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Experimental investigation and thermodynamic modeling of xenon clathrate hydrate stability conditions
Journal Pre-proof Experimental investigation and thermodynamic modeling of xenon clathrate hydrate stability conditions Ali Rasoolzadeh, Louwrens Aaldijk, Sona Raeissi, Alireza Shariati, Cor J. Peters PII:
S0378-3812(20)30074-1
DOI:
https://doi.org/10.1016/j.fluid.2020.112528
Reference:
FLUID 112528
To appear in:
Fluid Phase Equilibria
Received Date: 5 December 2019 Revised Date:
10 February 2020
Accepted Date: 13 February 2020
Please cite this article as: A. Rasoolzadeh, L. Aaldijk, S. Raeissi, A. Shariati, C.J. Peters, Experimental investigation and thermodynamic modeling of xenon clathrate hydrate stability conditions, Fluid Phase Equilibria (2020), doi: https://doi.org/10.1016/j.fluid.2020.112528. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
CRediT author statement
Ali Rasoolzadeh: Conceptualization, Writing-Original draft preparation, Software, Validation Louwrens Aaldijk: Conceptualization, Validation Sona Raeissi: Supervision, Writing-Review & Editing Alireza Shariati: Supervision, Methodology, Writing-Review & Editing Cor J. Peters: Conceptualization, Supervision
Experimental Investigation and Thermodynamic Modeling of Xenon Clathrate Hydrate Stability Conditions Ali Rasoolzadeha, Louwrens Aaldijkb, Sona Raeissia1, Alireza Shariatia, Cor J. Petersb,c a
Natural Gas Engineering Department, School of Chemical and Petroleum Engineering, Shiraz University, Mollasadra Avenue, Shiraz 71348-51154, Iran b Laboratory of Applied Thermodynamics and Phase Equilibria, Faculty of Applied Sciences, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands c Center for Hydrate Research, Department of Chemical and Biological Engineering, Colorado School of Mines, Golden, Colorado 80401, United States ABSTRACT In this work, the equilibrium conditions of xenon hydrates have been measured within wide pressure and temperature ranges. Various experimental equipment have been used for this purpose, namely the tensimeter, the Cailletet apparatus, and the high-pressure autoclave. A number of three-phase equilibrium data points were measured for liquid water-hydrate-vapor (Lw-H-V) and the ice-hydrate-vapor (I-H-V). It was concluded that there is a good consistency between the experimental data points measured in this work and those obtained by the other groups in the literature. A modified van der Waals-Platteeuw (vdW-P) model was used to predict the xenon hydrate stability conditions. The Kihara spherical-core potential function was used to represent the intermolecular forces between the water molecules and the xenon molecules in the cavities. The fugacity of xenon in the vapor/gas phase was computed using the Peng-Robinson (PR) EoS. The solubility of xenon in the liquid phase was calculated through the Krichevsky-Kasarnovsky equation. The investigated model had the ability to predict the xenon hydrate stability conditions with good accuracy within wide ranges of pressures and temperatures, resulting in an average absolute deviation (AAD) of about 0.61 K for (Lw-H-V) and 0.42 K for (I-H-V) equilibrium temperatures.
Keywords: Xenon hydrate, Clathrates, Equilibria, Hydrate-liquid-vapor, Modeling, van der Waals-Platteeuw
1. Introduction 1
Corresponding author. Tel.: +98-713-6133707. E-mail: [email protected].
1
For many years, gas hydrates have attracted the interest of scientists because of both their harmful effects and their useful applications. For example, the blockage of gas pipelines by gas hydrate formation results in serious problems such as sudden pressure drops in gas pipelines, serious safety problems, and expenses. However, gas hydrates can also be fruitful in applications like flow assurance, safety, energy recovery, cool storage applications, gas separations, gas transportation, carbon capture and storage (CCS), water desalination, separation of ionic liquids, and food and pharmaceutical engineering such as protein recovery and drug delivery systems. Furthermore, gas hydrates can be great sources of energy, and hydrogen hydrates are being considered as green fuels [1‒9]. The mechanism of gas hydrate formation is through the occupation of unstable empty cavities formed through the connection of water molecules (host molecules) by suitable component(s) (guest molecules) such as methane, ethane, propane, carbon dioxide, hydrogen sulfide, nitrogen, oxygen, hydrogen, argon, krypton, xenon, etc. The van der Waals forces between the host molecules and the guest molecules lead to the stabilization of the hydrate structure [1, 10‒11]. Depending on the chemical nature of the guest molecules, the pressure-temperature (P-T) conditions, and the size of the guest molecules, one of three structures will be formed: structure I (sI), structure II (sII), or structure H (sH) [1]. Methane, ethane, carbon dioxide, and xenon are gases which prefer to form sI hydrates, while propane, oxygen, nitrogen, and hydrogen prefer to form sII hydrates. The sH hydrate is formed in the presence of small molecules like hydrogen, methane, and xenon, as a help gas which occupies the small and medium cages, while the large cages accommodate larger molecules like methyl cyclohexane (MCH) and Methyl tert-butyl ether (MTBE) [1].
2
Methane is the most-commonly used hydrate former, both in the laboratories and in nature. Despite the acceleration in understanding the hydrate formation mechanisms and conditions, there is still a fundamental knowledge gap regarding hydrate formation. Gaining adequate knowledge on hydrate formation conditions and mechanisms, for example in permafrost and oceanic sediment conditions, can assist in the extraction of energy from the methane hydrate reservoirs existing in the deepest parts of the oceans and seafloors [1, 12]. However, reaching seafloor pressure and temperature conditions (very high pressures and low temperatures) in a laboratory is not an easy task. For example, at a temperature of 303.15 K, the equilibrium pressure of methane hydrate is about 72 MPa [1] which is rather high for experimentation. Because of this, other gases which require milder conditions to form hydrates can be used as alternatives to methane. Xenon is a good candidate for this purpose because, similar to methane, xenon forms sI hydrates [13], but at a temperature of 303.15 K, the xenon hydrate equilibrium pressure is only about 3.5 MPa, much lower than the corresponding methane hydrate equilibrium pressure. Furthermore, pure xenon hydrate can also be compared to sH hydrate since xenon can act as a help gas to decrease the hydrate formation pressures and tune the storage capacity of such hydrates [1]. In the laboratories, xenon hydrate has also been investigated as a storage media that can maintain the enzymatic activity of model proteins because xenon is a non-reactive and non-toxic noble gas which is not harmful to humans [14, 15]. Due to the interest, a number of scientific groups have performed experimental measurements to determine xenon hydrate stability conditions. Barrer and Edge carried out an experimental study on the kinetics and equilibria of xenon hydrates at very low temperatures [16]. Ewing and Ionescu measured the xenon hydrate dissociation conditions within a temperature range of (273.15-285.15) K and the pressure range of 3
(0.154-0.497) MPa [17]. Dyadin et al. experimentally studied the phase behavior of xenon hydrates up to very high pressures [18‒20]. Makogon et al. measured the sI xenon hydrate equilibrium conditions within a temperature range of (273.15 to 288.15) K and pressure range of (0.155 to 0.698) MPa. They also measured the stability conditions of sH hydrates with xenon as the help gas [21]. Ohgaki et al. experimentally investigated the three-phase coexistence curve of (hydrate-liquid xenon-gas) and (hydrate-liquid water-gas). They studied the (hydrate-liquid water-gas) phase boundaries of xenon hydrates within the temperature range of (290 to 320) K and pressures up to 70 MPa. They also evaluated the heat of hydrate dissociation for xenon hydrates using the Clausius–Clapeyron equation, which has an average value of 65 kJ.mol-1 [22]. Sugahara et al. experimentally investigated the thermodynamic stability boundaries of xenon hydrates at pressures up to 440 MPa within the temperature range of (324.40 to 345.30) K [23]. Using calorimetry measurements, Booker et al. studied the xenon hydrate dissociation conditions within the temperature range of (287.75 to 298.95) K up to pressures of 2 MPa [14]. While the number of investigations on this system in the literature are not small, the quantity of available xenon hydrate equilibrium data points is not great, with only seven data points available for xenon (I-H-V) equilibria. The goal of this contribution is to experimentally measure the xenon hydrate stability conditions of (Lw-H-V) and (I-H-V) over wide pressure and temperature ranges. A total of 87 (Lw-H-V) and 31 (I-H-V) equilibrium data points are measured and reported for xenon hydrates. The modified van der Waals-Platteeuw (vdW-P) model is then used to predict the xenon hydrate stability conditions.
