PHASE EQUILIBRIA OF CARBON DIOXIDE AND METHANE GAS-HYDRATES PREDICTED WITH THE MODIFIED ANALYTICAL S-L-V EQUATION OF STATE

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1 EPJ Web of Conferenes, 010 (01) DOI: /epjonf/ Owned by the authors, published by EDP Sienes, 01 PHASE EQUILIBRIA OF CARBON DIOXIDE AND METHANE GAS-HYDRATES PREDICTED WITH THE MODIFIED ANALYTICAL S-L-V EQUATION OF STATE Válav VINŠ 1, Andreas JÄGER 1, Jan HRUBÝ, Roland SPAN Abstrat: Gas-hydrates (lathrates) are non-stoihiometri rystallized solutions of gas moleules in the metastable water lattie. Two or more omponents are assoiated without ordinary hemial union but through omplete enlosure of gas moleules in a framework of water moleules linked together by hydrogen bonds. The lathrates are important in the following appliations: the pipeline blokage in natural gas industry, potential energy soure in the form of natural hydrates present in oean bottom, and the CO separation and storage. In this study, we have modified an analytial solid-liquid-vapor equation of state (EoS) [A. Yokozeki, Fluid Phase Equil. 3 (004)] to improve its ability for modeling the phase equilibria of lathrates. The EoS an predit the formation onditions for CO - and CH 4 -hydrates. It will be used as an initial estimate for a more ompliated hydrate model based on the fundamental EoSs for fluid phases. 1. INTRODUCTION TO GAS-HYDRATES Clathrates of natural gas, ommonly alled gas-hydrates, are non-stoihiometri solid solutions of a low moleular weight gas and water [1]. The gas-hydrates are beoming important in many areas of human ativities. Originally, the gas-hydrates were investigated due to the pipeline blokage in the natural gas industry. With inreasing energy onsumption, dereasing reserves of fossil fuels, and limate hanges, the gashydrates are being investigated for two following appliations: potential energy soure in the form of natural hydrates present in the oean bottom, and the arbon dioxide separation and storage []. The lathrate is a solid rystallized solution of the gaseous omponent in the thermodynamially metastable water lattie. In gas-hydrates, the gas moleule, alled the guest, is situated in the avity, referred also as the age or host, formed by a framework of water moleules linked together by hydrogen bonds. The guest-host interations are realized through van der Waals type dispersion fores. The lathrate beomes thermodynamially stable under given temperature and pressure if a ertain fration of avities is oupied by the guest moleules. The guest oupany of the avities stabilizes the empty water lattie. 1 Institute of Thermomehanis AS CR, v. v. i., Dolejškova 140/5, Prague 8, Czeh Republi (vins.valav@seznam.z, hruby@it.as.z) Institute for Thermodynamis, Ruhr-Universität Bohum, Universitätsstr. 150, Bohum, Germany (a.jaeger@thermo.ruhr-uni-bohum.de, roland.span@thermo.rub.de) This is an Open Aess artile distributed under the terms of the Creative Commons Attribution Liense.0, whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited. Artile available at or

