Predictive Method for the Change in Equilibrium Conditions of Gas

ARTICLE pubs.acs.org/IECR

Predictive Method for the Change in Equilibrium Conditions of Gas Hydrates with Addition of Inhibitors and Electrolytes Min-Kang Hsieh,† Yen-Ting Yeh,† Yan-Ping Chen,† Po-Chun Chen,‡ Shiang-Tai Lin,*,† and Li-Jen Chen*,† † ‡

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan Central Geological Survey, P.O. Box 968, New Taipei City 235, Taiwan

bS Supporting Information ABSTRACT: Here we present a predictive method for the change in the three-phase (vaporliquidhydrate) equilibrium condition of gas hydrates upon the introduction of organic inhibitors and electrolytes. The PengRobinsonStryjekVera (PRSV) equation of state, combined with the COSMO-SAC activity coefficient liquid model through the modified HuronVidal (MHV1) mixing rule, is used to describe the fluid phase, and the van der Waals and Platteeuw (vdWP) model is used to describe the hydrate crystalline phase. The temperature-dependent Langmuir absorption constants for the vdWP model are determined by fitting to the equilibrium condition of pure gas hydrates. Once determined, the method contains no adjustable binary interaction parameters and can be used for prediction of the phase behaviors of gas hydrates with additives that do not enter the cages of the clathrate hydrates (e.g., most inhibitors and electrolytes). We examined the accuracy of this method using five pure gas hydrates, five organic inhibitors, and nine electrolytes, and over ranges of temperature (259.0303.6 K) and pressure (1.37 1052.08 108 Pa). The average relative deviations in the predicted equilibrium temperatures are found to be 0.23% for pure gas hydrates, 0.72% with organic inhibitors, and 0.18% with electrolytes, respectively. We believe that this method is useful for many gas hydrate related engineering problems such as the screening of inhibitors for gas hydrates in flow assurance.

1. INTRODUCTION The phase behavior of waterhydrocarbon systems is a principal issue in natural gas/oil recovery and transportation. In particular, the formation of gas hydrates in oil and gas pipelines, which leads to blockage of flow, can be prevented by changing the phase transition conditions with the addition of electrolytes or organic inhibitors.19 On the other hand, natural gas (methane) hydrates found in the permafrost and ocean floor sediments are abundant1012 and have been considered as a potential energy resource.13 The distribution of methane hydrates in the natural environment can be estimated based on the stability of gas hydrates in saline water.11 An accurate predictive thermodynamic model could be very useful for understanding the inhibition effect of additives (e.g., alcohols, glycols, or electrolytes) on the hydrate formation conditions. A general approach to the modeling of vapor liquidhydrate three-phase equilibrium (VLHE) is the combination of an equation of state (EOS) and the van der Waals Platteeuw (vdWP) model.14 For example, Anderson and Prausnitz1 developed a molecularthermodynamic correlation for modeling the inhibition effect of methanol on hydrate formation. Fugacities of all components in the vapor phase are derived from the RedlichKwong EOS,15 and those in the liquid phase are calculated by the UNIQUAC model.16 Jager et al.2 used the modified Helgeson EOS17 combined with a Bromley activity model18 in aqueous phase including methanol and electrolytes, and then adopted a new hydrate fugacity model in the hydrate phase. Ma et al.3 used the PatelTeja EOS19 with the Kurihara mixing rule20 and the hydrate model proposed by Chen and Guo21 to predict hydrate formation conditions for systems containing inhibitors. Li et al.4 used the statistical associating fluid theory (SAFT)22 equation of state to predict the inhibition r 2011 American Chemical Society

effect of methanol and glycols on hydrate formation. Although these models provide successful predictions of hydrate systems with alcohols and glycol, they also contain many interaction parameters that must be determined by regression to experimental data such as vaporliquid equilibrium (VLE) data. Similarly, when electrolytes are added to water, most modeling methods require the fitting of the model parameters to some phase equilibrium data containing electrolytes. For example, Nasrifar et al.5 reported the inhibition effect of electrolytes (NaCl, KCl, and CaCl2 ) and single alcohol compounds (methanol, ethylene glycol, and glycerol) on the formation conditions of natural gas and CO2 hydrates based on the ValderramaPatelTeja (VPT) EOS23 with the equation of Patwarthan and Kumar.24 Javanmardi et al.6 used a modified AdachiLuSugie EOS25 with the modified DebyeH€uckel model26 to describe the equilibrium conditions of several gas hydrates with electrolytes (NaCl, KCl, CaCl2) and single alcohols (methanol and ethylene glycol). Mohammadi and Tohidi7 used the VPT EOS with the modified DebyeH€uckel model to show the change in the phase boundary of methane hydrates with the addition of NaCl and methanol. Using the same method, Mohammadi and Richon8 studied the inhibition effects of mixed electrolytes (NaCl and KCl) on CH4 hydrate formation. More recently, Bandyopadhyay and Klauda9 used the predictive SoaveRedlichKwong EOS27 combined with the modified UNIQUAC model (LIFAC model)28 to describe the inhibition Received: September 14, 2011 Accepted: December 20, 2011 Revised: December 12, 2011 Published: December 20, 2011 2456

dx.doi.org/10.1021/ie202103a | Ind. Eng. Chem. Res. 2012, 51, 2456–2469

Industrial & Engineering Chemistry Research

ARTICLE

effects of NaCl and CaCl2 on CH4, C2H6, C3H8, and CO2 hydrate formation. While the aforementioned approaches are capable of predicting the phase behavior of gas hydrates with additives, their applicability may be limited if the necessary experimental phase equilibrium data (e.g., VLE) for obtaining the binary interaction parameters are not available. Even when such data are available, it is often the case that the temperatures are very different from (often much higher than) the gas hydrate formation conditions. The extrapolative use of the fitted parameters to conditions beyond where the data are available may also lead to a reduction in accuracy. In this work, we present a new approach based on the modeling of fugacity in the calculation of the VLHE for pure gas hydrates. In this method, the fugacity of a species in the fluid phase is determined by the PengRobinsonStryjekVera (PRSV) EOS29 combined with the predictive COSMO-SAC activity coefficient model30,31 through the first order modified HuronVidal (MHV1) mixing rule.32 In the solid hydrate phase, the fugacity is determined based on the van der Waals and Platteeuw (vdWP) model. We examine its capability in modeling the three-phase coexisting conditions of five gas hydrates (CH4, C2H6, C3H8, CO2, and N2), and its predictive power for the change of equilibrium conditions upon addition of organic inhibitors (methanol (MeOH), ethanol (EtOH), ethylene glycol (EG), diethylene glycol (DEG), and glycerol) and various electrolytes (NaCl, NaBr, KCl, KBr, HCl, NaOH, CaCl2, MgCl2, CaBr2, and seawater). As a comparison, we also examined the prediction of the phase boundary by replacing the COSMO-SAC model with the UNIQUAC model, and by changing the mixing rule to the van der Waals one-fluid mixing rule. The predictions from the PRSV + MHV1 + COSMOSAC + vdWP approach, where no parameter fitting is necessary, is found to be similar to or more accurate than those from other methods (using the van der Waals (vdW) mixing rule or the UNIQUAC model), whose binary interaction parameters are obtained from VLE data. We believe that the PRSV + MHV1 + COSMOSAC + vdWP method is a powerful predictive tool for modeling the phase behavior of gas hydrates with addition of various inhibitors.

