Modeling the Dissociation Conditions of Carbon ... - ACS Publications

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Modeling the Dissociation Conditions of Carbon Dioxide + TBAB, TBAC, TBAF, and TBPB Semiclathrate Hydrates Ayako Fukumoto,† Patrice Paricaud,*,† Didier Dalmazzone,† Wassila Bouchafaa,† Thi Thu-Suong Ho,† and Walter Fürst† †

ENSTA-ParisTech, UCP, 828 Boulevard des Maréchaux, 91762 Palaiseau, France ABSTRACT: The thermodynamic approach developed by Paricaud [J. Phys. Chem. B 2011, 115, 288−299] is applied to predict the dissociation conditions of semiclathrate hydrates made with tetra-n-butyl ammonium bromide (TBAB), tetra-nbutyl ammonium chloride (TBAC), tetra-n-butyl ammonium fluoride (TBAF), and tetra-n-butyl phosphonium bromide (TBPB). The SAFT-VRE equation of state is used to describe the properties of fluid phases, and a good description of osmotic and mean activity coefficients of electrolyte solution is obtained. The temperature−composition diagrams of water + tetra-n-alkylammonium/alkylphosphonium salt binary systems are well described by the model. Group contribution methods are proposed to predict the fusion enthalpies and the congruent melting points of semiclathrate hydrates. The van der Waals and Platteeuw theory is combined with the model to calculate the dissociation conditions of carbon dioxide semiclathrate hydrates. The liquid−vapor−hydrate three phase lines can be accurately described over wide ranges of pressure and salt concentrations, by optimizing only one parameter per hydrate phase.

1. INTRODUCTION Semiclathrate hydrates1 are crystal compounds that exhibit cage-like structures formed of water molecules and anions of tetra-alkylammonium/alkylphosphonium salts. The cations of the salt are embedded in large cages, and small cages remain empty.2 Shimada et al.3 found that tetra-n-butyl ammonium bromide (TBAB) hydrate can entrap small gas molecules in the small cages, which are dodecahedral. The phase behaviors of semiclathrate hydrates are complex because of the broad variety of crystalline structures that are observed experimentally. An extensive review of the solid−liquid phase diagrams of tetraalkylammonium halide salt + water systems has been done by Dyadin and Udachin.4 Since gas molecules can be entrapped at nearly ambient temperature and at reasonable pressures, potential industrial applications of gas semiclathrate hydrates are carbon dioxide capture and storage, gas separation, and refrigeration. Despite the numerous experimental investigations on the equilibrium conditions of semiclathrate hydrates, few attempts have been made to model such systems.5−8 Among those attempts, Paricaud6 proposed a thermodynamic model to determine the dissociation conditions of CO2 + TBAB + water system. The model can describe both the solid−liquid phase diagram of salt + water mixtures and the dissociation condition of gas semiclathrate hydrates. In this study, we propose new parameters for the TBAB systems and extend the approach to other salts. The thermodynamic properties of the liquid phase are described with the SAFT-VRE (statistical associating fluid theory with variable range for electrolytes) equation of state. The Gibbs free © XXXX American Chemical Society

energy of gas semiclathrate hydrates is calculated by using the van der Waals and Platteeuw model,9 and by considering that the empty hydrate reference phases are the hydrate phases observed in the water + salt binary mixture. We have studied systems composed of water, carbon dioxide, and of the following salts: TBAB, tetra-n-butyl ammonium fluoride (TBAF), tetra-n-butyl ammonium chloride (TBAC), and tetra-n-butyl phosphonium bromide (TBPB). This paper is organized as follows: we first recall the main working equations of the model to determine the solid−liquid equilibria (SLE) in water + salt binary systems, and calculate liquid−vapor−hydrate (L−V−H) three-phase lines. The SAFT-VRE model for electrolyte solutions is briefly discussed. The presentation and discussion of the calculation results can be found in the last section.

2. COMPUTATIONAL METHODS The thermodynamic approach used in this work is the same as in ref 6. The main working equations are recalled here, and further details about the model and algorithms can be found elsewhere.6 2.1. SLE between the Electrolyte Solution and Hydrate Phases. According to Paricaud,6 the SLE condition for hydrates involving electrolytes is given by Special Issue: Modeling and Simulation of Real Systems Received: March 13, 2014 Accepted: July 29, 2014

A

dx.doi.org/10.1021/je500243k | J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Table 1. SAFT-VRE Pure Component Parametersa H2O CO2

m

σ/Å

ε/k/K

λ

sites

εHB/k/K

kHB/Å3

ref

1 2

3.036 3.1364

253.30 168.89

1.8000 1.5157

2H+2E no site

1365.92

1.0202

13,14 14

a m is number of segments. σ, ε, and λ are the segment diameter, the depth, and range of the square-well potential. εHB and kHB are the association energy and bonding volume.

Δgdis RT

=

where ni is the number of cages i per salt molecule, and ghyd,β is the Gibbs free energy of the reference empty semiclathrate hydrate phase. Ncav and Ng are the number of types of cavity and gas, respectively. In our approach the empty semiclathrate hydrates can have different types of structures depending on the concentration of the salt in the solution. The most stable hydrate phase is the phase that has the lowest Gibbs free energy (ghyd,F) at given T, P and composition. The occupancy fraction Yij of cavities i by gas molecules j can be written as

Δg 0 + νC ln(xCγC) + νA ln(xAγA ) RT + νw ln(x wγw ) = 0

(1)

where R is ideal gas constant, T the temperature, and νi, xi, γi are the stoichiometric coefficient, the mole fraction, and the activity coefficient of species i, respectively. The subscripts C, A, and w denote cations, anions, and water, respectively. Equation 1 was obtained from the minimization of the total Gibbs free energy under the constraint that the composition of the hydrate phase is fixed.6 For all the salts considered in this study, νC = νA = 1, and νw is the hydration number in the hydrate phase. Δgdis is the dissociation Gibbs free energy per mole of salt and is equal to 0 at equilibrium. By assuming that the difference between the heat capacities of solid and aqueous phases is negligible compared to the other terms, one can express Δg0 as

Cij = (2)

i=1

Ng

Ncav

=

∑ ni ln(1 − ∑ Yij) − i=1

j=1

Δh0 ⎛ T⎞ ⎜1 − ⎟ RT ⎝ T0 ⎠

Δg 0(T0 , P0) Δv (P − P0) − RT RT0 0

−

(7)