2 EPJ Web of Conferenes Most of the hydrate models are based on the theory by van der Waals and Platteeuw (vdwp) [1] ombining the statistial thermodynamis with the lassial thermodynamis. The vdwp theory was further improved by Parrish and Prausnitz [3] for pratial alulations of the hydrate formation onditions. Temporary hydrate models suh as that by Lee and Holder [4], Klauda et al. [5,6], Ballard and Sloan [7,8], managed to avoid most of the limiting assumptions of the original models by van der Waals and Platteeuw [1] and Parrish and Prausnitz [3]. However appliation of these models [4]-[8] is still rather ompliated and requires advaned omputational ability. On the other hand, Yokozeki [9] developed a simple analytial equation of state (EoS) that an provide qualitatively good estimate for the solid-liquid-vapor (S-L-V) equilibria. This EoS an simultaneously model all three phases and an be extended to simple mixtures suh as CO +CH 4. In the following two studies [10,11], Yokozeki demonstrated that his EoS may also be used for gas-hydrates modeling. The S-L-V EoS provided qualitatively good predition for CH 4 - and CO -hydrate formation onditions. The Yokozeki s EoS is therefore a relatively simple alternative to muh more ompliated hydrate models based on the vdwp theory. In this study, we modified the original Yokozeki s [9]-[11] S-L-V EoS to improve its preditive ability for CH 4 -and CO -hydrate phase equilibria. Moreover, there were found several small disrepanies in the Yokozeki s artiles [9]-[11] whih we also tried to fix in this work. The modified S-L-V EoS models quite aurately the formation onditions for both gas-hydrates over relatively wide ranges of temperature and pressure. It is planned to use this EoS as an initial estimate for a more ompliated hydrate model based on the vdwp theory ombined with the fundamental EoSs for fluid phases [1]-[14] and for the solid phase of pure omponents, i.e. water ie Ih [15] and dry ie [16].. ANALYTICAL EQUATION OF STATE FOR SOLID-LIQUID-VAPOR Yokozeki s [9] analytial EoS represents a physially reasonable extension of the original van der Waals [17] fluid EoS. The final form of the S-L-V EoS is given as follows RT v d a p (1) vb v v with volumetri parameters 0 bd. Figure 1 shows a p-log(v) diagram for arbon dioxide obtained from the S-L-V EoS (1). The EoS is ompared with the ubi Peng- Robinson [18] EoS for fluid phases. Redued parameters of the S-L-V EoS are defined in the following manner p p p p ar a, b r b, r, dr d, () RT ZRT ZRT ZRT where the attration parameter a r and the volumetri parameter b r are onsidered temperature dependent n arta0 a1t Texp at T (3) m brtb0 b1exp bt T. (4) In the modified S-L-V EoS, the pure omponent parameters a to d were taken from Yokozeki [9] for CO and from Yokozeki [11] for H O and CH 4. Values for the redued parameters, oeffiients, and exponents in equations () to (4) an be found in the original artiles by Yokozeki [9,11] p.

3 EFM11 Figure 1: Shemati pv diagram for CO alulated from the S-L-V EoS (1) and the Peng-Robinson EoS Table 1 ompares the temperature and pressure at the triple point alulated from the S- L-V EoS for vapor (V) liquid (L) solid (I) phase equilibria with the tabulated values. The water parameters provided in [10] must ontain some kind of inauray as they did not result in the satisfatory triple point. The alulated temperature 58.3 K and pressure kpa differed strongly from the tabulated values in this ase. T [K] T(EoS) [K] dt [%] p [kpa] p(eos) [kpa] dp [%] CH CO H O Table 1: Comparison of the triple point onditions for pure substanes alulated by the analytial S-L-V EoS with the tabulated values Mixtures an be treated with a help of the van der Waals / Lorentz-Berthelot mixing rules [9] in the S-L-V EoS. The mixture parameters a to d an be alulated from the mole frations x in the following way i N N N N N i j 1 ij i j, i i, i i, i i. (5) a aa k x x b bx x d d x i j i1 i1 i1 In equation (5), N = for a binary mixture and k ij stands for the binary interation parameter. For simple mixtures of normal ompounds suh as CO +CH 4, the binary interation parameter may be onsidered onstant, i.e. independent of omposition or temperature [9]. However as Yokozeki showed in his other studies [10,11], the aqueous solutions annot be suffiiently modeled with onstant k ij. For mixtures ontaining water, the binary interation parameter kij has to be onsidered omposition dependent. Therefore in equation (5), Kij given as follows kij has to be replaed with the variable interation parameter p.3

