1

Phase Equilibria with Hydrate Formation in

H2O + CO2 Mixtures Modeled with Reference

Equations of State

Andreas JÄGER1, Václav VINŠ2,*, Johannes GERNERT1,

Roland SPAN1, Jan HRUBÝ2

1Thermodynamics, Ruhr-Universität Bochum, Universitätsstr. 150, 44780 Bochum, Germany 2 Institute of Thermomechanics AS CR, v. v. i., Dolejškova 1402/5, 182 00 Prague 8, Czech Republic

Abstract

Formation of gas hydrates is an important feature of the water–carbon dioxide

system. An accurate description of thermodynamic properties of this system

requires a consistent description of both fluid (liquid, vapor, and supercritical

fluid) and solid states (ice, dry ice, and hydrates) and of their respective phase

equilibria. In this study, we slightly modified and refitted the gas hydrate model

by A.L. Ballard and E.D. Sloan [Fluid Phase Equil. 194 (2002) 371-383] to

combine it with highly accurate equations of state (EoS) in form of the Helmholtz

energy and Gibbs energy for other phases formed in the water–carbon dioxide

system. The mixture model describing the fluid phases is based on the IAPWS-95

formulation for thermodynamic properties of water by W. Wagner and A. Pruß [J.

Phys. Chem. Ref. Data 31 (2002) 387-535] and on the reference EoS for CO2 by

R. Span and W. Wagner [J. Phys. Chem. Ref. Data 25 (1996) 1509-1596]. Both

pure-fluid equations are combined using newly developed mixing rules and an

excess function explicit in the Helmholtz energy. Pure-component solid phases

were modeled with the IAPWS formulation for water ice Ih by R. Feistel and W.

Wagner [J. Phys. Chem. Ref. Data 35 (2006) 1021-1047] and with the dry ice EoS

by A. Jäger and R. Span [J. Chem. Eng. Data 57 (2012) 590-597]. Alternatively,

the hydrate model was combined with the GERG-2004 EoS [O. Kunz, R.

* Corresponding author, email: vins.vaclav@seznam.cz, telephone: +420 266 053 152, fax: + 420

286 584 695

2

Klimeck, W. Wagner, M. Jaeschke, GERG Technical Monograph 15, VDI Verlag

GmbH, Düsseldorf, 2007]. Since the gas hydrate model uses the fugacity of the

gas component in the coexisting phase as an input variable, the accuracy of the

predicted phase equilibria was significantly improved by using highly accurate

EoSs for coexisting phases. The new hydrate model can be used in a temperature

range of 150 ÷ 295 K and at pressures up to 500 MPa. Together with the models

describing the fluid and pure solid phases it allows for the desired accurate and

consistent description of all phases and phase equilibria including, e.g., flash

calculations into two and three phase regions.

Keywords: carbon dioxide, gas hydrate, modeling, phase equilibrium,

reference equation of state

1. Introduction

The binary system of water and carbon dioxide is a mixture important both from a

scientific and from an engineering point of view. Many studies are being

published especially with relevance to environmental and geological processes,

natural gas industry, and carbon dioxide separation and storage (CCS)

technologies. The water–carbon dioxide (H2O+CO2) mixture shows complex

phase equilibria including liquid-liquid immiscibility, interrupted critical curves,

and formation of gas hydrates. The H2O+CO2 system can form a total of six

different phases: vapor (V), water-rich liquid (Lw), carbon dioxide-rich liquid

(Lc), water ice (Iw), dry ice (Ic), and gas hydrates (H). A qualitative description of

the H2O+CO2 phase equilibria was provided, for example, by Diamond [1] or

Longhi [2].

Gas hydrates are non-stoichiometric solid solutions of two or more

components forming a so called clathrate structure [3]. The clathrate is in general

a structure in which one of the components (host) forms cages enclosing

molecules of another component (guest). The crystalline host lattice is a

thermodynamically metastable phase which is stabilized by the presence of guest

molecules in its cavities. The clathrate becomes thermodynamically stable under

given temperature and pressure if a certain fraction of the cavities is occupied by

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the guest molecules. Gas hydrates usually form one of three crystal structures;

namely the cubic structure I (sI), the cubic structure II (sII), and the hexagonal

structure (sH). The form of the hydrate structure depends on the gas, i.e. guest,

contained in the hydrate cavities.

Gas hydrates were originally investigated due to the risk of pipeline

blockage during natural gas transport. However, increasing energy consumption,

decreasing reserves of fossil fuels, and climate change provide other important

challenges for which hydrates shall be further investigated; namely for Carbon

Capture and Storage (CCS) technologies and as a potential energy source in the

form of methane hydrates.

Most models that describe thermodynamic properties and phase equilibria

of gas hydrates are based on the approach by van der Waals and Platteeuw [4].

Using the Langmuir adsorption theory, van der Waals and Platteeuw (vdWP)

managed to evaluate the chemical-potential difference of water in the metastable

empty hydrate β-lattice and in the hydrate lattice stabilized by the presence of

guest molecules, H Hw w w

. Parrish and Prausnitz [5] modified the vdWP

model for practical calculations of the hydrate dissociation pressure both for a

pure guest and for guest mixtures. They used a reference hydrate for which a

chemical-potential difference could be determined from available experimental

data for the dissociation pressure at reference conditions (T0, p0). The chemical-

potential difference of the reference hydrate was then applied as a base for

equilibrium calculations for other types of hydrates. However, the hydrate models

by van der Waals and Platteeuw [4] and by Parrish and Prausnitz [5] neglected

some important properties of the gas hydrates, e.g., the lattice distortion caused by

different guest types and the influence of non-spherical guest molecules on the

guest-host interactions. Succeeding researchers such as Holder et al. [6,7], Klauda

et al. [8,9,10], and Sloan et al. [3,11,12] managed to improve the original vdWP

model and to overcome most of its limitations.

In the present study, we chose the hydrate model developed by Ballard

and Sloan [3,11,12] because of its sound physical background. The model of

Ballard and Sloan allows for a prediction of phase equilibria for various types of

gas hydrates forming all three common structures, i.e. sI, sII, and sH. This model

is also implemented in the CSMGem code [12] used for calculating hydrate

formation conditions in many practical applications.

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Our main goal was to prepare an accurate hydrate model based on Ballard

and Sloan’s [11] approach that can be applied to pure CO2 hydrates forming

structure sI. Unlike other hydrate models, the new CO2 hydrate model was used in

conjunction with the reference Helmholtz energy and Gibbs energy equations of

state (EoS) for the fluid phases and the pure solid phases present in the H2O+CO2

mixture. The fluid phases can be modeled with the reference EoSs, i.e. the

IAPWS-95 formulation [13] for water and the reference equation for carbon

dioxide [14], in combination with a newly developed mixture model relying on

adjusted mixing rules for the reducing parameters and an excess function explicit

in the Helmholtz energy – this combination is referred to as EOS-CG [15]

(“Equation of state for combustion gases, i.e. EOS-CG, and combustion gas like

mixtures” was the title of the E.ON IRI project under which the development of

the model by Gernert and Span was started). Alternatively, the GERG-2004 [16]

EoS developed primarily for the description of natural gases was used. This model

contains simplified but still accurate EoS for H2O and CO2 and a mixing model

relying only on adjusted mixing rules for the system H2O+CO2. Properties of

water ice Ih can be computed from the IAPWS formulation [17,18] and properties

of dry ice are represented by a new EoS by Jäger and Span [19].

2. Hydrate phase model

Unlike other hydrate models, e.g., by Holder et al. [6,7], working with the

chemical-potential difference of water in the hydrate empty β-lattice and in the

pure water phase (ice or liquid water), Lw or Iww w w , Ballard and Sloan

[11] directly expressed the chemical potential of water in the gas hydrate Hw in

the following manner

Hw w ,, , , ln 1 ,J i i J J

i J

T p f g T p RT v C T p f

. (1)

In equation (1), fJ is the fugacity of gas of kind J, wg denotes the Gibbs energy of

water in the metastable empty β-lattice, i.e. in a hypothetical phase, R is the

universal gas constant (8.314472 J·mol-1·K-1), vi denotes the number of cavities of

type i per one host (water) molecule, and Ci,J is the Langmuir constant. Since fJ is

calculated from the fluid property model, there is always a direct interaction

between the model used to describe the fluid phases and the hydrate model.

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Van der Waals and Platteeuw [4] derived an important relation for the

Langmuir’s isotherm describing localized adsorption without mutual interaction

of the adsorbed guest molecules.

