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Experimental Data Assessment Test for Composition of Vapor Phase in Equilibrium with Gas Hydrate and Liquid Water for Carbon Dioxide + Methane or Nitrogen + Water System Ali Eslamimanesh,† Saeedeh Babaee,‡ Amir H. Mohammadi,*,†,§ Jafar Javanmardi,‡ and Dominique Richon†,§ MINES ParisTech, CEP/TEP - Centre Énergétique et Procédés, 35 Rue Saint Honoré, 77305 Fontainebleau, France Department of Chemical Engineering, Shiraz University of Technology, 71555-313, Shiraz, Iran § Thermodynamics Research Unit, School of Chemical Engineering, University of KwaZulu-Natal, Howard College Campus, King George V Avenue, Durban 4041, South Africa † ‡

S Supporting Information *

ABSTRACT: Accurate knowledge of the compositions of the equilibrium phases in the systems containing gas hydrates is essential for many hydrate-based separation processes. Unfortunately, there are limited sets of such experimental data available in the literature partly due to the difficulties in measurements of the compositions of the phases in equilibrium with gas hydrate. Consequently, satisfactory accuracy of the measurements may not be obvious. Therefore, reliability of the corresponding data should be checked prior to their further applications. In this article, we present a thermodynamic assessment test (consistency test) based on the area test approach for the experimental compositional data of vapor phase in equilibrium with gas hydrate + liquid water for the carbon dioxide + methane or nitrogen + water system. The van der Waals and Platteeuw (vdW-P) solid solution theory is used to model the hydrate phase, and the Valderrama−Patel−Teja equation of state (VPT-EoS) along with the nondensity dependent (NDD) mixing rule is applied to deal with the fluid phases. The results show that only one of the studied experimental data sets seems to be thermodynamically consistent, and the rest of the data seem to be either not fully consistent or inconsistent.

1. INTRODUCTION Gas hydrate formation and dissociation is a reversible process in which pressurized gas and water combine to form a solid, called gas hydrate or clathrate hydrate.1 In gas hydrates, the gas molecules are trapped in water cavities that are composed of hydrogen-bonded water molecules.1 Considerable researches have been devoted in the last decades to examine potential industrial applications of gas hydrates.1−4 Examples are natural gas storage and transportation, carbon dioxide (CO2) capture from industrial and flue gases, CO2 sequestration, steam reforming processes, hydrogen (H2) storage, water desalination, refrigeration systems, food industry, and so forth.1−9 Moreover, carbon dioxide + methane or nitrogen + water systems are one of the major systems in CO2 capture processes.1−13 Thermodynamic models on the basis of accurate experimental equilibrium data are needed to reliably predict gas hydrate thermodynamic properties for potential industrial applications. As most of the existing models have been developed for hydrocarbon systems, model parameters must be reconsidered for clathrate hydrates containing carbon dioxide using reliable phase equilibrium data.1−6 In addition, any deviation in the measurement of hydrate phase equilibrium properties will lead to significant errors in predictions of the models. Consequently, measuring accurate experimental data on the phase behavior of mixed clathrate hydrates containing CO2 is of great significance. However, several error sources in experimental measurements, including calibration of pressure transducers, temperature probes, and detectors of gas © 2012 American Chemical Society

chromatographs, possible errors during the measurements of phase equilibria especially those dealing with gas hydrates, improper design of the equipment, and so forth, may result in the generation of erroneous experimental data or at least data with high uncertainty. To check and validate existing thermodynamic models or developing new ones with respect to accurate calculation or estimation of vapor phase composition of carbon dioxide in equilibrium with gas hydrate + water in gaseous systems containing carbon dioxide (e.g., mixture of carbon dioxide + methane or nitrogen + water), reliable experimental data sets are required. This work, which is among our efforts to assess the validity of experimental phase equilibrium data, aims at testing the thermodynamic consistency of the corresponding literature data using a theoretically correct method.

2. THERMODYNAMIC CONSISTENCY TEST The thermodynamically exact “Gibbs-Duhem equation”14−18 is generally applied to analyze thermodynamic consistency of experimental phase equilibrium data. As a matter of fact, if the values of the activity/fugacity coefficients of all of the components in the mixture do not satisfy this relation within an acceptable deviation, the experimental data are suspected to Received: Revised: Accepted: Published: 3819

October 27, 2011 January 12, 2012 January 17, 2012 January 17, 2012 dx.doi.org/10.1021/ie202465r | Ind. Eng. Chem. Res. 2012, 51, 3819−3825

