Cij =

Cij =

(5)

Cijf j 1 + ∑j Cijf j

(6)

4π kBT

∫0

R

ij −w(r ) yz 2 zzr dr expjjj j kBT zz k {

(7)

i Bij y expjjjj zzzz T kT {

Aij

(8)

where Aij and Bij are fitted parameters related to guest type j in cavity type i. Otherwise, the difference of the chemical potential between water in the state of an empty hydrate lattice and the solid phase (eq 3) is calculated as follows: Δμwice

(1)

RT

=

ij Δμ0 yz jj w zz − jj z j RT zz k {T0 , P0

∫T

T

0

ij Δhw yz jjj 2 zzz dT + k RT {

∫P

P

0

ij ΔVw yz jj zz dP k RT {

(9)

Δμ0w

where is the reference state of the chemical potential difference between water in an empty hydrate lattice and pure water at 273.15 K and 0 MPa. The chemical potential difference between water of an empty hydrate lattice and the liquid phase can be expressed as follows: ij Δμ0 yz = jjjj w zzzz − j RT z RT k {T0 , P0

(2)

ΔμwL

where (3)

− ln ϕx w

and ΔμwH = μwβ − μwH

z {

where T is the temperature, kB is the Boltzmann constant, w(r) is the cell potential function of component j in cavity type i in the radial distance from the center of the molecule, and R is the radius of the cell. The cell potential function is based on the Kihara potential, and can be found elsewhere.12,72 Moreover, Parrish and Prausnitz proposed a more straightforward expression as a function of the temperature for calculating the Langmuir adsorption coefficients

where is the chemical potential of water in the λ phase, i.e., the liquid or ice phase, and μH w is the chemical potential of water in the hydrate phase (H). However, it is not clear how to describe the reference state of water in the hydrate phase. To overcome this, van der Waals and Platteeuw2 proposed to include a hypothetical empty hydrate phase (β) as a reference state. Therefore, the equilibrium condition can be rewritten as

Δμwλ = μwλ − μwβ

j=1

where f j is the fugacity of component j in the vapor phase; in this work, it is calculated by the PR EoS71 that will be described later. Cij is the Langmuir constant that defines the water−gas interaction in the hydrate lattice; it is a temperature dependent adsorption constant that is specific for the type i cavities and the j gas molecules. It is calculated as follows:

μλw

Δμwλ = ΔμwH

N

where νi is the number of cavities of type i per water molecule in the hydrate phase, i.e., 1 and 2 are the small and large cavities, respectively; θij stands for the fractional occupancy of the guest molecule j in the type i cavities; and N is the number of different types of guest molecules. The fractional cavity occupancy is calculated by using the Langmuir adsorption theory.

2. DESCRIPTION OF THE THERMODYNAMIC MODEL 2.1. Model of the Hydrate Phase Equilibria. The phase equilibrium conditions for hydrate formation are given by the equality of the chemical potentials of each component in all the phases in equilibrium. The vdWP model only considers a phase change from water in the aqueous phase (or ice phase) to hydrate phase. μwλ = μwH

yz

∑ θijzzzzz

∫T

T

0

ij Δhw yz jjj 2 zzz dT + k RT {

∫P

P

0

ij ΔVw yz jj zz dP k RT {

(10)

where Δhw and ΔVw are the difference of the molar enthalpy and the molar volume between the empty lattice and ice or

(4) 12371


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The α correlation proposed by Melhem et al.75 is given by

liquid water, respectively. The difference of enthalpy is given by72 Δhw =

Δhw0

+

∫T

α = exp[α0(1 − Tr) + α1(1 − Tr1/2)]2

T

(14)

ΔCp ,w dT

(11)

0

where α0 and α1 are fitting parameters, with optimized values of 0.883 and 0.039, respectively. The translation volume technique is used for improving the saturated liquid density without affecting the vapor pressure. In this study, a translation function suggested by Mathias et al.76 was used:

and ΔCp,w = 38.12 − 0.141(T − 273.15)

T > T0

(12)

Δh0w

where is the reference enthalpy between the empty lattice and the ice or liquid water. Table 1 shows the thermodynamic

V corr = V EoS + c corr

ref property

phase

structure I

structure II

Δμ0w [J/mol] Δh0w [J/mol] Δh0w [J/mol] ΔV [cm3/mol] ΔV [cm3/mol]

liquid liquid ice liquid ice

1263.0 −4620.5 1389 3.0 4.598

883.8 −4984.5 1025 3.4 4.998

where fc is a function of the critical compressibility factor proposed by Uscanga-Hernández77 as follows: fc = Zc − (ZcEoS + s)

where

reference properties and the difference of the molar volume for structure I and structure II. The last term in eq 10 includes the gas solubility of component j in water solution, where ϕ is the osmotic coefficient (activity coefficient of water) and xw is the molar fraction of water. The osmotic coefficient was set constant to 1 when it is not in the presence of inhibitors, and it will be further described in detail. The concentration of the water was calculated for the hydrate formation as follows:72

ZEoS c

ZcEoS =

is the calculated compressibility factor given by 1 (1 − 0.0777960739039) 3

(18)

where χ0 and χ1 are fitted parameters, with values of −0.0076 and −0.0114, respectively. The bulk modulus, δ, is given by

where xj is the hydrocarbon molar fraction, and it is calculated as follows:

δ=−

fj Hkw exp(PV̅ /RT )

(17)

According to the correction term of the critical compressibility factor, s is expressed as χ Tr ≡ T /Tc s = χ0 + 1 Tr (19)

x w = 1 − xj

∞

(15)

EoS

where V and V are the corrected molar volume and the molar volume obtained by the PR EoS,71 respectively, whereas c is a translation function expressed as Ä É RTc ÅÅÅÅ ij 0.41 yzÑÑÑÑ zzÑÑ c= Ås + fc jj Pc ÅÅÅÇ (16) k 0.41 + δ {ÑÑÖ

Table 1. Thermodynamic Reference Properties and Difference of Molar Volume for Structures I and II of Gas Hydrates72

xj =

Tr ≡ T /Tc

(13)

(V EoS)2 ij ∂P yz zz ·jj RT k ∂V EoS {

(20)

where

where f j is the gas fugacity of a guest molecule of type j, and Hkw is the temperature-dependent Henry’s constant expression proposed by Krichevsky and Kasarnovsky.73 This work aims to predict the hydrate dissociation pressure, at a given temperature, in the three-phase equilibrium conditions, IHV or LwHV, using the vdWP model in conjunction with the thermodynamic model for aqueous electrolyte solutions with partial ionization (AES-PI).70 An iterative procedure to perform the hydrate equilibrium calculations is required; therefore, we solve the eq 5 and eq 9 or eq 10 by using the accelerated successive substitution method74 instead of the bisection method to improve convergence. When the convergence condition of eq 2 is satisfied, the hydrate formation is reached. 2.2. Modified AES-PI Equation of State. As was mentioned above, to calculate the osmotic coefficient and the mean ionic activity coefficient, the reader is referred to ́ Macias-Salinas et al.70 In this work, we use an improved implementation for the AES-PI EoS that has recently been developed in the water volume calculation, more specifically in the osmotic coefficient. The water volume accuracy increased by using the alpha (α) correlation suggested by Melhem et al.75 and a new translation volume in the PR-EoS.