,,

,1i K K

i Ki J J

J

C f

C f

(2)

In this relation, ,i K is the probability of finding a molecule of type K in a cavity

of type i. For the pure CO2 hydrate, i.e. for just one guest component, J equals

unity. One unit cell of the hydrate structure sI consists of two small and six large

cavities and it contains 46 water molecules in total. Consequently the number of

cavities per one water molecule, vi, equals 1/23 and 3/23 for small and large

cavities, respectively. According to Ballard and Sloan [11], the Gibbs energy of

water in the empty β-lattice can be expressed as follows

0 0

w w ww,0

20

, ,d d

pT

T p

g T p h T V T pgT p

RT RT RT RT

, (3)

where w,0g is the Gibbs energy at the reference conditions given by the

temperature T0 and the pressure p0. In our model, we set T0 = 273.15 K and

p0 = 2.0 MPa. In equation (3), wh and wV are, respectively, the enthalpy and the

molar volume of the empty β-lattice at the reference pressure p0. The enthalpy wh

can be determined from the isobaric heat capacity at the reference pressure as

0

w w,0 w dT

p

T

h T h c T T . (4)

The Gibbs energy and the enthalpy of the empty β-lattice at reference conditions,

w,0g and w,0h , are parameters adjusted to the experimental data for the hydrate

phase equilibria; more details are provided in section 4.1.

2.1 Heat capacity of the empty β-lattice

The isobaric heat capacity of the empty β-lattice, wpc , is usually assumed being

almost the same as the heat capacity of pure water ice Ih [11]. Since the hydrate

structure sI has a cubic crystal structure and not the hexagonal configuration as

water ice Ih, we considered also the heat capacity data for cubic ice measured by

Yamamuro et al. [20] in our study. Comparison of these data with the heat

capacity of pure water ice Ih predicted by the IAPWS formulation for water ice Ih

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[18] showed that both heat capacities differ by no more than 1% in the

temperature interval between 100 K and 164 K, i.e. in the relevant region for

which the cubic-ice data is available. The heat capacity of the empty β-lattice

could, therefore, be assumed to be the same as that for water ice Ih. The following

correlation for wpc was applied in our CO2 hydrate model

I -1 -2 -1 -1w w 0.12814 J mol K 2.74566 J mol K p pc c T . (5)

Equation (5) was fitted to data computed from the IAPWS EoS for water ice Ih

[18] at the reference pressure p0 = 2.0 MPa. Correlation (5) can be used in the

temperature range between 100 K and 300 K, which corresponds to the

temperature range relevant for CO2 hydrate formation.

2.2 Molar volume of the hydrate phase

The present model is intended to predict pure CO2 hydrate only. Therefore, the

CO2 hydrate could be taken as the reference hydrate for structure sI instead of

methane hydrate, which is usually considered in other models for this purpose [5].

In such a case, one does not have to consider lattice distortion caused by different

guest-types. Consequently, the lattice parameter of the empty CO2 hydrate β-

lattice is equal to the lattice parameter of the filled CO2 hydrate.

H2 2CO COa a (6)

Equation (6) represents a significant simplification of the hydrate model. The

lattice parameter Ha could be correlated to the experimental data for the CO2

hydrate measured mostly by X-ray diffraction. The following correlation for the

lattice parameter was considered in our model

2 3 1 0aH0 1 0a 2 0a 3 0a

2 0a

exp 11

p pa a T T T T T T

p p

. (7)

The reference lattice parameter a0 = 12.00223 Å corresponds to an averaged

experimental value measured by Huo et al. [21] at T0a = 275.4017 K and

p0a = 2.91 MPa. Thermal expansion coefficients α1 to α3 were determined by

fitting experimental data for the lattice parameter of the CO2 hydrate [21-25].

These coefficients have the following values: α1 = 1.16115084·10-4 K-1,

α2 = 2.56995816·10-7 K-2, α3 = 1.50098381·10-10 K-3. The thermal expansion given

by equation (7) increases with temperature. Fig. 1 shows the temperature variation

of the CO2 hydrate lattice parameter at low pressures, i.e. at 0ap p . The results

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of equation (7) are compared to the correlated experimental data by Ikeda et al.

[22,23], Udachin et al. [24], Huo et al. [21], and Hester et al. [25]. As can be seen,

the experimental data are available only up to a temperature of 276 K. Above this

point, the temperature dependence of the lattice parameter is extrapolated. This

fact may cause some deviations especially for the LwLcH three-phase equilibrium

at temperatures between 283 K and 294 K. Besides, equation (7) is compared with

other correlations for the temperature dependence of the lattice parameter in Fig.

1. The new correlation is in quite good agreement with the fit reported by Ikeda et

al. [23]. On the other hand, the correlation by Udachin et al. [24] lies somewhat

aside in the temperature range between 100 K and 200 K as their data are slightly

off the data sets of other authors. The correlation by Tse et al. [26], used in many

hydrate models, is significantly shifted to higher values of the lattice parameter as

it was derived from the data measured with ethylene oxide hydrate rather than

with CO2 hydrate. The difference between ethylene oxide hydrate and CO2

hydrate proves lattice distortion by different guest-types.

The pressure dependence of the lattice parameter in equation (7) differs

from the form proposed by Ballard and Sloan [11]. Ballard and Sloan considered a

constant value of the linear isothermal compressibility equal to 1.0·10-11 Pa-1 for

CO2 hydrate in their model. Correlation (7) uses a functional form for the

compressibility that is monotonically decreasing, reaching zero at infinite

pressure. The values of the coefficients κ1 and κ2, provided in section 4.1., were

evaluated from fitting CO2 hydrate formation data and from the high pressure X-

ray diffraction data by Hirai et al. [27]. The lattice parameter change (caused by

the pressure increase from 200 MPa to 400 MPa) predicted by equation (7) is in

good agreement with experimental values by Hirai et al. [27], measured at

temperatures between 230 K and 278 K. Results of equation (7) were also

compared with other models for the isothermal compressibility of gas hydrates,

e.g., with the potential model for methane hydrate by Docherty et al. [28]. At

lower pressures, equation (7) gives a larger compressibility of CO2 hydrate than

the model by Docherty et al. [28], while at pressures above 230 MPa the CO2

hydrate compressibility becomes much smaller than that for methane hydrate [28].

This result is in agreement with the original model by Ballard and Sloan [12]

(CSMGem) which uses an even ten times smaller compressibility for CO2 hydrate

than for methane hydrate at high pressures.

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Molar volumes of the empty hydrate β-lattice and the stable hydrate can

be determined from equations (8) and (9), respectively, where Nw = 46 for hydrate

structure sI.

3

w AV wV a N N (8)

3

AVH

, w1 i i Ji J

a NV

v N

(9)

2.3 Cell potential function and Langmuir constant

The Langmuir constant ,i JC is an important quantity characterizing the cavity

occupancy by the guest (gas) molecules as described by equation (2) and,

consequently, the molar composition of the hydrate. Using the Lennard-Jones and

Devonshire cell theory, van der Waals and Platteeuw [4] calculated the Langmuir

constant as follows:

2,

B B0

4exp di J

w rC T r r

k T k T

. (10)

In equation (10), kB denotes the Boltzmann constant (1.3806488·10-23J·K-1), r is

the radial coordinate, and w(r) is the cell potential over an approximately

spherically symmetric cavity. Equation (10) corresponds to a simplified case with

the water molecules “smeared” over the cavity wall. In the original hydrate model

by van der Waals and Platteeuw [4], a contribution to the potential energy due to

the interaction of a guest molecule with an element of the cavity wall was

determined with the Lennard-Jones potential. However, most of the succeeding

hydrate models [5,7,8,11,29,30] use the Kihara potential with a spherical hard

core rather than the Lennard-Jones potential. The Kihara potential defining the

interaction energy φ between a guest molecule and a water molecule in the cavity

wall is given in the following way

hc

12 6

hchc hc

, 2

4 , 22 2

r r a

r r ar a r a

. (11)

The distance parameter σ defines zero potential energy, the energy parameter ε

corresponds to the strongest attraction between the guest molecule and the cage

element, and ahc is the hard core radius. Values of the Kihara potential parameters

9

σ, ε, and ahc are usually determined by two different approaches: one can either

use second virial coefficient and gas viscosity data for a pure substance, or

correlate the Kihara parameters to hydrate formation data. Unfortunately, the two

methods give significantly different values, particularly for the energy parameter

ε. For example, Martín and Peters [30] considered the Kihara parameters derived

from second virial coefficient and gas viscosity data. Their value for the energy

parameter ε was determined from a standard Lorentz-Berthelot mixing rule, i.e.

B 2 2 B/ CO H O / 219 Kk k , where ε/kB (CO2) = 469.7 K and

ε/kB (H2O) = 102.134 K. On the other hand, Parrish and Prausnitz [5] correlated

the Kihara parameters to hydrate formation data. Their value for the energy

parameter describing interactions between the CO2 molecule and the water cage is

ε/kB = 169.09 K. We consider the Parrish and Prausnitz [5] method more

convenient for gas hydrates modeling since the water molecules are considered

being “smeared” over the cavity wall in the vdWP model [4] – see equation (10).

We therefore correlated the Kihara potential parameters to the hydrate formation

data in a similar way as, e.g., Parrish and Prausnitz [5], Ballard and Sloan [11], or

Yoon et al. [29].