Industrial & Engineering Chemistry Research

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where f is the fugacity, i refers to the ith component in the mixture, subscript w stands for water, and superscripts V, L, and H denote the vapor, liquid, and hydrate phases, respectively. The Valderrama modification of the Patel and Teja equation of state (VPT-EoS)32 with the nondensity-dependent (NDD) mixing rules33 is used to calculate the compressibility factor, fugacity coefficients, and the mole fractions of components in the liquid and vapor phases, and the solid solution theory of van der Waals−Platteeuw34 is applied to determine the fugacity of water in the hydrate phase. 2.2.a. Fluid Phase Model. The VPT-EoS32 is believed to be a strong tool for modeling systems containing water and polar compounds.33 This equation of state is written as follows:29−32

be thermodynamically inconsistent. This is mainly because of various probable errors during experimental works, especially those dealing with high pressure, low temperature, low concentrations of particular species in the mixtures, timeconsuming phase transitions, compositional gradients, hysteresis, and so forth. The consistency (or data assessment) tests generally include the following methods:14−23 the “Slope Test”, the “Integral Test”, the “Differential Test”, and the “TangentIntercept Test”. Good reviews of these methods can be found elsewhere.17 Valderrama and co-workers19−23 have already performed several thermodynamic consistency tests on experimental phase equilibrium data of various systems. Very recently, we applied almost the same method on important systems in the petroleum industry13,24−27 including water content of methane in equilibrium with gas hydrate, liquid water or ice,24 sulfur content of hydrogen sulfide vapor,25 solubility data of carbon dioxide and methane with water inside and outside gas hydrate formation region,13 solubility of waxy paraffins in natural gas systems,26 and diamondoids solubility in gaseous system.27 2.1. Methodology. The “Gibbs-Duhem”14−18 equation for a binary mixture at constant temperature can be written in terms of fugacity coefficients and compressibility factors of the vapor or gas phase as follows:13,19−28

∫

1 dp = py2

∫ (Z −11)φ

2

dφ2 +

∫

p=

RT a − v−b v(v + b) + c(v − b)

(5)

where R is the universal gas constant, T is temperature, v is molar volume, and a = a ̅ α(Tr)

(1 − y2 ) dφ y2 (Z − 1)φ1 1

(6)

a̅ =

ΩaR2Tc 2 Pc

(7)

b=

Ω bRTc pc

(8)

ΩcRTc pc

(9)

(1)

where Z is the compressibility factor of the gas mixture, p denotes the pressure, d is the derivative symbol, y stands for the mole fractions of particular species in the vapor or gas phase, and φ stands for the fugacity coefficient of the vapor or gas phase. In this expression, subscripts 1 and 2 refer to components 1 and 2 in the related phase, respectively. The properties φ1, φ2, and Z can be calculated using an appropriate thermodynamic model. In eq 1, the left-hand side can be designated by AP, and the right-hand side can be expressed as Aφ. If a set of data is supposed to be consistent, AP should be equal to Aφ within the acceptable defined deviation. To set the margins of error, a percent area deviation (ΔAi %) between experimental and calculated values is defined as:13,19−27 ⎡ Aφ − AP ⎤ i⎥ ΔA i % = 100⎢ i ⎢⎣ ⎥⎦ A Pi

c=

where the alpha function is given as α(Tr) = [1 + F(1 − TrΨ)]2

where Ψ = 0.5 and the coefficient F is given by F = 0.46286 + 3.58230(ωZc) + 8.19417(ωZc)2

where i refers to the data set number. The maximum values accepted for these deviations regarding the proposed systems are discussed later. 2.2. Thermodynamic Model. To evaluate the parameters for the consistency test, that is, Z, φ1, and φ2, a previously checked gas hydrate thermodynamic model29−31 can be applied along with appropriate numerical methods for flash calculations in order to reach convergence of the algorithms for the investigated systems at the conditions of interest. The general phase equilibrium criteria that is the equality of fugacities of each component throughout all phases is considered to model the phase behavior as follows:1,29−31 (3)

f wV = f wL = f wH

(4)

(11)

The subscripts c and r in the preceding equations denote the critical and reduced properties, respectively, and ω is the acentric factor. Besides, the coefficients Ωa, Ωb, and Ωc are calculated by

(2)

fi V = fi L

(10)

Ωa = 0.66121 − 0.76105Zc

(12)

Ω b = 0.02207 + 0.20868Zc

(13)

Ωc = 0.57765 − 1.87080Zc

(14)

where Zc is the critical compressibility factor. Avlonitis29 relaxed the constraints on F and Ψ for water in order to improve the predicted vapor pressure and saturated volume for these compounds: F = 0.72318,

Ψ = 0.52084

(15)

31

Later, Tohidi-Kalorazi relaxed the alpha function for water, αw(Tr), using experimental water vapor pressure data in the range of 258.15−374.15 K, in order to improve the predicted water fugacity: α w (Tr) = 2.4968 − 3.0661Tr + 2.7048Tr 2 − 1.2219Tr 3 (16) 3820

dx.doi.org/10.1021/ie202465r | Ind. Eng. Chem. Res. 2012, 51, 3819−3825


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The fugacity of water in the hydrate phase, f wH, is given by29−31,35