2. Experiments
4
The specifications of the materials used in this work are presented in Table 1. No further purification was applied to xenon. Table 1 In this work, a wide pressure range was selected to measure the phase stability of xenon hydrates. For this purpose, experimental measurements were conducted using three different experimental approaches. For the pressure range of (0.051 to 0.405) MPa, that covers all the experimental (I-H-V) data and several of the (Lw-H-V) data, a tensimeter was utilized, which consists of a measuring vessel, a gas storage tank, and a mercury pressure gauge. The heart of the measuring vessel is a stainless steel cylinder which is sealed on top with an O-ring Teflon disc. The temperature is digitally read with the aid of a quartz thermometer with an accuracy of ±0.01 K. The details of this device are presented in an earlier publication [24]. The experimental tests in the pressure range of (0.304 to 10.132) MPa were conducted with the aid of the Cailletet apparatus, which works according to the synthetic method. With this apparatus, equilibrium data points can be measured in the temperature range of (250 to 450) K and pressures up to 15 MPa. A platinum resistance thermometer measures the temperature with a maximum error of ±0.02 K. The pressure is determined by a dead-weight pressure gauge with an accuracy of 0.03% of the reading. Mercury is used as a pressure transmitting and sealing fluid in the Cailletet apparatus. Xenon hydrates may be formed by reducing the solution temperature at constant pressure. Then, by increasing the temperature at a slow rate (0.1 K.h-1) at constant pressure, the hydrate crystals are decomposed. The hydrate formation step is involved with meta-stabilities, therefore, the temperature is increased very slowly until the ice is completely melted. Further increase in temperature results in the beginning of hydrate dissociation. The temperature at which the hydrate crystals disappear is taken as the hydrate equilibrium temperature. Since the xenon hydrate is denser than ice and water phases [25], it can visually be recognized from the other phases. A microscope placed in front of the 5
equilibrium cell, together with an adjustable light source, both assist in providing higher visual clarity. The details of the Cailletet apparatus were presented in previous studies [24, 26‒29]. Finally, for the pressure range of (10.132 to 405.300) MPa, a high-pressure autoclave apparatus was used to perform the xenon hydrate stability experiments. The autoclave is equipped with sapphire windows and the contents are mixed using a magnetic stirrer. A platinum resistant thermometer records temperatures with an accuracy of ±0.01 K. Measurement and control of the pressure is done using a pressure gauge with an accuracy of 1% of the reading. The details of this device are presented elsewhere [24, 30‒32].
3. Thermodynamic modeling In this work, the modified van der Waals-Platteeuw (vdW-P) model [1, 33‒36] was used to calculate xenon hydrate phase equilibria. The basis of the vdW-P model is the equality of the chemical potential of water in the hydrate, aqueous liquid/ice, and the vapor phases. Fig. 1 compares the vapor pressures of xenon [37] and water [38] within a specific temperature range. Fig. 1 Since the vapor pressures of water are much lower than those of xenon, the contribution of water in the vapor phase is often neglected [1, 33]. Therefore, the hydrate stability conditions can be obtained by the equating the chemical potential of water in the hydrate and the aqueous liquid/ice phases [1, 33]: ,
=
/
,
1
A hypothetical unstable empty hydrate lattice ( two phases: 6
) is considered as a reference state for the
−
=
−
/
2
Eq. (2) can be written in the following form: ∆
,
/
=∆
,
3
The left-hand side of Eq. (3) is the water chemical potential difference between the empty and filled hydrate lattices and the right-hand side is the water chemical potential difference between the empty hydrate and the aqueous liquid/ice phase [1, 33]. The water chemical potential difference between the empty and occupied hydrate lattices is determined using solid solution theory, and by assuming the encapsulation of the guest molecules in the hydrate cavities following the ideal Langmuir adsorption theory [1, 33, 34]: ∆
&' '( )* + ,-
=
!"
In Eq. (4),
and
respectively, .
value of /0 and
ln 1 +
!"
# $ 4
are the universal gas constant and the hydrate equilibrium temperature,
is the number of small or large cavities per water molecule which has the 0
/0
for the small and large cages of sI, respectively.
is the Langmuir
constant related to the occupation of component "i" in the cavity "m". # is the fugacity of the
hydrate former in the vapor phase and its formulation is as follows: # = 1 2 5 where φi is the fugacity coefficient of component i and takes into account the non-idealities of the hydrate former in the vapor/gas phase and is determined by an equation of state. In this work, the Peng-Robinson EoS [39] was applied to calculate the fugacity of hydrate former in the vapor/gas phase.
stands for the intermolecular forces between the guest and the host
molecules, and can be calculated using an appropriate potential function:
7
Ŕ )
−; < . 44 7 exp < ?< 6 = 56 56 >
In Eq. (6), 56 represents Boltzmann’s constant, ; < is the potential function to describe the intermolecular forces between the guest and host molecules, Ŕ is the cavity radius, A
indicates the molecular-core radius, and < is the distance between the guest molecule and the center of the cavity [1]. The Kihara potential function [40, 41] is used here to describe the intermolecular forces between the guest and the host molecules, as follows: E ∗ ". E∗ 0 ; < = 4B CD G −D G H 7 < − 2A < − 2A E ∗ = E − 2A 8 In Eq. (7), A represents the host-guest molecular core radius, B is the maximum attractive
potential, and E ∗ is the water-guest core distance at zero potential (balance between the
attraction and repulsion forces). To take into account all of the interactions between the guest molecule in the cavity and all of the water molecules in the cavity wall, the average form of the spherical core Kihara potential function is used [41]: E ∗".
; < = 2KB C "" DL Ŕ < L& =
">
1 < A CD1 − − G O Ŕ Ŕ
+
&
A
Ŕ
L G− ""
− D1 +
<
E ∗0
ŔM <
A − G Ŕ Ŕ
DL / + &
A
Ŕ
L M GH 9
H 10
In Eq. (9), K is the coordination number. In Eq. (10), O is a constant and has the values of 4, 5, 10, or 11, as indicated in Eq. (9). The water chemical potential difference between the empty hydrate and the aqueous liquid phase is computed through the following equation [1, 34]:
8
/
∆
=
>
∆
>
>
T
−7 TU
∆ℎ
.
/
S
? +7 >
∆R
/
? − VW X Y
When ice is the coexisting phase, the term VW X Y
11
is omitted. In Eq. (11), ∆
>
is the water
chemical potential difference between the empty hydrate lattice and the aqueous liquid/ice phase at the reference conditions (
>
= 273.15 [ AW?
>
=
)+
, ∆ℎ
/
indicates the
difference of water molar enthalpy between the empty hydrate lattice and the aqueous liquid/ice phase, ∆R
/
is the difference of the molar volume of water between the empty
hydrate lattice and the aqueous liquid/ice phase, X represents the mole fraction of water in
the aqueous liquid phase, and Y is the activity coefficient of water in the aqueous liquid phase. ∆ℎ ∆ℎ
/
is a temperature-dependent parameter having the following relation:
= ∆ℎ
>
In Eq. (12), ∆ℎ
>
/
T
+ 7 ∆ TU
S\
? 12
is the water molar enthalpy difference between the empty hydrate lattice
and the aqueous liquid/ice phase at the reference conditions, and ∆
S\
is the water molar heat
capacity difference between the empty hydrate lattice and the aqueous liquid/ice phase and has a linear relation to temperature: ∆
S\
= −38.12 + 0.141
−
>
13
Table 2 presents the thermodynamic reference parameters for the sI hydrate, as well as the geometry of the cages [1, 34]. Table 2
9
The solubility of xenon in water is very small, however, to reach greater precision in the calculations, the Krichevsky-Kasarnovsky equation is applied to determine the solubility of xenon in water [42, 43]: #], ln D G = ln^_],, X],
)+,` a
+
R],
b
−
)+,`
-)+
14
In Eq. (14), #], is the fugacity of xenon in the vapor phase, X], is the mole fraction of xenon
in the aqueous liquid phase. _],, (solvent), R],
dilution, and
b
)+,`
is the Henry's constant for xenon (solute) in water
is the partial molar volume of xenon in the aqueous liquid phase at infinite )+,`
-)+
is the saturation pressure of water.
The fugacity of xenon in the vapor phase is calculated using the PR EoS [39]. Table 3 exhibits the critical properties, acentric factor, and the Kihara parameters for xenon. Table 3 The Henry's constant for xenon in water is calculated using the following correlation [43, 45]: ln^_],,
188.78 188.78 . G − 36.855 D1 − G + ln 1.01325 15 )+,` a = 39.273 D1 −
In Eq. (15), _],,
)+,`
is given in bars and
is in K. Since the solution temperature is lower
than the solvent’s critical temperature, it is reasonable to assume R],
independent [43]. R],
b
is taken to be 46 cme . mole
"
b
is pressure-
[46]. The saturation pressure of water
is calculated using the following relation [38]: ln
= 73.65 −
7258.20
− 7.30 ln
+ 4.17 × 10
0 .