The melting point of semiclathrate phase is calculated by performing a vapor−liquid−hydrate three-phase equilibrium calculation at given pressure and global composition, and neglecting the mass of the semiclathrate hydrate phase. Vapor− liquid equilibrium is solved with a flash calculation for fixed P, composition, and with initial value of T, to obtain the compositions of the vapor and liquid phases. Then, T is changed until eq 7 is satisfied. 2.3. SAFT-VRE Equation of State. The thermodynamic properties of vapor phases and electrolyte solutions are determined with the SAFT-VRE model, which was proposed by Galindo et al.12 In this approach, molecules i are modeled as chains of mi hard spheres of diameter σi, and ions are modeled as charged hard spheres. The Helmholtz free energy A is given by

(3)

A Aideal Aseg Achain Aassoc AMSA = + + + + NkT NkT NkT NkT NkT NkT

Ng

(8)

ideal

where N is the number of molecules, A the ideal term, and Aseg and Achain are the energy contributions due to interactions between segment and chain formation, respectively. Aassoc is the

∑ Yij) j=1

(6)

νC ln(xCγC) + νA ln(xAγA ) + νw ln(x wγw )

2.2. Thermodynamic Model for Gas Semiclathrate Hydrates. The van der Waals and Platteeuw9 proposed a thermodynamic model (vdw-P model) for gas hydrates, which is based on the following conditions: gas molecules cannot distort cavities; each cavity can only entrap one gas molecule; the interactions between entrapped gas molecules can be neglected; cavities are spherical. By combining the vdw-P model with the approach presented in section 2.1, one can express the Gibbs free energy of the semiclathrate hydrate phase per mole of salt (ghyd,F), which is partially filled of gas molecules, as6 g hyd,F = g hyd, β + RT ∑ ni ln(1 −

⎛ ε cell ⎞ 4π cell ij ⎟ V ij exp⎜⎜ ⎟ kT kT ⎠ ⎝

Here, is the free volume of gas molecule j inside cavity i, εcell is the depth of the square-well cell potential, and k the ij Boltzmann constant. The equilibrium condition can be rewritten as

∂ (νC ln(xCγC) + νA ln(xAγA ) ∂T

Ncav

(5)

Vcell ij

where Δh0 is an enthalpic parameter fitted to the SLE coexistence curve of the salt hydrate; Δν0 is a parameter that expresses the effect of pressure on the melting points of the hydrate phase; T0 is the melting point of the hydrate at congruent melting, and at atmospheric pressure P0 = 0.101325 MPa. By combining eq 1 and 2, one can calculate the melting temperature of semiclathrate hydrates at given pressure P and salt concentration in the solution. The constant Δg0(T0,P0) is determined by applying eq 1 and eq 2 at the stoichiometric composition.6 The dissociation enthalpy per mole of salt (Δhdis) is calculated as Δhdis = −T2∂(Δgdis/T)/∂T, where Δgdis is given by eq 1 and 2. Δhdis is not equal to Δh0 but the two values are close, so one can use the experimental fusion enthalpy as a first guess for Δh0. At P = P0, Δhdis is given by

+ νw ln(x wγw ))

N

1 + ∑k =g 1 Cik fk

where f j is the fugacity of gas molecules j in the fluid phases calculated with the SAFT-VRE equation of state. By approximating the cell potentials by square-well intermolecular potentials, one can express the Langmuir constant Cij as10,11

Δg 0 Δg 0(T0 , P0) Δh0 ⎛ T ⎞ Δv 0 = (P − P0) + ⎜1 − ⎟ + RT RT ⎝ T0 ⎠ RT RT0

Δhdis = Δh0 − RT 2

Cijf j

Yij =

(4) B



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association contribution. Water molecules are modeled as hard spheres with four association sites that describe hydrogenbonding: there are two donor sites H and two sites E, which represent hydrogen atoms and the lone pairs of the oxygen, respectively. Further details concerning the water model used in this work can be found elsewhere.6,13,14 The parameters for H2O and CO2 used in this study are reported in Table 1. The term AMSA takes into account the Coulombic interactions between ions, while the dispersion forces between ions are neglected. For the sake of simplicity, we have assumed that all electrolytes are fully dissociated. The water−ion solvation interactions are described by a short-range square-well potential of range λw‑ion, and depth εw‑ion. The temperature dependency of εw‑ion is given by ⎛1 1 ⎞ ⎜ ⎟ εw − ion /k = εw(298) − − ion / k + C ⎝T 298.15 ⎠

Figure 1. Mean activity coefficients of TBAB, TBAC, and TBAF + H2O solutions at T = 298.15 K and atmospheric pressure. The symbols denote the experimental data for different salts: □, TBAB, ref 16; △, TBAC, ref 16; ○, TBAF, ref 17. The solid lines are calculated with the SAFT-VRE model.

(9)

ε(298) w−ion

where is fitted on experimental mean activity coefficients and osmotic coefficients, and SLE of salt solutions. The C parameter is adjusted on the SLE data of salt + water binary systems. The interactions between the segments of chains i and j are modeled by a square-well of depth εij, diameter σij, and range λij. For εij and σij, the Lorentz−Berthelot combining rules are used with a binary parameter kij. For λij, a simple combining rule is applied with another binary parameter lij. εij =

εiiεjj (1 − kij),

σij = (σii + σjj)/2

(10)

λij = (σiiλii + σjjλjj)(1 − lij)/(σii + σjj)

(11)

Further details about the SAFT-VRE approach can be found elsewhere.12,15 Figure 2. Osmotic coefficients of TBAB, TBAC, and TBAF + H2O solutions at T = 298.15 K and atmospheric pressure. The symbols denote the experimental data for different salts: □, TBAB, ref 16; △, TBAC, ref 16; ○, TBAF, ref 17. The solid lines are predicted from the SAFT-VRE model.