Nonpolar−nonpolar binary interactions in fluid mixtures are described by applying the classical mixing rules as follows: a=

∑ ∑ yi yj aij i

b=

j

∑ yi bi ∑ yi ci

(19)

i

aij = (1 − k ij) aiaj

(20)

Δμwβ− H = μwβ − μwH = RT ∑ vm ln(1 +

where kij is the standard binary interaction parameter. For polar−nonpolar interaction, however, the classical mixing rules are not satisfactory, and therefore more complicated mixing rules are necessary. In this work, the NDD mixing rules developed by Avlonitis29 are applied to describe mixing in the a-parameter, as mentioned earlier: a = aC + a A

m

api =

Cjm(T ) =

∑ yp2 ∑ yi apilpi i

(23)

0 l pi = l pi − l1pi(T − T0)

(24)

(25)

⎛ w(r ) ⎞ 2 exp⎜ − ⎟r d r ⎝ kT ⎠

(29)

where δN̅ =

⎡

⎤ RT ⎥ − dV − ln Z ⎢⎜ ⎟ V ⎥⎥ V ⎢⎝ ∂ni ⎠ T , V , n ⎦ ⎣ j≠i

∫

∞

(30)

The fugacity coefficient of each component in all fluid phases is derived straightforwardly from the following relation:13,29−31 1 ln φi = RT

∫0

⎡ (σ*)12 ⎛ α ⎞ (σ*)6 ⎛ α ⎞⎤ w(r ) = 2z ε⎢ 11 ⎜δ10 + δ11⎟ − 5 ⎜δ4 + δ5⎟⎥ ⎢⎣ R̅ r ⎝ R̅ ⎠ R̅ ⎠⎥⎦ R̅ r ⎝

where p is the index of polar components and l represents the binary interaction parameter for the asymmetric term. Using the above EoS32 and the associated mixing rules, the fugacity of each component in fluid phases is calculated from: fi = yi φip

4π kT

where k is the Boltzmann’s constant. The function w(r) is the spherically symmetric cell potential in the cavity, with r measured from the center, and depends on the intermolecular potential function chosen for describing the encaged gas−water interaction. In this work, the Kihara37 potential function is applied to evaluate the Langmuir constant as follows:1,29−31,38

(22)

apai

j

where vm is the number of cavities of type m per water molecule in the unit hydrate cell and f j is the fugacity of the hydrate former j. Cjm is the Langmuir constant, which accounts for the gas−water interaction in the cavity. Numerical values for the Langmuir constant can be calculated by choosing a model for the guest−host interaction:1,29−31,36

(21)

p

∑ Cjmf j ) (28)

where aC is given by the classical quadratic mixing rules (eqs 17 and 20). The term aA corrects for asymmetric interaction which cannot be efficiently accounted for by the classical mixing rules:29 aA =

(27)

where f wβ is the fugacity of water in the empty hydrate lattice. In eq 27, Δμwβ−H is the chemical potential difference of water between the empty hydrate lattice (μwβ ) and the hydrate phase (μwH) and is obtained from the van der Waals and Platteeuw expression:1,29−31,36

(18)

i

c=

⎛ β− H ⎞ β exp⎜ − Δμw ⎟ f wH = f w ⎜ ⎟ RT ⎝ ⎠

(17)

∞ ⎢⎛ ∂p ⎞

⎡ −N −N ⎤ 1 ⎢⎛⎜ r r α ⎞ ̅ ⎛⎜ α⎞ ̅ 1− − ⎟ − 1+ − ⎟ ⎥ ⎝ N̅ ⎢⎣⎝ R̅ R̅ ⎠ R̅ R̅ ⎠ ⎥⎦ (31)

In the two preceding equations, z is the coordination number of the cavity (the number of oxygen molecules at the periphery of each cavity), ε would be characteristic energy, α is the radius of spherical molecular core, R̅ stands for the cavity radius, and N̅ is an integer equal to 4, 5, 10, or 11. Also, σ* = σ − 2α, where σ is the collision diameter.29−31,38 The fugacity of water in the empty hydrate lattice, f wβ , is given by29−31,36

(26)

where V is the total volume and n is the number of moles. 2.2.b. Hydrate Phase Model. The van der Waals and Platteeuw statistical thermodynamic model,34 based on ideal solid solution theory, is used to model the gas hydrate phase equilibria, as mentioned earlier. The model, which is similar to the Langmuir gas adsorption theory, considers the guest molecule to move around in a spherical cavity constructed of water molecules. Each cavity contains one guest molecule, and there is no interaction between the encaged molecules.34 Furthermore, the presence of the guest molecule in the cavity does not distort the hydrate crystal lattice.34