16
In Eq. (16), the pressure unit is Pa and temperature is in K.
10
4. Results and discussion In this work, 87 data points for the (Lw-H-V) phase boundary, and 31 data points for the three-phase equilibrium of the type (I-H-V) were measured. Table 4 presents the various experimental P-T phase equilibrium data points of xenon hydrates. Table 4 Fig. 2 compares the experimental results of xenon hydrate equilibria measured in this work with those of the literature. Fig. 2 It is clear from Fig. 2 that there is good agreement between the phase boundary data points measured in this work and those obtained by other groups in the literature. The detailed statistical comparison of the experimental xenon hydrate equilibrium data measured in this work and those obtained by the other groups is presented in the associated supplementary file. Fig. 2a validates the accuracy of the three different facilities used in this work in the various pressure ranges, and gives further experimental evidence of the phase boundary of the type Lw-H-V. It is interesting to note that there is no discontinuity of the data measured by the three devices, providing one continuous smooth curve, further highlighting the accuracy of the measurements. Fig. 2b extends the literature temperature and pressure ranges for the three-phase equilibria of the type I-H-V to higher temperatures and pressures than currently available in literature. By comparing the results of Fig. 2 to the hydrate equilibrium data of methane [1], it is evident that xenon requires milder conditions (lower pressures and higher temperatures) to form its hydrate in comparison with methane. Therefore, xenon can be a good alternative to methane for gas hydrate application purposes, such as gas storage and transportation or as a media to maintain the enzymatic activity of proteins.
11
For modeling the equilibrium conditions of xenon hydrates, it is necessary to initially calculate the solubility of xenon in the aqueous phase. Although the solubility of xenon in water is very low, to reach higher precision, the solubility of xenon in water is calculated using the Krichevsky-Kasarnovsky (K-K) equation. Fig. 3 compares the experimental data points of xenon solubility in water [47] and the modeling results. Fig. 3 In Fig. 3, the temperature range is (285.85 to 344.85 K), the xenon mole fraction in the liquid phase range is (0.00004 to 0.00011), and the pressure is atmospheric [47]. Following the calculation of xenon solubility in water, in order to determine whether the interactions of the guest molecules in the adjacent cavities with one another are important or not, the Boltzmann probability factor is calculated for the xenon hydrate. This indicates the possibility of finding a molecule within the cavity and can be calculated as follows [1]: −r < hiVjklAWW m
neighboring xenon molecules in the adjacent cavities, and therefore, do not distort the hydrate lattice The modified van der Waals-Platteeuw model was then used to calculate both types of xenon hydrate phase transitions measured in this work, i.e., three phase (Lw-H-V) and (I-H-V) equilibria. The errors of the model were estimated as both average absolute relative deviation (AARD%) and average absolute deviation (AAD) in the equilibrium temperatures, as follows, tt u % = ttu [ =
y
100 × Owlnx< i# uAjA
1 × Owlnx< i# uAjA
y
z]{
z]{
−
z]{
−
|)} y
18
|)} y
19
Table 5 presents the calculated results. Table 5 The AAD (K) and AARD (%), covering all of the experimental data points, are 0.556 K and 0.187, respectively, which indicates a high level of modeling accuracy for a system which contains high-pressure data points. This is particularly accurate when considering that no optimizing parameters were used in the model. Fig. 5 provides a comparison between the experimental data measured in this work and the corresponding modeling results (with no optimizations). Fig. 5 It is clear that the model can also accurately predict the slopes and curvatures of the stability conditions of xenon hydrates, regarding both types of (Lw-H-V) and (I-H-V) equilibria. The high level of modeling accuracy is due to several reasons: (1) Xenon is not small enough to
13
allow double or multiple occupancies at high pressures (as opposed to nitrogen, oxygen, argon, krypton, and hydrogen). (2) As is evident from Fig. 4, xenon has the tendency to move, rotate, and vibrate near the cavity center, therefore, the xenon molecule in the cavity has no or very little impact on the xenon molecules in the adjacent cavities. (3) The water molar volume difference between the empty hydrate lattice and the liquid/ice phase has very little dependency on the pressure and temperature, and assuming this property to have a constant value (Table 2) is quite an acceptable approximation. (4) The Kihara potential functions and the Kihara parameters used in this work (from Ref. 1) are thoroughly capable of representing the intermolecular interactions between the xenon and the water molecules. (5) The K-K equation used in this work is highly accurate to calculate the solubility of xenon in water.
5. Conclusions In this work, xenon hydrate equilibrium conditions were experimentally and theoretically investigated. The measurements were carried out in three different facilities with complementary operational ranges, consisting of a tensimeter, the Cailletet apparatus, and a high-pressure autoclave. A total of 87 data points for the (Lw-H-V) three-phase equilibria in the pressure range of (0.15 to 376.00) MPa and the temperature range of (273.15 to 343.75) K, and 31 data points for the (I-H-V) three-phase equilibria in the pressure range of (0.1 to 0.15) MPa and the temperature range of (263.61 to 272.72) K were measured. To model the system, the modified vdW-P approach was used to predict the xenon hydrate stability conditions. For this purpose, the fugacity of xenon in the vapor phase was calculated using the PR EoS, and the Langmuir constants were calculated using the Kihara potential function.
14
The modeling results showed that the model can accurately predict the xenon hydrate stability temperatures with the AAD of about 0.56 K for all of the data points.
Acknowledgments Ali Rasoolzadeh is thankful to Shiraz University for the postdoctoral fellowship.
15
Nomenclature μ : Chemical potential of water in the hydrate phase μ
/
: Chemical potential of water in the liquid/ice phase
∆μ : Chemical potential difference of water between the hydrate phase and the empty hydrate lattice /
∆μ : Chemical potential difference of water between the liquid/ice phase and the empty hydrate lattice R : Number of cavities of type m per water molecule in the hydrate lattice •
: Langmuir constant of the guest "i" in the cavity of type "m"
# : Fugacity of component "i" in the vapor phase 5: Boltzmann's constant Ŕ: Cavity radius
A: Molecular core radius
; < : The potential model E ∗ : Collision diameter
B: Minimum potential energy K: Coordination number
∆ℎ : Water molar enthalpy difference between the liquid or ice phase and the empty hydrate lattice
∆R : Water molar volume difference between the liquid or ice phase and the empty hydrate lattice X : Mole fraction of water in the water-rich phase
Y : Activity coefficient of water in the water-rich phase
∆ S\ : Water molar heat capacity difference between the liquid or ice phase and the empty hydrate lattice _],,
b
)+,` :
Henry's constant for xenon (solute) in water (solvent)
R], : Partial molar volume of xenon in the aqueous liquid phase at infinite dilution )+,`