3. RESULTS AND DISCUSSION 3.1. Thermodynamic Properties of Salt Solutions. The salt parameters σions, ε(298) w−ion/k, and λw‑ion are listed in Table 2. Table 2. SAFT-VRE Salt Parameters for TBAB, TBAC, TBAF, and TBPBa salts

σions/Å

(298) εw−ion /k/K

C/K2

λw‑ion

ref

TBAB TBAC TBAF TBPB

3.85 3.55 4.15 3.85

596 450 630 596

0.0 1.0·105 −1.0·105 4.0·105

1.1 1.2 1.22 1.1

6, this work this work this work this work

behaves as a strong electrolyte as the osmotic coefficients increases dramatically when the molality is increased. The C parameter for TBAF was fitted to better describe the SLE of TBAF + water mixture. One can observe that the experimental osmotic coefficients of TBAB and TBAC exhibit several extrema and inflection points that could not be captured by the model. This behavior can be explained by the formation of ion pairs which are not taken into account by the model. However, the accuracy of the model is sufficient for the purpose of this work, which is the calculation of SLE at low salt concentrations. 3.2. Solid−Liquid Equilibria in Quaternary Ammonium/Phosphonium Salt + H2O Binary Systems. The SLE between semiclathrate hydrate phases and solutions of quaternary ammonium/phosphonium salts are determined by combining eqs 1 and 2. The parameters Δh0 and T0 are determined to get the best agreement with SLE experimental data. For all salts, we assume that Δv0 = −30 cm3·mol−1, which was the value previously determined for TBAB.6 The dissociation enthalpy at congruent melting are calculated with eq 3. Average absolute deviation (AAD) between the calculation and experimental data are defined as AAD = ∑ | Tcal − Texp|/nAAD, where Tcal and Texp are temperature obtained

a σions, ε(298) w−ion, and λw‑ion are the ionic diameter of ions, and the depth and range of the SW potential between water and ions. C describes the temperature dependency of εw‑ion.

They were determined by fitting the experimental activity coefficients and osmotic coefficients at T = 298.15 K reported by Lindenbaum and Boyd16 for TBAC, and by Wen and Saito17 for TBAF. The SAFT-VRE parameters for TBAB were previously obtained.6 Here, the C parameter for TBAB has been fixed to 0, to get a better description of SLE data. There is no activity coefficient data for TBPB in the literature. Since TBPB is rather similar to TBAB with respect to ion size, we used the same salt parameters for both salts, apart from the C parameter. As shown in Figure 1 and Figure 2, the calculated activity coefficients and osmotic coefficients are in good agreement with the experimental data. One can observe that TBAF C



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from the calculation and experiments, respectively; nAAD is the number of data. 3.2.1. TBAB + H2O. The phase behavior of the TBAB + H2O binary system was investigated by several researchers, mainly because TBAB is nontoxic and produced in large quantities. Nakayama18 found a hydrate phase of hydration number vw = 24 (H24) with a congruent melting temperature of 12.9 °C. Dyadin and Udachin4 reported four types of hydrates of hydration numbers vw = 24, 26, 32, and 36, and congruent melting points at (12.4, 12.2, 11.6, and 9.5) °C, respectively. Oyama et al.19 found only two types of hydrates of hydration number vw = 26 (H26) and 38 (H38) with congruent melting points at 12.0 °C and 9.9 °C. Most authors now agree that only two types of hydrates can be observed for TBAB. Although the existence and stability of type B (H38) has been confirmed experimentally by Shimada et al.,20 the hydration number of type A is still not clear, because the hydration number was approximately determined as the maximum of the SLE curve by Oyama et al.19 We have previously studied the TBAB+H2O system and proposed parameters by considering the H38 and H26 hydrates. In this study, we propose two sets of parameters corresponding to two possible hydration numbers for type A hydrate. In the first case, we consider two hydrates of hydration number vw = 38 and 26 (H38−26). In the second case, we consider the hydrates H38 and H24 (H38−24). The SLE phase diagrams corresponding to these two cases are shown in Figure 3a,b, and the corresponding Δh0 and T0 parameters are reported in

Table3. A good agreement is obtained between experimental data19,21−23 and the calculated curves for both cases (H38−26 Table 3. Hydrate Parameters Used To Compute the SLE of the Salt + H2O Systema salts

vw

T0/K

Δh0/kJ·mol−1

TBAB

38 26 24 32 30 24 32 29 38 32 24

282.6 285.0 285.2 287.5 288.1 287.8 299.8 300.9 282.2 282.3 281.9

200 150 145 170.5 154 130 210 199 195 185 150

TBAC

TBAF TBPB

a

vw is the hydration number. T0 is the temperature at congruent melting at atmospheric pressure. Δh0 is the enthalpic parameter.

and H38−24). However, the calculated phase diagrams in the case of H38−24 is in better agreement with experimental data than in the case of H38−26. The triple point (liquid−type B− type A) is predicted at T = 279.88 K, wTBAB = 0.103 for the case H38−26, and at T = 280.41 K, wTBAB = 0.118 for the case H38−24. The predicted dissociation enthalpies are (199.2, 149.1, and 144.0) kJ·mol−1 for H38, H26, and H24, respectively. The predicted values for H38 and H26 hydrates are close to the experimental data,19,23,24 as shown in Table 4. Table 4. Experimental and Calculated Enthalpy of Dissociation of Semi-clathrate Hydratesa salt

vw

−1 Δhcal dis/kJ·mol

−1 Δhexp dis /kJ·mol

TBAB

38 26 24 32 30 24 32 29 38 32 24

199.2 149.1 144.0 179.1 163.1 141.1 213.0 202.3 196.2 186.3 151.4

201,19 21924 151,24 152,23 15320

TBAC

TBAF TBPB

17929 157,29 16428 12829 20330 17430 18736

exp vw is the hydration number. Δhcal dis and Δhdis are the calculated and experimental fusion enthalpy, respectively.

a

3.2.2. TBAC + H2O. Aladko and Dyadin25 found three different types of hydrate phases for the TBAC + H2O binary system, with hydration number vw = 32, 30, and 24, and congruent melting points at (14.7, 15.1, and 15.1) °C, respectively. We calculated the SLE curves for the TBAC + H2O binary system by considering the three hydrate phases with the same hydration numbers vw = 32, 30, and 24 (H32−30−24). The optimized parameters Δh0 and T0 are listed in Table 3. The calculated SLE curve and experimental data from the literature25−28 are compared in Figure 4. The model can accurately describe the melting temperatures of the semiclathrate hydrate, except for the data from Sun et al.,26 which are at lower temperatures. The triple point is predicted at T =