∂P RT = − EoS ∂V EoS (V − b)2 +

2a(V EoS + b) [V EoS(V EoS + b) + b(V EoS − b)]2

(21) 77

For details, the reader is referred to Uscanga-Hernández. According to the modified calculation of the osmotic coefficient, it is expressed as ϕ=1−

1000 ln(γw ) υmM w

(22)

where υ is the total number of cations and anions, m is the molality, Mw is the molar mass of water, and γw is the water activity coefficient. 2.3. Peng−Robinson Equation of State. The PR EoS has found wide acceptance in the petroleum and gas industries, as well as in academia, due to its simplicity and ability to estimate the phase behavior of either nonpolar or slightly polar systems. However, it is not accurate with strongly polar and associating fluids such as water. Table 2 shows the critical properties and the acentric factors of pure components needed as model inputs. For the details of fugacity calculation with PR EoS, the reader is referred to the Peng and Robinson EoS.71 12372


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Industrial & Engineering Chemistry Research Table 2. Critical Properties of Pure Components78 component

Tc [K]

Pc [MPa]

ω

methane ethane propane water

190.56 305.32 369.83 647.14

4.599 4.872 4.248 22.064

0.011 0.099 0.152

consider only the occupancy of small and large cages, which is essential for calculating the hydrate dissociation pressure. Besides, Sarupria and Debenedetti81 used molecular dynamics simulations and concluded that the dissociation rate depends on the fractional occupancy of each cavity type and it cannot be described by considering only the cavity behavior. Therefore, in this work, we use the Parrish and Prausnitz approach, to calculate the Langmuir constants and the fractional occupancy expression for determining the cavity occupancies. Regarding the reported Aij and Bij parameters for hydrocarbon type j in cavity type i, we found several sets of values in the literature. According to Karakatsani and Kontogeorgis,10 “A straightforward comparison of different parameter sets is not always possible because of the different data sets used and the different method used for characterizing the fitting.” Thus, we decided to adjust the Aij and Bij parameters to equilibrium phase data for methane, ethane, and propane in the large cavities. Regarding the small cavity for methane, we use the Aij and Bij values reported by Munck et al.82 to decrease the number of fitted parameters. For modeling purposes of hydrate dissociation conditions, we present in Table 3 the gas compounds considered in this study along with the corresponding number of experimental points, hydrate structure, equilibrium phase, temperature and pressure ranges, as well as AADP, and the source of experimental data. As shown in Table 3, the

3. ADJUSTABLE PARAMETERS OF THE MODEL 3.1. Determination of Langmuir Constant Parameters. According to Sun and Duan57 and Holder et al.79 the parameters of the Kihara potential for evaluating the Langmuir constants are empirically regressed from experimental data of phase equilibrium. As a result, these parameters do not agree with the second virial coefficient and viscosity data. Additionally, Aij and Bij parameters from the Parrish and Prausnitz approach are also fitted parameters to experimental data. As stated by Barkan and Sheinin,80 “this approach is the most effective, since in this case, we obtain optimal (but not real) potential parameters of the heterogeneous gas-water interactions, what allows a considerable degree of smoothing of all the errors caused by choice of a true function for intermolecular interaction, of rules for combination of potential parameters.” Furthermore, as pointed out by Karakatsani and Kontogeorgis,10 the Parrish and Prausnitz approach can

Table 3. Experimental Data and AADP for Methane, Ethane, and Propane Hydrates n-alkane

N

hydrate structure

phase

T range [K]

P range [MPa]

AADP [%]

source

methane

11 30 10 18 10 14

I I I I II II

IHV LwHV IHV LwHV IHV LwHV

244.2−272.2 273.6−301.5 244.9−271.9 273.7−285.4 245.0−272.1 273.25−278.36

0.97−2.47 2.67−65.43 0.12−0.44 0.51−2.30 0.04−0.16 0.17−0.54

2.80 3.56 2.77 4.47 0.70 2.92

83 84 83 85 83 86

ethane propane

Table 4. A and B Parameters for Langmuir Constants Calculation structure I small cavity

structure II large cavity

small cavity

large cavity

n-alkane

Aij × 103b [K/MPa]

Bb [K]

Aij × 103 [K/MPa]

B [K]

Aij [K/MPa]

B [K]

Aij × 103 [K/MPa]

B [K]

methane ethane propane

7.13 0.0 0.0

3187 0.0 0.0

62.41 3.91 0.0

3012.86 4407.06 0.0

NC 0.0 0.0

NC 0.0 0.0

NC NC 4.99

NCa NC 5271.24

a

NC = not calculated. bMunck et al.82

Figure 1. Experimental and calculated fractional cage occupancies for methane hydrate: (a) small cage; (b) large cage. 12373


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Figure 2. Experimental and calculated dissociation pressures of IHV and LwHV phases: (a) methane hydrate; (b) ethane hydrate; (c) propane hydrate.

Table 5. Aqueous Electrolyte Solutions Considered in This Study electrolyte

σ+Pa [Å]

σ−Pa [Å]

ΔhG+⊕b [kJ/mol]

ΔhG−⊕b [kJ/mol]

NaCl

1.90

3.62

−401.3

−307.2

KCl CaCl2

2.66 1.98

3.62 3.62

−328.1 −1583.4

−307.2 −307.2

P [MPa]

mmax [mol/kg]

0.1, 20, 40, 60, 80

6.00

89

0.1 0.1

4.00 3.50

90, 91 92, 93

T [K] 273.15, 283.15, 293.15, 298.15, 303.15 298.15, 323.15, 353.15 298.15, 382.15, 473.15

source for γ± and ϕ

a

Pauling.94 bFriedman et al.95

Table 6. Optimized Parameters for AES-PI EoS at 298.15 K and 0.1 MPa AAD [%] electrolyte

N

σ+ [Å]

σ− [Å]