where Tc is the critical temperature, Pc is the critical pressure, and k0 = 0.378893 + 1.4897153ω 0.17131848ω2 + 0.0196554ω3, with ω being the acentric factor. The parameter k1 for each component in this study is taken from Stryjek and Vera.29,33 When the PRSV EOS is used for a mixture, some mixing rule describing the composition dependence of variables a and b is necessary. One commonly used method is the van der Waals onefluid mixing rule am ¼

∑i ∑j xi xj aij

ð5Þ

bm ¼

∑i xi bi

ð6Þ

where aij ¼

∑

∑

where CMHV1 = 0.53 is a constant. The volume parameter bm is the same as in eq 6. The MHV1 mixing rule does not contain the binary interaction parameter kij; therefore, the EOS can become predictive for mixtures if a predictive liquid model is used for Gex LM. 2.2. Fugacity from the PRSV EOS. The fugacity of PRSV EOS for a chemical species in a mixture is given as follows ln

2.1. The PengRobinsonStryjekVera Equation of State. The PRSV EOS describes the pressurevolumetemperature relation of a fluid as

RT aðTÞ V _b V _ ðV _ þ bÞ þ bðV _ bÞ

b ¼ 0:077796

k ¼ k0 þ k1

RTc Pc rffiffiffiffiffi! T T 1 þ 0:7 Tc Tc

f̅ i ðT, P, _xÞ 1 ∂Nbm Pbm ¼ ðZ 1Þ ln Z bm ∂Ni T, Nj6¼i RT xi P 2 ! 2 am 6 1 ∂N am þ pffiffiffi 4 2 2bm RT Nam ∂Ni

ð1Þ

where P is the pressure, R is the gas constant, T is the temperature, and V is the molar volume. The variables a and b are species dependent parameters and are determined from the critical properties of a pure fluid as " rffiffiffiffiffi!#2 R 2 Tc 2 T 1 þ k 1 ð2Þ aðTÞ ¼ 0:457235 Tc Pc ð3Þ

ð7Þ

and kij is the binary interaction parameter that can be obtained by fitting to experimental phase equilibrium data. Another popular approach is the MHV1 mixing rule,32 which determines the composition dependence of variable a by requiring the excess Gibbs free energy from the EOS to be the same as that from a liquid model, which after some manipulations becomes C am ai ¼ xi bRT bi RT i " # C G 1 bm _ ex LM þ xi ln ð8Þ þ MHV1 C RT bi i

2. THEORY

P¼

pffiffiffiffiffiffiffi ai aj ð1 kij Þ ¼ aij

"

T, Nj6¼i

# pffiffiffi V _ þ ð1 2Þbm 1 ∂Nbm pffiffiffi ln bm ∂Ni T, Nj6¼i V _ þ ð1 þ 2Þbm #

ð9Þ where N is the total number of moles and Ni is the number of moles of species i. The expression of derivatives, (1/N)(∂N2 am/ ∂Ni) and ∂Nbm/∂Ni, depends on the mixing rule used. For the vdW mixing rule they are ! 1 ∂N 2 am ¼2 xj aij ð10Þ N ∂Ni j

∑

T, Nj6¼i

ð4Þ 2457

∂Nbm ∂Ni

¼ bi

ð11Þ

T, Nj6¼i dx.doi.org/10.1021/ie202103a |Ind. Eng. Chem. Res. 2012, 51, 2456–2469


ARTICLE

When the MHV1 mixing rule is considered, eq 10 becomes 1 ∂N 2 am N ∂Ni

!

T, Nj6¼i

bi bRT bm ¼ am þ MHV1 ln γi þ ln C bm bi # ai bi þ 1 þ am bm bi

(

∑j xjlj θτ

∑j

θi ¼

with interaction coefficient cES = 6525.69 + (1.4859 108)T2, and chb ðσ tm , σ sn Þ 8 o 1 > 2 t s > > > 4013:78=ðkcal 3 mol 3 Α 3 e Þ if s ¼ t ¼ OH and σ m σ n < 0 > o > < 932:31=ðkcal 3 mol1 3 Α 3 e2 Þ if s ¼ t ¼ OT and σ tm σ sn < 0 ¼ o > 1 > 2 t s > > 3016:43=ðkcal 3 mol 3 Α 3 e Þ if s ¼ OH, t ¼ OT, and σ m σ n < 0 > > :0 otherwise

ln γi ¼ ln γi COSMOSAC þ ln γi PDH

and uij ujj RT

nhb, OH, OT

∑s ∑σ

psi ðσ sm Þ½ln ΓsS ðσ sm Þ ln Γsi ðσsm Þ

m

þ ln

Φi z θi Φi þ qi ln þ li 2 xi Φi xi

ð18Þ γCOSMOSAC , i

ri and qi are the normalized volume and surface area parameters for species i, uij is an energy parameter for interaction between species i and species j, and z is the coordination number whose value is set to 10 in this work. In the UNIQUAC model, τij and τji are two adjustable parameters whose values are obtained by fitting to experimental VLE data. The second liquid model considered is a predictive one, the COSMO-SAC activity coefficient model:30 ln γi ¼ ni

ð17Þ

The COSMO-SAC model was originally developed for mixtures of nonelectrolytes. Recently, Hsieh and Lin31 extended the method to electrolyte solutions by inclusion of the long-range interactions via the PitzerDebyeH€uckel model.35