Figure 3. Temperature−composition diagrams of the TBAB + H2O mixture, calculated by considering the hydrate phases H38 and H26 (a), and the hydrate phases H38 and H24 (b). The composition is expressed in terms of the TBAB weight fraction (wTBAB). The symbols denote the experimental data: □, ref 21; ▽, vw = 38 from ref 19; △, vw = 26 from ref 19; ◇, ref 22; ○, ref 23. The solid lines are the calculated SLE curves. AAD between the calculation and experimental data from ref 21 for H38−26 and H38−24 are 0.50 K and 0.33 K, respectively. D



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ionic SAFT-VRE parameters are only available for molalities below 1.6 mol·kg−1, which corresponds to a TBAF weight fraction of 0.295. Additional measurements of activity coefficients at higher salt concentrations would be very useful to further improve the model. The calculated dissociation enthalpies are 213.0 and 202.3 kJ·mol−1 for H32 and H29, hydrates. 3.2.4. TBPB + H2O. The phase diagram of the TBPB+H2O binary system is predicted by using the same SAFT-VRE parameters as those for the TBAB + H2O system (section 3.1). The C parameter is fitted on SLE data. Only one type of hydrate phase is observed experimentally, but different values of hydration numbers have been reported in the literature: Dyadin and Udachin4 and Zhang et al.34 proposed 32, while Muromachi et al.35 recently found a hydration number of 38. As a result, we have considered two possible cases for TBPB. We first calculated the phase diagram by only considering the hydrate phase H32 of hydration number 32. The corresponding SLE curve is shown in Figure 6a, and compared to experimental

Figure 4. Temperature−composition diagram of the TBAC + H2O mixture, calculated by considering the hydrate phases H32, H30, and H24. The composition is expressed in terms of the TBAC weight fraction (wTBAC). The symbols denote the experimental data: □, ref 27; △, ref 26; ◇, ref 25; ○, ref 28. The solid lines are the calculated SLE curves. AAD between the calculation and experimental data from ref 25 is 0.63 K.

285.1 K, wTBAC = 0.162 for the three phases L−H32−H30, and at T = 287.8 K, wTBAC = 0.396 for L−H30−H24. The calculated dissociation enthalpies are (179.1, 163.1, and 141.1) kJ·mol−1 for the H32, H30, and H24, hydrates, respectively. Rodionova et al.29 and Nakayama28 reported 179 kJ·mol−1 and 164 kJ·mol−1 for the experimental dissociation enthalpy of H32 and H30, respectively, which are close to our predicted value (Table 4). 3.2.3. TBAF + H2O. The phase diagram of the TBAF + H2O binary system was studied by Dyadin and Udachin4 and Rodionova et al.30 They both found two hydrate phases and proposed the hydration numbers vw = 32.3 and 28.6, and vw = 32.8 and 29.7, respectively. We have also assumed that only two types of hydrate phases can form in the binary system, and considered the hydration numbers vw = 32 and 29. The parameters Δh0 and T0 for TBAF hydrates are reported in Table 3. The SLE phase diagram of the TBAF + water mixture is depicted in Figure 5. The model well describes the experimental data reported by Nakayama,18 Dyadin et al.,31 Sakamoto et al.,32 and Lee et al.33 Note that the experimental osmotic and mean activity coefficients used to determine the

Figure 6. Temperature−composition diagrams of the TBPB + H2O mixture, calculated by considering the hydrate phase H32 (a) and the hydrate phases H38 and H24 (b). The composition is expressed in terms of the TBPB weight fraction (wTBPB). The symbols denote the experimental data: □, ref 2; △, ref 36; ◇, ref 37; ○, ref 38; ▽, ref 34. The solid line are the calculated SLE curves. AAD between the calculation and experimental data from ref 34 for H32 and H38−24 are 0.42 and 0.37 K.

data.2,34,36,37 The calculated SLE curve agrees well with the data except for the data of Mayoufi et al.36 that are at lower temperatures than the others. Lin et al.38 explained that the gap was caused by the method used to determine the melting temperature. We then considered a second case involving two hydrate phases: H38 and H24. From the data of Zhang et al.34 shown in Figure 6, it seems indeed that there is a phase transition around wTBPB = 0.4. The temperature composition T−x diagram calculated with both H38 and H24 hydrates is

Figure 5. Temperature−composition diagram of the TBAF + H2O mixture, calculated by considering the hydrates phases H32 and H29. The composition is expressed in terms of the TBAF weight fraction (wTBAF). The symbols denote the experimental data: △, ref 18; □, ref ; ◇, ref 32; ○, ref 33. The solid lines are the calculated SLE curves. AAD between the calculation and experimental data from ref 31 is 0.68 K. E



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Δs 0 pre = νwsw0 + sC0 + sA0

shown in Figure 6b. The hydrate parameters are listed in Table 3. The L−H38−H24 triple point is predicted at T = 281.89 K, wTBPB = 0.421. In both cases, the T−x phase diagram is well described. The calculated dissociation enthalpies are (196.2, 186.3, and 151.4) kJ·mol−1 for the H38, H32, and H24 hydrates, respectively (Table 4). The calculated value for H32 is close to the experimental data obtained by Mayoufi et al.36 (187 kJ·mol−1). 3.3. Group Contribution Methods To Predict the Hydrate Parameters. We propose a group contribution method to predict Δh0 and T0 for a given hydrate phase of known hydration number vw. The correlations can be used to predict phase diagrams, fusion enthalpies, and congruent melting points, as long as the compositions of the hydrate phases are known. All group contributions are reported in

s0w, s0C, Δs0pre

(14)

s0A

and are determined by minimizing the error between and Δh0/T0. The congruent melting point T0‑pre can be estimated with an average absolute deviation of 0.3 K for all systems considered in this study. The comparisons between the hydrate parameters obtained by fitting each binary system and the values predicted with the group contribution are shown in Figure 7. It should be noted that the correlation for T0 should

Table 5. Group Contributions To Predict the Hydrate Parameters T0‑pre, Δh0pre, Δs0pre, and the Dissociation Enthalpies Δhdis‑pre from eqs 12 to 14a species

h0i /kJ·mol−1

hHi /kJ·mol−1

s0i /kJ·mol−1

H2O TBA+ TBP+ F− Br− Cl−

3.929 22.760 27.767 62.227 27.936 13.368

3.944 24. 657 32.068 63.275 24.685 21.795

0.0139 0.0619 0.0852 0.1974 0.1142 0.0577

The congruent melting point T0 is predicted from T0‑pre = Δh0pre/ Δs0pre. Δh0pre and Δs0pre are the enthalpic and entropic parameters. h0i and hHi are the enthalpic and entropic contributions of species i. a