Kd [mol/kg]

k+,w

F

γ±

ϕ

ΔhG⊕

NaCl KCl CaCl2

29 19 15

3.25 3.67 3.50

4.63 4.47 4.76

33.06 288.56 39148.98

0.00108 0.00073 −0.43581

18.8 13.1 15.9

0.13 0.47 2.20

0.19 2.19 2.18

0.06 1.39 × 10−10 2.25

The correlating Aij and Bij parameters are summarized in Table 4. To validate the obtained values of the fitted parameters, we calculate each cage occupancy by using the expression of the fractional cavity occupancy (eq 6) for methane only, with the purpose of comparison to experimental data.88 Figure 1 shows the measured and calculated fractional occupancies of small and large cavities for methane hydrate. As can be seen in Figure 1a, there exists a great deal of scatter among the experimental occupancies for small cavities. The proposed model slightly overestimated the occupancy fraction of small cavity; notwithstanding

experimental data cover a moderate temperature range (244.2− 301.5 K) and a wide pressure range (0.04−65.43 MPa). The following objective function was minimized by the simplex optimization procedure of Nelder and Mead:87 ij Piexp − Picalc yz zz ∑ jjjj zz exp P i i=1 k { N

Fobj =

2

(23)

Pexp and Pcalc stand for the experimental and calculated pressures, respectively; whereas N is the number of experimental points. 12374


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this fact, the predictions are between the experimental error bars. The experimental and calculated occupancy fractions of large cavity present a satisfactory agreement, as demonstrated in Figure 1b. Figure 2 depicts the comparison between the experimental and calculated dissociation pressures of IHV and LwHV phases for methane, ethane, and propane over a moderate range of temperature and a wide range of pressure. The present modeling approach provides a good fitting of experimental dissociation pressure data in the IHV and LwHV phases for the three light hydrocarbon hydrates under study, except at high pressure (Figure 2a,b). These discrepancies can be attributed to

Table 7. Optimized Parameters for the AES-PI EoS at Different Temperatures and 0.1 MPa AAD [%] electrolyte

N

T [K]

Kd [mol/kg]

F

γ±

ϕ

NaCl

10 10 10 10 19 19 6 6

273.15 283.15 293.15 303.15 323.15 353.15 382.15 473.15

2.41 × 1011 1.51 × 1011 155.89 29.13 6.98 4.13 20.59 1

23.3 21.3 4.3 18.1 12.8 11.3 0.4 29.4

1.36 0.34 0.06 0.30 0.21 0.12 6.18 17.93

0.72 0.37 0.24 0.32 2.93 3.49 2.82 13.89

KCl CaCl2

Figure 3. Experimental and predicted coefficients for NaCl aqueous solution at different temperatures: (a) mean ionic activity coefficients; (b) osmotic coefficients.

Table 8. Optimized Kd and f Parameters of Methane, Ethane, and Propane Hydrates Inhibited with Salt Aqueous Solutions AAD [%] n-alkane

salt

N

molality [mol/kg]

Kd [mol/kg]

F

T range [K]

P range [MPa]

γ±

ϕ

methane

NaCl

6 5 5 5 10 9 9 6 7 4 4 4 5 3 4 6 6 4 4 4 4 5 5 5

2.27 3.53 4.69 5.43 2.08 3.54 4.84 0.71 1.49 1.02 1.53 2.22 1.90 3.02 4.28 0.71 1.49 1.59 1.90 3.02 4.28 1.49 2.37 3.35

579.98 1996.81 3819.58 6349.31 857.51 3381.99 27632.42 890.79 58.42 11807 × 103 54360 × 103 18120 × 103 591.70 1355.46 3016.11 574.91 607.83 5580 × 103 594.89 1048.23 717.59 77.15 526.38 9.83

24.0 24.4 23.6 23.5 17.7 18.8 19.6 13.3 13.8 7.8 7.9 8.0 23.3 23.5 24.0 13.3 13.9 7.9 25.5 25.8 26.0 14.0 14.2 14.7

268.3−278.05 263.35−274.95 261.85−272.85 263.05−268.65 279.16−295.47 274.4−290.37 270.66−284.92 271.6−283.2 270.1−281.5 278.4−282.3 273.5−278.8 270.1−273.5 273.7−280.4 272.7−277.1 266.25−271.45 271.4−282.3 270.4−281.6 272.3−278.5 270.05−272.15 266.15−268.15 260.95−263.05 271.05−273.45 269.05−271.25 266.45−267.55

2.69−7.55 2.39−8.57 2.94−11 4.78−9.55 7.51−64.6 7.92−64.14 7.85−64.03 2.71−8.69 2.78−8.82 6.88−10.22 4.92−9.7 6.3−10.29 0.883−2.165 1.082−2.151 0.689−1.524 0.47−1.85 0.5−2.11 1.1−2.52 0.241−0.421 0.221−0.379 0.2−0.331 0.23−0.42 0.22−0.39 0.23−0.34

1.55 2.97 2.65 1.71 2.19 3.24 3.75 0.16 0.28 3.83 6.44 9.51 0.81 0.87 1.40 0.15 0.29 6.48 0.35 0.55 0.77 0.07 0.10 0.09

0.65 1.14 0.89 1.24 1.04 1.26 1.25 1.67 2.86 0.19 0.48 1.23 0.39 0.37 0.74 1.67 2.80 0.62 0.18 0.57 1.61 2.81 3.64 5.85

KCl CaCl2

ethane

NaCl

KCl

propane

CaCl2 NaCl

KCl

12375

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98

99 100

101, 102

99 99 100

100


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Industrial & Engineering Chemistry Research Table 8. continued AAD [%] n-alkane

salt

N

molality [mol/kg]

Kd [mol/kg]

F

T range [K]

CaCl2

4 4 136

1.15 1.62

12872 × 103 6990 × 103

8.0 8.0

269.65−271.35 266.35−267.75

overall

P range [MPa] 0.25−0.37 0.23−0.35 average maximum

γ±

ϕ

3.26 4.99 2.25 9.51

0.92 1.40 1.44 5.85

source 100

Figure 4. Contributions of AES-PI EoS to osmotic coefficient (a) and mean ionic activity coefficient (b) at 2.27 m and osmotic coefficient (c) and mean ionic activity coefficient (d) at 5.43 m for NaCl aqueous solution of the methane hydrate at moderate pressures.