∑j

li ¼ ðz=2Þðri qi Þ ðri 1Þ

ln τij ¼

n

cES ðσtm þ σ sn Þ2 chb ðσ tm , σsn Þðσtm σ sn Þ2

ð13Þ

xi qi xj qj

∑σ

) ΔWðσ sm , σ tn Þ RT

ΔWðσ tm , σsn Þ ¼

where xi ri , xj rj

∑t

ptj ðσ tn Þ Γsj ðσ tn Þ exp

where ΔW(σm,σn) is the electrostatic interaction between two segments of charge density σm and σn.

k

Φi ¼

nhb, OH, OT

ð16Þ

3

∑i ∑ θj kijτkj 75

∑j

ln Γsj ðσ sm Þ ¼ ln

ð12Þ

where γi is the activity coefficient of species i in the liquid phase, calculated from liquid activity coefficient model. 2.3. Liquid Activity Coefficient Models. The excess Gibbs free energy needed in the MHV1 mixing rule is obtained from a liquid activity coefficient model. Two liquid models are considered here. The first is a correlative model, i.e., the UNIQUAC activity coefficient model: Φi z θi Φi þ qi ln þ li ln γi ¼ ln 2 xi Φi xi 2 ! 6 þ qi 41 ln θj τji

non-hydrogen bonding (nhb) atoms. Γj(σm) is the activity coefficient of segment with a charge density of σm in solution j (j is i for the pure liquid or S for the mixture), and is determined from

∑j xjlj

ð14Þ

where ni = Ai/aeff is the number of surface segments contained in species i, Ai is the surface area of species i, aeff is the effective segment area, and pi(σm) is the σ-profile of a substance i. The σ-profile is the probability of finding a surface segment with a screening charge density σm on the surface of species i. The σ-profile reflects the electronic nature of a chemical and is determined from first principles solvation calculation.34 pS(σm) is the σ-profile of the mixture

The first term on the right-hand side of eq 17, considers the short-range interaction in the liquid phase and is calculated from eq 14 with a modification of the calculation of segment activity coefficient Γj(σm) to include of ion effects (see ref 31 for details). ln Γtj ðσ tm Þ ¼

ln

8 nonhb, hb, > > > < ion, iongrp > > > :

∑s ∑σ n

9 > > > = s t ΔWðσ m , σ n Þ s s s s þ ln Γj ðσ n Þ pj ðσ n Þ exp > RT > > ;

ð19Þ

The second term on the right-hand side of eq 17, γPDH , i considered the long-range interaction in liquid phase of species i, is given by " ! 1000 1=2 2zi 2 PDH ¼ Aϕ ln γi lnð1 þ FIx 1=2 Þ Ms F !# zi 2 Ix 1=2 2Ix 3=2 þ ð20Þ 1 þ FIx 1=2

ð15Þ

where Ms is the average molecular weight of solvent, Ix is the ionic strength Ix = (1/2)∑i xizi2, and zi is the net charge of ion i. Aϕ is the DebyeH€uckel constant, and is defined as !3=2 1 2πNA ds 1=2 Qe 2 ð21Þ Aϕ ¼ 3 1000 εs kT

The superscript s indicates surface from hydrogen bonding atoms, such as the hydroxyl group (OH) and any others (OT), or

where ds is the average density of solvent, NA is the Avogadro constant, Qe is the charge of an electron, εs is the average dielectric constant of solvent, and k is the Boltzmann constant.

xi Ai psi ðσÞ ∑ i psS ðσÞ ¼ ∑i xi Ai

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ARTICLE

F is the closest approach parameter, given by F¼

∑i xi ri

ð22Þ

0:134A_

where ri is the radius of ion i and the radii for each ion are taken from the literature without further adjustment; xi* is its solventfree mole fraction. Since the temperature considered in this work may go below the freezing point of water, we used the following equation for the density of water:36,37

8 > ½999:83952 þ 16:94518ðT 273:15Þ 7:98704 > > > > 103 ðT 273:15Þ2 4:617046 > > > > 105 ðT273:15Þ3 þ 1:05563 107 ðT273:15Þ4 2:805425 > > > < 1010 ðT273:15Þ5 =½1 þ 1:687985ðT273:15Þ T > 273:3 K dw ¼ > > > > > 18015:28=½18:0182 1:407964 103 ðT 273:15Þ > > > > > > > : þ 1:461418 104 ðT 273:15Þ2 T < 273:3 K

ð23Þ The dielectric constant of water is determined as follows:38 εw ¼ 78:3055 3:18404 101 ðT 298:15Þ þ 3:84093 104 ðT 298:15Þ2

ð24Þ

2.4. Fugacity from the van der WaalsPlatteeuw Model. The original van der WaalsPlatteeuw model14 was developed to describe the chemical potential of water in the solid hydrate phase. Chen and Guo,39 and more recently, Klauda and Sandler40 revised the original vdWP equations to describe the fugacity of water, !! f̅ wH ðT, P, yi Þ ¼ fwβ ðT, PÞ exp

∑m νm ln

1

∑l θml

ð25Þ where νm is the number of cages of type m per water molecule in the hydrate lattice; fwβ is the fugacity of a hypothetical empty hydrate lattice, given by 0Z P 1 β V ðT, PÞ dP B sat w C B Pw C ð26Þ f̅ wβ ðT, PÞ ¼ Pwsat ϕsat ðTÞ exp B C w RT @ A β where ϕsat w is assumed to be unity, Vw is the molar volume of the sat empty lattice, and Pw is the saturated vapor pressure of empty hydrate. Sloan and co-workers41 proposed the following correlation equations for sI and sII hydrates:

ln Pwsat, β ¼ 17:440

6003:9 T

for sI hydrates

ð27Þ

ln Pwsat, β ¼ 17:332

6017:6 T

for sII hydrates

ð28Þ

The occupancy of each cage by a guest is given by θml ¼

Cml ðTÞ f̅ l V ðT, P, yl Þ 1 þ Cml ðTÞ fl ðT, P, yl Þ

∑l

ð29Þ

where fl is the fugacity of species l in the vapor phase, yl is the mole fraction of component l in the vapor phase, and Cml(T) is the Langmuir adsorption constant for guest l in m type cage. In this work, the following simple temperature dependence of Cml with two fitting parameters, Aml and Bml, is used: Aml Bml exp Cml ¼ ð30Þ T T In this work, the values of Aml and Bml are determined by fitting to the three-phase coexisting conditions of pure gas hydrates.