Table 5. The parameter Δh0 of a given hydrate phase of hydration number vw can be predicted by Δh0 pre = νwhw0 + hC0 + hA0

h0w,

h0C,

(12)

h0A

where and are the contributions for water, the alkylammonium/alkylphosphonium cation, and the anion. The three parameters are determined by minimizing the error between Δh0pre and the hydrate parameters Δh0 fitted to SLE data of salt + water binary mixtures. The average deviation between Δh0pre and Δh0 is 1.3% for all systems considered in this study. A similar group contribution method can also be employed to predict the fusion enthalpy of a hydrate: Δhdis − pre = νwhwH + hCH + hAH

hHw ,

hHC ,

Figure 7. Comparison between the hydrate parameters (T0, Δh0) and the correlations (eq 12 to 14). The lines correspond to f(x) = x.

be used with care, because the phase diagrams of alkylammonium/alkylphosphonium hydrates are within a very narrow temperature range, and a very accurate prediction of T0 is required to get a good description of the phase diagram for both binary and ternary systems in the presence of CO2. We used our correlations to predict the hydrate parameters for TBPC, as a blind test. Since the crystallographic data of TBPC39 are similar to those of TBAC (ref 29), it is reasonable to assume a hydration number of 30 for TBPC, which corresponds to a TBPC weight fraction of 0.35. The predicted values for TBPC, T0−pre = 284 K, Δh0pre = 159 kJ·mol−1, compare very well with the experimental values of Sakamoto et al.39 (283.45 K, 162.0 kJ·mol−1). 3.4. Dissociation Conditions of Quaternary Ammonium/Phosphonium Salts + CO2 Hydrates. The liquid− vapor−hydrate (L−V−H) three-phase lines of CO2 + water + tetra-n-alkyl ammonium/phosphonium salts have been modeled by combining the van de Waals and Platteeuw term with the thermodynamic model for salt + binary systems.6 The

(13)

hHA

where and are determined by minimizing the error between Δhdis‑pre and the experimental values of Δhdis. The average deviation is 1.3 %. The correlations (12) and (13) provide reasonable predictions and can be used if no data are available. In both cases the contribution for water is about 4 kJ· mol−1, which is lower than the fusion enthalpy of pure water (6 kJ·mol−1). The contribution of the anions increases as the size of the anion decreases, showing that small anions stabilize the semiclathrate hydrate phases more than large anions. The contribution of the TBP+ cation to the fusion enthalpy is higher than that of TBA+ cation. By analogy with a pure component, we propose to calculate the congruent melting temperature T0‑pre as T0‑pre = Δhpre0/ Δspre0, where Δh0pre is given by eq 12; Δs0pre is also calculated as the sum of contributions s0w, s0C, and s0A, as F



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SAFT-VRE binary parameters (kij = −0.76138 and lij = 0.16732) characterizing the H2O + CO2 interactions were previously determined14 by fitting VLE data in the temperature range from 278.2 K to 318.2 K and for pressures up to 8 MPa. When there are salts in the liquid phase, the solubility of CO2 can be predicted by neglecting the dispersion interactions between ions and CO2 molecules.6 The L−V−H three-phase lines of CO2 + water + salt ternary systems were determined by adjusting the parameters Vcell ij and to the experimental dissociation conditions. As mentioned εcell ij before, the free volume Vcell depends on the size of gas ij molecules and on the type of cages. In this study, CO2 is always encapsulated in 512 cavities for all semiclathrate hydrates. Consequently, we have assumed that the free volume parameter cell (Vcell ij ) for CO2 is the same for all hydrate phases, and εij is the only cell parameter adjusted per system. In this work, the cell potential parameters for TBAB are thus different from the parameters previously proposed.6 3.4.1. TBAB + H2O + CO2. The unit cell of type B TBAB semiclathrate hydrate of hydration number 38 (H38) consists of six D cages of type 512, four T cages of type 512 62, and four P cages of type 512 6320,24 (6D4T4P). Each unit cell of type B hydrate contains two TBA+ cations and two Br− anions, and each TBA+ cation is located at the center of a cavity consisting of 2T2P cages.20 Hence, the number ni of empty 512 cages per TBAB molecule is equal to 3 for H38. For type A hydrate with hydration number 26 (H26), the unit cell consists of a combination of 10, 16, and 4 cages of type 512, 512 62, and 512 63, respectively (10D16T4P).24 Rodionova et al.24 showed by single-crystal X-ray analysis that not only T and P cavities, but also D cavities can include TBA+ cations; one TBA+ cation can be contained inside a cavity composed of 3TP cages, while two cations TBA+ can be contained in a cavity composed of 4T cages and 4D cages. A unit cell of 10D16T4P cavities can contain six TBA+ cations in four 3TP cavities and one 4T4D cavity. Besides, there are six D empty cages, so the number ni of empty 512 cages per TBAB molecule is set equal to 1 for H26. We also assumed that the number of empty 512 cages is 1 for H24. The L−V−H three-phase lines for the CO2 + TBAB + H2O system have been fitted to experimental data,40−43 considering separately the H38−26 and H38−24, and are shown in Figures 8 and 9. In these figures the solid and dashed lines denote type A (H24 or H26) and type B (H38) hydrate phases, cell respectively. The parameters Vcell ij and εij optimized on the experimental curves are listed in Table 6. In our previous paper,6 we used different values for Vcell ij but the same value for for all hydrates phases. We also used ni = 2 empty cages per εcell ij TBAB molecules for type A hydrate, while we use ni = 1 in this study. Note that there was a misprint in our previous paper: the value of εcell ij should be 2750 K for both hydrate phases, not 2882 K. In this paper, we decided to use the same value Vcell ij = 0.0052 3 Å for all hydrate phases since the cages have all the same structure. This value corresponds to the parameter Aij = 4πVcell ij / k = 4.73·10−3 K·MPa−1 when the Langmuir constant is expressed as Cij = Aij/T exp(Bij/T). Parrish and Prausnitz10 and Munck et al.44 reported the values Aij = 8.97·10−3 and Aij =2.44·10−3 K·MPa−1 for the Langmuir constant of CO2 in the small cavities of structures II and I, and our value is of the same order. In this paper, the values of the cell potential well depth εcell ij are determined by fitting the experimental L−V−H lines at different concentrations of the salts. Note that the lines are very cell is used inside an sensitive with respect to εcell ij , since εij