-)+
: Saturation pressure of water
16
References [1] E.D. Sloan, C.A. Koh, Clathrate Hydrates of Natural Gases, 3rd Ed., CRC Press, Baca Raton, 2007. [2] E.D. Sloan, Fundamental principles and applications of natural gas hydrates, Nature 426 (2003) 353‒359. [3] H. Moeini, M. Bonyadi, F. Esmaeilzadeh, A. Rasoolzadeh, Experimental study of sodium chloride aqueous solution effect on the kinetic parameters of carbon dioxide hydrate formation in the presence/absence of magnetic field, J. Nat. Gas Sci. Eng. 50 (2018) 231‒ 239. [4] E.D. Sloan, Natural Gas Hydrates in Flow Assurance, Gulf Professional Publishing, 2010. [5] A. Rasoolzadeh, J. Javanmardi, A.H. Mohammadi, An experimental study of the synergistic effects of BMIM-BF4, BMIM-DCA and TEACl aqueous solutions on methane hydrate formation, Pet. Sci. 16(2) (2019) 409‒416. [6] A. Eslamimanesh, A.H. Mohammadi, D. Richon, P. Naidoo, D. Ramjugernath, Application of gas hydrate formation in separation processes: A review of experimental studies, J. Chem. Thermodyn. 46 (2012) 62‒71. [7] I. Chatti, A. Delahaye, L. Fournaison, J.P. Petitet, Benefits and drawbacks of clathrate hydrates: a review of their areas of interest, Energy Convers. Manage. 46(9-10) (2005) 1333‒ 1343. [8] C.A. Koh, E.D. Sloan, A.K. Sum, D.T. Wu, Fundamentals and applications of gas hydrates, Annu. Rev. Chem. Biomol. Eng. 2 (2011) 237‒257. [9] K. Ogata, T. Tsuda, S. Amano, S. Hashimoto, T. Sugahara, K. Ohgaki, Hydrogen storage in trimethylamine hydrate: Thermodynamic stability and hydrogen storage capacity of hydrogen + trimethylamine mixed semi-clathrate hydrate, Chem. Eng. Sci. 65(5) (2010) 1616‒1620. [10] J.B. Klauda, S.I. Sandler, Phase behavior of clathrate hydrates: a model for single and multiple gas component hydrates, Chem. Eng. Sci. 58(1) (2003) 27‒42. [11] A. Rasoolzadeh, A. Shariati, Hydrogen hydrate cage occupancy: A key parameter for hydrogen storage and transport, Fluid Phase Equilib. 494 (2019) 8‒20. [12] X. Fu, W.F. Waite, L. Cueto‐Felgueroso, R. Juanes, Xenon hydrate as an analogue of methane hydrate in geologic systems out of thermodynamic equilibrium, Geochem., Geophys., Geosyst. 20(5) (2019) 1‒11. [13] C. Sanloup, H.K. Mao, R.J. Hemley, High-pressure transformations in xenon hydrates, PNAS 99(1) (2002) 25‒28. [14] R.D. Booker, C.A. Koh, E.D. Sloan, A.K. Sum, E. Shalaev, S.K. Singh, Xenon hydrate dissociation measurements with model protein systems, J. Phys. Chem. B 115(34) (2011) 10270‒10276. [15] J. Shin, S. Han, S.P. Kang, Investigation of xenon and natural gas hydrate as a storage medium to maintain the enzymatic activity of the model proteins, Korean J. Chem. Eng. 33(4) (2016) 1345‒1351. 17
[16] R.M. Barrer, A.V.J. Edge, Gas hydrates containing argon, krypton and xenon: kinetics and energetics of formation and equilibria, Proc. R. Soc. London, Ser. A 300(1460) (1967) 1‒ 24. [17] G.J. Ewing, L.G. Ionescu, Dissociation pressure and other thermodynamic properties of xenon-water clathrate, J. Chem. Eng. Data 19(4) (1974) 367‒369. [18] Y.A. Dyadin, E.G. Larionov, T.V. Mikina, L.I. Starostina, Clathrate hydrate of xenon at high pressure, Mendeleev Commun. 2(6) (1996) 44‒45. [19] Y.A. Dyadin, E.G. Larionov, D.S. Mirinskij, T.V. Mikina, E.Y. Aladko, L.I. Starostina, Phase diagram of the Xe–H2O system up to 15 kbar, J. Inclusion Phenom. Mol. Recognit. Chem. 28(4) (1997) 271‒285. [20] Y.A. Dyadin, E.G. Larionov, T.V. Mikina, L.I. Starostina, Clathrate formation in KrH2O and Xe-H2O systems under pressures up to 15 kbar, Mendeleev Commun. 7(2) (1997) 74‒76. [21] T.Y. Makogon, A.P. Mehta, E.D. Sloan, Structure H and structure I hydrate equilibrium data for 2, 2-dimethylbutane with methane and xenon, J. Chem. Eng. Data 41(2) (1996) 315‒ 318. [22] K. Ohgaki, T. Sugahara, M. Suzuki, H. Jindai, Phase behavior of xenon hydrate system, Fluid Phase Equilib. 175(1‒2) (2000) 1‒6. [23] K. Sugahara, T. Sugahara, K. Ohgaki, Thermodynamic and Raman spectroscopic studies of Xe and Kr hydrates, J. Chem. Eng. Data 50(1) (2005) 274‒277. [24] L. Aaldijk, Monovariante gashydraatevenwichten in het stelsel xenon-water, Ph.D. Thesis, Delft University of Technology, The Netherlands, 1971. [25] S. Takeya, A. Hachikubo, Structure and density comparison of noble gas hydrates encapsulating xenon, krypton and argon, ChemPhysChem 20(19) (2019) 2518‒2524. [26] A.M. Schilderman, S. Raeissi, C.J. Peters, Solubility of carbon dioxide in the ionic liquid 1-ethyl-3-methylimidazolium bis (trifluoromethylsulfonyl) imide, Fluid Phase Equilib. 260(1) (2007) 19‒22. [27] S. Raeissi, C.J. Peters, Carbon dioxide solubility in the homologous 1-alkyl-3methylimidazolium bis (trifluoromethylsulfonyl) imide family, J. Chem. Eng. Data 54(2) (2008) 382‒386. [28] S. Raeissi, C.J. Peters, Bubble-point pressures of the binary system carbon dioxide + linalool, J. Supercrit. Fluids 20(3) (2001) 221‒228. [29] S. Raeissi, C.J. Peters, High pressure phase behaviour of methane in 1-butyl-3methylimidazolium bis (trifluoromethylsulfonyl) imide, Fluid Phase Equilib. 294(1-2) (2010) 67‒71. [30] P.J. Smits, C.J. Peters, J. de Swaan Arons, High-pressure phase behaviour of {xCHF3 + (1− x) H2O} and {x1CHF3 + x2NaCl + (1− x1− x2) H2O}, J. Chem. Thermodyn. 29(12) (1997) 1517‒1525. [31] P.J. Smits, R.J.A. Smits, C.J. Peters, J. de Swaan Arons, High pressure phase behaviour of {xCF4 + (1− x) H2O}, J. Chem. Thermodyn. 29(1) (1997) 23‒30.
18
[32] P.J. Smits, R.J.A. Smits, C.J. Peters, J. de Swaan Arons, High pressure phase behaviour of {x1CF4 + x2NaCl + (1− x1− x2) H2O)}. J. Chem. Thermodyn. 29(4) (1997) 385‒393. [33] J.H. Van der Waals, J.C. Platteeuw, Clathrate solutions. Adv. Chem. Phys, 2 (1958) 1‒ 57. [34] W.R. Parrish, J.M. Prausnitz, Dissociation pressures of gas hydrates formed by gas mixtures, Ind. Eng. Chem. Process Des. Dev. 11(1) (1972) 26‒35. [35] A. Rasoolzadeh, A. Shariati, Considering double occupancy of large cages in nitrogen and oxygen hydrates at high pressures, Fluid Phase Equilib. 434 (2017) 107‒116. [36] G.D. Holder, S.P. Zetts, N. Pradhan, Phase behavior in systems containing clathrate hydrates: a review, Rev. Chem. Eng. 5 (1988) 1‒70. [37] A. Michels, T. Wassenaar, Vapour pressure of liquid xenon, Physica 16(3) (1950) 253‒ 256. [38] B.E. Poling, G.H. Thomson, D.G. Friend, R.L. Rowley, W.V. Wilding, in: R.H. Perry, D.W. Green (Eds.), Chemical Engineers Handbook, 2008. [39] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fundam. 15(1) (1976) 59‒64. [40] T. Kihara, The second virial coefficient of non-spherical molecules, J. Phys. Soc. Japan 6(5) (1951) 289‒296. [41] V. McKoy, O. Sinanoglu, Theory of dissociation pressures of some gas hydrates, J. Chem. Phys. 38(12) (1963) 2946‒2956. [42] I.R. Krichevsky, J.S. Kasarnovsky, Thermodynamical calculations of solubilities of nitrogen and hydrogen in water at high pressures, J. Am. Chem. Soc. 57(11) (1935) 2168‒ 2171. [43] J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, 3rd Ed., Prentice Hall, 1998. [44] National Institute of Standards and Technology (NIST). Chemistry Webbook, SRD 69, 2018. [45] B.B. Benson, D. Krause Jr, Empirical laws for dilute aqueous solutions of nonpolar gases, J. Chem. Phys. 64(2) (1976) 689‒709. [46] R.A. Pierotti, Aqueous solutions of nonpolar gases, J. Phys. Chem. 69(1) (1965) 281‒ 288. [47] A.S. Kertes, H.L. Clever, Solubility Data Series. 2: Krypton, Xenon and Radon: Gas Solubilities, International Union of Pure and Applied Chemistry (IUPAC). Analytical Chemistry Division, Commission on Equilibrium Data, Subcommittee on Solubility Data, Pergamon, 1979.