́ proposed by Macias-Salinas et al.70 Table 5 presents the three electrolytes considered in this study along with their corresponding Pauling crystal ionic diameters (σ+P and σ−P)94 free energies of hydration (ΔhG⊕) at 298.15 K and 0.1 MPa,95 temperature and pressure ranges, and maximum molalities at which experimental data were measured as well as the sources for experimental mean ionic activity coefficients and osmotic coefficients. The simplex optimization procedure of Nelder and Mead87 was employed to minimize the objective function ́ proposed by Macias-Salinas et al.70 for fitting the experimental data at 298.15 K and 0.1 MPa. The simultaneous fitting of the measured γ± and ΔhG⊕ values resulted in the adjusted parameters given in Table 6. It can be seen that the adjusted values of the ionic diameters (σ+ and σ−) are higher than the Pauling ́ crystal ionic diameters (σ+P and σ−P). According to Macias70 Salinas et al. the increase in the ionic diameter indicates presumably the effect of hydration in the ionic diameter. It can

the Langmuir constants since they could be pressure dependent as stated by Sloan and Kho1 and Jiang and Adidharma.11 3.2. AES-PI EoS Parameters. The AES-PI EoS was proposed from an analytical expression of the Helmholtz free energy. Such an equation of state contains three contributions: (a) a charge−charge interaction term given by the mean spherical approximation (MSA) expression, (b) a discrete solvent term to consider the short-range interactions between uncharged particles based on the PR EoS, and (c) an ioncharging term described by the Born continuum-solvent model.70 Then, according to the original AES-PI EoS,70 one of their contributions uses the PR EoS to take into account the short-range interactions between a single electrolyte and water. However, the improvement in the water volume, as mentioned in section 2.2, modified those interactions, and therefore the parameter values of the AES-PI EoS. Accordingly, we calculate the five parameters of the model at 298.15 K and 0.1 MPa as 12376


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Figure 5. Contributions of AES-PI EoS to osmotic coefficient (a) and mean ionic activity coefficient (b) for NaCl aqueous solution of methane hydrate at 3.54 m and high pressures.

Figure 6. IHP for NaCl aqueous solution of methane hydrate in the small cavity at different molalities and moderate pressures (IHV phase).

and water (k+,w). As can be seen, these values are lower for NaCl and KCl solutions than for CaCl2 solution. This variation suggests that the geometric mean rule71 describes better the NaCl and KCl solutions than the CaCl2 solution. The resulting adjusted values of the correction factor, F, are high enough to suggest that the use of the dispersion theory alone significantly underestimates the value of the short-range attraction parameter for ionic species.70 The values of AAD for γ±, ϕ, and

also be observed in Table 6 that the increase in the ionic diameter is higher for cations than for anions, which is associated with the higher number of alkali metal ions, that have been experimentally evidenced.70 The optimum Kd values reported in Table 6 show a complete dissociation for the three single aqueous electrolyte solutions considering that those values are higher than unity. Table 6 also shows the optimized values of the short-range interaction parameter between cation 12377


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Figure 7. IHP for NaCl aqueous solution of methane hydrate in the large cavity at different molalities and moderate pressures (IHV phase).

ΔhG⊕ demonstrate that the ability of the AES-PI EoS to represent the experimental data is quite good. The AES-PI EoS considers that the parameter of the dissociation constant (Kd) and the correction factor for the attraction parameter of ionic species in the PR contribution (F) are temperature dependent. Accordingly, the effective ionic diameters σ+ and σ− and the binary interaction parameter between the cation and water (k+,w) were also set constant. The AES-PI EoS has been proven to obtain the mean ionic and osmotic coefficients over a wide range of salt molalities, a pressure of 0.1 MPa, and temperatures higher than 298.15 K with quite good predictions; however, it has not been validated at higher pressures and at temperatures lower than 298.15 K. To apply this approach, it is necessary to have access to experimental data of mean ionic activity coefficients and osmotic coefficients under typical conditions of the deepwater environment. To the best of our knowledge, such data cannot be found in the scientific literature. To overcome the lack of experimental data, we extrapolate the measured values of mean ionic activity coefficients and osmotic coefficients by using Newton’s interpolation with not finite divided differences.96 The extrapolated data is labeled in the paper as experimental data, and is given in Table 7 along with the fitted parameters used in the AES-PI EoS. As shown in Table 7, the Kd values decrease as the temperature increases. Such behavior is a result of the decrease in the dielectric constant of the solution. However, the F values do not show a clear trend; they change as the temperature is varied. Our interest is to obtain the correct tendency of the mean ionic activity coefficients and osmotic

coefficients as a function of the temperature to extrapolate them under conditions of the deepwater ambient. Figure 3 shows the experimental and calculated mean ionic activity coefficients (Figure 3a) and the experimental and calculated osmotic coefficients (Figure 3b) as a function of molality, from 0.1 to 6 m, at 273.15, 283.15, 293.15, 298.15, and 303.15 K for the single aqueous solution of NaCl. The AES-PI EoS accurately predicts the experimental mean ionic activity coefficients. However, the model overpredicts the observed mean ionic activity coefficient at 273.15 K and at 2 m, whereas it underpredicts at molalities higher than 5 m (Figure 3a). Figure 3b presents a reasonable agreement between the experimental and calculated osmotic coefficients, even though the AES-PI EoS underpredicts in molality values higher than 5 m at 273.15 K. Table 8 shows the fitted parameters used in the AES-PI EoS for the methane hydrate inhibited by NaCl aqueous solution. As seen in Table 8, the Kd values increase as the temperature and pressure ranges increase, even though the molalities are similar for the methane hydrate inhibited by NaCl aqueous solution. The Kd values also increase as the molalities increase, but the temperature and pressure ranges decrease in the ethane hydrate−NaCl aqueous solution system. However, such adjusted values do not show any tendency as the molalities increase and the temperature and pressure ranges decrease for the propane hydrate−NaCl aqueous solution system. The obtained Kd values with the KCl solution do not present any behavior for the methane, ethane, and propane hydrates, being that the molalities increase as the temperature and pressure ranges decrease in all three systems. In Table 8, we can also 12378


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Figure 8. IHP for NaCl aqueous solution of methane hydrate in the large cavity at different molalities and high pressures (LwHV phase).