3. COMPUTATIONAL DETAILS The three-phase (vaporliquidhydrate) coexisting condition of gas hydrate is determined based on the equality of fugacity of water among all three phases. f̅ Vw ðT, P, yi Þ ¼ f̅ Lw ðT, P, xi Þ ¼ f̅ H w ðT, P, yi Þ

ð31Þ

The vdWP model is used to describe the fugacity of water, fH w (T,P,yi), in the solid hydrate phase. Three different approaches were used to determine the fugacity in the fluid phases: (1) the PRSV EOS with vdW mixing rule, denoted as PRSV + vdW, (2) the PRSV EOS with the correlative UNIQUAC model through the MHV1 mixing rule, denoted as PRSV + MHV1 + UNIQUAC, and (3) the PRSV EOS with the predictive COSMOSAC model through the MHV1 mixing rule, denoted as PRSV + MHV1 + COSMOSAC. The binary interaction parameter, kij (eq 7), needed in PRSV + vdW is assumed to have the following temperature dependency: kij ¼ αij T þ βij

ð32Þ

In the PRSV + MHV1 + UNIQUAC approach, the interaction parameters τij and τji are assumed to be temperature dependent. uij ujj hij τij ¼ exp ¼ exp gij þ ð33Þ RT T where αij, βij, gij, and hij are parameters whose values are adjusted to minimize the root-mean-square error (RMSE) in the calculated composition of VLE. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Ndata calc expt ð34Þ ðx xj Þ2 RMSE ¼ Ndata j ¼ 1 j

∑

where Ndata is the number of VLE data used. The optimized values for αij, βij, gij, and hij are provided in Tables S1 and S2 of the Supporting Information. No parameter fitting is required in the PRSV + MHV1 + COSMOSAC approach. The inputs for this method are the σ-profiles of each component. The procedure of obtaining the σ-profiles can be found elsewhere34 and is not reproduced here. The values of Aml and Bml (eq 30) for the Langmuir adsorption constants in the vdWP model for each guest in each type of cage are determined by fitting to the VLHE of pure gas hydrates using the PRSV + vdW + vdWP approach. The objective function used was vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u u 1 Ndata Pjcalc Pjexpt 2 ð35Þ RMSE ¼ t expt Ndata j ¼ 1 Pj

∑

where P is the equilibrium pressure at some given temperature, and Ndata is the number of VLHE data used. The optimal values of these two parameters for each gas hydrate are shown in Table 1. 2459



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Table 1. Values of Langmuir Constants (eq 30) in the vdWP Model for Five Gas Molecules in Single Gas Clathrate Hydrates large cage 6

small cage Aml 106

Aml 10

structure

no. of

guest

type

params

(K/kPa)

Bml (K)

CH4

sI

4

14.5250

3380.65

8.6119

3388.96

C2H6

sI

2

43.4154

3799.54

C3H8

sII

2

4.3258

5430.62

CO2 N2

sI sII

4 4

138.5086 184.6114

3208.49 2901.03

4.4963 0.4384

2869.89 2896.03

(K/kPa)

Bml (K)

The formation conditions of gas hydrate with inhibitive additives can be directly predicted by using the Langmuir adsorption constant from Table 1. [Note that all the additives considered in this work are assumed to be absent from the hydrate phase. Therefore, the Langmuir adsorption constants for the additives are not needed.] Three methods are used for systems containing organic inhibitors: (1) PRSV + vdW + vdWP (eqs 9, 10, 11, and 25), (2) PRSV + MHV1 + UNIQUAC + vdWP (eqs 9, 11, 12, 13, and 25), and (3) PRSV + MHV1 + COSMOSAC + vdWP (eqs 9, 11, 12, 14, and 25). For systems containing electrolytes, however, complications emerge because the critical properties and acentric factor needed for the PRSV EOS are not available for electrolytes. To circumvent the problem, we take advantage of the fact that the electrolytes do not enter the vapor phase under the conditions of interest in this work. Therefore, the fugacity of species in the vapor phase can be determined based on the same methods outlined above, and the fugacity in the liquid phase is determined from the activity coefficient as f̅ wL ðT, P, _xÞ ¼ xw γw fwL ðT, PÞ

Figure 1. Flowchart for calculating the three-phase coexisting condition of gas hydrates.

ð36Þ

where f Lw is the fugacity of pure water and is determined from the PRSV EOS. The activity coefficient in eq 36 is determined from the modified COSMO-SAC model for electrolytes (eq 17). This approach is referred to as PRSV + MHV1 + COSMOSAC(ion) + vdWP (eqs 9, 11, 12, 18, and 25). The procedure of determining the three-phase coexisting condition is illustrated in Figure 1. For a given temperature T and liquid phase composition x with an initial guess for the equilibrium pressure P, the VLE is first determined via isothermal flash calculations.42 The fluid phase fugacities are then used in the calculation of water fugacity in the hydrate phase from the vdWP model. If the water fugacity in the hydrate phase equals that in the fluid phase, then the equilibrium pressure is found. Otherwise, a new pressure is generated and the calculation is repeated until the equality of water fugacity is attained. It should be noted that the Gibbs degree of freedom for gas hydrates with one additive is 2 (F = C + 2 P = 3 + 2 3). In other words, the three-phase coexisting temperature (or pressure) can be determined when the pressure (or temperature) and the concentration of the additive in the liquid phase is specified. The concentration of additive in this work is the concentration in the liquid feed. The absolute average relative deviations in temperature (AARD-T) and pressure (AARD-P) are used to measure the quality of predictive ability in this work. AARD-T and AARD-P are

Figure 2. Three-phase coexisting condition of CH4 (circles),4559 C2H6 (inverted triangles),45,46,50,6063 C3H8 (squares),46,52,54,6468 CO2 (diamonds),46,55,59,65,6976 and N2 (triangles)48,49,7779 gas hydrates. The solid lines are predictions from the PRSV + vdW + vdWP method, the dashed lines are predictions from the PRSV + MHV1 + UNIQUAC + vdWP method, and the dotted lines are predictions from the PRSV + MHV1 + COSMOSAC + vdWP method. Note the PRSV + vdW + vdWP method and the PRSV + MHV1 + UNIQUAC + vdWP method are indistinguishable in this work.