Figure 8. Liquid−vapor−hydrate (L−V−H) three-phase lines for the CO2 + TBAB + H2O system calculated by considering the hydrate phases H38 and H26. The symbols denote the experimental data at different TBAB weight fractions: ◇, wTBAB = 0.03, ref 41 □, wTBAB = 0.05 from ref 42 ○, wTBAB = 0.05 from ref 43, respectively; ▷, ▽, and △ are at wTBAB = 0.1 from ref 42, ref 43, and ref 40; respectively; ◁, wTBAB = 0.19, ref 43. The lines, from the left to right, are the predictions of the model at wTBAB = 0.03, 0.05, 0.1, and 0.19. The second figure is an enlargement of the region around the phase change from H26 (solid line) to H38 (dashed line), at wTBAB = 0.19.

Figure 9. Liquid−vapor-hydrate (L−V−H) three-phase lines for the CO2 + TBAB + H2O system calculated by considering the hydrate phases H38 and H26 (a), and the hydrate phases H38 and H24 (b). The symbols denote experimental data at different TBAB weight fractions: blue ○, wTBAB = 0.32, ref 43; green ◇ and green △ are at wTBAB = 0.4 from ref 23 and ref 40; red □, wTBAB = 0.55, ref 43. The blue, green, and red curves are calculated by the model at wTBAB = 0.32, 0.4, and 0.55. The solid lines and dashed lines indicate type A (H26 or H24) and type B (H38) hydrates.

exponential function to compute the Langmuir constants; εcell ij is the well depth of the square-well potential describing the effective attractive interactions between gas molecules and the G



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3.4.2. TBAC + H2O + CO2. The structure of TBAC semiclathrate hydrate was studied by Rodionova et al.29 by Xray single-crystal analysis. They observed that the unit cell of TBAC semiclathrate hydrate consists of 10D16T4P cavities for H32 and H30, similarly to TBAB H26 hydrate. They also reported two types of TBA+ cation inclusions: the cation is contained in a cavity composed of either 3TP cages or 4T cages. As a result, the unit cell can contain up to 5 TBA+ cations and has 10 empty D cages. Thus, the number ni of D empty cages per TBAC molecule is equal to 2 for H32 and H30. For H24 hydrate, we assumed that the number ni of empty cages is also 1, comparable to TBAB H24. The L−V−H lines calculated at different TBAC concentrations are shown in Figure 10. The experimental data of Li et

Table 6. Additional Hydrate Parameters Used To Compute the L−V−H Three-Phase Lines of the Salt + H2O + CO2 Systema salts

vw

ni

3 Vcell ij /Å

εcell ij /k/K

TBAB

38 26 24 32 30 24 32 29 38 32 24

3 1 1 2 2 1 2 1.33 3 2 1

0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052 0.0052

3040 3040 3040 3040 2950 3130 2700 2800 3040 3200 3340

TBAC

TBAF TBPB

a vw and ni is the hydration number and number of D cages per salt cell molecule. Vcell ij is free volume of gas molecule j inside cavity i. εij is the depth of the square-well cell potential.

surrounding molecules (water molecules composing the 512 cage + other water molecules and ions surrounding the cage). As can be seen in Figure 8, for low TBAB concentrations, the model predicts that type B hydrate (H38) is always more stable than type A hydrate (H24 or H26). For wTBAB = 0.19, type A hydrate is more stable at low pressures, then the model predicts a change of phase from type B to type A, as the pressure is increased. The quadruple point at wTBAB = 0.19 is predicted around T = 284.16 K, P = 0.37 MPa for the L−V−H38−H26 phases, and at T = 284.02 K, P = 0.35 MPa for L−V−H38−H24. The predicted change of slope of the P−T curve due to the phase change from type A to type B is rather small and at low pressures, so it would be difficult to observe it experimentally if no structure analysis is performed. The prediction of type B in the presence of CO2 at wTBAB = 0.19 is in agreement with experimental observations: as shown by Ye and Zhang,43 the CO2 + TBAB semiclathrates at wTBAB = 0.1 and at wTBAB = 0.19 have the same morphology (type B), but the morphology of type A hydrate in the TBAB + water binary mixture at wTBAB = 0.19 is different. In Figure 9a,b, the L−V−H lines at high TBAB weight fractions (0.32, 0.4, and 0.55), are fitted to the experimental data reported by Deschamps and Dalmazzone,23 Ye and Zhang,43 and Arjmandi et al.40 For wTBAB = 0.55 the new set of parameters leads to a much better description of the data compared to our previous calculations.6,43 The model predicts a change from type A to type B hydrate (L−V−H38−H26) at wTBAB = 0.55 and P = 1.5 MPa, and deviations from data are observed between the model and experimental data at pressures lower than 1.5 MPa. However, the calculated metastable L−V− H38 three-phase line at pressures below the quadruple point goes through the experimental data. Moreover, the experimental three-phase line at low pressures tends to the melting point of the metastable type B (H38) hydrate in the water + TBAB binary mixture at the same concentration, but not to the melting point of type A hydrate. Thus, it is likely that only type B being formed at wTBAB = 0.55 may be due kinetic effects if type B forms faster than type A. It would be necessary to analyze the structure of the gas hydrate phase to confirm our observations. The AAD between the calculation and the experimental data for H38−26 and H38−24 are 0.34 K and 0.35 K, respectively.