The cell potential w(r), required in equation (10), is defined as a sum of

all gas–water interactions in one cell (shell). According to McKoy and Sinanoglu

[31], w(r) can be determined for the Kihara potential in the following way

12 6

10 11 4 5hc hc11 5

2a a

w r zR r R R r R

, (12)

where R is the shell radius, z denotes the coordination number, i.e. the number of

water molecules in the shell, and δ is given by equation (13), with N = 4, 5, 10,

and 11.

hc hc11 1

N N

N a ar r

N R R R R

(13)

Most of the hydrate models [7,8,30] based on the vdWP theory evaluate the

Langmuir constant given by equation (10) by considering three water-shells

proposed by John and Holder [32]. John and Holder detected that the second and

the third layer of water molecules have a significant influence on the interaction of

gas molecule with the water cage. However, contributions of the second and the

third water-shell are much smaller than that of the first shell.

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Ballard and Sloan [11] suggested another way of water-shell definition

called a multi-layered shell. They considered only the first water-shell, however

split into several layers. Single crystal X-ray diffraction data were used to define

separate layers both for the small and the large cavities of the hydrate structures sI

and sII. Table 1 summarizes values for the radius Ri and the coordination number

zi for the water-shells considered both in John and Holder’s [32] and Ballard and

Sloan’s [11] approach. Besides the three water-shells (JH) and the multi-layered

shell (BS), we tested also an additional water-shell definition in this study. The

multi-layered first shell defined by Ballard and Sloan [11] was combined

(COMB) with the second and the third shells given by John and Holder [32]. The

Langmuir constant predicted by all three approaches was also compared with a

simple temperature correlation, independent of pressure, for ,i JC developed by

Klauda and Sandler [9] (KS). A comparison of all four models for the Langmuir

constant calculated at a constant pressure of 3.0 MPa is shown in Fig. 2. The

Kihara potential parameters σ and ε correlated to the CO2 hydrate formation data

have following values: three water-shells (JH) model σ = 2.927 Å, ε/kB = 153.6 K,

multi-layered shell (BS) model σ = 2.946 Å, ε/kB = 175.4 K, and combined shell

(COMB) model σ = 2.961 Å, ε/kB = 155.4 K. The hard core radius has a constant

value of 0.6805 Å in all cases. As can be seen in Fig. 2, the Langmuir constant

predicted for both the small and the large cavity has a comparable temperature

dependency for all models. Only the temperature correlation by Klauda and

Sandler [9] provides lower values of Ci,J for the small cavity. This correlation

represents a simplified, pressure independent case. Therefore, it is not

recommended for precise hydrate equilibria modeling.

With the proper set of Kihara potential parameters, all four approaches for

the Langmuir constant calculation can predict the hydrate formation conditions

with approximately the same accuracy. Nevertheless, compared to other models

the KS model does not reproduce the high pressure equilibrium LwLcH well as it

does not take into account the pressure dependence. The cage occupancy i

modeled by BS, JH, and COMB approaches does not differ by more than 3.5%

from each other. On the other hand, KS overestimates the hydration number, i.e.

the number of water molecules per one gas molecule in the hydrate unit cell, since

the small cavities are two times less occupied than in case of other models.

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The pressure dependency of the Langmuir constant can be defined by

varying the shell radius Ri. Ballard and Sloan [11] suggested making the shell

radius proportional to the lattice parameter change, i.e.

H

,00

,,i i

a T pR T p R

a . (14)

The tabulated values for the shell radius ,0iR given in Table 1 correspond to a

reference lattice parameter of a0 = 12.03 Å. According to equation (14), the

pressure and temperature dependent shell radius can be expressed from the lattice

parameter aH determined from equation (7). Consequently, the Langmuir constant,

originally given by equation (10), becomes dependent on both temperature and

pressure in this case.

1 ,, , 2

,B B0

, ,4, exp d

JR T p an i Jn

i J

w r T pC T p r r

k T k T

(15)

In equation (15), n stands for n-th water-shell or n-th water-shell layer depending

on the water-shell definition employed.

Although most of the investigated definitions of water-shells, combined

with the appropriate set of Kihara potential parameters, provided comparable

results for the Langmuir constant, we decided to use the multi-layered water shell

by Ballard and Sloan [11] in the CO2 hydrate model. The evaluation of the Kihara

potential parameters is discussed in section 4.1 in detail.

3. Equations of state used for other phases

As far as the authors know, none of the existing hydrate models has been used in

combination with reference equations of state for the fluid phases. For example,

Klauda and Sandler [8] used a modified UNIFAC model, Bandyopadhyay and

Klauda [10] and Yoon et al. [29] used the predictive Soave-Redlich-Kwong

(PSRK) EoS, Martín and Peters [30] used a cubic-plus-association (CPA) EoS and

Ballard and Sloan [11] used a combination of fugacity / activity models for the

fluid phases. A more detailed description of the EoSs used by Ballard and Sloan

[11] is provided by Jager et al. [33]. Jäger and Span [19] showed that the EoS

used for the fluid phase(s) has a significant impact on calculated solid-fluid phase

equilibria, even if different accurate multiparameter EoSs were considered. This is

particularly true when modeling hydrate formation, since the fluid phase fugacity

12

of the guest is an input to the hydrate model and is thus directly affecting the

model.

In this work, highly accurate EoSs explicit in the reduced Helmholtz

energy [13-16] and in the reduced Gibbs energy [17-19] were used in combination

with the previously described modified model for CO2 hydrates. The EoSs were

evaluated using a software package developed at the Thermodynamics institute at

Ruhr-University Bochum. The phase equilibrium algorithm implemented in that

package is a modified form of the algorithm of Michelson (see [34], [35]) that is

to be published. This algorithm was extended such that also equilibria with solid

phases and equilibria of three or four phases can be predicted. A necessary

requirement for the calculation of caloric properties of phase equilibria is that the

equations used for the different phases have the same or consistent reference

points. In this study, two different sets of reference points have been used: Either

the reference points of the GERG-2004 EoS (as it is implemented in the GERG

software; see [16]), or the standard reference points of the commercial software

REFPROP 9.0 [36]. In the following section, all equations of state used are briefly

described with their respective reference points.

3.1 IAPWS-95, reference equation of state for H2O [13]

The reference state for the IAPWS-95 EoS was chosen such that the internal

energy becomes u = 0 J·mol-1 and the entropy becomes s = 0 J·mol-1·K-1 for the

saturated liquid at the triple point (Ttr = 273.16 K; ptr,cal = 0.000611654753216

MPa). This corresponds to setting the enthalpy and entropy at 300 K and 1 kPa to

the values given in Table 2.

3.2 Reference equation of state for CO2 [14]

The reference point of the CO2 EoS was set for the saturated liquid at

T = 273.15 K. The reference point is given in Table 3.

3.3 Mixing rules for the reference equations of state for CO2 and H2O

In the GERG EoS, an extended corresponding states model for mixtures is used to

describe the system CO2+H2O. The standard mixing rules implemented

correspond to Lorentz-Berthelot mixing rules. With focus on natural gases and

other mixtures, the general mixing rules have been modified by Kunz et al. [16].

13

For normal mixture behavior and / or mixtures with a limited number of available

experimental data, up to four parameters might be fitted. For mixtures with

complex behavior and / or extensive sets of available experimental data, an

additional departure function can be fitted. In this way the description of mixtures

can be improved beyond the level of accuracy common for extended

corresponding states approaches. On the basis of these modified mixing rules,

Gernert and Span [15] developed new mixing rules for a number of binary

systems relevant for CCS applications, including the system CO2+H2O. These

mixing rules have been used in this work in order to describe the mixture of CO2

and H2O in combination with the reference EoSs for the pure fluids.

This set of EoS, reference states, and mixing rules will further be referred

to as “EOS-CG”.

3.4 GERG equations of state and mixing rules [16]

When using the GERG-2004 equations in REFPROP 9.0 [36], the reference

points are set in the same way as for the reference EoSs. That means that the

reference point for the H2O equation used in GERG is equal to the reference point

of the IAPWS-95 EoS given in Table 2. Note that since the GERG water equation

differs from the reference equation, setting the reference point according to Table

2 does not result in u = 0 J·mol-1 and s = 0 J·mol-1·K-1 for the saturated liquid at

the triple point. When using REFPROP 9.0, the reference point of the CO2

equation used in the GERG is also the same as the one given in Table 3. However,

the numerical values are slightly different. This is due to a different vapor

pressure and saturated liquid density calculated by the GERG CO2 equation and a

different molar mass used for the CO2 equation of the GERG-2004. The reference

point is provided in Table 4. This setup will further be referred to as “GERG”.

The GERG software by Kunz et al. [16] uses different reference states for

the EoSs than REFPROP 9.0 [36]. In this software, the reference state of every

pure substance is defined at T = 298.15 K and p = 0.101325 MPa. For this

temperature and pressure, the reference density is calculated by the ideal gas law.