19
Table 1 The specifications of materials used in this work Material
Supplier
Molar.mass (g.mol-1)
Purity (mole fraction)
xenon
Philips
131.293
0.9999
water
TU Delft
18.053
distilled
20
Table 2 The thermodynamic reference properties and the geometry of cages for the sI hydrate [1, 34] Property
Value
∆
1263.60
∆ℎ ∆ℎ
>
(J.mol-1)
>
(J.mol-1)
-4858.90
>
(J.mol-1)
1150.60
∆R ∆R
(cm3.mol-1)
4.60
(cm3.mol-1)
3.00
Structure
I
Cavity description
512
51262
Cavity
Small
Large
Number of cavities per unit cell
2
6
Average cavity radius (Å)
3.95
4.33
Coordination number
20
24
21
Table 3 Critical properties, acentric factor, and Kihara parameters for xenon [1, 44] Component
xenon
€
Tc (K)
Pc (MPa)
[44]
[44]
[44]
289.73
5.842
0.0036
22
a (Å) [1]
E ∗ (Å) [1]
ε/k (K) [1]
0.2357 3.32968 193.708
Table 4 The experimentally measured P-T data points of xenon hydrate equilibrium conditions T/K 273.15 a 274.43 a 276.67 a 280.62 a 284.66 a 284.05 b 291.25 b 294.73 b 297.11 b 299.13 b 302.88 b 307.00 b 310.85 c 313.96 c 318.34 c 321.46 c 323.41 c 325.85 c 328.58 c 331.65 c 337.11 c 341.45 c
P / MPa 0.15300 a 0.17468 a 0.21805 a 0.32839 a a
0.49426 b 0.47 b 0.97 1.37 b 1.75 b 2.17 b 3.29 b 5.30 b 9.91 c 26.58 c 55.02 c 78.56 c 94.25 c 113.86 c 144.54 c 180.14 c 257.36 c 334.42 c
263.61 a 0.10203 a 266.71 a 0.11713 a 268.67 a 0.12605 a 270.04 a 0.13294 a 270.67 a 0.13821 a 271.15 a 0.14044 a 271.87 a 0.14490 a 272.43 a 0.14854 a Standard uncertainties: a
T/K 273.49 a 274.75 a 277.66 a 281.66 a 285.66 a 285.17 b 292.24 b 295.11 b 297.99 b 300.05 b 303.77 b 308.15 b 311.21 c 314.83 c 319.44 c 321.72 c 324.45 c 326.69 c 329.48 c 333.43 c 338.18 c 342.74 c 264.79 a 266.79 a 268.76 a 270.17 a 270.76 a 271.15 a 271.87 a 272.60 a
Lw-H-V P / MPa T/K a 0.15949 273.56 a a 0.18016 274.83 a 0.24075 a 278.67 a 0.36416 a 282.66 a 0.55253 a 282.18 b 0.52 b 286.06 b b 1.06 293.33 b 1.42 b 296.06 b b 1.93 298.36 b 2.37 b 300.89 b b 3.61 304.93 b 6.37 b 309.35 b c 11.87 311.99 c c 31.48 316.13 c 62.86 c 319.80 c c 79.34 322.48 c 103.13 c 324.88 c c 121.71 327.40 c 153.09 c 330.17 c c 203.87 334.66 c 276.06 c 339.29 c c 358.24 343.75 c I-H-V 0.10670 a 265.69 a 0.11602 a 267.68 a 0.12605 a 269.61 a 0.13507 a 270.65 a 0.13882 a 270.77 a 0.14044 a 271.69 a 0.14540 a 272.08 a 0.15037 a 272.72 a
For tensimeter: u(T) = is 0.01 K and u(P)= 0.025 kPa. 23
P / MPa 0.15979 a 0.18178 a 0.26851 a 0.40419 a 0.39 b 0.59 b 1.19 b 1.58 b 1.99 b 2.63 b 4.24 b 7.82 b 14.81 c 39.33 c 63.97 c 86.40 c 106.02 c 129.55 c 162.41 c 221.75 c 292.96 c 376.01 c
T/K 273.82 a 275.74 a 279.56 a 283.62 a 283.28 b 289.07 b 294.15 b 296.11 b 298.86 b 302.08 b 305.77 b 310.15 b 312.51 c 317.28 c 320.49 c 323.02 c 325.08 c 328.06 c 330.88 c 335.93 c 340.30 c
P / MPa 0.16405 a 0.19789 a 0.29526 a 0.44654 a 0.43 b 0.78 b 1.29 b 1.57 b 2.09 b 3.00 b 4.68 b 8.82 b 18.73 c 47.18 c 70.71 c 91.17 c 109.05 c 137.49 c 168.78 c 239.48 c 310.81 c
0.11065 a 0.12088 a 0.13030 a 0.13740 a 0.13871 a 0.14419 a 0.14642 a 0.15077 a
265.70 a 267.73 a 269.73 a 270.66 a 271.09 a 271.77 a 272.30 a
0.11095 a 0.12149 a 0.13253 a 0.13831 a 0.14104 a 0.14378 a 0.14814 a
b
For Cailletet: u(T) = is 0.02 K and u(P)= 0.01 MPa.
c
For autoclave: u(T) = is 0.01 K and u(P)= 0.05 MPa.
Table 5 AAD and AARD% of the thermodynamic model for the different types of xenon hydrate equilibria Experimental data
NP*
AAD (K)
AARD%
(Lw-H-V)
87
0.606
0.199
(I-H-V)
31
0.416
0.153
Total
118
0.556
0.187
*NP is the number of data points
24
List of Figures Fig. 1. Comparison between the experimental vapor pressures of xenon and water [37, 38] Fig. 2. Comparison between the experimental xenon hydrate stability conditions of this work to the data available in the literature. (a): Lw-H-V, (b): I-H-V Fig. 3. Comparison between the experimental data points on the solubility of xenon in water to the modeling results at atmospheric pressure [47] Fig. 4. The Boltzmann probability factor as a function of the xenon distance from the cavity center at 300 K Fig. 5. Comparison between the experimental data of xenon hydrate and the modeling results
25
10
1 Xenon Water Psat / MPa
0.1
0.01
0.001
0.0001 270
275
280
285
290
T/K Fig. 1. Comparison between the experimental vapor pressures of xenon and water [37, 38]
26
1000
Lw-H-V (This work) Lw-H-V [17] Lw-H-V [18-20] Lw-H-V [21] Lw-H-V [22] Lw-H-V [23] Lw-H-V [14]
P / MPa
100
10
1
0.1 270
280
290
300
310
320
330
340
350
360
T/K (a)
1 I-H-V (This work) I-H-V [16]
P / MPa
I-H-V [21]
0.1
0.01 210
220
230
240
250
260
270
280
T/K (b) Fig. 2. Comparison between the experimental xenon hydrate stability conditions of this work to the data available in the literature. (a): Lw-H-V, (b): I-H-V 27
390
Model Experimental data [47]
370
T/K
350 330 310 290 270 250 0
0.00002 0.00004 0.00006 0.00008
0.0001
0.00012 0.00014
xenon mole fraction in water Fig. 3. Comparison between the experimental data points on the solubility of xenon in water to the modeling results at atmospheric pressure [47]
28
6.E-16 Small cavity Large cavity
Boltzmann Probability Factor
5.E-16
4.E-16
3.E-16
2.E-16
1.E-16
0.E+00 0.0
0.5
1.0
1.5
2.0
2.5
Distance from the center of the cavity (Å) Fig. 4. The Boltzmann probability factor as a function of the xenon distance from the cavity center at 300 K
29
1000 Lw-H-V experimental data (This work) I-H-V experimental data (This work) Lw-H-V model I-H-V model
P / MPa
100
10
1
0.1 250
270
290
310
330
350
T/K
Fig. 5. Comparison between the experimental data of xenon hydrate and the modeling results
30
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
S0378-3812(20)30074-1
DOI:
https://doi.org/10.1016/j.fluid.2020.112528
Reference:
FLUID 112528
To appear in:
Fluid Phase Equilibria
Received Date: 5 December 2019 Revised Date:
10 February 2020
Accepted Date: 13 February 2020
Please cite this article as: A. Rasoolzadeh, L. Aaldijk, S. Raeissi, A. Shariati, C.J. Peters, Experimental investigation and thermodynamic modeling of xenon clathrate hydrate stability conditions, Fluid Phase Equilibria (2020), doi: https://doi.org/10.1016/j.fluid.2020.112528. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.