between the experimental and calculated osmotic coefficients are greater. This is because the AES-PI model poorly represents the observed ϕ values at lower temperatures as molalities decrease; i.e., the model overpredicts the values of osmotic coefficients. However, we consider that the overpredicted values are in a reasonable range because of the variation of their values is in the second significant figure (see Figures 4b and 5b). Also, the predictive capability of the model for calculating the ϕ values at high salt concentration is good. The values of the contributions of the AES-PI EoS are constant as the pressure increases for ϕ and γ±, as seen in Figures 4 and 5. Additionally, the major contribution of the terms is the Born contribution for γ±, irrespective of the range of pressure, temperature, and salt concentration (see Figures 4a,c and 5a). On the other hand, the values are similar among them at the lowest molality for ϕ (Figure 4b), although, as the salt concentration increases, the values of MSA and Born contributions change (Figure 4d). This behavior indicates that each term in the AES-PI EoS contributes to the total value of γ± and ϕ. Figure 4a shows that the major contribution to γ± comes from the Born term, which describes the ion hydration. Additionally, its contribution is predominant over others in the γ±, as the salt molality is greater in the solution (Figure 4a). Accordingly, the values of the Born term are lower for ϕ. It can also be observed in Figures 4 and 5 that the values of the PR term are close to unity; such behavior suggests that the short-range forces are not significant at these concentrations. These intermolecular

observe that the hydrate−CaCl2 aqueous solution systems show the highest Kd values regardless of the molalities or the temperature or pressure ranges. On the other hand, the optimum Kd values suggest that the AES-PI EoS predicts a complete dissociation for all the electrolyte solutions with degrees of dissociation very close to unity irrespective of their molalities, temperature, and pressure ranges. Most strong electrolytes dissociate completely in water at ambient temperature.70 It can also be found in Table 8 that the F values are almost constant according to their salt inhibitor regardless of the molality, temperature, and pressure ranges except at pressures higher than 15 MPa, such as methane hydrate inhibited with NaCl aqueous solution. It can also be seen in Table 8 that the ability of the AES-PI EoS to represent the experimental data of γ± and ϕ is quite good at conditions of the deepwater ambient, with overall AAD values of 2.25 and 1.44% for γ± and ϕ, respectively, being the highest values for the CaCl2 aqueous solution. Figures 4 and 5 depict a comparison between the experimental and calculated mean ionic activity and osmotic coefficients as a function of pressure. These figures present the MSA, PR, and Born contributions as a function of pressure for the NaCl aqueous solution of the methane hydrate at different molalities. Figures 4a,c and 5a show that the AES-PI equation is capable of describing the calculated γ± in a molality range from 2.27 to 5.43 mol/kg. It can be observed in Figures 4b,d and 5b that, as the salt concentration increases, the agreement 12379


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Figure 9. IHP for CaCl2 aqueous solution of methane hydrate in the large cavity at different molalities.

Firoozabadi,104 who studied the effect of NaCl on the gas hydrate by using molecular dynamics, showed that the ions disrupt some hydrogen bonds of the gas hydrate because of the ionic hydration. Consequently, the dipoles of water molecules are aligned to ions by electrostatic interactions causing destabilization in the hydrogen bonds of the hydrate. On the other hand, Wang et al.105 developed a numerical model based on the lattice Boltzmann method in order to simulate the cyclopentane hydrate formation in brine. This model considers a hydration reaction at the hydrate surface, which uses a water molecule. In the context of the AES-PI EoS, the hydration of ions is included by the Born contribution. This contribution is based on the continuum-hydration model developed by Born.106 It states that the ions are initially uncharged species; however, these species receive a permanent electric charge and become ions. In other words, the ions are considered as charged hard spheres existing in a continuum having a uniform dielectric constant. Additionally, the adjusted ionic diameters (σ+ and σ−) contain the effect of ionic hydration. However, such a hydration effect is calculated at 298.15 K and 0.1 MPa because the experimental free energies of hydration are reported at such conditions. We suggest that the ionic hydration effect is a process that takes place as the conditions of the hydrate inhibition change, and not only at 298.15 K and 0.1 MPa. Such conditions could represent the ionic hydration limit. Accordingly, we set the adjusted ionic diameters to remain constant at the conditions mentioned earlier.

forces become important at much higher concentrations, i.e., near saturation. The MSA contribution is significant at molalities less than 1 m, although this term presents the major contribution for ϕ. Thus, the AES-PI EoS is able to capture the small contribution of the PR term within the range of moderate salt molality. The poor agreement between the experimental and obtained osmotic coefficients could be attributed to the Born contribution. We consider that the Born contribution is an important term, even though the values of the PR and MSA contributions are greater than the values of Born term for ϕ. This is because it takes into account the ion hydration. However, such a term is not considered completely the solvation effect at lower molalities and temperatures for the osmotic coefficients only. Accordingly, we have investigated this issue with a chemical term. This term handles the ion−solvent interactions by means of a chemical equilibrium approach based on a stepwise ion solvation.103 It is important to highlight that all the studied systems in this work, inhibited with aqueous electrolyte solutions, show that the Born contribution is dominant in the AES-PI EoS under different conditions of the deepwater ambient. 3.3. Estimation of the Hydration Process Parameters for Ions. It is well-known that the inhibition effect of the salt in aqueous solution is due to the ion formation, which results in a stronger ion−water molecule interaction than the water− water molecule interactions per se. Therefore, the salt ions avoid hydrate formation. According to Jiménez-Á ngeles and 12380


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Figure 10. c and d parameters for methane hydrate in the presence of NaCl aqueous solution at moderate pressures (IHV phase).

Figure 11. c and d parameters for methane hydrate in the presence of NaCl aqueous solution at high pressures (LwHV phase).

Figure 12. c and d parameters for methane hydrate in the presence of CaCl2 aqueous solution.

In a first attempt to consider the ionic hydration process (IHP) in the calculation of the hydrate dissociation pressure at different molalities of aqueous salt solutions, we proposed an optimized parameter in eq 24 to represent the ionic hydration process. Such an adjusted parameter multiplies the Langmuir constants as follows: IHPCij = Cij IHPp

where IHPCij is the Langmuir constant considering the ionic hydration process related to guest type j in cavity type i, Cij is a Langmuir constant, and IHPp is the parameter of the ionic hydration process. We used the Langmuir constants considering that these constants represent the hydrate formation. According to Jiménez-Á ngeles and Firoozabadi104 the ionic hydration causes the hydrogen bond breaking of the hydrate, so to take into account this phenomena, we included the IHPp

(24) 12381


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Table 9. Constants for Calculating the Ionic Hydration Process of Methane, Ethane, and Propane According to eq 26 c

d

n-alkane

electrolyte

c1 × 102 [kg2/mol2]

c2 × 101 [kg/mol]

c3

d1 × 105 [kg2/mol2]

d2 × 104 [kg/mol]

d3 × 103

methane

NaCl

5.10 5.43 0.0 30.19 8.30 0.0 0.0 26.81 26.12 0.0

−6.11 5.95 −1.78 −13.49 −8.22 −2.38 0.0 −21.42 −31.81 −1.04

1.89 1.72 2.00 1.52 2.17 2.56 0.07 4.28 10.07 0.17

3.92 −1.45 0.0 99.58 28.24 0.0 0.0 −1235.5 −20.96 0.0

−4.25 1.47 −0.60 −35.25 −3.03 −3.12 0.0 1013.2 348.27 −5.59

1.27 −0.37 −0.30 2.99 −5.68 −1.56 −0.05 −20.73 −34.27 0.81

ethane

propane

KCl CaCl2 NaCl KCl CaCl2 NaCl KCl CaCl2

Figure 13. Experimental and calculated pressures for methane hydrate in the presence of NaCl aqueous solution at different molalities and moderate pressures (IHV phase), according to eq 26.