Finally, the entropy and enthalpy for this reference ideal-gas state are set to zero.

The integration constants corresponding to the reference point of each fluid in the

GERG EoS are provided in the work of Kunz et al. [16] (page 471 – 473, table

A3.1). We note that for the system CO2+H2O GERG-2004 and GERG-2008 [37]

14

are identical. The original GERG setup is referred to as “ORGE” in the following

text.

3.5 Equation of state for ice Ih

Feistel and Wagner [18] developed an equation for the common hexagonal water

ice Ih, which is explicit in the Gibbs energy. They demonstrated that sublimation

and melting equilibria of pure H2O might be predicted very accurately using this

equation for ice Ih in combination with the reference equation for fluid water. In

order to predict these phase equilibria, the reference state of the ice equation has

to be linked to the reference state of the equation of the fluid phase. This can be

realized using the following two conditions given, e.g., by Tillner-Roth [38]

S V Ltr tr tr tr tr tr( , ) ( , ) ( , )g T p g T p g T p , (16)

fus

S L tr trtr tr tr tr

tr

( , )( , ) ( , )

h T ps T p s T p

T

. (17)

Equation (16) defines the phase equilibrium condition at the triple point. Using

this condition together with the enthalpy of fusion fush at the triple point, one

can set the entropy of the solid phase to match equation (17). Feistel and Wagner

provide information on how their integration constants have to be set in order to

connect the ice equation to the IAPWS-95 EoS. In this work, we additionally

calculated the value of the constants when connecting the ice equation to the

GERG water EoS using the reference point of the GERG-software (ORGE) and

the reference point used in REFPROP 9.0 (GERG). Values for the ice equation

constants g00 and s0 defined in the original ice Ih paper [18] are provided in Table

5 for all fluid EoSs considered in this study.

3.6 Equation of state for dry ice

An equation for solid carbon dioxide (dry ice Ic) explicit in the Gibbs energy was

recently published by Jäger and Span [19]. The authors demonstrated that this

equation in combination with the reference EoS for CO2 [14] is capable of

accurately predicting the melting and sublimation phase equilibria of pure CO2.

This equation might also be connected to arbitrary equations for fluid CO2 using

the conditions given as equations (16) and (17). Values for the integration

constants g0 and g1 that connect the dry ice equation to the respective pure fluid

equation used in this work are given in Table 6. Trusler [39] recently published an

15

EoS for dry ice explicit in the Gibbs energy. The representation of the

experimental data by this equation is comparable to the results of Jäger and Span

[19]. However, the EoS of Trusler is valid over a larger temperature and pressure

range and could be interesting for studies at even higher pressures.

4. Phase equilibria with pure CO2 hydrate

An overview of the various three-phase equilibria investigated in the present study

is given in Fig. 3. The three-phase lines and quadruple points presented in this

figure have been calculated using EOS-CG or GERG as indicated. The

experimental data available is also displayed. Note that the three-phase lines

VLwLc and VLwIw are three-phase lines of the system CO2+H2O without a

hydrate phase present. These equilibria have also been predicted by using EOS-

CG or GERG. The quality of the predictions for the three-phase lines containing

hydrate when using EOS-CG or GERG will be discussed in detail in section 5.

Basically three different sets of equations had to be solved in order to

calculate the three-phase equilibria presented. For equilibrium lines of two fluid

phases in combination with a hydrate phase (VLwH, VLcH, and LwLcH) the set

of equations is as follows

FP1 FP1 FP2 FP2c c

FP1 FP1 FP2 FP2w w

FP1 FP1 H FP1 FP1w w c

( , , ) ( , , ) 0

( , , ) ( , , ) 0

( , , ) ( , , ( , , )) 0

T p x T p x

T p x T p x

T p x T p f T p x

. (18)

In equation (18), stands for the chemical potential of the indicated phase and

component, the subscript c stands for CO2, and the subscript w for water. The

superscripts indicate the phases (FP1 means the first fluid phase, FP2 the second

fluid phase and H stands for the gas hydrate. As the two fluid phases might also

be liquids, the fluid phases are just indicated as 1 and 2 rather than V and L.). f

denotes the fugacity of the indicated phase and component. Solving this system of

equations at given pressure, which was our main way of calculating the phase

equilibria, the unknowns are the temperature and the compositions of the fluid

phases. Using the closing condition that the sum over all compositions equals 1, it

follows that for each phase the three independent variables that need to be

calculated are the temperature T, the mole fraction of CO2 in fluid phase 1 (FP1)

FP1cx and the mole fraction of CO2 in fluid phase 2 (FP2) FP2

cx .

16

For equilibria containing solid H2O or CO2, the assumption was made that

these phases are formed by pure components. Thus, the composition of these

phases is known and the set of equations can be reduced. For three-phase

equilibria of hydrate, solid H2O and a fluid phase (VHIw) the set of equations to

be solved is as follows

FP1 FP1 Iww w

FP1 FP1 H FP1 FP1w w c

( , , ) ( , ) 0

( , , ) ( , , ( , , )) 0

T p x T p

T p x T p f T p x

. (19)

Again at given pressure the unknowns are the temperature T and the mole fraction

of CO2 in fluid phase 1 (FP1), FP1cx . For equilibria of hydrate, solid CO2 and a

fluid phase (VHIc, LcHIc, and LwHIc) the set of equations is

FP1 FP1 Icc c

FP1 FP1 H FP1 FP1w w c

( , , ) ( , ) 0

( , , ) ( , , ( , , )) 0

T p x T p

T p x T p f T p x

. (20)

With given pressure, the unknowns of this set of equations are again the

temperature T and the mole fraction of CO2 in fluid phase 1 (FP1). All sets of

equations presented can also be solved at given T for unknown p.

In case of four-phase equilibria (quadruple points) the unknowns are T, p,

and the compositions of all fluid phases involved. Each of the quadruple points

presented in this work can be calculated using a combination of equations (18) to

(20).

4.1 Fitting of the hydrate model

There are several parameters in the CO2 hydrate model that have to be correlated

to hydrate formation data. In the present study, we tried to collect all experimental

data available for the three-phase equilibria and for the quadruple points of the

CO2 hydrate. A brief summary of the collected data is given in section 4.2.

The fluid EoSs together with the pure solid EoSs described above were

used to calculate the two-phase equilibria of the phases being in equilibrium with

the hydrate phase at the three-phase equilibria lines. The chemical potential of

water, w , and the fugacity of CO2, fc, were calculated from the two-phase

equilibria at the temperatures and pressures corresponding to the experimental

data for the hydrate three-phase equilibria, i.e. for VHIw, VLwH, VLcH, and

LwLcH equilibria using the respective fluid models. The calculated sets of

temperatures, pressures, fugacities of CO2, and chemical potentials of water at a

17

given three-phase equilibrium were used to determine the unknown parameters of

the hydrate phase model. The chemical potential of water in the hydrate phase Hw

given by equation (1) was correlated to the chemical potential of water in other

phases w . The Levenberg-Marquardt optimization method was employed to

correlate the CO2 hydrate model to the hydrate formation conditions. In the

Levenberg-Marquardt method, the following objective function was minimized

exp 2H

w, w,i1

MINn

ii

with α = V, Lw, or Iw, (21)

with nexp = 314. The hydrate model was correlated to the VHIw, VLwH, and

LwLcH three-phase equilibrium data. The data for VLcH did not have to be

considered in the optimization procedure as this three-phase equilibrium was

predicted with a very good accuracy by all model configurations investigated in

this study. The Levenberg-Marquardt optimization method could be used to fit up

to ten parameters at once. However, only two to five parameters had to be

optimized in our case.

Table 7 shows final values for the Kihara potential parameters and

coefficients for the isothermal compressibility used in the CO2 hydrate model.

Values for the energy parameter ε/kB and the hard core radius ahc were taken from

the work of Ballard and Sloan [11] as their multi-layered definition of the water-

shell, marked as BS, was employed in our model. The distance parameter σ is

slightly different from the value provided by Ballard and Sloan [11] (2.97638 Å).

The Kihara potential parameters and the isothermal compressibility are properties

of the hydrate phase. Values of these quantities shall therefore be independent of

the EoS-types used for modeling the phases in equilibrium with the hydrate phase.

A single set of values for the Kihara parameters and the isothermal

compressibility, provided in Table 7, was used for all types of EoSs.

As already mentioned, various accurate EoSs, namely EOS-CG, GERG,

and ORGE, were used for modeling other phases than hydrate in this study.

Unfortunately, different sets of EoSs usually differ from each other in the

reference state definition. Consequently, the hydrate model, respectively its

reference conditions, has to be individually correlated to each set of EoSs used.

The reference Gibbs energy w,0g and the reference enthalpy w,0h of the empty β-

lattice were fitted for all EoSs considered using the Levenberg-Marquardt

18

optimization method. Values for w,0g and w,0h are given in Table 8. The values

for EOS-CG and GERG are similar since these two sets of EoSs use similar

definitions of the reference point.