4 EPJ Web of Conferenes We note that the Kij K ij kk x x kxkx ij ji i j ij i ji j. (6) parameter onsidered in this study has different omponent-indies in the denominator than the parameter employed by Yokozeki [10,11]. Definition of by equation (6) should be more onsistent with the usual form of the van Laar s mixing rule [19]. 3. PHASE EQUILIBRIUM OF A BINARY MIXTURE Phase equilibrium of a multiomponent system is generally defined by equality of hemial potentials of eah omponent in all phases present. This ondition an be transformed to equality of fugaities whih is a more ommon ase in the EoS-based alulations. At a onstant pressure, the fugaity-based equilibrium ondition an be defined only in terms of the fugaity oeffiients i and the mole frations x i H H L L V V xi i xi i xi i. (7) Equation (7) is an example for the three phase equilibrium of the hydrate (H), water-rih liquid (L ), and vapor phases. In our alulations, we would like to use the analytial S-L- V EoS (1) as an initial estimate for the mole frations at the given system temperature and pressure. Therefore, we diretly solve the equilibrium ondition given by equation (7) instead of using the graphially-based ommon tangent method proposed by Yokozeki [11]. In suh a ase, the equilibrium ondition (7) is solved by a multidimensional Newton-Raphson optimization method applied on the following set of equations L L H H x1 1 x11 0 L L H H x x 0. (8) V V H H x11 x11 0 V V H H x x 0 The pressure is usually set as the independent variable and the temperature and the L H mole frations x 1, x 1, and x V 1 are found as the unknown quantities. Mole frations of the seond omponent in a binary mixture are obtained from the simple ondition x 1 x1. Aording to Yokozeki [9], the fugaity oeffiient in equation (8) an be determined as follows 1 b ibdbid bd di v lni ln 1 bln 1 ln b v v b vb (9) 1 id bid b a 1 ln Z b v vb xi vrt where the omposition derivative of the attration parameter a an be derived from equations (5) and (6) in the following way a kk ij ji kx ij i kji xj kk ij ji xi xj aa i j 1 Kijxj aa i jxx i j. (10) xi j kxkx ij i ji j To assure good onvergene of the phase equilibrium algorithm and to improve its numerial auray, the S-L-V EoS (1) was redued aording to the first omponent ritial point (subsript 1), i.e. Kij p.4

5 EFM11 T v d a p Z v b v v Z 1 1. (11) The redued quantities and the mixture parameters are defined as follows p Z 1RT1 T p, vv, T, (1) p p T RT Z RT Z RT Z RT a a b b d d p p p p ,,,. (13) The pure-omponent attration parameters redued aording to the first omponent s ritial point are given in the following manner T p. (14) 1 a1r a1r, ar ar T 1 p The volumetri parameters for a pure omponent redued aording to the first omponent ritial point are defined as follows ZTp 1 e 1r e1 r, e r er Z 1T1p with e b,, d. (15) In equations (14) and (15), the redued parameters of a pure omponent a r, b r, r, and dr are given by equation (). The fugaity oeffiient an be alulated from the redued EoS (11) in the following way 1 b irbdb irdbdd ir lni ln 1 bln 1 b v v b v 1 ird b irdb a 1 pv ln ln Z 1 vb b v vb xi vtz 1 T. (16) The omposition derivative of parameter a, i.e. a xi, an be transformed to its redued form a xi, required in equation (16), by replaing the pure omponent parameters a i and a j in equation (10) with their redued values a ir and aj r given by equation (14). The phase equilibrium algorithm based on the set of equations similar to equation (8) an be used for two- and three-phase equilibria of a pure omponent or for two-, three-, and four-phase equilibria of a binary mixture. 4. HYDRATE PHASE MODEL The analytial S-L-V EoS an be relatively simply used for normal mixtures, forming just one solid phase, as demonstrated by Yokozeki [9] for a binary mixture of CO +CH 4. However the appliation on the hydrate forming systems requires some additional modifiations. The S-L-V EoS must provide a desription of two different solid phases; namely the pure-omponent ie, e.g. water ie Ih or dry ie of CO, and the solid twoomponent solution, i.e. the gas-hydrate. Yokozeki introdued empirial orretion oeffiients desribing the differene between the pure-omponent ie phase and the hydrate phase. The volumetri parameter b of water was orreted in his CO -and CH 4 - hydrate models. In the first ase, the parameter b was redued for the water ie [10], while in the seond ase the parameter b was inreased for the water in the hydrate phase [11]. Unfortunately, we found that neither of these two approahes provided satisfatory results. The option of onsidering two different types of water ie, i.e. one for the pure water and one for the binary mixture forming gas-hydrate, is not appropriate in p.5