Figure 6 shows the behavior of the ionic hydration process as a function of pressure for the methane hydrate with NaCl aqueous solution at 2.27, 3.53, 4.69, and 5.43 m. Figure 6 shows that all values of IHP are positive and they become more positive as the salt concentration increases. Moreover, as the molality increases, the values of IHP present a linear trend. In this case, the IHP values varied from 0.209 to 0.216, from 0.096 to 0.104, from 0.038 to 0.043, and from 0.02 to 0.021, for 2.27, 3.52, 4.68, and 5.48 m, respectively. In Figure 6, it is also shown that, as the NaCl concentration increases, the range of the IHP values decreases. This behavior could be attributed

parameter in eq 24. However, we found a rather poor agreement between the measured hydrate dissociation pressures and these calculated values with the hydrate inhibition approach. Therefore, we realized that a single parameter was not enough to give a good representation of the experimental data of the inhibited hydrate with aqueous salt solutions for any of the hydrates studied in this work. To follow the behavior of the ionic hydration process as a function of the pressure, we adjusted the IHPp parameter for each hydrate dissociation pressure datum, i.e., N = 1, by minimizing eq 23. This behavior is depicted in Figures 6 and 7. 12382


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Figure 14. Experimental and calculated pressures for methane hydrate in the presence of NaCl aqueous solution at different molalities and high pressures (LwHV phase), according to eq 26.

depicts that the IHP values are positive and remain positive as the pressure increases at the higher molalities (3.54 and 4.84 m), whereas they decrease as the pressure increases at the lowest molality, 2.08 m. In other words, the trend of the IHP values at the lowest molality is different from that of the highest molality. We also observed that the IHP values varying from 1.04 to 0.96, from 0.38 to 0.41, and from 0.15 to 0.17 decrease as the molality increases. It is important to notice that the IHP values are smaller for the methane hydrate system in the IHV phase than in the LwHV phase. An examination of Figure 8 shows that there exists a great deal of scatter among the IHP values. A slightly linear trend is observed in parts a and b of Figure 8 at the molalities of 2.08 and 4.84 m, respectively. The scattering observed of the IHP may be caused by the Langmuir constants per se since it is a non-pressure-dependent parameter as stated by Sloan and Kho1 and Jiang and Adidharma.11 Figure 9 presents the IHP values as a function of pressure for the methane hydrate dissociation pressures inhibited with the CaCl2 aqueous solution in a moderate pressure range, 4.92− 10.29 MPa. As can be seen, the IHP values are positive, and such values remain positive as the pressure increases. We also observed that the range of the IHP values is from 0.69 to 0.72, from 0.205 to 0.207, and from 0.022 to 0.024 at 1.02, 1.53, and 2.22 m, respectively. We can also notice that the linear trend of the IHP values increases as the molality increases, although these values decrease as the molality increases. This latter behavior is observed in all the studied systems in this work. The

to that the hydration limit of ions is reaching up through the ionic diameters. Figure 7 depicts the behavior of IHP as a function of pressure from the methane hydrate dissociation pressure for the large cavity inhibited with the aqueous NaCl solution at 2.27, 3.53, 4.69, and 5.43 m. As can be seen, the behavior of the IHP as a function of pressure for the large cavity is similar to the IHP for the small cavity, even though the IHP values for the large cavity are higher than those values for the small cavity. This is probably because the large cavity has more hydrogen bonds than the small cavity. Therefore, the IHP could be associated with the probability that it exists in a large cavity more than a small one. The range of the IHP values for the large cavity is from 1.12 to 1.15, from 0.51 to 0.55, from 0.202 to 0.227, and from 0.103 to 0.114. Due to the linear trend of the behavior of the IHP values as a function of pressure, we correlate such values using a straight-line equation: IHPc = c + dP

(25)

where IHPc is the correlation of the ionic hydration process for the small and large cavity, c and d are correlation coefficients whereas P is the pressure. Figure 8 shows the behavior of the IHP as a function of pressure using the inhibitor of NaCl aqueous solution. Note that these systems are over the widest range of pressure of all the light hydrocarbon hydrates inhibited with aqueous salt solutions studied in this work (7.51−64.6 MPa). Figure 8 12383


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Figure 15. Experimental and calculated pressures for methane hydrate in the presence of CaCl2 aqueous solution at different molalities, according to eq 26.

Table 10. Experimental Temperature and Pressure Data for Methane, Ethane, and Propane Hydrates Inhibited with Aqueous Electrolyte Solutions n-alkane

salt

N

molality [mol/kg]

T range [K]

P range [MPa]

AADP [%]

methane

NaCl

6 5 5 5 10 9 9 6 7 4 4 4 5 3 4 6 6 4 4 4 4 5 5 5

2.27 3.53 4.69 5.43 2.08 3.54 4.84 0.71 1.49 1.02 1.53 2.22 1.90 3.02 4.28 0.71 1.49 1.59 1.90 3.02 4.28 1.49 2.37 3.35

268.3−278.05 263.35−274.95 261.85−272.85 263.05−268.65 279.16−295.47 274.4−290.37 270.66−284.92 271.6−283.2 270.1−281.5 278.4−282.3 273.5−278.8 270.1−273.5 273.7−280.4 272.7−277.1 266.25−271.45 271.4−282.3 270.4−281.6 272.3−278.5 270.05−272.15 266.15−268.15 260.95−263.05 271.05−273.45 269.05−271.25 266.45−267.55

2.69−7.55 2.39−8.57 2.94−11 4.78−9.55 7.51−64.6 7.92−64.14 7.85−64.03 2.71−8.69 0.278−0.882 6.88−10.22 4.92−9.7 6.3−10.29 0.883−2.165 1.082−2.151 0.689−1.524 0.47−1.85 0.5−2.11 1.1−2.52 0.241−0.421 0.221−0.379 0.2−0.331 0.23−0.42 0.22−0.39 0.23−0.34