4.2 Experimental data for CO2 hydrates

A fairly large number of data sets exists for CO2 hydrates. The three-phase

equilibrium lines VLwH, VLcH, VHIw, and LwLcH have been measured by

several authors [40-70]. An overview on the data sets used for fitting is provided

in Table 9. If the authors gave information on measurement uncertainties in

temperature and pressure or on the quality (purity) of the CO2 and H2O used, the

respective information is also provided in Table 9. The given uncertainties are

mostly uncertainties of the sensors and thus not equal to the whole measurement

uncertainty. The given AAD values indicate that the data sets mostly agree among

each other well. Unfortunately, the only two data sets for the high pressure

LwLcH equilibrium, where the temperature maximum of the hydrate formation is

reached, do not agree. Manakov et al. [42] measured up to 10 K higher hydrate

formation temperatures than Nakano et al. [44]. These data sets will be discussed

in more detail in section 5.

5. Results and discussion

Results for the different three-phase equilibrium lines as seen in Fig. 3 will be

discussed in detail in the following. The respective equilibria have been calculated

using the three different setups (EOS-CG, GERG, and ORGE); for comparisons

the results of the CSMGem code [12] are also plotted. As already indicated, the

present model is only capable of predicting pure CO2 hydrates to this point,

whereas the CSMGem code can predict various hydrates and also hydrate

mixtures. When the GERG EoS is used, the only relevant difference between the

two setups GERG and ORGE is the reference point. Thus, the results for GERG

and ORGE are practically identical (using the respective reference point of the

hydrate model); therefore only the results for GERG are shown in figures.

However, due to the fact that a fitting routine is used in order to determine the

reference point related constants of the hydrate model, GERG and ORGE provide

slightly different numerical values.

19

The different three-phase equilibrium lines have been calculated between

the quadruple points. The quadruple points VLwHIw, VLcLwH, VLcHIc, and

LwLcHIc have been calculated. The results for GERG and EOS-CG are

summarized in Table 10. The experimental data available for quadruple points is

given in Table 10. A discussion of the various quadruple points possible for the

mixture CO2+H2O can be found in [42]. The results of EOS-CG for the quadruple

points agree very well with the literature data. Also the GERG setup yields good

results. However, the lower quadruple point VLwHIw predicted by GERG is

located at a pressure about 20% and a temperature about 2 K higher than the

prediction using the EOS-CG setup, which again agrees very well with literature

data. A closer investigation showed that the melting and sublimation equilibria of

pure water might be predicted fairly well using the GERG water EoS and the

water ice EoS by Feistel and Wagner [18]. To further investigate the difference in

the prediction of the quadruple point VLwHIw, various combinations of the

mixing rules and pure-fluid EoSs used have been tested. These investigations have

shown that the prediction of this quadruple point strongly depends on the mixing

rules, rather than on the equations used for the pure substances. If, e.g., the GERG

equations for H2O and CO2 are used in combination with the mixing rules by

Gernert and Span [15], the predicted quadruple point (VLwHIw) is almost the

same as the one calculated with the EOS-CG setup. On the other hand, if the

reference equations for H2O and CO2 are used in combination with the GERG

mixing rules, the quadruple point will be predicted almost the same as with the

GERG setup. Further investigating this fact we used the standard Lorentz-

Berthelot mixing rules in combination with the reference and GERG pure

substance EoSs. Regardless of the pure fluid equations used, the quadruple point

predicted is at approximately 272 K and 1.1 MPa. Thus, the conclusion is that the

prediction of the hydrate equilibria depends much more on the mixing rules than

on the equations for the pure fluids. For all combinations of mixing rules and pure

fluid EoSs the hydrate model had to be refitted to the experimental data.

The average absolute deviation (AAD) according to

,exp ,cal1

1AAD=

n

i ii

T Tn

(22)

for all three-phase equilibria calculated is given in Table 11. Besides the results

for GERG and EOS-CG with the water-shells defined by Ballard and Sloan [11],

20

i.e. BS, other water-shell definitions have been used for fitting the hydrate model

(JH, KS, and COMB – see section 2.3). The resulting AAD when the hydrate

model is used with different water-shells in combination with EOS-CG is also

given in Table 11 for comparison. As can be seen, the results when using the

water-shell definition of BS, JH, or COMB are comparable with regard to the total

AAD of all data fitted. In case of JH and COMB the prediction of the three-phase

line VLwH is slightly better than for BS. However, better representation of VLwH

using the models presented always resulted in a worse description of either VHIw

or LwLcH equilibria. This effect can also be seen when using BS, JH, or COMB.

BS water-shell definition was used in the end, since the results for VHIw are

better in this case compared to JH and COMB. For LwLcH, VLcH, and the

overall representation of the data, almost no differences between BS, JH, and

COMB are noticeable. Using the temperature correlation for the Langmuir

constant by Klauda and Sandler (KS), the overall results are significantly worse.

Due to the quadruple point problem already discussed, GERG is particularly not

capable of representing the VHIw equilibrium properly. These relatively bad

results of the GERG model had to be expected due to two reasons:

1) GERG with its pure component EoSs and mixing rules has been

developed to very accurately predict typical natural gas mixtures. The precise

description of H2O+CO2 mixtures lies beyond its original scope. For instance, the

CO2 solubility in liquid water is strongly underestimated by GERG, whereas

correct values are obtained with EOS-CG [15].

2) Only the reference conditions w,0g and w,0h were correlated for the

GERG and ORGE setups. Values for the Kihara potential parameters and the

compressibility constants were considered being the same as for EOS-CG.

However, fitting all parameters of the hydrate model to GERG did not result in a

much better representation of the data. LwLcH might be improved, but only with

VHIw getting worse and vice versa. The quadruple point VLwHIw is always

predicted at the higher pressure and temperature using GERG.

The AAD of EOS-CG for each data source is summarized in Table 9 for

the three-phase equilibrium data.

The results for the VLwH equilibrium are shown in Fig. 4 and in the

deviation plot Fig. 5. GERG and EOS-CG results are comparable whereas

CSMGem predicts the upper quadruple point (VLwLcH) at a temperature that is

21

about 0.4 K lower than the respective temperature of EOS-CG and GERG. This

difference in lower quadruple point prediction has already been discussed.

However, the prediction of the lower quadruple point of CSMGem lies between

the values of EOS-CG and GERG. The experimental data can be predicted mostly

within an uncertainty in temperature of 0.5 K, the average absolute deviation for

GERG and EOS-CG can be taken from Table 11.

In Fig. 6 results for the VHIw equilibrium are shown. EOS-CG yields the

best prediction of the entire three-phase line, whereas GERG gives the worst

results, due to the erroneous prediction of the quadruple point by GERG. The

results of CSMGem again lie between GERG and EOS-CG. Fig. 7 shows

deviations in predicted equilibrium temperatures. While GERG systematically

deviates from the data, EOS-CG describes the data without significant systematic

deviations. Even the low pressure and low temperature data is well represented by

EOS-CG. Most data are predicted within a deviation range of 0.4 K and +3 K /

–9 K by EOS-CG and GERG, respectively.

Results for the LwLcH equilibrium are presented in Fig. 8. Temperature

deviations for this equilibrium are plotted in Fig. 9. As already mentioned, there is

a disagreement between the two high pressure data sets by Nakano et al. [44] and

by Manakov et al. [42]. Manakov et al. conducted a series of experiments with

different molar ratios of CO2 / H2O. For low CO2 contents, the hydrate formation

temperatures measured by Manakov et al. agreed fairly well with the experimental

data by Nakano et al. [44] and Takenouchi and Kennedy [47]. Manakov et al. [42]

speculate that only their data presented in Fig. 8 correspond to a stable LwLcH

equilibrium and that the data measured by the other authors correspond to a

metastable hydrate phase. Manakov et al. [42] suggested that for very high

pressures the large cavities of the CO2 hydrate might be multiply occupied with

reference to the data presented by Circone et al. [71]. Since one of the basic

assumptions of the hydrate model considered in this work is that every cage is

only occupied by up to one CO2 molecule, considering the multiple cage

occupancy would require a modification of the entire hydrate model. It was

therefore not possible to fit the data of Manakov et al. [42] using the present

model. Forcing the LwLcH equilibria to fit these data had a strong influence on

the representation of the other three-phase equilibrium lines and led to an

unreasonable behavior regarding the molar volume of the hydrate. Furthermore,

22

the other experimental data on the LwLcH three phase line [41, 43, 44, 47, 50, 53,

52] are qualitatively consistent and fit well to the results obtained with EOS-CG.

Consequently, the data of Manakov et al. [42] and the concept of multiple

occupancy were not considered for the new hydrate model. We note that the high

pressure region should be subject to further investigation even though no other

study states that CO2 hydrates can show multiple cage occupancy. Most of the

considered consistent LwLcH data are represented by EOS-CG and GERG within

about 0.5 K. At higher pressures the results for GERG get slightly worse, as

GERG predicts the temperature maximum about 2 K higher than the data of

Nakano et al. [44].