defined as 100% % AARD T ¼ Ndata

100% % AARD P ¼ Ndata 2460

∑i

T calc T expt i i Tiexpt

ð37Þ

∑i

Pcalc Pexpt i i Piexpt

ð38Þ



ARTICLE

Table 2. Comparison of Average Absolute Relative Deviations from Different Methods in the Prediction of the Equilibrium Condition (VLHE) of Single Gas Hydrate Systems AARD-P (%) guest

no. of data pts

temp range (K)

41

Sloan

43

9

Zhang (linear)

Klauda

method I

AARD-T (%) a

method II

b

method III

c

method IIIc

CH4

904559

273.2303.6

3.44

4.25 (e300 K)

2.65

4.77

4.90

5.04

C2H6

5145,46,50,6063

273.4287.4

10.94

2.77

3.51

1.96

1.94

1.93

0.05

C3H8 CO2

5146,52,54,6468 11546,55,59,65,6976

273.2278.2 271.6282.8

3.32 3.22

3.09 0.89

3.41 2.79

2.63 0.95

2.62 1.11

2.61 10.12

0.05 0.32

N2

5648,49,7779

273.2299.7

4.30

11.26

5.82

6.66

6.66

6.77

0.40

overall

363

4.54

3.90

3.41

3.16

3.23

6.14

0.23

0.20

a

Method I is the PRSV + vdW + vdWP method. b Method II is the PRSV + MHV1 + UNIQUAC + vdWP method. c Method III is the PRSV + MHV1 + COSMOSAC + vdWP method.

Table 3. Comparison of Occupancies of CO2 and CH4 in Single Gas Hydrates at Equilibrium (VLHE) Conditions literature experimental methods guest CH4

temp (K)

occupancya

this workb

Sum et al.80 c

Subramanian81

modeling methods Tse et al.82

CSMGem83

CSMHYD83

273.2

θL

0.967

0.971

273.2

θS

0.947

0.92

273.2

θL/θS

1.021

1.055

1.156

1.053

1.146

1.118

literature experimental methods guest CH4

temp (K)

occupancya

this workb

80

Sum et al.

modeling methods

Subramanian81

Tse et al.82

CSMGem83

CSMHYD83

276.7

θL

0.972

0.973

276.7

θS

0.955

0.866

276.7

θL/θS

1.018

1.124

literature experimental methods guest CO2

temp (K)

occupancya

84

modeling methods

this workb

Davidson

Subramanian81

Tse et al.82

CSMGem83

CSMHYD83

273.2

θL

0.985

273.2

θS

0.388

273.2

θL/θS

2.542

8.333

1.908

1.347

a The symbols θL and θS stand for the occupancy of gas molecule in the large and small cages, respectively. b The results of this work are calculated from the PRSV + MHV1 + COSMOSAC + vdWP method. c The experimental temperature is 273.7 K.

4. RESULTS AND DISCUSSION 4.1. Pure Gas Hydrates. Figure 2 shows the prediction of the three-phase coexisting conditions (VLHE) of five different pure gas hydrates (CH4, C2H6, C3H8, CO2, and N2 hydrates) from three approaches (PRSV + vdW + vdWP, PRSV + MHV1 + UNIQUAC + vdWP, and PRSV + MHV1 + COSMOSAC + vdWP). The binary interaction parameters kij needed in PRSV + vdW + vdWP and τij needed in PRSV + MHV1 + UNIQUAC + vdWP are determined from VLE data of water and the gas molecules. [Their values are provided in the Supporting Information.] The Langmuir adsorption parameters (Aml and Bml) are fitted to reproduce the experimental VLHE data based on the PRSV + vdW + vdWP model. [Their values are listed in Table 1.]

From Figure 2 the PRSV + vdW + vdWP model accurately correlates the VLHE data over a wide range of conditions. Larger deviations are found at high temperatures (e.g., T > 291 K for N2 and T > 299 K for CH4). The Langmuir adsorption parameters obtained from PRSV + vdW + vdWP are used in the other two methods without further parametrization. Therefore, the results from PRSV + MHV1 + UNIQUAC + vdWP and PRSV + MHV1 + COSMOSAC + vdWP represent their predictive power for the three-phase coexisting conditions. As can be seen in Figure 2, the predicted phase boundary from the three methods are nearly identical, indicating that the Langmuir adsorption parameters (listed in Table 1) are transferable between different methods. The AARD-P values from the three methods are compared in Table 2. The accuracies from the three 2461



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Table 4. Comparison of the Accuracy of Different Methods in the Prediction of Gas Hydrate with Organic Inhibitors AARD-T (%) guest

inhibitor

CH4

MeOH

inhibitor concn (wt %) 4.250

5371,8587

232.8286.4

117019930

0.90

1.25

2.02

EtOH

515

1388,89

271.1284.7

298013670

0.34

0.26

0.43

EG

6.950

3985,87

263.4287.1

242016380

0.44

0.71

0.77

DEG

6.650

349092

265.0283.2

203314500

0.98

glycerol

1050

2293,94

264.2288.5

272020530

0.34

0.66

0.92

1.14

MeOH

550

3171,95,96

237.5281.9

4172820

0.51

0.79

1.48

EtOH EG

510 530

889 1496,97

273.6282.0 269.0281.0

5902230 4902380

0.24 0.09

0.14 0.13

0.29 0.16

DEG

7.125

2290,91

272.0285.6

7032800

0.14

glycerol

1040

1493

264.5286.5

4703600

0.09

0.36

0.52

0.62

MeOH

535

4871,96,98

248.0275.6

137492

0.15

0.20

0.38

EtOH

515

4189,99

268.5276.6

200510

0.30

0.11

0.37

EG

1050

3791,96,99

264.7275.8

198470

0.11

0.16

0.18

DEG glycerol

7.329.9 1020

4491,96,99 2993,99

268.2277.0 269.5276.8

178500 190520

0.19 0.05

0.19

0.16

0.25

MeOH

1050

3271,85,100

232.6277.3

13503720

0.78

0.94

2.14

EtOH

520

2090,100

268.1280.1

13103450

0.50

0.21

0.92

EG

1030

2275,100

264.1279.8

11503950

0.17

0.32

0.72

DEG

7.430

2890,100

262.8282.4

13304060

0.64

glycerol

1040

4693,100,101

264.7270.9

12504130

0.46

0.52 0.42 (358 pts)

0.56 0.52 (358 pts)

0.96 0.72 (597 pts)

overall CO2

method IIIc

press. range (kPa)

overall C3H8

method IIb

temp range (K)

overall C2H6

method Ia

no. of data pts

overall overall

597

a Method I is the PRSV + vdW + vdWP method. b Method II is the PRSV + MHV1 + UNIQUAC + vdWP method. c Method III is the PRSV + MHV1 + COSMOSAC + vdWP method.