⎛ β− I/L ⎞ β = f I/L exp⎜ Δμw ⎟ fw w ⎜ RT ⎟ ⎝ ⎠ 3821

(32)



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where f wI/L is the fugacity of pure ice or liquid water and the quantity inside the parentheses is given by the following equation:29−31,36 Δμwβ− I/L RT

=

μwβ (T , p) RT

−

Table 3. Binary Interaction Parameters between the Investigated Gases Using the VPT-EoS32 with NDD Mixing Rule33,42

μwI/L (T , p)

kij

RT

β− I/L T Δh w Δμw0 dT + = − RT0 T0 RT 2 β− I/L P Δvw dp RT P0

∫

∫

μwβ

(33)

where and are the chemical potential of the empty hydrate lattice and of pure water in the ice (I) or the liquid (L) state, respectively. p is the equilibrium pressure and T0 is the absolute temperature at the ice point. Δμw0 is the reference chemical potential difference between water in the empty hydrate lattice and pure water in the ice phase at 273.15 K.29−31,36 Δhwβ−I/L and Δvwβ−I/L are molar enthalpy and volume differences between an empty hydrate lattice and ice or liquid water. Δhwβ−I/L is given by the following equation:29−31,36 T

∫T

0

ΔCp w dT

where is the enthalpy difference between the empty hydrate lattice and ice, at the ice point and zero pressure. The heat capacity difference between the empty hydrate lattice and the pure liquid water phase is also temperature dependent, and the following equation is used:29−31,36

Furthermore, the heat capacity difference between hydrate structures and ice is set equal to zero.35 2.2.c. Model Parameters. Table 1 shows the physical properties of the compounds studied in this study. The binary

Zcc

ωd

water methane nitrogen carbon dioxide

22.055 4.599 3.394 7.382

647.13 190.56 126.10 304.19

0.2294 0.2862 0.2917 0.2744

0.3449 0.0115 0.0403 0.2276

a

Critical pressure. bCritical temperature. cCritical compressibility factor. dAcentric factor.

interaction parameters between the species of the investigated systems for the VPT-EoS32 with NDD mixing rule33 are reported in Tables 2 and 3. Moreover, the applied values of the Kihara37 potential function parameters are shown in Table 4. Table 2. Binary Interaction Parameters between the Investigated Gases and Water Using the VPT-EoS32 with NDD Mixing Rule33,42 H2O(i) gas

kij = kji

lij0

lij1

CH4(j) CO2(j) N2(j)

0.5028 0.1965 0.4792

1.8180 0.7232 2.6575

0.0049 0.0024 0.0064

compound

α,a Å

σ*,b Å

ε/k,c K

methane carbon dioxide nitrogen

0.3834 0.6805 0.3525

3.1650 2.9818 3.0124

154.54 168.77 125.15

(36)

where superscripts pred and exp refer to the predicted and experimental values, respectively. It is shown that that the majority of the ARD% values of the model results used in this work are less than 20% from the experimental data. Therefore, the model is generally acceptable for the data assessment test. If a set of data seems to be consistent, a percent area deviation (ΔAi%) between experimental and predicted values (defined by eq 2) must be within an acceptable range. For determination of the acceptable percentages of the two evaluated areas of deviations from each other, the error propagation can be performed on the existing experimental data. This is normally done using the general equation of error propagation,39 considering the temperature and mole fractions of carbon dioxide in vapor phase as the independent measured variables. The error in the calculated areas, EA, and the percent error, EA%, are calculated as follows:13,19−27

Table 1. Physical Properties of the Investigated Compounds42 Tc,b K

0.035 −0.036 0

|y pred − yiexp | ARD% = 100 i yiexp

(35)

pc,a MPa

N2(j)

0.092 0 −0.036

2.3. Consistency Criteria. The deviations of the thermodynamic model results (mole fractions in the vapor phase) should lie within a defined acceptable range. In this work, the accepted absolute relative deviations for the vapor phase mole fraction predictions (defined by the following equation) are considered to be between 0 and 20% according to capabilities of the thermodynamic model29−31 for this purpose:

(34)

compound

CO2(j)

0 0.092 0.035

a The radius of the spherical molecular core. bσ* = σ − 2α, where σ is the collision diameter. cε is the characteristic energy, and k is the Boltzmann’s constant.