As can be seen in Fig. 1, experimental data on the lattice parameter of the

CO2 hydrates is only available for temperatures up to 276 K. Therefore, the lattice

parameter and consequently the molar volume of the hydrate have to be

extrapolated in the temperature region of the LwLcH equilibrium. Since the model

provides qualitatively correct behavior for hydrate formation at the LwLcH

equilibrium, the lattice parameter given by equation (7) seems to be extrapolated

in a reasonable way.

Results for the VLcH equilibrium are given in Fig. 10 and Fig. 11. As

already mentioned in the description of the fitting process, the VLcH data have

not been used in fitting the model. EOS-CG and GERG predict these data very

well, mostly within 0.2 K.

In Fig. 12, the predicted hydrate composition Hcx over pressure is shown

for all three-phase lines and quadruple points. Except for the VHIw equilibrium,

EOS-CG and GERG give almost the same hydrate composition. The results for

VHIw have different trends since for EOS-CG the CO2 content in the hydrate

phase increases with decreasing temperature, while for GERG it decreases. For

the LwLcH equilibrium, the mole fraction of CO2 with increasing pressure

asymptotically approaches the maximum CO2 content in the hydrate structure sI

with all cavities filled, Hcmax ( ) 8 / 54 0.14815x . The hydrate composition,

given in terms of the hydration number n, has been compared to the available

experimental data [24, 54, 72-76] and to the results of the simulation by Sun and

Duan [77]. Unfortunately, the available experimental data for the hydration

number of CO2 hydrates are rather scattered. As can be seen in Table 12, the

EOS-CG model is in good agreement with the data by Kumar et al. [76] and

23

results of the simulation by Sun and Duan [77], who use angle-dependent ab initio

potentials of molecules instead of Kihara potentials.

Numerical values for the most important three-phase equilibrium lines

(VHIw, VLwH, VLcH, and LwLcH) are provided in a supplementary file of this

article; namely in Tables S1 to S8. The data might be used for comparison of the

presented CO2 hydrate model with other hydrate models.

Besides the three-phase equilibria, also data on two-phase equilibria with

hydrates, namely VH, LwH, and LcH, have been measured by some authors [61,

78-83]. These works provide the composition of the respective fluid phase in

equilibrium with the gas hydrate. The present model was not fitted to these data.

The results for EOS-CG for these two-phase equilibria are also given in a

supplementary file (Tables S9, S10, and S11). As can be seen, the calculated

compositions of the fluid phases agree well with experimental data for the two-

phase equilibria. The most recent data by Chapoy et al. [83], which became

available after the development of the presented hydrate model was completed,

are represented well, too.

6. Conclusions

In this study, the hydrate model by Ballard and Sloan [11] was modified for the

accurate prediction of the complex phase equilibria in the water-carbon dioxide

system connected with gas hydrate formation. The model was combined with the

reference EoSs (EOS-CG) for both H2O and CO2, with mixing rules describing

the fluid phases of the mixture, and with equations for solid phases of these

components. Application of the highly accurate EoSs helped to accurately

describe the CO2 hydrate formation data available.

The CO2 hydrate model was correlated to a fairly large set of

experimental data including the lattice parameter and the conditions at hydrate

formation. Parameters of the model, i.e. the isothermal compressibility of the

hydrate, the reference conditions, and the Kihara potential parameters were

correlated to three-phase equilibrium data using the Levenberg-Marquardt

algorithm. As a partial result, various models for the cell potential function were

investigated. The definition of the water-shells by Ballard and Sloan [11] was

found to give results similar to the John and Holder [32] approach, if appropriate

sets of the Kihara interaction parameters are used. Combining the modified

24

classical statistical thermodynamic model for gas hydrate with the most accurate

equations of state available for the system CO2+H2O, hydrate formation

conditions can be predicted very accurately over the whole temperature and

pressure range (temperature range from 150 K to 295 K and pressures up to

500 MPa).

The CO2 hydrate model was also tested with the GERG-2004 EoS

developed mainly for natural gases. The results of the CO2 hydrate model

combined with the GERG EoS are slightly worse than those of EOS-CG;

particularly for the VHIw equilibrium as the GERG mixing rules are not able to

accurately predict the quadruple point VLwHIw. Nevertheless, the CO2 model can

be successfully used also with the GERG EoS in the region relevant for most

industrial applications, i.e. at temperatures from 240 K to 293 K and pressures up

to 200 MPa. The average absolute deviation (AAD) for the predicted temperature

along the three-phase equilibrium lines is equal to 0.179 K and 0.835 K for the

EOS-CG and GERG EoSs, respectively, considering 405 experimental data points

in total. Selected results of the model both with the EOS-CG and GERG EoS are

given in Tables S1 to S11 as Supplementary Material to this article.

Acknowledgements

The authors are grateful to the International Association for the Properties of Water and Steam

(IAPWS) which supported Václav Vinš’s stay at the Ruhr-University Bochum. The project has

also been supported by the following grants: E.ON Ruhrgas Contract “Calculation of Complex

Phase Equilibria”, the Czech Science Foundation grants No. GPP101/11/P046, GAP101/11/1593,

and the institutional support RVO:61388998.

List of symbols

a lattice parameter (Å = 10-10 m)

ahc hard core radius (Å)

BS multilayered water shell [11]

C Langmuir constant (Pa-1)

COMB combined shells

cp heat capacity (J mol-1 K-1)

f fugacity (Pa)

g Gibbs energy (J mol-1)

h enthalpy (J mol-1)

25

JH three-water shells [32]

kB Boltzmann constant (J K-1)

KS Klauda and Sandler fit [9]

N number of molecules

n hydration number

NAV Avogadro constant (mol-1)

p pressure (Pa)

r radial coordinate (Å)

R universal gas constant (J mol-1 K-1)

Ri radius of shell i (Å)

s entropy (J mol-1 K-1)

T temperature (K)

u inner energy (J mol-1)

v cavity number

V molar volume (m3·mol-1)

w cell potential (J)

z coordination number

Greek letters

α parameter of thermal expansion

ε potential well (J)

θ cavity occupancy

κ isothermal compressibility (Pa-1)

σ distance parameter (Å)

µ chemical potential (J mol-1)

φ potential energy (J)

Subscripts

a lattice

c carbon dioxide

cal calculated

exp experimental

i water shell index

J, K gas component (J = K = 1)

26

n n-th water shell or layer

tr triple point

w water

0 reference condition

Superscripts

FP fluid phase

fus fusion

H hydrate

I ice

L liquid

S solid

α general phase except hydrate

β empty lattice

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31

FIGURES

0 50 100 150 200 250 30011.8

11.85

11.9

11.95

12

T, K

a, 1

0−10

m

Tse et al. (1987) − FIT (EtO)Ikeda et al. (1999) − DPIkeda et al. (2000) − DPIkeda et al. (2000) − FITUdachin et al. (2001) − DPUdachin et al. (2001) − FITHuo et al. (2005) − DPHester et al. (2007) − DPthis work, eq. (7)

Fig. 1 Temperature dependence of the lattice parameter a of CO2 hydrate as predicted by

experimental data by Ikeda et al. [22,23], Udachin et al. [24], Huo et al. [21], Hester et al.

[25], by the correlation used in this work and by correlations by Tse et al. [26], Ikeda et

al. [23], and Udachin et al. [24] (DP – experimental data point, FIT – temperature

correlation).

140 160 180 200 220 240 260 280 30010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

T, K

Ci,J

, Pa−

1

John and Holder (1982)Ballard & Sloan (2002)Combined shellsKlauda & Sandler (2003)

Fig. 2 Langmuir constant for small (dashed lines) and large (solid lines) cavities predicted from

the three water-shells model by John and Holder [32], the multi-layered shell model by

Ballard and Sloan [11], the combined shell definition, and by the temperature correlation

by Klauda and Sandler [9]

32

210 220 230 240 250 260 270 280 290 300−0.5

0

0.5

1

1.5

2

2.5

T, K

log 10

(p),

MP

a

LcHIc

VHIc

VLcH

VHIw VLwIw

VLwH

VLwLc

LwLcH

LwHIcExperimentGERGEOS−CGQ−point

Fig. 3 Phase equilibrium diagram for H2O+CO2 mixture determined with EOS-CG [15] and

with GERG-2004 [16] for the fluid phases, the IAPWS equation for water-ice Ih [17,18],

the equation for dry ice by Jäger and Span [19], and the new CO2-hydrate model.