Figure 3. Three-phase coexisting condition of CH4 hydrates in methanol aqueous solutions. The experimental data with different methanol concentrations are shown in open71 or closed87 triangles (10 wt %), open71 or closed87 squares (20 wt %), and open86 or closed87 diamonds (50 wt %). The experimental data without methanol are shown in circles as reference. The solid lines are predictions from the PRSV + vdW + vdWP method, the dashed lines are from the PRSV + MHV1 + UNIQUAC + vdWP method, and the dotted lines are from the PRSV + MHV1 + COSMOSAC + vdWP method.

methods are quite similar, except that a larger error is observed for CO2 hydrates from PRSV + MHV1 + COSMOSAC + vdWP.

Figure 4. Three-phase coexisting condition of CH4 hydrates in glycerol aqueous solutions. The experimental data at different glycerol concentrations are shown in inverted triangles (10 wt %),93 squares (25 wt %),94 and diamonds (40 wt %)93 and triangles (50 wt %).94 The experimental data without glycerol are shown in circles as reference. The solid lines are predictions from the PRSV + MHV1 + COSMOSAC+ vdWP method.

Also presented in Table 2 are the accuracies of the same gas hydrates from the work of Sloan,41 Zhang et al.,43 and Bandyopadhyay and Klauda.9 The accuracies of all these methods are quite similar (about 4%), with PRSV + vdW + vdWP being the most accurate model (3.16%). The PRSV + MHV1 + COSMOSAC+ vdWP model is the least accurate among all the 2462



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Table 5. Comparison of the Accuracy of Different Methods in the Prediction of Methane Hydrate with Electrolytes AARD-T (%) electrolyte

no. of data pts

molality range

temp range (K)

9153,88,102104

this worka

AARD-P (%) this worka

Klauda9

0.905.43

261.8299.1

0.36

9.72

8.10c

NaBr

105

13

0.511.08

272.5280.0

0.11

3.07

KCl

13104

0.711.49

270.1283.2

0.16

4.69

KBr

12105

0.440.93

272.8285.5

0.09

2.64

CaCl2

39103,104

0.473.12

259.9284.4

0.17

5.20

9.45d

105,106

NaCl

MgCl2

32

0.321.85

270.8286.4

0.16

5.16

CaBr2

10105

0.260.88

272.9284.8

0.07

2.17

KCl + NaCl NaCl + CaCl2

31107 24107

0.974.73 0.832.69

264.4281.5 266.0281.8

0.09 0.06

2.67 1.77

seawaterb

4107

277.0283.9

0.05

1.48

overall

269

0.20

5.72

a

Prediction from the PRSV + MHV1 + COSMOSAC(ion) + vdWP method. b The composition of seawater is NaCl (2.394 wt %), MgCl2 (0.508 wt %), Na2SO4 (0.401 wt %), CaCl2 (0.115 wt %), KCl (0.069 wt %), KBr (0.011 wt %), NaF (0.009 wt %), and SrCl2 (0.002 wt %).107 c The 45 NaCl data points with molality range 1.133.54 m and temperature range 263.4299.1 K are used. d The nine CaCl2 data points with molality range 0.471.02 m and temperature range 279.9284.4 K are used.

Table 6. Comparison of the Accuracy of Different Methods in the Prediction of CO2 Hydrate with Electrolytes AARD-T (%)


Klauda9

0.20 0.26

6.73 8.73

12.7c

272.6283.1

0.12

4.38

273.3283.0

0.18

6.21

0.282.25

259.5280.9

0.17

5.42

4.47d

MgCl2

106

15

0.321.17

272.5281.6

0.33

10.83

KCl + NaCl

17108

0.984.06

262.9279.9

0.2

6.26

NaCl + CaCl2

22108

0.843.78

259.0279.2

0.21

6.34

seawaterb overall

1273 289

272.1281.5

0.25 0.21

8.34 6.86

electrolyte

no. of data pts

molality range

temp range (K)

NaCl KCl

14372,104,108 25104,108

0.405.47 0.412.36

262.7282.1 269.0281.1

HCl

1670

0.101.04

NaOH

1470

0.101.04

CaCl2

25104,108

a

this work

Prediction from the PRSV + MHV1 + COSMOSAC(ion) + vdWP method. b The composition of seawater for five data points is NaCl (2.391 wt %), MgCl2 (0.507 wt %), Na2SO4 (0.401 wt %), CaCl2 (0.115 wt %), KCl (0.069 wt %), KBr (0.011 wt %), NaF (0.005 wt %), and SrCl2 (0.002 wt %) and for another seven data points is NaCl (2.35 wt %), MgCl2 (0.498 wt %), Na2SO4 (0.392 wt %), and CaCl2 (0.11 wt %).73 c The 26 NaCl data points with molality range 0.533.01 m and temperature range 265.4280.9 K are used. d The eight CaCl2 data points with molality range 0.280.48 m and temperature range 271.1280.9 K are used. a

Table 7. Comparison of the Accuracy of Different Methods in the Prediction of C2H6 Hydrate with Electrolytes AARD-T (%)

this worka

Klauda9

0.12

4.53

6.35b

269.5282.3 267.2283.0

0.05 0.14

1.88 5.31

2.803.78

267.1276.5

0.25

9.77

2.54

266.9267.1

0.27

5109

2.11

268.1276.4

0.08

2.72

KCl + NaCl + CaCl2

5109

2.29

269.0278.4

0.21

7.75

KCl + NaCl + CaCl2 + KBr

5109

2.44

269.4278.8

0.17

6.36

overall

82

0.14

5.26

electrolyte

no. of data pts

molality range

temp range (K)

NaCl

20104,109,110

0.904.28

265.4280.4

KCl CaCl2

104,109

17 16104,109

0.711.88 0.471.59

KCl + NaCl

9109

NaCl + CaCl2

5109

KCl + CaCl2

this worka

AARD-P (%)

10.3

a Prediction from the PRSV + MHV1 + COSMOSAC(ion) + vdWP method. b The eight NaCl data points with molality range 1.893.01 m and temperature range 272.7280.4 K are used.