6 EPJ Web of Conferenes our point of view. On the other hand, the seond option with inreased volume of the hydrate phase ompared to the water ie seems quite reasonable. Nevertheless, we deteted some errors in the phase equilibrium alulated with suh modified hydrate phase. The ommon tangent method used for the verifiation of our results did not work properly for the hydrate phase in this ase. In our gas-hydrate model, we onluded with another option proposed by Yokozeki [11] for the hydrate phase modifiation. Instead of orreting the volumetri parameter b, we dereased the attration parameter a for the gas-hydrate. The interations of moleules in the hydrate phase are weaker than in the water ie. Consequently, the attration parameter a had to be lowered in the following manner ahohydrate a HO ie a, with a (17) In equation (17), a is an empirial oeffiient whose value was found aording to the fat that the gas amount in both the CO - and CH 4 -hydrates should not exeed the maximum value of %. The maximum mole fration orresponds to fully oupied avities by gas moleules in the water lattie of si hydrate struture. We note that the original Yokozeki s CH 4 -hydrate model [11] did not fulfill this ondition for temperatures above 316 K. 5. BINARY INTERACTION PARAMETERS FOR AQUEOUS SOLUTIONS The S-L-V EoS onsiders two sets of binary interation parameters: one for the fluid phase, i.e. vapor-liquid equilibrium (VLE), and one for the hydrate phase, i.e. solid solution. The binary interation parameter k in equation (6) is usually taken as temperature dependent for the aqueous solutions. The k ij parameter was obtained by orrelating the S-L-V EoS to the available experimental data for the VLE and for the hydrate formation onditions. ij Figure : Binary interation parameter k 1 for hydrate phase of CH 4 (1) + H O() mixture; phases: H hydrate, L liquid, V vapor, I water ie The fluid phases of the methane-water mixture were modeled with the same binary interation parameter as by Yokozeki [11]. The k ij parameter was alulated from the following linear temperature funtion k T a at. (18) ij 0 1 K p.6

7 EFM11 Sine the hydrate phase was modeled in a different way than by Yokozeki [11] see setion 4 the k ij parameter for the solid phase had to be found separately. The modified S-L-V EoS was orrelated to the experimental data points for the hydrate formation onditions, i.e. the three phase equilibrium lines at whih the gas-hydrate is present, seleted from the databases by Sloan and Koh [] and Kroenlein et al. [0]. For brevity we do not provide full referenes for all experimental data onsidered in our study but only the abbreviations used by Sloan and Koh [] and Kroenlein et al. [0]. The binary interation parameter for the CH 4 -hydrate was fitted to the data by Roberts et al. (1940), Deaton and Frost (1946), Marshall et al. (1964), Falabella (1975), Adisasmito et al. (1991), Makogon and Sloan (1994), and Dyadin and Aladko (1996). Figure shows the temperature dependene of the binary interation parameter k 1 for CH 4 -hydrate; methane(1) and water(). The parameter k 1 was orrelated with the third-order polynomial funtion 3 k1 Ta0 a1 a a3, (19) where T K 00K. Table summarizes values of the oeffiients a 0 to a 3 for the binary interation parameter for the VLE and hydrate phase of the methane-water binary mixture. CH 4 + H O VLE CH 4 H-phase k 1 * k 1 * k 1 k 1 T < 7 K T a E E E a E E E+0 - a E E+0 - a E E+01 - * Taken from Yokozeki [11] Table : Coeffiients of the temperature orrelation for the binary interation parameter for the methane(1)-water() mixture Figure 3: Binary interation parameter k 1 for the VLE of CO (1) + H O() mixture determined from the solubility data [1],[] For the CO + H O system, the binary interation parameters had to be orrelated both for the VLE and the solid phase as another ombination of the pure omponent p.7