Δhw0

ΔC pw = −38.12 + 0.141(T − T0)

CH4(j)

Table 4. Kihara37 Potential Parameters Used in the Thermodynamic Model29−31,41

μwI/L

β− I/L = Δh 0 + Δh w w

gas CH4(i) CO2(i) N2(i)

⎡ ∂Aφj ⎤ ⎡ ∂A φj ⎤ ⎥ΔT + ⎢ ⎥Δy EA = ⎢ ⎣ ∂T ⎦ ⎣ ∂y ⎦

(37)

⎡ E EA % = 100⎢ A ⎢⎣ Aφj

(38)

⎤ ⎥ ⎥⎦

where subscript j refers to the jth calculated area. We have assumed maximum uncertainties of ±0.02 K for the experimental temperature (the interval of the confidence for estimating the uncertainties has been considered to be 0.95; therefore, the expanded uncertainties (Uc) in the reported temperatures are ±0.02 K) and ±0.001 mol fraction for the 3822



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experimental compositional data of vapor phase (the interval of the confidence for estimating the uncertainties has been considered to be 0.95; therefore, the expanded uncertainties (Uc) in the reported mole fractions are ±0.001). However, these uncertainties depend on the method of experimental measurements. The maximum acceptable errors are much more dependent on the uncertainty of solubility measurements, and one can also neglect the first right-hand side term of eq 37. As a result of the fact that analytical derivatives are not so easy regarding the expression of the (VPT-EoS)32 with NDD mixing rules33 and the applied equations for hydrate phase, the partial derivatives of the two preceding equations have been evaluated using the central finite difference40 method. The subsequent results show that the ΔAi% value should be between 0 and about 20%. Therefore, the range [0,20]% has been established as the acceptable error range of calculated areas for probable thermodynamically consistent data. The thermodynamic consistency test criteria are applied through the following steps:13,24−27 1. Check that the percentage Δy2 is not outside of the margins of errors [0,20]%. If it is so, eliminate the weak predictions until the absolute relative deviations of the results from experimental values are within the acceptable range. 2. If the model correlates the data within the acceptable error ranges of the compositional data and the assessment test is fulfilled for all points in the data set, the proposed model is reliable and the data seem to be thermodynamically consistent. 3. In the case that the model correlates the data acceptably and the area test is not accomplished for most of the data set (about 75% of the areas), the applied model is reliable; however, the experimental data seem to be thermodynamically inconsistent. 4. In the case that the model acceptably correlates the data and some of the area deviations (equal to or less than 25% of the areas) are outside the error range [0,20]%, the applied method declares the experimental values as probable not fully consistent.

Table 5. Final Results of Thermodynamic Consistency Test on the Investigated Experimental Data set no. 1 3 4 5 6 7 8 9 10 11 14 15

TRa b

TI TI TI TI TI TI NFCd,e NFCd,e TI TI TCf TI

refc 3 43 43 43 43 44, 45 44, 45 44, 45 46 47 6 6

a

Test result. bTI: probable thermodynamically inconsistent data. Sources of experimental data. dNFC: probable not fully consistent data. eThe values of the integrals have been evaluated after elimination of about 25% of the data in the corresponding data set. fTC: probable thermodynamically consistent data. c

what should be considered regarding the future data on this system.13,24−27 It is worth it to point out that the three-phase compositional data of Belandria et al.6,43 are the first comprehensive data reported in the literature to deal with the compositions of vapor + hydrate + aqueous phases. As already mentioned, such data are indeed rare. In order to generate these kinds of data reported by Belandria et al.6,43 (the data have been produced at CEP/TEP laboratory), we faced some technical problems (refer to the original article for observing the designed experimental apparatus and the pursued measurements procedure). This may be one of the reasons that some of these data seem to be inconsistent. Furthermore, measurements of data (such as those presented in the work of Belandria et al.6,43) are not experimentally easy and can be subjected to nonnegligible uncertainties. However, we cannot say that we are completely sure about the validity of percentages (shares) of the uncertainties on area deviations, as they are modeldependent, to declare the data to be really inconsistent. Nevertheless, it is recommended to reduce the sources of the uncertainties by some precautions like more careful calibrations of the instruments, more precise measurements, more careful design of the apparatuses, and so forth. Second, not all of the uncertainties and errors in the measurements are originated from the calibrations or design of the apparatuses. These errors may come from the performance of the person(s) who measure(s) the data or the operational conditions of the laboratory that may not be constant during the measurements. That is why we were very interested to perform such consistency tests, which are generally ignored by the researchers who are mainly concerned in the modeling issues. Another factor to consider is that the data on which the thermodynamic consistency test applied herein should be reported as isotherms because the main assumption in development of the employed expression (eq 1) is similar to that assumed in developing the original “Gibbs-Duhem equation”14−18 at constant temperature.13,24−27 This fact assigns some limitations to choose the experimental data sets for the consistency test especially for scarce compositional data of the vapor phase in equilibrium with gas hydrates for mixtures of CO2 + methane or nitrogen + water. Consequently, selection