CRediT author statement
Ali Rasoolzadeh: Conceptualization, Writing-Original draft preparation, Software, Validation Louwrens Aaldijk: Conceptualization, Validation Sona Raeissi: Supervision, Writing-Review & Editing Alireza Shariati: Supervision, Methodology, Writing-Review & Editing Cor J. Peters: Conceptualization, Supervision
Experimental Investigation and Thermodynamic Modeling of Xenon Clathrate Hydrate Stability Conditions Ali Rasoolzadeha, Louwrens Aaldijkb, Sona Raeissia1, Alireza Shariatia, Cor J. Petersb,c a
Natural Gas Engineering Department, School of Chemical and Petroleum Engineering, Shiraz University, Mollasadra Avenue, Shiraz 71348-51154, Iran b Laboratory of Applied Thermodynamics and Phase Equilibria, Faculty of Applied Sciences, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands c Center for Hydrate Research, Department of Chemical and Biological Engineering, Colorado School of Mines, Golden, Colorado 80401, United States ABSTRACT In this work, the equilibrium conditions of xenon hydrates have been measured within wide pressure and temperature ranges. Various experimental equipment have been used for this purpose, namely the tensimeter, the Cailletet apparatus, and the high-pressure autoclave. A number of three-phase equilibrium data points were measured for liquid water-hydrate-vapor (Lw-H-V) and the ice-hydrate-vapor (I-H-V). It was concluded that there is a good consistency between the experimental data points measured in this work and those obtained by the other groups in the literature. A modified van der Waals-Platteeuw (vdW-P) model was used to predict the xenon hydrate stability conditions. The Kihara spherical-core potential function was used to represent the intermolecular forces between the water molecules and the xenon molecules in the cavities. The fugacity of xenon in the vapor/gas phase was computed using the Peng-Robinson (PR) EoS. The solubility of xenon in the liquid phase was calculated through the Krichevsky-Kasarnovsky equation. The investigated model had the ability to predict the xenon hydrate stability conditions with good accuracy within wide ranges of pressures and temperatures, resulting in an average absolute deviation (AAD) of about 0.61 K for (Lw-H-V) and 0.42 K for (I-H-V) equilibrium temperatures.
Keywords: Xenon hydrate, Clathrates, Equilibria, Hydrate-liquid-vapor, Modeling, van der Waals-Platteeuw
1. Introduction 1
Corresponding author. Tel.: +98-713-6133707. E-mail: [email protected].
1
For many years, gas hydrates have attracted the interest of scientists because of both their harmful effects and their useful applications. For example, the blockage of gas pipelines by gas hydrate formation results in serious problems such as sudden pressure drops in gas pipelines, serious safety problems, and expenses. However, gas hydrates can also be fruitful in applications like flow assurance, safety, energy recovery, cool storage applications, gas separations, gas transportation, carbon capture and storage (CCS), water desalination, separation of ionic liquids, and food and pharmaceutical engineering such as protein recovery and drug delivery systems. Furthermore, gas hydrates can be great sources of energy, and hydrogen hydrates are being considered as green fuels [1‒9]. The mechanism of gas hydrate formation is through the occupation of unstable empty cavities formed through the connection of water molecules (host molecules) by suitable component(s) (guest molecules) such as methane, ethane, propane, carbon dioxide, hydrogen sulfide, nitrogen, oxygen, hydrogen, argon, krypton, xenon, etc. The van der Waals forces between the host molecules and the guest molecules lead to the stabilization of the hydrate structure [1, 10‒11]. Depending on the chemical nature of the guest molecules, the pressure-temperature (P-T) conditions, and the size of the guest molecules, one of three structures will be formed: structure I (sI), structure II (sII), or structure H (sH) [1]. Methane, ethane, carbon dioxide, and xenon are gases which prefer to form sI hydrates, while propane, oxygen, nitrogen, and hydrogen prefer to form sII hydrates. The sH hydrate is formed in the presence of small molecules like hydrogen, methane, and xenon, as a help gas which occupies the small and medium cages, while the large cages accommodate larger molecules like methyl cyclohexane (MCH) and Methyl tert-butyl ether (MTBE) [1].
2
Methane is the most-commonly used hydrate former, both in the laboratories and in nature. Despite the acceleration in understanding the hydrate formation mechanisms and conditions, there is still a fundamental knowledge gap regarding hydrate formation. Gaining adequate knowledge on hydrate formation conditions and mechanisms, for example in permafrost and oceanic sediment conditions, can assist in the extraction of energy from the methane hydrate reservoirs existing in the deepest parts of the oceans and seafloors [1, 12]. However, reaching seafloor pressure and temperature conditions (very high pressures and low temperatures) in a laboratory is not an easy task. For example, at a temperature of 303.15 K, the equilibrium pressure of methane hydrate is about 72 MPa [1] which is rather high for experimentation. Because of this, other gases which require milder conditions to form hydrates can be used as alternatives to methane. Xenon is a good candidate for this purpose because, similar to methane, xenon forms sI hydrates [13], but at a temperature of 303.15 K, the xenon hydrate equilibrium pressure is only about 3.5 MPa, much lower than the corresponding methane hydrate equilibrium pressure. Furthermore, pure xenon hydrate can also be compared to sH hydrate since xenon can act as a help gas to decrease the hydrate formation pressures and tune the storage capacity of such hydrates [1]. In the laboratories, xenon hydrate has also been investigated as a storage media that can maintain the enzymatic activity of model proteins because xenon is a non-reactive and non-toxic noble gas which is not harmful to humans [14, 15]. Due to the interest, a number of scientific groups have performed experimental measurements to determine xenon hydrate stability conditions. Barrer and Edge carried out an experimental study on the kinetics and equilibria of xenon hydrates at very low temperatures [16]. Ewing and Ionescu measured the xenon hydrate dissociation conditions within a temperature range of (273.15-285.15) K and the pressure range of 3
(0.154-0.497) MPa [17]. Dyadin et al. experimentally studied the phase behavior of xenon hydrates up to very high pressures [18‒20]. Makogon et al. measured the sI xenon hydrate equilibrium conditions within a temperature range of (273.15 to 288.15) K and pressure range of (0.155 to 0.698) MPa. They also measured the stability conditions of sH hydrates with xenon as the help gas [21]. Ohgaki et al. experimentally investigated the three-phase coexistence curve of (hydrate-liquid xenon-gas) and (hydrate-liquid water-gas). They studied the (hydrate-liquid water-gas) phase boundaries of xenon hydrates within the temperature range of (290 to 320) K and pressures up to 70 MPa. They also evaluated the heat of hydrate dissociation for xenon hydrates using the Clausius–Clapeyron equation, which has an average value of 65 kJ.mol-1 [22]. Sugahara et al. experimentally investigated the thermodynamic stability boundaries of xenon hydrates at pressures up to 440 MPa within the temperature range of (324.40 to 345.30) K [23]. Using calorimetry measurements, Booker et al. studied the xenon hydrate dissociation conditions within the temperature range of (287.75 to 298.95) K up to pressures of 2 MPa [14]. While the number of investigations on this system in the literature are not small, the quantity of available xenon hydrate equilibrium data points is not great, with only seven data points available for xenon (I-H-V) equilibria. The goal of this contribution is to experimentally measure the xenon hydrate stability conditions of (Lw-H-V) and (I-H-V) over wide pressure and temperature ranges. A total of 87 (Lw-H-V) and 31 (I-H-V) equilibrium data points are measured and reported for xenon hydrates. The modified van der Waals-Platteeuw (vdW-P) model is then used to predict the xenon hydrate stability conditions.
2. Experiments
4
The specifications of the materials used in this work are presented in Table 1. No further purification was applied to xenon. Table 1 In this work, a wide pressure range was selected to measure the phase stability of xenon hydrates. For this purpose, experimental measurements were conducted using three different experimental approaches. For the pressure range of (0.051 to 0.405) MPa, that covers all the experimental (I-H-V) data and several of the (Lw-H-V) data, a tensimeter was utilized, which consists of a measuring vessel, a gas storage tank, and a mercury pressure gauge. The heart of the measuring vessel is a stainless steel cylinder which is sealed on top with an O-ring Teflon disc. The temperature is digitally read with the aid of a quartz thermometer with an accuracy of ±0.01 K. The details of this device are presented in an earlier publication [24]. The experimental tests in the pressure range of (0.304 to 10.132) MPa were conducted with the aid of the Cailletet apparatus, which works according to the synthetic method. With this apparatus, equilibrium data points can be measured in the temperature range of (250 to 450) K and pressures up to 15 MPa. A platinum resistance thermometer measures the temperature with a maximum error of ±0.02 K. The pressure is determined by a dead-weight pressure gauge with an accuracy of 0.03% of the reading. Mercury is used as a pressure transmitting and sealing fluid in the Cailletet apparatus. Xenon hydrates may be formed by reducing the solution temperature at constant pressure. Then, by increasing the temperature at a slow rate (0.1 K.h-1) at constant pressure, the hydrate crystals are decomposed. The hydrate formation step is involved with meta-stabilities, therefore, the temperature is increased very slowly until the ice is completely melted. Further increase in temperature results in the beginning of hydrate dissociation. The temperature at which the hydrate crystals disappear is taken as the hydrate equilibrium temperature. Since the xenon hydrate is denser than ice and water phases [25], it can visually be recognized from the other phases. A microscope placed in front of the 5
equilibrium cell, together with an adjustable light source, both assist in providing higher visual clarity. The details of the Cailletet apparatus were presented in previous studies [24, 26‒29]. Finally, for the pressure range of (10.132 to 405.300) MPa, a high-pressure autoclave apparatus was used to perform the xenon hydrate stability experiments. The autoclave is equipped with sapphire windows and the contents are mixed using a magnetic stirrer. A platinum resistant thermometer records temperatures with an accuracy of ±0.01 K. Measurement and control of the pressure is done using a pressure gauge with an accuracy of 1% of the reading. The details of this device are presented elsewhere [24, 30‒32].