8 EPJ Web of Conferenes parameters a to d were used for this mixture than by Yokozeki [10]. The k 1 parameter for VLE was orrelated to the Henry s law data provided by Crovetto [1] and Anderson []. The Henry s law onstant k H was alulated from the following equation L L f 1 1 x1p V L V kh lim lim p1 ( T, p, x1 0), (0) x10 x x10 1 x1 V where p states for the vapor pressure of the pure water at the given temperature. Figure 3 shows the binary interation parameter k 1 determined for the modified S-L-V EoS. As an be seen, the new orrelation differs from the original Yokozeki s dependene [10]. The k 1 parameter was determined from the experimental data by Wendland et al. [3] measured for the three phase equilibrium line V-L 1 -L (L 1 arbon dioxide rih liquid). The binary interation parameter for the hydrate phase was fitted to the various three phase equilibrium lines inluding V-H-I, V-H-L 1, V-H-L, and H-L 1 -L equilibrium. The experimental data for the CO -hydrate formation onditions were again taken from the databases by Sloan and Koh [] and Kroenlein et al. [0]. Following data sets were onsidered in our study: Deaton and Frost (1946), Larson (1955), Takenouhi and Kennedy (1965), Vlahakis et al. (197), Falabella (1975), Ng and Robinson (1985), Ohgaki et al. (1993), Nakano et al. (1998b), Wendland (1999), Moojier-van den Heuvel et al. (001), Hahikubo et al. (00), and Mohammadi et al. (005). Figure 4: Binary interation parameter k 1 for hydrate phase of CO (1) + H O() mixture; phases: H hydrate, I water ie, L1 CO -rih liquid, L H O-rih liquid, V vapor CO + H O VLE CO H-phase k 1 k 1 k 1 k 1 T < 57 K T a E E E E a E E E E+04 - a E+04 - a E+03 - a E+03 - Table 3: Coeffiients of the temperature orrelation for the binary interation parameter for the arbon dioxide(1)-water mixture() p.8

9 EFM11 Figure 4 shows the k 1 parameter depending on temperature. The data for the V-H-L 1 equilibrium are rather inonsistent and lie aside from the other three phase lines. These data were therefore not used in the k 1 orrelation. As it is shown in the following setion, negleting the k 1 parameter for V-H-L 1 equilibrium did not result in any signifiant loss of auray of the modified S-L-V EoS. The parameter k 1 for the CO -hydrate phase was orrelated with the fourth-order polynomial funtion 3 4 k1 Ta0 a1 a a3 a4, (1) where again T K 00K. Values of the oeffiients a 0 to a 4 for the binary interation parameter for the VLE and hydrate phase of the arbon dioxide-water system are listed in Table RESULTS The hydrate formation onditions predited by the modified S-L-V EoS were ompared with the experimental data. Figure 5 shows the log(p)-t diagram for the methane-water mixture. As an be seen, the EoS ahieved quite good agreement with experiments over wide temperature and pressure ranges of K and MPa, respetively. Figure 5: Phase diagram of the water-methane mixture. Comparison of seleted experimental data with results of the modified S-L-V EoS Like the original S-L-V EoS by Yokozeki [11], the modified EoS predited two different quadruple points. The experimentally verified V-H-L-I point at T = K and p =.744 MPa and the seond theoretial V-H-L-mI point at the high pressure 08.4 MPa and temperature K. The pure water ie exists only to a maximum pressure of approximately 6.0 MPa [11]. The three phase equilibrium line H-L-mI alulated at high pressures is therefore onsidered as the metastable equilibrium with the metastable water ie (mi). However as far as we know, the seond quadruple point V-H-L-mI was p.9

10 EPJ Web of Conferenes not experimentally verified and the dashed three-phase line H-L-mI should therefore be onsidered only as a hypothetial solution. The predited mole fration of methane in the hydrate phase varying with temperature and pressure is plotted in Figure 6. The data points were alulated at the experimental temperatures and pressures of the various three-phase equilibrium lines. The mole fration of CH 4 did not exeed the physially defined maximum value of %. Figure 6: Mole fration of methane in the gas-hydrate along four different three-phase equilibrium lines predited by the modified S-L-V EoS Figure 7: Phase diagram of the water-arbon dioxide mixture. Comparison of seleted experimental data with results of the modified S-L-V EoS Figure 7 shows the log(p)-t diagram for the arbon dioxide-water system. The modified S-L-V Eos is again in a good agreement with the experimental hydrate formation data in p.10