1.03 1.29 2.65 2.24 2.33 2.58 1.89 0.68 0.76 1.39 1.66 1.36 0.86 1.83 1.20 0.99 0.54 1.18 1.65 2.60 1.94 0.62 1.53 2.03

KCl CaCl2

ethane

NaCl

KCl

propane

CaCl2 NaCl

KCl

12384

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97

98 99

100, 101

98 98 101

101


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Industrial & Engineering Chemistry Research Table 10. continued n-alkane

salt

N

molality [mol/kg]

T range [K]

CaCl2

4 4 136

1.15 1.62

269.65−271.35 266.35−267.75

overall

P range [MPa] 0.25−0.37 0.23−0.35 average maximum

AADP [%] 0.87 1.77 1.52 2.65

source 101

Figure 16. Experimental and calculated dissociation pressures of methane hydrate: (a) NaCl aqueous solution at moderate pressures (IHV phase); (b) NaCl aqueous solution at high pressures (LwHV phase); (c) KCl aqueous solution; (d) CaCl2 aqueous solution.

calculations of this methane hydrate system, we observed that this value is not sensitive to such calculations, even though the d parameter is a function of molality and pressure. Such behavior indicates that the influence of pressure is greater than that of molality when correlating the dissociation pressure data for this methane hydrate system inhibited with NaCl aqueous solution at moderate pressures, although the values of the d parameter are small. We therefore excluded the obtained IHP value at 2.27 m. Accordingly, we obtained a good fit to a quadratic equation as a function of molality. We thus fitted the ionic hydration process to the following suggested correlation as a function of pressure and molality:

methane hydrate system and the other light hydrocarbon hydrate systems inhibited with the different salt solutions were correlated to a straight line with eq 25. We then correlated the dissociation pressures of the light hydrocarbon hydrate systems as a function of pressure for each molality in order to know the behavior of the c and d parameters as a function of the molality. Figures 10, 11, and 12 show the behavior of the c and d parameters as a function of the molality for the methane hydrate in the presence of the NaCl aqueous solution at moderate pressures (IHV phase) and high pressures (LwHV phase) and in the presence of the CaCl2 aqueous solution, respectively. As shown in Figure 10, the values of c and d parameter are positive, and they are less positive as the molality increases. The range of the values for c parameter is from 0.08 to 0.79, whereas the range of the values for the d parameter is from 5.55 × 10−5 to 2.62 × 10−4. Figure 10 also presents that the values of the c parameter accurately fit a quadratic equation, whereas the values of the d parameter also exhibit the behavior of a quadratic equation, except at the molality of 2.27 m. Such behavior can be attributed to that the IHP values showed a slight scatter at 2.27 m than at higher molalities. However, after an extensive testing of

IHPc = c1m2 + c 2m + c3 + (d1m2 + d 2m + d3)P

(26)

where c1, c2, c3, d1, d2, and d3 are correlation constants, m is molality, and P is pressure. It is important to mention that this is the only light hydrocarbon hydrate system that exhibited such behavior in the d parameter since all other systems presented a reasonable fit to the quadratic equation for both parameters. Figure 11 depicts the behavior of the c and d parameters as a function of molality. As shown in Figure 11, the values of the 12385


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Figure 17. Experimental and calculated dissociation pressures of ethane hydrate: (a) NaCl aqueous solution at moderate pressures (IHV phase); (b) NaCl aqueous solution at high pressures (LwHV phase); (c) KCl aqueous solution; (d) CaCl2 aqueous solution.

c parameter are positive, and they are less positive as the molality increases. The d values are negative, but they increase their values up to positive values as the molality increases. The range of the c and d parameters are from 0.11 to 0.72 and from −3.92 × 10−6 to 1.26 × 10−4, respectively. Note that the range of the c parameter is similar for the phases IHV and LwHV of the methane hydrate as mentioned above. It can also be observed that Figure 11 presents a satisfactory agreement with a quadratic equation as a function of the molality. Accordingly, we used eq 26 to calculate the IHP. Figure 12 presents the behavior of the c and d parameters for the methane hydrate inhibited with the CaCl2 aqueous solution. As shown in Figure 12, the values of the c and d parameters are positive, except for the d parameter at the molality of 1.53 m. Also, both parameters accurately adjust to the quadratic equation as a function of the molality. Therefore, these parameters also fit eq 26. On the other hand, the negative value of the d parameter could be attributed to that the IHP values exhibit a slight scatter at 1.53 m (see Figure 9). The ranges of the c and d parameters are from 0.46 to 1.47 × 10−2 and from 4.57 × 10−4 to 5.18 × 10−5, respectively. Table 9 shows the adjusted c1, c2, c3, d1, d2, and d3 constants of the IHP correlation (eq 26) for methane, ethane, and propane hydrates inhibited with aqueous electrolyte solutions. Figures 13−15 show the obtained values with the IHPc correlation (eq 26) as a function of experimental and calculated

pressures. As seen in Figures 13−15, the calculated pressures are in good agreement with the experimental ones when we describe the IHP with the IHPc correlation, which is as a function of pressure and molality. This means that such a correlation is able to fit the IHP to a straight-line equation as given by eq 26. It is important to note that the calculated values with the IHPc correlation (eq 26) were used for the hydrates of large and small cavities. Figures 13−15 also present that the adjusted values of the IHP with the suggested correlation are smaller than the optimized values for large cavity, but they are larger than the adjusted values for the smaller cavity (see Figures 6 and 7). Figure 13 shows that the IHP increases as the pressure increases and it decreases as the molality increases, whereas Figures 14 and 15 present a different behavior in the IHP. An examination of Figures 14a,b and 15b reveals that the IHP decreases as the pressure increases; i.e., the slope of the straight line changes from positive to negative. We suggest that this behavior could be attributed to the inclusion of the hydrated chloride or sodium ions into the hydrate structure. According to Tung et al.,107 the chlorine or sodium ions replace one of the water molecules from the structure cavity of the hydrate. Accordingly, such a replacement causes the rearrangement of the water molecules from the cavity to better solvate the chloride or sodium ions affecting the size and shape of the hydrate structure; namely, such a rearrangement of the water molecules produces a distortion in the hydrate structure. 12386


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Figure 18. Experimental and calculated dissociation pressures of propane hydrate: (a) NaCl aqueous solution at moderate pressures (IHV phase); (b) NaCl aqueous solution at high pressures (LwHV phase); (c) KCl aqueous solution; (d) CaCl2 aqueous solution.