3. RESULTS AND DISCUSSION The investigated experimental ranges of pressures and molar fractions of carbon dioxide in the vapor phase at various temperatures available in the open literature are presented in the Supporting Information. It is indicated in this material that the applied thermodynamic model29−31 in this work results in generally reliable predictions of vapor phase compositions for the investigated equilibrium conditions (except some of the data sets). Table 5 reports the final results of the thermodynamic consistency test for molar compositions of CO2 in the vapor phase in equilibrium with gas hydrate and liquid water. As can be seen, only one of the treated data sets seems to be thermodynamically consistent. Additionally, the results of such a test introduce a procedure to select the experimental data by which a thermodynamic model can be reliably tuned and the optimal values of the model parameters can be obtained satisfactorily.13,24−27 The probable thermodynamically inconsistent data (sometimes the probable not fully consistent data) used for tuning of the models may bring about inaccurate predictions of the model in further applications, and the cause of such deviations may not be easily figured out.13,24−27 This is 3823



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for the CO2 + nitrogen or methane + water system is drastic, especially for CO2 capture and sequestration processes. In this work, a thermodynamic consistency test was applied on the related isothermal experimental data sets available in open literature. The VPT-EoS32 with NDD mixing rules33 was applied to model fluid phases while the hydrate phase was modeled using the van der Waals−Platteeuw34 solid solution theory. The consistency test was based on the area test approach derived from the original “Gibbs-Duhem equation”14−18 at constant temperature. The results show that except for one of the studied experimental phase equilibrium data, all of the data seem to be either not fully consistent or inconsistent. In addition, the results indicate that the measurements of such data must be done very accurately to be able to use them in tuning of future models for predictions of vapor phase compositions at different equilibrium conditions. It is also revealed that new experimental procedures or techniques can be developed to avoid obtaining high uncertainties in the generated experimental data, although it needs a comprehensive and detailed comparison between the existing experimental methods.

of a proper thermodynamic model for this purpose is not an easy task due to the fact that there is not enough data to absolutely validate the model performance. Therefore, we have herein applied a known thermodynamic model29−31 for prediction of phase equilibria of systems containing gas hydrates, which has been already checked over wide ranges of dissociation conditions of many gas hydrate systems or mutual solubility of various gases in the phases present,13,24−27 and also can, generally, represent the corresponding experimental data for the consistency test. One way of solving the problem of scarcity of data may be generating more data in a statistical form using statistical software.13,24−27 The generated data can be treated as pseudoexperimental.13,24−27 Though, this is doubtful and seems to be incorrect for the data in the hydrate formation region because there is possibility of structure change of the clathrate hydrate, and this would result in inaccurate generation of the pseudoexperimental data.13,24−27 Apart from that, it is not recommended to generate such data based on doubtful data, which have not been yet theoretically checked for consistency.13,24−27 In addition, the performed phase equilibrium data assessment test is inevitably model-dependent. If there is a way that we could measure directly the required parameters for the consistency test including the fugacity coefficients, we could have employed this method only based on experimental measurements (we are sure at the moment that it is impossible). This conclusion is valid even at low pressures, when we can assume with high confidence that the fugacity coefficients and compressibility factor are unity. Another conclusion is that the thermodynamic consistency tests may provide only rough information about the data quality. The user must be very careful in keeping or removing these treated data from the database merely depending on the consistency test results. We recommend that the user keep all the data defined as fully consistent and some of the data declared as not fully consistent with the help of his/her own skills and experiences on the related subject. In the final analysis, a significant point should not be omitted from our discussion. In this work, we have studied almost all of corresponding isothermal phase equilibrium data available in open literature. However, we are well aware that perhaps not all of the experimental data are fully trustable from an experimental point of view. This may be due to the inaccuracy of the used experimental techniques. Furthermore, calibration of the pressure transducers, temperature probes, and gas chromatograph detectors are significant factors in defining the uncertainties of the experimental data, as mentioned earlier. As a matter of fact, the results also suggest that new experimental techniques may lead to obtaining more reliable compositional data. However, we should be aware that PVT methods may have limitations. Nondestructive methods like RAMAN spectroscopy may contribute to more promising results. However, these kinds of techniques are expensive. In the present work, as we were interested in defining only the data quality, consequently, we have focused on consistency tests while our objective has not been a comparison between the different experimental methods. More meticulous investigations should be made on the validity of the applied experimental techniques in future works.

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ASSOCIATED CONTENT

* Supporting Information S

There are five Supporting Information files including the ranges of the studied experimental data in the first XLS file, the details of the model results and the required parameters for the consistency test for both of the investigated systems in the second, third, and fourth XLS files, and the detailed results of the performed consistency test in the fifth one. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: + (33) 1 64 69 49 70. Fax: + (33) 1 64 69 49 68. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support of the ANR (Agence Nationale de la Recherche) through SECOHYA project and OSEM (Orientation Stratégique des Ecoles des Mines) are gratefully acknowledged. A.E. wishes to thank MINES ParisTech for providing a Ph.D. scholarship. The authors are grateful to Prof. José O. Valderrama for the fruitful discussions on the issue.