3. Thermodynamic modeling In this work, the modified van der Waals-Platteeuw (vdW-P) model [1, 33‒36] was used to calculate xenon hydrate phase equilibria. The basis of the vdW-P model is the equality of the chemical potential of water in the hydrate, aqueous liquid/ice, and the vapor phases. Fig. 1 compares the vapor pressures of xenon [37] and water [38] within a specific temperature range. Fig. 1 Since the vapor pressures of water are much lower than those of xenon, the contribution of water in the vapor phase is often neglected [1, 33]. Therefore, the hydrate stability conditions can be obtained by the equating the chemical potential of water in the hydrate and the aqueous liquid/ice phases [1, 33]: ,
=
/
,
1
A hypothetical unstable empty hydrate lattice ( two phases: 6
) is considered as a reference state for the
−
=
−
/
2
Eq. (2) can be written in the following form: ∆
,
/
=∆
,
3
The left-hand side of Eq. (3) is the water chemical potential difference between the empty and filled hydrate lattices and the right-hand side is the water chemical potential difference between the empty hydrate and the aqueous liquid/ice phase [1, 33]. The water chemical potential difference between the empty and occupied hydrate lattices is determined using solid solution theory, and by assuming the encapsulation of the guest molecules in the hydrate cavities following the ideal Langmuir adsorption theory [1, 33, 34]: ∆
&' '( )* + ,-
=
!"
In Eq. (4),
and
respectively, .
value of /0 and
ln 1 +
!"
# $ 4
are the universal gas constant and the hydrate equilibrium temperature,
is the number of small or large cavities per water molecule which has the 0
/0
for the small and large cages of sI, respectively.
is the Langmuir
constant related to the occupation of component "i" in the cavity "m". # is the fugacity of the
hydrate former in the vapor phase and its formulation is as follows: # = 1 2 5 where φi is the fugacity coefficient of component i and takes into account the non-idealities of the hydrate former in the vapor/gas phase and is determined by an equation of state. In this work, the Peng-Robinson EoS [39] was applied to calculate the fugacity of hydrate former in the vapor/gas phase.
stands for the intermolecular forces between the guest and the host
molecules, and can be calculated using an appropriate potential function:
7
Ŕ )
−; < . 44 7 exp < ?< 6 = 56 56 >
In Eq. (6), 56 represents Boltzmann’s constant, ; < is the potential function to describe the intermolecular forces between the guest and host molecules, Ŕ is the cavity radius, A
indicates the molecular-core radius, and < is the distance between the guest molecule and the center of the cavity [1]. The Kihara potential function [40, 41] is used here to describe the intermolecular forces between the guest and the host molecules, as follows: E ∗ ". E∗ 0 ; < = 4B CD G −D G H 7 < − 2A < − 2A E ∗ = E − 2A 8 In Eq. (7), A represents the host-guest molecular core radius, B is the maximum attractive
potential, and E ∗ is the water-guest core distance at zero potential (balance between the
attraction and repulsion forces). To take into account all of the interactions between the guest molecule in the cavity and all of the water molecules in the cavity wall, the average form of the spherical core Kihara potential function is used [41]: E ∗".
; < = 2KB C "" DL Ŕ < L& =
">
1 < A CD1 − − G O Ŕ Ŕ
+
&
A
Ŕ
L G− ""
− D1 +
<
E ∗0
ŔM <
A − G Ŕ Ŕ
DL / + &
A
Ŕ
L M GH 9
H 10
In Eq. (9), K is the coordination number. In Eq. (10), O is a constant and has the values of 4, 5, 10, or 11, as indicated in Eq. (9). The water chemical potential difference between the empty hydrate and the aqueous liquid phase is computed through the following equation [1, 34]:
8
/
∆
=
>
∆
>
>
T
−7 TU
∆ℎ
.
/
S
? +7 >
∆R
/
? − VW X Y
When ice is the coexisting phase, the term VW X Y
11
is omitted. In Eq. (11), ∆
>
is the water
chemical potential difference between the empty hydrate lattice and the aqueous liquid/ice phase at the reference conditions (
>
= 273.15 [ AW?
>
=
)+
, ∆ℎ
/
indicates the
difference of water molar enthalpy between the empty hydrate lattice and the aqueous liquid/ice phase, ∆R
/
is the difference of the molar volume of water between the empty
hydrate lattice and the aqueous liquid/ice phase, X represents the mole fraction of water in
the aqueous liquid phase, and Y is the activity coefficient of water in the aqueous liquid phase. ∆ℎ ∆ℎ
/
is a temperature-dependent parameter having the following relation:
= ∆ℎ
>
In Eq. (12), ∆ℎ
>
/
T
+ 7 ∆ TU
S\
? 12
is the water molar enthalpy difference between the empty hydrate lattice
and the aqueous liquid/ice phase at the reference conditions, and ∆
S\
is the water molar heat
capacity difference between the empty hydrate lattice and the aqueous liquid/ice phase and has a linear relation to temperature: ∆
S\
= −38.12 + 0.141
−
>
13
Table 2 presents the thermodynamic reference parameters for the sI hydrate, as well as the geometry of the cages [1, 34]. Table 2
9
The solubility of xenon in water is very small, however, to reach greater precision in the calculations, the Krichevsky-Kasarnovsky equation is applied to determine the solubility of xenon in water [42, 43]: #], ln D G = ln^_],, X],
)+,` a
+
R],
b
−
)+,`
-)+
14
In Eq. (14), #], is the fugacity of xenon in the vapor phase, X], is the mole fraction of xenon
in the aqueous liquid phase. _],, (solvent), R],
dilution, and
b
)+,`
is the Henry's constant for xenon (solute) in water
is the partial molar volume of xenon in the aqueous liquid phase at infinite )+,`
-)+
is the saturation pressure of water.
The fugacity of xenon in the vapor phase is calculated using the PR EoS [39]. Table 3 exhibits the critical properties, acentric factor, and the Kihara parameters for xenon. Table 3 The Henry's constant for xenon in water is calculated using the following correlation [43, 45]: ln^_],,
188.78 188.78 . G − 36.855 D1 − G + ln 1.01325 15 )+,` a = 39.273 D1 −
In Eq. (15), _],,
)+,`
is given in bars and
is in K. Since the solution temperature is lower
than the solvent’s critical temperature, it is reasonable to assume R],
independent [43]. R],
b
is taken to be 46 cme . mole
"
b
is pressure-
[46]. The saturation pressure of water
is calculated using the following relation [38]: ln
= 73.65 −
7258.20
− 7.30 ln
+ 4.17 × 10
0 .
16
In Eq. (16), the pressure unit is Pa and temperature is in K.
10
4. Results and discussion In this work, 87 data points for the (Lw-H-V) phase boundary, and 31 data points for the three-phase equilibrium of the type (I-H-V) were measured. Table 4 presents the various experimental P-T phase equilibrium data points of xenon hydrates. Table 4 Fig. 2 compares the experimental results of xenon hydrate equilibria measured in this work with those of the literature. Fig. 2 It is clear from Fig. 2 that there is good agreement between the phase boundary data points measured in this work and those obtained by other groups in the literature. The detailed statistical comparison of the experimental xenon hydrate equilibrium data measured in this work and those obtained by the other groups is presented in the associated supplementary file. Fig. 2a validates the accuracy of the three different facilities used in this work in the various pressure ranges, and gives further experimental evidence of the phase boundary of the type Lw-H-V. It is interesting to note that there is no discontinuity of the data measured by the three devices, providing one continuous smooth curve, further highlighting the accuracy of the measurements. Fig. 2b extends the literature temperature and pressure ranges for the three-phase equilibria of the type I-H-V to higher temperatures and pressures than currently available in literature. By comparing the results of Fig. 2 to the hydrate equilibrium data of methane [1], it is evident that xenon requires milder conditions (lower pressures and higher temperatures) to form its hydrate in comparison with methane. Therefore, xenon can be a good alternative to methane for gas hydrate application purposes, such as gas storage and transportation or as a media to maintain the enzymatic activity of proteins.