Figure 10. Liquid−vapor−hydrate (L−V−H) three-phase lines for the CO2 + TBAC + H2O system, calculated by considering the hydrate phases H32, H30, and H24. The symbols denote the experimental data at different TBAC weight fractions: □, wTBAC = 0.0434, ref 42; ▷, wTBAC = 0.06, ref 46; △, wTBAC = 0.0875, ref 42; ◁, wTBAC = 0.15, ref 46; ○ and ◇ are at wTBAC = 0.34 from ref 45 and ref 46, respectively; + wTBAC = 0.44, ref 46. The lines are the predictions of the model. The dotted, dashed, and solid lines indicate the hydrate phases H24, H30, and H32, respectively.

al.,42 Makino et al.,45 and Shi et al.46 were used to determined εcell ij . The AAD between the calculation and the experimental data is 0.42 K. The dotted, solid, and dashed lines denote the three-phase lines for H24, H30, and H32, respectively. At lower concentration of TBAC, only H32 is observed for pressures up to 5 MPa. At wTBAC = 0.34, a phase change from H32 to H30 is observed; the quadruple point of L−V−H32−H30 is predicted at T = 291.6 K, P = 1.7 MPa. Another phase change is predicted at wTBAC = 0.44; the quadruple point of L−V−H30−H24 is predicted at T = 291.6 K, P = 2.4 MPa. The corresponding cell parameters Vcell ij and εij parameters are listed in Table 6. 3.4.3. TBAF + H2O + CO2. McMullan et al.47 studied the structure of TBAF semiclathrate hydrates. They reported that the structure of TBAF·32.8H2O is tetragonal-I, which corresponds to the cage composition 10D16T4P in a unit cell.4 The number of empty cages per TBAF molecule is set equal to 2 for H32, by assuming that the inclusions of TBA+ cations in cavities are the same as those in H32 TBAC hydrate. Komarov et al.48 studied the structure of TBAF·29.7H2O, which was found as cubic, and corresponds to cage composition of 16D48T in a unit cell. They found that the TBA+ cation is located in a cavity composed of 4T cages, so the unit cell can contain 12 cations with 16 D cages remaining empty. Thus, we set ni = 16/12 ≈ 1.33 for H29. H



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The L−V−H lines for the CO2 + TBAF + H2O system have been fitted to the literature data42,49,50 to determine the cell potential parameter εijcell. As can be seen in Figure 11,

Figure 11. Liquid−vapor−hydrate (L−V−H) three-phase lines for the CO2 + TBAF + H2O system, calculated by considering the hydrate phases H32 and H29. The symbols denote the experimental data at different TBAF weight fractions: +, wTBAF = 0.02, ref 49; ◇, wTBAF = 0.05, ref 49; □, wTBAF = 0.041, ref 42; △, wTBAF = 0.083, ref 42; ▽, wTBAF = 0.104, ref 50; ▷, wTBAF = 0.15, ref 49; ◁, wTBAF = 0.31, ref 50; ○, wTBAF = 0.448, ref 50. The lines are the predictions of the model. The solid lines correspond to the H29 phase, and the dashed lines indicate the H32 phase. Figure 12. Liquid−vapor−hydrate (L−V−H) three-phase lines for the CO2 + TBPB + H2O system calculated by considering the hydrate phase H32 (a), and the hydrate phases H38 and H24 (b). The symbols denote the experimental data at different TBPB weight fractions: △, wTBPB = 0.05, ref 51; ◇, □, and ○ are at wTBPB = 0.1 from ref 34, ref 36, and ref 51, respectively; ◁ and ▷ are at wTBPB = 0.2 from ref 34 and ref 36. The curves, from the left to right, are calculated by the model at wTBPB = 0.05, 0.1, and 0.2. The dashed lines indicate the hydrate phase H32.

reasonable fittings are obtained over the salt concentration range (AAD = 0.60 K). Only H32 is observed at low TBAF concentrations for pressures up to 5 MPa, while H29 is observed at high concentrations. At wTBAF = 0.448, the model is able to capture the change of behavior and the destabilization of the hydrate by further adding salt; however, some deviations between the model and experimental data are observed, which can be explained by the fact that the activity coefficients of the species are not well predicted in the liquid phase. Further experimental measurements at higher concentrations are necessary to obtain more reliable ionic and cell parameters. The parameter εcell ij is significantly lower for CO2 + TBAF + H2O systems than for other salts. This could be explained by the fact that the F− anions affect the structure and distort the D cages more than the other anions. 3.4.4. TBPB + H2O + CO2. As mentioned previously in this paper, we have considered two cases for TBPB systems. We first assume that there is only one hydrate phase H32 of hydration number 32, according to Dyadin and Udachin4 and Zhang et al.34 Then, we consider the case where the H38 and H24 hydrates can be observed. According to Muromachi et al.,35 the number ni of empty cages per TBPB molecule is 3 for H38. By analogy with TBAC and TBAF systems, we assumed that the number of empty cages is equal to 2 for H32, and 1 for H24. We first calculated the three-phase lines of CO2 + TBPB + H2O systems for wTBPB = 0.05, 0.1, and 0.2 (Figure 12a). We used the experimental data from Zhang et al.,34 Mayoufi et al.,36 and Shi et al.51 to determine the cell potential parameter εcell ij for H32 (case 1) and H38−24 hydrate (case 2). The value of εcell ij is larger than that obtained for TBAC and TBAB hydrates (Table 6). The different values of εcell ij between different hydrate phases and salts can be explained by the fact that D empty cages do not have always the same shape. Moreover, the interactions between the encaged gas molecules and the ions may change depending on the type of ions and the relative positions of ions and molecules.

In the second case, the value εcell ij for H24 hydrate has been determined by fitting the L−V−H line at wTBPB = 0.5 and 0.6. The calculated lines at wTBPB = 0.35, 0.371, 0.5, and 0.6 are compared to experimental data34,36,52 in Figure 13a,b. The calculated lines are in good agreement with the data except with the data of Shi et al.51 at wTBPB = 0.6. Shi et al.51 measured an equilibrium point at T = 288.2 K, P = 1.77 MPa, while Mayoufi et al.36 obtained a point at T = 286.2 K, P = 1.7 MPa. Different experimental methods were used by the two groups: Mayoufi et al.36 used the DSC method, while Shi et al.51 used the isochoric pressure-search method. In the second case, the model predicts a phase change from H24 to H38 as pressure is increased at wTBPB = 0.5 and 0.6, and the L−V−H38−H24 quadruple point is predicted at around T = 286.9 K, P = 1.4 MPa at wTBPB = 0.5 and T = 287.6 K, P = 3.2 MPa at wTBPB = 0.6. In both cases (H32 and H38−24), the model can accurately describe the experimental dissociation temperatures over the whole salt concentration and pressure ranges. The AAD between the calculation and the experimental data for H32 and H38−24 are 0.41 K and 0.36 K, respectively. Additional crystal structure analysis in the presence of gas would be useful to decide which case to use.