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REFERENCES

(1) Sloan, E. D.; Koh, C. A. Clathrate Hydrates of Natural Gases, 3rd ed.; Taylor & Francis Group: Boca Raton, 2008. (2) Adisasmito, S.; Frank, R.; Sloan, D. Hydrates of carbon dioxide and methane mixtures. J. Chem. Eng. Data 1991, 36, 68−71. (3) Ohgaki, K.; Takano, K.; Sangawa, H.; Matsubara, T.; Nakano, S. Methane exploitation by carbon dioxide from gas hydrates - phase equilibria for CO2-CH4 mixed hydrate system. J. Chem. Eng. Jpn. 1996, 29, 478−483. (4) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D.; Naidoo, P.; Ramjugernath, D. Application of gas hydrate formation in separation processes: A review of experimental studies. J. Chem. Thermodyn. 2012, 46, 62−71. (5) Belandria, V.; Mohammadi, A. H.; Richon, D. Phase equilibria of clathrate hydrates of methane + carbon dioxide: New experimental data and predictions. Fluid Phase Equilib. 2010, 296, 60−65.

4. CONCLUSION Requirement of reliable experimental data of vapor compositions of CO2 in equilibrium with gas hydrates and liquid water 3824



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(28) Bertucco, A.; Barolo, M.; Elvassore, N. Thermodynamic consistency of vapor-liquid equilibrium data at high pressure. AIChE J. 1997, 43, 547−554. (29) Avlonitis D. Thermodynamics of gas hydrate equilibria. Ph.D. Thesis, Department of Petroleum Engineering, Heriot-Watt University, Edinburgh, U.K., 1992. (30) Mohammadi, A. H.; Anderson, R.; Tohidi, B. Carbon monoxide clathrate hydrates: equilibrium data and thermodynamic modeling. AIChE J. 2005, 51, 2825−2833. (31) Tohidi-Kalorazi, B. Gas hydrate equilibria in the presence of electrolyte solutions. Ph.D. Thesis, Department of Petroleum Engineering, Heriot-Watt University, Edinburgh, U.K., 1995. (32) Valderrama, J. O. A generalized Patel-Teja equation of state for polar and non-polar fluids and their mixtures. J. Chem. Eng. Jpn. 1990, 23, 87−91. (33) Avlonitis, D.; Danesh, A.; Todd, A. C. Prediction of VL and VLL equilibria of mixtures containing petroleum reservoir fluids and methanol with a cubic EoS. Fluid Phase Equilib. 1994, 94, 181−216. (34) van der Waals, J. H.; Platteeuw, J. C. Clathrate solutions. Adv. Chem. Phys. 1959, 2, 1−57. (35) Anderson, F. E.; Prausnitz, J. M. Inhibition of gas hydrates by methanol. AIChE J. 1986, 32, 1321−1333. (36) Holder, G. D.; Corbin, G.; Papadopoulos, K. D. Thermodynamic and molecular properties of gas hydrates from mixtures containing methane, argon and krypton. Ind. Eng. Chem. Fundam. 1980, 19, 282−286. (37) Kihara, T. On Isihara-Hayashida’s theory of the second virial coefficient for rigid convex molecules. J. Phys. Soc. Jpn. 1953, 8, 686− 687. (38) McKoy, V.; Sinanoglu, O. Theory of dissociation pressures of some gas hydrates. J. Chem. Phys. 1963, 38, 2946−2956. (39) Mickley, H. S.; Sherwood, T. K.; Reed, C. E. Applied Mathematics in Chemical Engineering; McGraw Hill: New York, 1957. (40) Constantinides, A.; Moustofi, N. Numerical Methods for Chemical Engineers with MATLAB Applications; Prentice Hall PTR: Prentice Hall, NJ, 1999. (41) Sloan, E. D., Jr. Clathrate Hydrates of Natural Gases, 2nd ed.; Taylor & Francis Group: Boca Raton, FL, 1998. (42) Danesh, A. PVT and Phase Behaviour of Petroleum Reservoir Fluids, 1st ed.; Elsevier: Amsterdam, Netherlands, 1998. (43) Belandria, V.; Eslamimanesh, A.; Mohammadi, A. H.; Théveneau, P.; Legendre, H.; Richon, D. Compositional analysis and hydrate dissociation conditions measurements for carbon dioxide + methane + water system. Ind. Eng. Chem. Res. 2011, 50, 5783−5794. (44) Seo, Y. T.; Kang, S. P.; Lee, H.; Lee, C. S.; Sung, W. M. Hydrate phase equilibria for gas mixtures containing carbon dioxide: A proof-of concept to carbon dioxide recovery from multicomponent gas stream. Korean J. Chem. Eng. 2000, 17, 659−667. (45) Kang, S. P.; Lee, H.; Lee, C. S.; Sung, W. M. Hydrate phase equilibria of the guest mixtures containing CO2, N2 and tetrahydrofuran. Fluid Phase Equilib. 2001, 185, 101−109. (46) Seo, Y. T.; Lee, H. Structure and guest distribution of the mixed carbon dioxide and nitrogen hydrates as revealed by X-ray diffraction and 13C NMR spectroscopy. J. Phys. Chem. B 2004, 108, 530−534. (47) Bruusgaard, H.; Beltran, J.; Servio, P. Vapor-liquid water-hydrate equilibrium data for the system N2 + CO2 + H2O. J. Chem. Eng. Data 2008, 53, 2594−2597.