270 272 274 276 278 280 282 2841

1.5

2

2.5

3

3.5

4

4.5

5

T, K

p, M

Pa

Deaton & Frost (1946)Unruh & Katz (1949)Larson (1955)Robinson & Mehta (1971)Vlahakis et al.(1972)Ng & Robinson (1985)Adisasmito et al.(1991)Dholabhai et al.(1993)Ohgaki et al.(1993)Chun et al.(1996)Komai (1997)Fan & Guo (1999)Wendland et al.(1999)Fan et al.(2000)Servio & Englezos (2001)Mooijer−van den Heuvel et al.(2001)Seo et al.(2001)Hachikubo et al.(2002)Mohammadi et al.(2005)Yasuda & Ohmura (2008)Chapoy et al.(2011)GERGEOS−CGCSMGem

Fig. 4 Three phase equilibrium line for vapor + water-rich liquid + hydrate (VLwH) phase

equilibrium. Results of the hydrate model combined with EOS-CG and GERG versus

experimental data.

33

1 1.5 2 2.5 3 3.5 4 4.5−1

0

1

Tex

p − T

EoS

, K GERG − VLwH

1 1.5 2 2.5 3 3.5 4 4.5−1

0

1

Tex

p − T

EoS

, K

p, MPa

EOS−CG − VLwH

Deaton & Frost (1946)

Unruh & Katz (1949)

Larson (1955)

Robinson & Mehta (1971)

Vlahakis et al.(1972)

Ng & Robinson (1985)

Adisasmito et al.(1991)

Dholabhai et al.(1993)

Ohgaki et al.(1993)

Chun et al.(1996)

Komai (1997)

Fan & Guo (1999)

Wendland et al.(1999)

Fan et al.(2000)

Servio & Englezos (2001)

Mooijer−van den Heuvel et al.(2001)

Seo et al.(2001)

Hachikubo et al.(2002)

Mohammadi et al.(2005)

Yasuda & Ohmura (2008)

Chapoy et al.(2011)

Fig. 5 Deviations in temperature calculated by GERG and EOS-CG for the VLwH three phase

equilibrium from the experimental data.

34

140 160 180 200 220 240 260 280−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

T, K

log 10

(p),

MP

a

Larson (1955)Miller & Smythe (1970)Adamson et al.(1971)Falabella (1975)Schmitt (1986)Wendland et al.(1999)Hachikubo et al.(2002)Yasuda & Ohmura (2008)Mohammadi & Richon (2009)Fray et al.(2010)GERGEOS−CGCSMGem

Fig. 6 Three phase equilibrium line for the vapor + hydrate + water-ice (VHIw) phase

equilibrium. Results of the hydrate model combined with EOS-CG and GERG versus

experimental data.

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−10

−5

0

5

Tex

p − T

EoS

, K

GERG − VHIw

−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−10

−5

0

5

Tex

p − T

EoS

, K

log10

(p), MPa

EOS−CG − VHIw

Larson (1955)

Miller & Smythe (1970)

Adamson et al.(1971)

Falabella (1975)

Schmitt (1986)

Wendland et al.(1999)

Hachikubo et al.(2002)

Yasuda & Ohmura (2008)

Mohammadi & Richon (2009)

Fray et al.(2010)

Fig. 7 Deviations in temperature calculated by GERG and EOS-CG for the VHIw three phase

equilibrium from the experimental data.

35

280 285 290 295 300

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

T, K

log 10

(p),

MP

a

Unruh & Katz (1949)Takenouchi & Kennedy (1965)Robinson & Mehta (1971)Ng & Robinson (1985)Ohgaki et al.(1993)Ohgaki & Hamanaka (1995)Chun et al.(1996)Nakano et al.(1998)Fan & Guo (1999)Wendland et al.(1999)Mooijer−van den Heuvel et al.(2001)Seo et al.(2001)Manakov et al.(2009)Chapoy et al.(2011)GERGEOS−CGCSMGem

Fig. 8 Three phase equilibrium line for water-rich liquid + carbon dioxide-rich liquid + hydrate

(LwLcH) phase equilibrium. Results of the hydrate model combined with EOS-CG and

GERG EoS versus experimental data.

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6−3

−2

−1

0

1

Tex

p − T

EoS

, K

GERG − LwLcH

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6−3

−2

−1

0

1

Tex

p − T

EoS

, K

log10

(p), MPa

EOS−CG − LwLcH

Takenouchi & Kennedy (1965)

Ng & Robinson (1985)

Ohgaki et al.(1993)

Ohgaki & Hamanaka (1995)

Nakano et al.(1998)

Fan & Guo (1999)

Mooijer−van den Heuvel et al.(2001)

Manakov et al.(2009)

Chapoy et al.(2011)

Fig. 9 Deviations in temperature calculated by GERG and EOS-CG for the LwLcH three phase

equilibrium from the experimental data.

36

250 255 260 265 270 275 280 285 2902

2.5

3

3.5

4

4.5

5

T, K

p, M

Pa

Unruh & Katz (1949)Larson (1955)Robinson & Mehta (1971)Vlahakis et al.(1972)Ohgaki et al.(1993)Chun et al.(1996)Fan & Guo (1999)Wendland et al.(1999)Mooijer−van den Heuvel et al.(2001)GERGEOS−CG

Fig. 10 Three phase equilibrium line for vapor + carbon dioxide-rich liquid + hydrate (VLcH)

phase equilibrium. Results of the hydrate model combined with EOS-CG and GERG

versus experimental data.

2.5 3 3.5 4 4.5

−0.2

0

0.2

Tex

p − T

EoS

, K GERG − VLcH

2.5 3 3.5 4 4.5

−0.2

0

0.2

Tex

p − T

EoS

, K

p, MPa

EOS−CG − VLcH

Larson (1955)

Vlahakis et al.(1972)

Ohgaki et al.(1993)

Fan & Guo (1999)

Wendland et al.(1999)

Fig. 11 Deviations in temperature calculated by GERG and EOS-CG for the VLcH three phase

equilibrium from the experimental data.

37

−0.5 0 0.5 1 1.5 2 2.5

13.2

13.4

13.6

13.8

14

14.2

14.4

14.6

14.8

log10

(p), MPa

x H(C

O2),

%

Q1

Q2

Q3

Q4

VHIw

VLwH

LwLcH

VLcH

LcHIcVHIc

LwHIc

Q−point EOS−CGQ−point GERGEOS−CGGERG

Fig. 12 Carbon dioxide mole fraction in the hydrate phase predicted by new CO2 hydrate model

combined with EOS-CG and GERG.

38

TABLES

Table 1. Water-shell radii Ri and coordination numbers zi for hydrate structure sI provided for the three

water-shells by John and Holder [32] and the multi-layered water-shell by Ballard and Sloan

[3,11], respectively. Values correspond to the lattice parameter a0 = 12.03 Å.

Cavity R1 (Å) z1 R2 (Å) z2 R3 (Å) z3 R4 (Å) z4

John and Holder [32] Small 3.875 20 6.593 20 8.056 50 - - Large 4.152 21 7.078 24 8.285 50 - -

Ballard and Sloan [3,11] Small 3.83 8 3.96 12 - - - - Large 4.47 8 4.06 8 4.645 4 4.25 4

Table 2. Reference point used for the IAPWS-95 reference equation for H2O [13]. The reference point

is the same as the one used in the software REFPROP 9.0 [36].

p (MPa) T (K) h (J·mol-1) s (J·mol-1·K-1)

0.001 300 45957.191490119709 164.00522417832

Table 3. Reference point used for the reference equation for CO2 [14]. The reference point is the same

as the one used in the software REFPROP 9.0 [36].

p (MPa) T (K) h (J·mol-1) s (J·mol-1·K-1)

3.48514075779576 273.15 8801.96 44.0098

Table 4. Reference point used in the software REFPROP 9.0 [36] for the GERG CO2 equation [16].

p (MPa) T (K) h (J·mol-1) s (J·mol-1·K-1)

3.48501391776228 273.15 8801.90 44.0095

Table 5. Integration constants of the ice Ih equation [18] when using the different equations for the

fluid phase.

Equation of state used Constant g00 (J kg-1) Constant s0 (J kg-1·K-1)

EOS-CG -632020.233335886 -3327.33756492168

GERG -637966.096687019 -3350.95149898262

ORGE -3185921.68104710 -10312.6873872297

39

Table 6. Integration constants of the dry ice equation [19] when using the different equations for the

fluid phase.

EoS used Constant g0 Constant g1

EOS-CG -2.6385478 4.5088732

GERG -2.6377698 4.5082221

ORGE -6.0236092 19.006871

Table 7. Kihara potential parameters and coefficients for the isothermal compressibility used in the

CO2 hydrate model.

σ (Å) ε/kB (K) ahc (Å) κ1 (Pa-1) κ2 (Pa-1)

2.9463252 175.405 0.6805 1.3937770·10-10 5.2228914·10-9

Table 8. Reference Gibbs energy and reference enthalpy for the empty β-lattice of the CO2 hydrate.