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Table 8. Comparison of the Accuracy of Different Methods in the Prediction of C3H8 Hydrate with Electrolytes AARD-T (%) electrolyte NaCl

no. of data pts

molality range

temp range (K)

5766,67,104,111,112

0.444.28

261.0276.6

this worka


Klauda9

0.15

9.28

8.66c

KCl

25

0.713.35

266.5276.2

0.15

9.78

CaCl2

32104,111113

0.471.62

263.0276.2

0.13

8.46

KCl + NaCl

5113

2.69

265.2269.1

0.05

2.84

NaCl + CaCl2

24110,112,113

0.842.07

265.9275.1

0.09

5.67

KCl + CaCl2

7113

1.98

266.3270.1

0.05

2.97

KCl + NaCl + CaCl2

13112,113

0.833.42

261.9274.5

0.12

7.54

seawaterb overall

4112 167

0.54

272.0276.1

0.06 0.13

3.63 7.95

104,111,112

a

Prediction from the PRSV + MHV1 + COSMOSAC(ion) + vdWP method. b The composition of seawater is NaCl (2.6436 wt %), MgCl2 (0.2560 wt %), Na2SO4 (0.3310 wt %), CaCl2 (0.0713 wt %), KCl (0.0740 wt %), and SrCl2 (0.0020 wt %).112 c The nine NaCl data points with molality range 0.553.01 m and temperature range 266.2276.4 K are used.

methods mainly because of its poor performance in CO2 hydrates. We found that the PRSV + MHV1 + COSMOSAC model underestimates the solubility of CO2 in liquid water, while PRSV + vdW and PRSV + MHV1 + UNIQUAC can accurately correlate the solubility data. This indicates that COSMO-SAC is less accurate for mixtures containing CO2. Since this method does not require input of any binary interaction parameters, and therefore can be used even when no VLE data are available, its performance in the prediction of the coexisting conditions is quite satisfactory. The occupancy of the guest in the cages of clathrate hydrate can be estimated based on the Langmuir constants (eq 29). Table 3 compares the calculated occupancy of CH4 and CO2 in sI hydrates from various models with experimental data. It can be seen that the occupancy determined on the basis of the Langmuir constants from this work agrees reasonably well with the experiment and other modeling methods. 4.2. Gas Hydrates with Organic Inhibitors. One advantage of the proposed models is their ability to provide predictions of the coexisting conditions of gas hydrates with the presence of additives as long as they do not enter into the cages of the hydrate. Table 4 shows the average absolute relative deviations (AARDs) in the predicted equilibrium temperature for CH4, C2H6, C3H8, and CO2 hydrates with the addition of methanol (MeOH), ethanol (EtOH), ethylene glycol (EG), diethylene glycol (DEG), or glycerol solution from this model. The AARD-T values are 0.42, 0.52, and 0.72% from the PRSV + vdW + vdWP method, the PRSV + MHV1 + UNIQUAC + vdWP method, and the PRSV + MHV1 + COSMOSAC + vdWP method, respectively. Although the prediction errors from the PRSV + MHV1 + COSMOSAC + vdWP method are slightly higher than those from the others, the predicted inhibition effect and tendency of all inhibitors added to the gas hydrate systems are correct. Furthermore, because of the extrapolative use of parameters determined from experimental VLE data for diethylene glycol (DEG) and glycerol, the equilibrium conditions of fewer systems (358 data points) can be determined from the PRSV + vdW + vdWP and PRSV + MHV1 + UNIQUAC + vdWP methods. The PRSV + MHV1 + COSMOSAC + vdWP method does not rely on experimental VLE data and can be used for all systems (597 data points) considered in this work.

Figure 5. Three-phase coexisting condition of CH4 and CO2 hydrates in aqueous NaCl solutions. The NaCl concentrations in aqueous solutions for CH4 hydrates53 are shown in filled circles (11.7 wt %), filled squares (17.1 wt %), filled triangles (21.5 wt %), and filled diamonds (24.1 wt %). For CO2 hydrates experimental data108 are shown in open circles (3.0 wt %), open squares (5.0 wt %), open triangles (10.0 wt %), open diamonds (15.0 wt %), and stars (20.0 wt %). The dashed and solid lines are predicted results of, respectively, CH4 and CO2 hydrates from the PRSV + MHV1 + COSMOSAC(ion) + vdWP method.

Figure 3 compares the prediction of the inhibition effect (lowering the equilibrium temperature under constant pressure) of methanol on the methane hydrate formation condition. This system is taken as an example because the prediction errors are found to be larger (see Table 4). It is observed that the deviation between prediction and experiment increases with the concentration of methanol in water. One possible reason is that equilibrium temperatures of the gas + H2O + MeOH hydrate systems are significantly lower than those of pure gas hydrate equilibrium data, from which the Langmuir constant parameters are determined. Thus, larger deviations in methanol-containing systems may be a result of extrapolation of the Langmuir constants at low temperatures. As a second example, Figure 4 shows the predictions of the methane hydrate equilibrium conditions in the presence of glycerol. Because of the lack of appropriate binary interaction parameters, both the PRSV + vdW + vdWP and PRSV + MHV1 + UNIQUAC + vdWP methods fail to describe 2464



Figure 6. Three-phase coexisting condition of CH4 and CO2 hydrates in NaCl and CaCl2 mixed electrolytes at different concentrations. The electrolyte concentrations for CH4 hydrates107 are shown in filled circles (3.0 wt % NaCl + 3.0 wt % CaCl2), filled squares (6.0 wt % NaCl + 3.0 wt % CaCl2), filled triangles (10.0 wt % NaCl + 3.0 wt % CaCl2), filled diamonds (10.0 wt % NaCl + 6.0 wt % CaCl2), crosses (3.0 wt % NaCl + 10.0 wt % CaCl2), and stars (6.0 wt % NaCl + 10.0 wt % CaCl2). For CO2 hydrates experimental data108 are shown in open circles (2.0 wt % NaCl + 8.0 wt % CaCl2), open squares (3.0 wt % NaCl + 3.0 wt % CaCl2), open triangles (5.0 wt % NaCl + 14.7 wt % CaCl2), open diamonds (8.0 wt % NaCl + 2.0 wt % CaCl2), and pluses (15.0 wt % NaCl + 5.0 wt % CaCl2). The dashed and solid lines are predicted results of, respectively, CH4 and CO2 hydrates from the PRSV + MHV1 + COSMOSAC(ion) + vdWP method.