(6) Belandria, V.; Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Gas hydrate formation in carbon dioxide + nitrogen + water system: Compositional analysis of equilibrium phases. Ind. Eng. Chem. Res. 2011, 50, 4722−4730. (7) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. An improved Clapeyron model for predicting liquid water−hydrate−liquid hydrate former phase equilibria. Chem. Eng. Sci. 2011, 66, 1759−1764. (8) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Thermodynamic model for predicting phase equilibria of simple clathrate hydrates of refrigerants. Chem. Eng. Sci. 2011, 66, 5439−5445. (9) Javanmardi, J.; Moshfeghian, M. Energy consumption and economic evaluation of water desalination by hydrate phenomenon. Appl. Therm. Eng. 2003, 23, 845−857. (10) Mohammadi, A. H.; Eslamimanesh, A.; Richon, D. Nitrous Oxide Clathrate Hydrates. In Advances in Chemistry Research; James, C., Ed.; Taylor, Nova Science Publishers, Inc: New York, 2011; Vol. 11. (11) Chapoy, A.; Mohammadi, A. H.; Chareton, A.; Tohidi, B.; Richon, D. Measurement and modeling of gas solubility and literature review of the properties for the carbon dioxide-water system. Ind. Eng. Chem. Res. 2004, 43, 1794−1802. (12) Mohammadi, A. H.; Richon, D. Development of predictive techniques for estimating liquid water-hydrate equilibrium of waterhydrocarbon system. J. Thermodynamics 2009, 1−12. (13) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Thermodynamic consistency test for experimental solubility data in carbon dioxide/methane + water system inside and outside gas hydrate formation region. J. Chem. Eng. Data 2011, 56, 1573−1586. (14) Prausnitz, J. M.; Lichtenthaler, R. N.; Gomez de Azevedo, E. Molecular Thermodynamics of Fluid Phase Equilibria; Prentice-Hall, Inc.: NJ, 1999. (15) Smith, J. M.; Van Ness, H. C.; Abbott, M. M. Introduction to Chemical Engineering Thermodynamics, 6th ed.; New York: McGrawHill, 2003. (16) Van Ness, H. C.; Abbott, M. M. Classical Thermodynamics of Non-electrolyte Solutions; McGraw-Hill: New York, 1982. (17) Raal, J. D.; Mühlbauer, A. L. Phase Equilibria: Measurement and Computation; Taylor & Francis: Washington, DC, 1998. (18) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (19) Valderrama, J. O.; Alvarez, V. H. A versatile thermodynamic consistency test for incomplete phase equilibrium data of highpressure gas−liquid mixtures. Fluid Phase Equilib. 2004, 226, 149−159. (20) Valderrama, J. O.; Robles, P. A. Thermodynamic consistency of high pressure ternary mixtures containing a compressed gas and solid solutes of different complexity. Fluid Phase Equilib. 2006, 242, 93−102. (21) Valderrama, J. O.; Zavaleta, J. Thermodynamic consistency test for high pressure gas−solid solubility data of binary mixtures using genetic algorithms. J. Supercrit. Fluids 2006, 39, 20−29. (22) Valderrama, J. O.; Reátegui, A.; Sanga, W. E. Thermodynamic consistency test of vapor-liquid equilibrium data for mixtures containing ionic liquids. Ind. Eng. Chem. Res. 2008, 47, 8416−8422. (23) Valderrama, J. O.; Faúndez, C. A. Thermodynamic consistency test of high pressure gas−liquid equilibrium data including both phases. Thermochim. Acta 2010, 499, 85−90. (24) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Thermodynamic consistency test for experimental data of water content of methane. AIChE J. 2011, 57, 2566−2573. (25) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Thermodynamic consistency test for experimental data of sulfur content of hydrogen sulfide. Ind. Eng. Chem. Res. 2011, 50, 3555−3563. (26) Mohammadi, A. H.; Eslamimanesh, A.; Richon, D. Wax solubility in gaseous system: thermodynamic consistency test of experimental data. Ind. Eng. Chem. Res. 2011, 50, 4731−4740. (27) Eslamimanesh, A.; Mohammadi, A. H.; Richon, D. Experimental data assessment test for diamondoids solubility in gaseous system. J. Chem. Eng. Data 2011, 56, 2655−2659. 3825


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