are not able to reasonably fit the linear IHPc (eq 25) (see Figures 14c and 15c). Accordingly, the c (0.116) and d (2.03 × 10−6) values obtained from eq 25 at 4.84 m are not the correct values. However, such values were fitted to a quadratic equation as a function of molality (see Figure 11). Then, it results that the IHPc correlation (eq 26) cannot describe the behavior of IHP values adequately at the highest molality. In other words, they are not able to correctly match the suggested linear correlation, eq 26. An exemplification of the above-mentioned system is presented as the following: Figure 8 shows the behavior of the IHP for the methane hydrate at the molality of 4.84 m. This is the IHP increases as the pressure increases. However, such behavior is reached up to 42.26 MPa. After this IHP datum, the following values decrease as the pressure increases. It results that such an IHP value is not able to correctly match the suggested linear correlation (eq 25) and eq 26. Consequently, the methane hydrate dissociation pressure exhibits a slightly underprediction at 64.03 MPa, as can be seen in Figure 16b. This example illustrates the strong effect of the IHPc correlation in predicting the dissociation pressures for the light hydrocarbon hydrates investigated in this work. Ethane hydrate dissociation pressures in NaCl, KCl, and CaCl2 aqueous solutions are shown in Figure 17. The present approach gives an accurate representation of the experimental dissociation pressures in salt aqueous solutions, as demonstrated by Figure 17. It can also be seen in Figure 17 that these dissociation pressures are lower than those of methane hydrate

4. CALCULATION OF LIGHT HYDROCARBON HYDRATE DISSOCIATION PRESSURES The present modeling approach was applied to predict and calculate the inhibition of light hydrocarbon hydrates (e.g., methane, ethane, and propane) in the presence of single NaCl, KCl, and CaCl2 solutions. Table 10 lists the molality of the aqueous electrolyte solutions, the temperature and pressure ranges, the AADP, and the source of the experimental data for methane, ethane, and propane hydrates. It can be seen from Table 10 that the calculated dissociation pressures for methane hydrate were only predicted at IHV and LwHV phases. It can also be seen from Table 10 that the AADP values vary from 0.62 up to 2.65. Figures 16−18 compare the measured dissociation pressures with those calculated in this work for the light hydrocarbon hydrate inhibited in the presence of NaCl, KCl, and CaCl2 aqueous solutions at the different conditions of pressure, temperature, and molality. Figure 16 shows the inhibition effect of NaCl, KCl, and CaCl2 solutions on the dissociation pressures for methane hydrate. In Figure 16, an excellent agreement is observed between experimental and calculated dissociation pressures for the methane hydrate. Figure 16 also shows that there is a slight underprediction of the methane hydrate dissociation pressure at the highest molality and pressure (see Figure 16b,d). Such an underprediction is because the optimized IHP values (see Figures 8c and 9c) from eq 24 12387


Article


Hkw = temperature-dependent Henry’s constant IHPc = defined in eq 26 IHPp = defined in eq 24 ki,j = binary interaction parameter in the PR EoS contribution kB = Boltzmann constant Kd = dissociation constant m = molality of the salt aqueous solution M = molar mass of water P = pressure R = universal gas constant s = defined in eq 19 T = temperature V = molar volume x = mole fraction Z = compressibility factor

(see Figure 16). Figure 18 depicts a comparison between the experimental and calculated propane hydrate dissociation pressures in the presence of the three aqueous solutions (NaCl, KCl, and CaCl2) considered in this work. As shown by Figure 18, the present modeling approach satisfactorily represents the observed dissociation pressures. It can also be seen that the propane hydrate exhibits the lowest dissociation pressures of the light hydrocarbon hydrates (see Figures 16 and 17).

5. CONCLUSIONS The modified AES-PI equation of state coupled with the van der Waals model was applied to calculate and predict the hydrate dissociation pressures for methane, ethane, and propane in the presence of single NaCl, KCl, and CaCl2 aqueous solutions under different conditions of pressure, temperature, and molality. The following conclusions can be drawn from this work: 1. The improved water volume in the AES-PI EoS increased the accuracy of the calculated mean ionic activity and osmotic coefficients, as well as the estimated free energies of hydration at 298.15 K and 0.1 MPa. 2. The modified AES-PI EoS reasonable calculated and predicted the osmotic and mean ionic activity coefficients under conditions of deepwater ambient. 3. The proposed approach properly depicts the inhibition effect at different salt concentrations of the light hydrocarbon hydrate dissociation pressures. 4. The present modeling approach is quite capable of representing the dissociation pressures for the light hydrocarbon hydrates by using a linear correlation for the ionic hydration process as a function of the pressure and molality.

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Subscripts

c = critical property 1 = pertains to small cavity 2 = pertains to large cavity w = water h = hydration condition i = cavity j = gas molecule r = reduced property + = cation − = anion ± = mean ionic condition Superscripts

H = hydrate phase 0 = reference state P = Pauling crystal ionic diameter

AUTHOR INFORMATION

Greek Symbols

Corresponding Author

ω = acentric factor ν = number of cavities Δ = change in energy α = temperature-dependent correlation of parameter a β = hypothetical empty hydrate phase χ = defined in eq 19 δ = bulk modulus ϕ = osmotic coefficient γ = activity coefficient λ = phase μ = chemical potential θij = fractional occupancy σ = effective ionic diameter υ = total number of cations and anions ω = acentric factor

*E-mail: [email protected]. ORCID

Fernando García-Sánchez: 0000-0003-1615-8016 Ricardo Macías-Salinas: 0000-0002-0372-8190 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M. A. Zúñiga-Hinojosa and J. Martínez gratefully acknowledge the National Council for Science and Technology of Mexico (CONACyT) for providing financial support for this work. R. Macías-Salinas is also grateful for the financial support provided by the Instituto Politécnico National during the realization of this work.

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Abbreviations

NOMENCLATURE a = attractive parameter in the PR EoS Aij, Bij = fitted constants as defined in eq 8 b = covolume parameter in the PR EoS c = translation volume c = defined in eq 26 C = Langmuir constant d = defined in eq 26 f = fugacity fc = function of critical compressibility factor F = correction factor for the attraction parameter of ionic species in the PR EoS G⊕ = Gibbs free energy of hydration

AAD = absolute average deviation AES-PI = aqueous electrolyte solutions with partial ionization EoS = equation of state IHP = ionic hydration process IHV = ice−hydrate−vapor LwHV = liquid−hydrate−vapor MSA = mean spherical approximation contribution PR = Peng−Robinson PSRK = predictive Soave−Redlich−Kwong QM = quantum mechanics SAFT = statistical associating fluid theory SAFT2 = ion-based statistical associating fluid theory 12388


Article

Industrial & Engineering Chemistry Research SAFT-VR = statistical associating fluid theory for potentials of variable range vdWP = van der Waals and Platteeuw

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