the equilibrium conditions of this system. In contrast, the inhibition effect of glycerol on CH4 hydrate formation is very well predicted by the PRSV + MHV1 + COSMOSAC + vdWP method. This again demonstrates the excellent predictive power of the PRSV + MHV1 + COSMOSAC + vdWP model. 4.3. Gas Hydrate with Saline Inhibitors. The formation condition of gas hydrates, including CH4, C2H6, C3H8, and CO2, in single (NaCl, NaBr, KCl, KBr, HCl, NaOH, CaCl2, MgCl2, and CaBr2) and mixed (KCl + NaCl, NaCl + CaCl2, KCl + CaCl2, KCl + NaCl + CaCl2, KCl + NaCl + KBr +CaCl2, and seawater) electrolyte solutions can be been predicted from the PRSV + MHV1 + COSMOSAC(ion) + vdWP model. Tables 5 8 compare our predictions with experiment44 and a recent work by Bandyopadhyay and Klauda.9 The AARD-T values of CH4, C2H6, C3H8, and CO2 hydrates with electrolyte solutions from our model are 0.20, 0.14, 0.13, and 0.21%, respectively, and the AARD-P values are 5.72, 5.26, 7.95, and 6.86%. The predictions for CH4 hydrate with CaCl2 (5.20% AARD-P) and of the CO2 hydrate system with NaCl (6.73% AARD-P) from this model are found to be more accurate than those of Bandyopadhyay and Klauda (9.45 and 12.7% AARD-P, respectively). The accuracies of C2H6 and C3H8 hydrate systems with NaCl are similar from both models, but a larger range of temperature condition is considered in this work. The PRSV + MHV1 + COSMOSAC(ion) + vdWP model is found to be less accurate for CH4 hydrates with NaCl (9.72% AARD-P) and CO2 hydrates with CaCl2 (5.42% AARD-P) when compared to Bandyopadhyay and Klauda’s method (8.10 and 4.47% AARD-P, respectively). The larger deviations are also found in the pure methane gas hydrate calculation (see Figure 2 at pressure above 50 MPa), indicating that the error is a result of

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Figure 7. Three-phase coexisting condition of CH4 (circles),107 C3H8 (triangles),112 and CO2 (squares)73 hydrates in seawater. The prediction results from the PRSV + MHV1 + COSMOSAC(ion) + vdWP method are shown in solid lines.

the inaccuracy in the Langmuir absorption constants from the vdWP model. Figures 5, 6, and 7 show the predictions of the three-phase coexisting conditions of methane and CO2 hydrates in aqueous NaCl (single salt) solutions, NaCl + CaCl2 (two salts) solutions, and seawater (multiple electrolytes), respectively. It can be seen that the PRSV + MHV1 + COSMOSAC(ion) + vdWP model provides accurate predictions over a wide range of electrolyte concentrations.

5. CONCLUSION In this work, we investigate the prediction of the change in the phase boundary of a single gas hydrate upon the addition of inhibitors using the combination of the vdWP model for the solid phase and the PRSV EOS for the fluid phase. We have examined three versions of the PRSV EOS: the vdW mixing rule (which requires one binary interaction parameter for each pair of species), the MHV1 mixing rule with the UNIQUAC activity coefficient model (which requires two binary interaction parameters for each pair of species), and the MHV1 mixing rule with the COSMO-SAC activity coefficient model (which requires no binary parameter). We found that the Langmuir constants obtained from the PRSV + vdW + vdWP approach are transferable to other methods without loss of accuracy in describing the phase boundary of pure gas hydrates. When the binary interaction parameters can be obtained (e.g., from VLE data), the PRSV + vdW + vdWP and PRSV + MHV1 + UNIQUAC + vdWP methods can provide accurate predictions of the phase boundary of gas hydrates with additives. The accuracy from the PRSV + MHV1 + COSMOSAC + vdWP model is comparable to that of other methods, except for CO2 hydrates and high inhibitor concentrations. However, the PRSV + MHV1 + COSMOSAC + vdWP model can still be used when the other methods fail due to the lack of suitable binary parameters. When the recent version of the COSMO-SAC model is used that allows the inclusion of electrolytes in the system, we found that the PRSV + MHV1 + COSMOSAC(ion) + vdWP model can provide accurate predictions of hydrate formation in single electrolyte or mixed electrolyte solutions. Therefore, we believe that the PRSV + MHV1 + COSMOSAC(ion) + vdWP model is a powerful tool for studying the inhibitor effects on the change of three-phase equilibrium conditions of gas hydrates. 2465



’ ASSOCIATED CONTENT

bS

Supporting Information. The binary interaction parameters for PRSV + vdW and PRSV + MHV1 + COSMOSAC models are provided in Tables S1 and S2. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: 02-33661369 (S.-T.L.); 02-33663049 (L.-J.C.). E-mail: [email protected] (S.-T.L.); [email protected]. (L.-J.C.).

’ ACKNOWLEDGMENT This study was supported by the Ministry of Economic Affairs of Taiwan (Grants 97-5226903000-02-04, 98-5226904000-0404, 99-5226904000-04-04, and 100-5226904000-04-04) and the National Science Council of Taiwan (Grants NSC 100-2923-E002-003-MY3, NSC 100-3113-M-002-001, and NSC 98-2221-E002-087-MY3). ’ NOMENCLATURE A = surface area of species Aϕ = DebyeH€uckel constant a(T), b = species-dependent parameters aeff = effective segment area Cml(T) = Langmuir adsorption constant d = density f = fugacity G = Gibbs free energy Ix = ionic strength k = Boltzmann constant kij = binary interaction parameter M = average molecular weight N = number of moles NA = Avogadro’s constant n = number of surface segments P = pressure Qe = charge of an electron q = normalized surface area R = gas constant r = normalized volume T = temperature V = volume ΔW = electrostatic interaction x = molar concentration in liquid phase y = molar concentration in vapor phase Z = compressibility factor zi = net charge of ion i Greek Symbols

ε = dielectric constant θ = gas occupancy or surface area fraction ϕ = fugacity coefficient Γ = activity coefficient of segment γ = activity coefficient k = parameter of PR EOS for a(T) ω = acentric factor F = closest approach parameter σ = charge density τ = binary interaction parameter

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ν = number of cages of type per water molecule in the hydrate lattice Superscripts and Subscripts

c = critical point calc = calculation result ex = excess property expt = experimental result H = hydrate phase L = liquid phase LM = liquid model m = mixture s = solvent sat = saturation condition V = vapor phase w = water β = empty hydrate phase

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Predictive Method for the Change in Equilibrium Conditions of Gas

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