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CONCLUSIONS The thermodynamic approach proposed by Paricaud6 is used to determine the dissociation conditions of semiclathrate hydrates made from TBAB, TBAC, TBAF, and TBPB salts. The SAFTVRE equation of state12 is employed to compute the I



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empirical dependence of the Langmuir constants with respect to the salt concentration. More experimental data are required to further improve the modeling of such systems; one would especially need some experimental measurements of osmotic coefficients at different temperatures. Structural information on gas semiclathrate hydrates as well as storage capacities would also confirm the number of empty D cavities per salt molecules, and the type of phases predicted by the model.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Funding

The authors thank AREVA for financial support as part of the nuclear and engineering funding of Areva-ParisTech. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Fowler, D. L.; Loebenstein, W. V.; Pall, D. B.; Kraus, C. A. Some unusual hydrates of quaternary ammonium salts. J. Am. Chem. Soc. 1940, 62, 1140−1142. (2) Dyadin, Y. A.; Udachin, K. A. Clathrate polyhydrates of peralkylonium salts and their analogs. J. Struct. Chem. 1987, 28, 394−432. (3) Shimada, W.; Ebinuma, T.; Oyama, H.; Kamata, Y.; Takeya, S.; Uchida, T.; Nagao, J.; Narita, H. Separation of gas molecule using tetra-n-butyl ammonium bromide semi-clathrate hydrate crystals. Jpn. J. Appl. Phys. 2003, 42, L129−L131. (4) Dyadin, Y. A.; Udachin, K. A. Clathrate Formation in WaterPeralkylonium Salts Systems. In Clathrate Compounds, Molecular Inclusion Phenomena, and Cyclodextrins; Springer: Netherlands, 1984; Vol. 3, pp 61−72. (5) Mohammadi, A. H.; Belandria, V.; Richon, D. Use of an artificial neural network algorithm to predict hydrate dissociation conditions for hydrogen + water and hydrogen + tetra-n-butyl ammonium bromide + water systems. Chem. Eng. Sci. 2010, 65, 4302−4305. (6) Paricaud, P. Modeling the dissociation conditions of salt hydrates and gas semiclathrate hydrates: application to lithium bromide, hydrogen iodide, and tetra-n-butylammonium bromide + carbon dioxide systems. J. Phys. Chem. B 2011, 115, 288−299. (7) Kwaterski, M.; Herri, J.-M., Thermodynamic modelling of gas semi-clathrate hydrates using the electrolyte NRTL model. In 7th International Conference on Gas Hydrates, Edinburgh, Scotland, July 17-21, 2011. (8) Joshi, A.; Mekala, P.; Sangwai, J. S. Modeling phase equilibria of semiclathrate hydrates of CH4, CO2, and N2, in aqueous solution of tetra-n-butyl ammonium bromide. J. Nat. Gas Chem. 2012, 21, 459− 465. (9) van der Waals, J. H.; Platteeuw, J. C. Clathrate solutions. Adv. Chem. Phys. 1959, 2, 1−57. (10) Parrish, W. R.; Prausnitz, J. M. Dissociation pressures of gas hydrates formed by gas mixtures. Ind. Eng. Chem. Process Des. Develop. 1972, 11, 26−35. (11) Dufal, S.; Galindo, A.; Jackson, G.; Haslam, A. J. Modelling the effect of methanol, glycol inhibitors and electrolytes on the equilibrium stability of hydrates with the SAFT-VR approach. Mol. Phys. 2012, 110, 1223−1240. (12) Galindo, A.; Gil-Villegas, A.; Jackson, G.; Burgess, A. N. SAFTVRE: phase behavior of electrolyte solutions with the statistical associating fluid theory for potentials of variable range. J. Phys. Chem. B 1999, 103, 10272−10281. (13) Patel, B. H.; Paricaud, P.; Galindo, A.; Maitland, G. C. Prediction of the salting-out effect of strong electrolytes on water plus alkane solutions. Ind. Eng. Chem. Res. 2003, 42, 3809−3823.

Figure 13. Liquid−vapor−hydrate (L−V−H) three-phase lines for the CO2 + TBPB + H2O system, calculated by considering the hydrate phase H32 (a) and the hydrate phases H38 and H24 (b). The symbols denote the experimental data at different TBPB weight fractions: blue △, blue □, and blue ◇ are at wTBPB = 0.35 from ref 34, ref 36, and ref 52, respectively; (green □) and (green ○) are at wTBPB = 0.371 from ref 36 and ref 51, respectively; (red △) and (red □) are at wTBPB = 0.5 from ref 34 and ref 36, respectively; (pink □) and (pink ○) are at wTBPB = 0.6 from ref 36 and ref 51, respectively. The blue, green, red, and pink curves are calculated by the model at wTBPB = 0.35, 0.371, 0.5, and 0.6, respectively. The solid and dashed lines indicate the hydrate phases H24 and H38.

thermodynamic properties of electrolyte solutions and vapor phases. The SLE curves of salt + water binary systems can be well described by the model over a wide range of salt composition, by adjusting two parameters Δh0 and T0 for each hydrate phase. Correlations are proposed to predict Δh0 and T0. Such correlations would be very useful to model gas semiclathrate systems when a hydrate phase is metastable in the salt-water binary mixture, but become stable in the presence of gas molecules. They could also be used to predict the phase diagrams of mixed semiclathrate hydrate systems when the hydrate phase contains several types of cations and anions. The dissociation conditions of gas semiclathrate hydrates are calculated by combining the model for binary mixtures with the van der Waals and Platteeuw.9 By assuming that CO2 molecules are only encaged in D cavities, we used the same free volume Vcell of a CO2 molecule inside a D cavity for all salts. We ij calculated the L−V−H three-phase lines of the CO2 + salt + water ternary systems by only fitting the cell parameter εcell ij for each hydrate phase. A very good description of L−V−H threephase lines is obtained over wide ranges of salt concentrations and pressures. The model can be employed for both salt + water binary systems and gas semiclathrate hydrates, and can accurately describe the change of behavior as the salt concentration is increased above the congruent composition. Contrary to other models, our approach does not require any J



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