Fluid EoS w,0g (J·mol-1) w,0h (J·mol-1)

EOS-CG 1040.9261 -4888.4682

GERG 1098.5310 -5588.0489

ORGE -10545.779 -51490.182

40

Table 9. CO2 hydrate equilibrium data used for the hydrate model fitting (VLcH data was not used for fitting). The information about temperature (T) and pressure (p) uncertainties

is not the total uncertainty of the measurement, but only the uncertainty of the sensor. Total uncertainties can be substantially higher, e.g. due to temperature gradients. In the

last column the AAD in K of EOS-CG is given for each author for the respective three phase equilibrium.

Author Eq. Type T (K) p Purity of CO2Qualityof H2O

AAD (K) for EOS-CG

Falabella [40] VHIw 0.05 - - - 0.212

Fan and Guo [41] LwLcH, VLcH, VLwH 0.2 0.025 MPa 99.99% Destilled / Deionized 0.599, 0.378, 0.213

Manakov et al. [42] LwLcH 0.2 < 200 MPa: 0.5% > 200MPa: 1 % - - -

Mooijer-van den Heuvel et al. [43] LwLcH, VLwH 0.01 0.005 MPa 99.95% Demineralized 0.188, 0.224

Nakano et al. [44] LwLcH 0.02 1 MPa 99.99% Destilled 0.135

Ng and Robinson [45] LwLcH, VLwH 0.1 < 1.72 MPa : 2.1kPa> 1.72 MPa: 21 kPa - - 0.457, 0.314

Robinson and Mehta [46] VLwH - - 99.8% - 0.121

Takenouchi and Kennedy [47] LwLcH 0.5 - 1 0.2 Mpa - - 0.255

Unruh and Katz [48] VLwH - - 99.5% - 0.129

Chun et al. [49] VLwH 0.1 0.01 Mpa 99.9% Destilled 0.168

Ohgaki et al. [50] LwLcH, VLcH, VLwH 0.06 5 kPa 99.99% Destilled / Deionized 0.166, 0.169, 0.244

Wendland et al. [51] VLcH, VLwH, VHIw 0.05 - 0.1 5 kPa 99.95% twice destilled 0.049, 0.091, 0.352

Chapoy et al. [52] LwLcH, VLwH 0.1 7 - 8 kPa 99.995% Destilled / Deionized 0.139, 0.160,

Ohgaki and Hamanka [53] LwLcH - - - - 0.162

Larson [54] VLcH, VLwH, VHIw 0.1 0.07 MPa 99.9% - 0.037, 0.190, 0.544

Vlahakis et al. [55] VLcH, VLwH - - - - 0.038, 0.126

Deaton and Frost [56] VLwH - - - - 0.097

41

Table 9: continued

Author Eq. Type T (K) p Purity of CO2Quality of H2O

AAD (K) for EOS-CG

Adisasmito et al. [57] VLwH min. 0.2 0.10% 99.8% Double deionized 0.109

Dholabhai et al. [58] VLwH 0.17 ca. 20 kPa 99.99% Destilled /Deionized 0.038

Komai et al. [59] VLwH 0.2 0.05 MPa - - 0.806

Fan et al. [60] VLwH 0.2 0.025 MPa 99.99% Destilled / Deionized 0.127

Servio and Englezos [61] VLwH 0.1 max 10.3425 kPa - - 0.277

Hachikubo et al. [62] VLwH, VHIw - - - - 0.207, 0.508

Mohammadi et al. [63] VLwH 0.11 0.008 MPa - - 0.086

Seo et al. [64] VLwH 0.2 0.01 Mpa 99.9% 99.1% 0.111

Yasuda and Ohmura [65] VLwH, VHIw 0.1 < 1.1 MPa: 3kPa >1.1 MPa: 11kPa 99.995% Destilled / Deionized 0.271, 0.294

Fray et al. [66] VHIw 0.02 - 0.1 0.08 - 0.4 kPa 99.998% Destilled / Deionized 0.252

Miller and Smythe [67] VHIw - - - - 0.246

Adamson and Jones [68] VHIw - - - - 0.385

Schmitt [69] VHIw - - - - 0.309

Mohammadi and Richon [70] VHIw 0.1 5 kPa 99.998% Deionized 0.660

42

Table 10. Calculated and measured quadruple points. For each quadruple point the temperature, pressure and the mole fraction of CO2 in the phases present are given.

Point Phases T (K) p (MPa) xV(CO2) xLw(CO2) xLc(CO2) xH(CO2)

Q1 VLwHIw Diamond [1] 271.67 1.034 - - - -

Larson [54] 271.8 1.048 - - - -

EOS-CG 271.246 1.01711 0.999355 0.015319 - 0.132442

GERG 273.056 1.26294 0.999395 0.000104 - 0.133909

Q2 VLwLcH Diamond [1] 283.15 4.51 - - - -

Unruh and Katz [48] 283.1 4.502 - - - -

Robinson and Mehta [46] 283.3 4.468 - - - -

Vlahakis et al. [55] 283.15 4.509 - - - -

Ohgaki and Hamanaka [53] 283.22 4.5 - - - -

Fan and Guo [41] 283.1 4.65 - - - -

Wendland et al. [51] 282.91 4.46 - - - -

Mooijer-van den Heuvel et al. [43] 283.27 4.48 - - - -

Seo et al. [64] 283.26 4.53 - - - -

EOS-CG 283.313 4.51581 0.999257 0.026286 0.998717 0.139848

GERG 283.287 4.50269 0.999321 0.000335 0.997624 0.139842

Q3 VLcHIc Diamond [1] 216.56 0.518 - - - -

EOS-CG 216.591 0.51791 0.999996 - 0.999948 0.145228

GERG 216.589 0.51784 0.999997 - 0.999945 0.145228

Q4 LwLcHIc EOS-CG 292.152 481.607 - 0.042895 0.994703 0.148103

GERG 294.536 510.001 - 0.000033 0.999990 0.148114

43

Table 11. AAD (K) according to equation (22) for EOS-CG with different cell potential functions (see

section 2.3) and for GERG with BS cell potential function. The overall AAD is also given for

every setup.

3-Phase-Eq EOS-CG& BS

EOS-CG& JH

EOS-CG& COMB

EOS-CG & KS

GERG & BS

VHIw (56 Data pts.) 0.3759 K 0.4854 K 0.5066 K 0.8868 K 4.4845 K

VLwH(188 Data pts.) 0.1652 K 0.1292 K 0.1274 K 0.1974 K 0.2186 K

LwLcH(73 Data pts.) 0.2138 K 0.2213 K 0.2100 K 0.7883 K 0.4725 K

VLcH(88 Data pts.) 0.0577 K 0.0577 K 0.0577 K 0.0577 K 0.0745 Ka

Total(405 Data pts.) 0.1797 K 0.1795 K 0.1793 K 0.3689 K 0.8323 K

a GERG converged only at 83 data points for VLcH

44

Table 12. Comparison of the hydration number n predicted by EOS-CG+BS model with the values

provided in other experimental and simulation studies. Predicted cage occupancies θsmall and

θlarge are also given.

T (K) p (MPa) n Author ncalc θsmall θlarge

268 0.99 6.412 [77] 6.484 0.674 0.958 272 1 6.505 [77] 6.581 0.645 0.950 273.15 1.2 6.441 [77] 6.499 0.673 0.955 273.15 1.2 6.441 [77] 6.499 0.673 0.955 273.65 1.2 6.548 [77] 6.511 0.670 0.954

276 3.8 6.099 [77] 6.110 0.825 0.980

278 6.2 6.093 [77] 6.110 0.825 0.980

283 4.5 6.151 [77] 6.149 0.812 0.976

269.65 0.961 6.07 ± 0.04 [54] 6.544 0.655 0.953

276.4 6.205 6.267 [72]a 6.101 0.828 0.981

273.65 1.331 7.23 [73] 6.453 0.690 0.958

276 3.85 6.2 [24]a 6.109 0.825 0.980

273.15 1.258 6.21 [74] 6.472 0.683 0.957 283 4.5 5.75 ± 0.3 [75] 6.149 0.812 0.976 272 1.02 6.57 ± 0.3 [75] 6.569 0.649 0.951 274.15 1.377 6.4 ± 0.3 [75] 6.445 0.693 0.959 275.15 1.556 6.5 ± 0.3 [75] 6.404 0.709 0.961 276.15 1.76 6.2 ± 0.3 [75] 6.364 0.724 0.963 277.15 1.989 6 ± 0.3 [75] 6.328 0.738 0.966 278.15 2.249 5.9 ± 0.3 [75] 6.295 0.751 0.968 279.15 2.55 5.9 ± 0.3 [75] 6.263 0.764 0.969 280.15 2.906 5.9 ± 0.3 [75] 6.232 0.777 0.971 281.15 3.335 5.9 ± 0.3 [75] 6.203 0.789 0.973 282.15 3.858 5.7 ± 0.3 [75] 6.175 0.801 0.975

253.15 3.2 6.04 [76] 6.000 0.867 0.989 a measured with D2O