11 EFM11 this ase. The ranges of validity are 0 94 K in temperature and 0 00 MPa in pressure. The quadruple points were predited at the pressures of MPa and MPa and temperatures of 7.50 K and 8.73 K for V-L -H-I and V-L 1 -L -H, respetively. Both these results are omparable with the experimental values []. Figure 8: Mole fration of arbon dioxide in the gas-hydrate along four different three phase equilibrium lines predited by the EoS The predited mole fration of arbon dioxide in the hydrate phase did not exeed the maximum allowed value at any of the four three-phase lines; see Figure 8. The average predited value is around 10.5 % whih approximately orresponds to the situation when only the large avities of the water lattie are oupied. This result seems rather reasonable as the CO moleules are relatively large. It is therefore quite probable that the small avities are oupied only marginally. 7. CONCLUSION The analytial S-L-V EoS by Yokozeki [9]-[11] was examined for predition of the gashydrate formation onditions in this study. The EoS was modified to improve its preditive ability for the CH 4 - and CO -hydrates. The hydrate phase was modeled by inreasing the water attration parameter ompared to the pure water ie. Binary interation parameters for the VLE and the hydrate phase were orrelated to relatively large amount of the phase equilibrium experimental data. The omputational algorithm allows solving various types of the phase equilibria using the Newton-Raphson method applied on the ondition of equal fugaities. The modified S-L-V EoS ahieved quite good auray and an be used as an initial estimate for the hydrate formation onditions. The predited mole fration of gas in the hydrate phase did not exeed the maximum value of % in the onsidered temperature and pressure ranges. NOMENCLATURE a attration parameter [Jm 3 mol - ] a 0 4 oeffiient b,, d volumetri param. [m 3 mol -1 ] a orretion oeffiient [-] e general parameter [m 3 mol -1 ] f fugaity [Pa] k ij binary interation parameter [-] k H Henry s law onstant [Pa] p.11

12 EPJ Web of Conferenes K ij omposition dependent k ij [-] p pressure [Pa] R universal gas onstant [Jmol -1 K -1 ] T temperature [K] v molar volume [m 3 mol -1 ] x mole fration [-] Z ompressibility fator pv/rt [-] Greek letters fugaity oeffiient [-] dimensionless temperature [-] Subsripts ritial point ondition i, j index of a omponent r redued quantity Supersripts redued quantity H gas-hydrate I ie L liquid V vapor ACKNOWLEDGEMENTS The authors are grateful to the International Assoiation for the Properties of Water and Steam (IAPWS) whih supported Válav Vinš s stay at the Ruhr University in Bohum. The projet has also been partly supported by the Czeh Siene Foundation grants No. P101/11/P046, P101/11/1593, and by the Researh Plan of the Institute of Thermo- 8. REFERENCES [1] Van der Waals J.H. and Platteeuw J.C.: Clathrate solutions, Advan. Chem. Phys., 1959, 1-57 [] Sloan E.D. and Koh C.A.: Clathrate hydrates of natural gases, 3rd ed., CRC Press, Taylor & Franis group, 008 [3] Parrish W.R. and Prausnitz J.M.: Dissoiation pressure of gas hydrates formed by gas mixtures, Ind. Eng. Chem. Proess Des. Develop. 11, 197, 6-35 [4] Lee S.Y. and Holder G.D.: Model for gas hydrate equilibria using a variable referene hemial potential: Part 1, AIChE J. 48, 00, [5] Klauda J.B. and Sandler S.I.: Phase behavior of lathrate hydrates: a model for single and multiple gas omponent hydrates, Chem. Eng. Si. 58, 003, 7-41 [6] Bandyopadhyay A.A. and Klauda J.B.: Gas hydrate struture and pressure preditions based on an updated fugaity-based model with the PSRK equation of state, Ind. Eng. Chem. Res. 50, 011, [7] Ballard A.L. and Sloan Jr. E.D.: The next generation of hydrate predition: An Overview, J. Supermole. Chem., 00, [8] Ballard A.L. and Sloan Jr. E.D.: The next generation of hydrate predition I. Hydrate standard states and inorporation of spetrosopy, Fluid Phase Equil , 00, [9] Yokozeki A.: Analytial equation of state for solid-liquid-vapor Phases, International Journal of Thermophysis 4, 003, [10] Yokozeki A.: Solid liquid vapor phases of water and water arbon dioxide mixtures using a simple analytial equation of state, Fluid Phase Equil. 3, [11] Yokozeki A.: Methane gas hydrates viewed through unified solid-liquid-vapor equations of state, International Journal of Thermophysis 6, 005, p.1

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