11
For modeling the equilibrium conditions of xenon hydrates, it is necessary to initially calculate the solubility of xenon in the aqueous phase. Although the solubility of xenon in water is very low, to reach higher precision, the solubility of xenon in water is calculated using the Krichevsky-Kasarnovsky (K-K) equation. Fig. 3 compares the experimental data points of xenon solubility in water [47] and the modeling results. Fig. 3 In Fig. 3, the temperature range is (285.85 to 344.85 K), the xenon mole fraction in the liquid phase range is (0.00004 to 0.00011), and the pressure is atmospheric [47]. Following the calculation of xenon solubility in water, in order to determine whether the interactions of the guest molecules in the adjacent cavities with one another are important or not, the Boltzmann probability factor is calculated for the xenon hydrate. This indicates the possibility of finding a molecule within the cavity and can be calculated as follows [1]: −r < hiVjklAWW m
neighboring xenon molecules in the adjacent cavities, and therefore, do not distort the hydrate lattice The modified van der Waals-Platteeuw model was then used to calculate both types of xenon hydrate phase transitions measured in this work, i.e., three phase (Lw-H-V) and (I-H-V) equilibria. The errors of the model were estimated as both average absolute relative deviation (AARD%) and average absolute deviation (AAD) in the equilibrium temperatures, as follows, tt u % = ttu [ =
y
100 × Owlnx< i# uAjA
1 × Owlnx< i# uAjA
y
z]{
z]{
−
z]{
−
|)} y
18
|)} y
19
Table 5 presents the calculated results. Table 5 The AAD (K) and AARD (%), covering all of the experimental data points, are 0.556 K and 0.187, respectively, which indicates a high level of modeling accuracy for a system which contains high-pressure data points. This is particularly accurate when considering that no optimizing parameters were used in the model. Fig. 5 provides a comparison between the experimental data measured in this work and the corresponding modeling results (with no optimizations). Fig. 5 It is clear that the model can also accurately predict the slopes and curvatures of the stability conditions of xenon hydrates, regarding both types of (Lw-H-V) and (I-H-V) equilibria. The high level of modeling accuracy is due to several reasons: (1) Xenon is not small enough to
13
allow double or multiple occupancies at high pressures (as opposed to nitrogen, oxygen, argon, krypton, and hydrogen). (2) As is evident from Fig. 4, xenon has the tendency to move, rotate, and vibrate near the cavity center, therefore, the xenon molecule in the cavity has no or very little impact on the xenon molecules in the adjacent cavities. (3) The water molar volume difference between the empty hydrate lattice and the liquid/ice phase has very little dependency on the pressure and temperature, and assuming this property to have a constant value (Table 2) is quite an acceptable approximation. (4) The Kihara potential functions and the Kihara parameters used in this work (from Ref. 1) are thoroughly capable of representing the intermolecular interactions between the xenon and the water molecules. (5) The K-K equation used in this work is highly accurate to calculate the solubility of xenon in water.
5. Conclusions In this work, xenon hydrate equilibrium conditions were experimentally and theoretically investigated. The measurements were carried out in three different facilities with complementary operational ranges, consisting of a tensimeter, the Cailletet apparatus, and a high-pressure autoclave. A total of 87 data points for the (Lw-H-V) three-phase equilibria in the pressure range of (0.15 to 376.00) MPa and the temperature range of (273.15 to 343.75) K, and 31 data points for the (I-H-V) three-phase equilibria in the pressure range of (0.1 to 0.15) MPa and the temperature range of (263.61 to 272.72) K were measured. To model the system, the modified vdW-P approach was used to predict the xenon hydrate stability conditions. For this purpose, the fugacity of xenon in the vapor phase was calculated using the PR EoS, and the Langmuir constants were calculated using the Kihara potential function.
14
The modeling results showed that the model can accurately predict the xenon hydrate stability temperatures with the AAD of about 0.56 K for all of the data points.
Acknowledgments Ali Rasoolzadeh is thankful to Shiraz University for the postdoctoral fellowship.
15
Nomenclature μ : Chemical potential of water in the hydrate phase μ
/
: Chemical potential of water in the liquid/ice phase
∆μ : Chemical potential difference of water between the hydrate phase and the empty hydrate lattice /
∆μ : Chemical potential difference of water between the liquid/ice phase and the empty hydrate lattice R : Number of cavities of type m per water molecule in the hydrate lattice •
: Langmuir constant of the guest "i" in the cavity of type "m"
# : Fugacity of component "i" in the vapor phase 5: Boltzmann's constant Ŕ: Cavity radius
A: Molecular core radius
; < : The potential model E ∗ : Collision diameter
B: Minimum potential energy K: Coordination number
∆ℎ : Water molar enthalpy difference between the liquid or ice phase and the empty hydrate lattice
∆R : Water molar volume difference between the liquid or ice phase and the empty hydrate lattice X : Mole fraction of water in the water-rich phase
Y : Activity coefficient of water in the water-rich phase
∆ S\ : Water molar heat capacity difference between the liquid or ice phase and the empty hydrate lattice _],,
b
)+,` :
Henry's constant for xenon (solute) in water (solvent)
R], : Partial molar volume of xenon in the aqueous liquid phase at infinite dilution )+,`
-)+
: Saturation pressure of water
16
References [1] E.D. Sloan, C.A. Koh, Clathrate Hydrates of Natural Gases, 3rd Ed., CRC Press, Baca Raton, 2007. [2] E.D. Sloan, Fundamental principles and applications of natural gas hydrates, Nature 426 (2003) 353‒359. [3] H. Moeini, M. Bonyadi, F. Esmaeilzadeh, A. Rasoolzadeh, Experimental study of sodium chloride aqueous solution effect on the kinetic parameters of carbon dioxide hydrate formation in the presence/absence of magnetic field, J. Nat. Gas Sci. Eng. 50 (2018) 231‒ 239. [4] E.D. Sloan, Natural Gas Hydrates in Flow Assurance, Gulf Professional Publishing, 2010. [5] A. Rasoolzadeh, J. Javanmardi, A.H. Mohammadi, An experimental study of the synergistic effects of BMIM-BF4, BMIM-DCA and TEACl aqueous solutions on methane hydrate formation, Pet. Sci. 16(2) (2019) 409‒416. [6] A. Eslamimanesh, A.H. Mohammadi, D. Richon, P. Naidoo, D. Ramjugernath, Application of gas hydrate formation in separation processes: A review of experimental studies, J. Chem. Thermodyn. 46 (2012) 62‒71. [7] I. Chatti, A. Delahaye, L. Fournaison, J.P. Petitet, Benefits and drawbacks of clathrate hydrates: a review of their areas of interest, Energy Convers. Manage. 46(9-10) (2005) 1333‒ 1343. [8] C.A. Koh, E.D. Sloan, A.K. Sum, D.T. Wu, Fundamentals and applications of gas hydrates, Annu. Rev. Chem. Biomol. Eng. 2 (2011) 237‒257. [9] K. Ogata, T. Tsuda, S. Amano, S. Hashimoto, T. Sugahara, K. Ohgaki, Hydrogen storage in trimethylamine hydrate: Thermodynamic stability and hydrogen storage capacity of hydrogen + trimethylamine mixed semi-clathrate hydrate, Chem. Eng. Sci. 65(5) (2010) 1616‒1620. [10] J.B. Klauda, S.I. Sandler, Phase behavior of clathrate hydrates: a model for single and multiple gas component hydrates, Chem. Eng. Sci. 58(1) (2003) 27‒42. [11] A. Rasoolzadeh, A. Shariati, Hydrogen hydrate cage occupancy: A key parameter for hydrogen storage and transport, Fluid Phase Equilib. 494 (2019) 8‒20. [12] X. Fu, W.F. Waite, L. Cueto‐Felgueroso, R. Juanes, Xenon hydrate as an analogue of methane hydrate in geologic systems out of thermodynamic equilibrium, Geochem., Geophys., Geosyst. 20(5) (2019) 1‒11. [13] C. Sanloup, H.K. Mao, R.J. Hemley, High-pressure transformations in xenon hydrates, PNAS 99(1) (2002) 25‒28. [14] R.D. Booker, C.A. Koh, E.D. Sloan, A.K. Sum, E. Shalaev, S.K. Singh, Xenon hydrate dissociation measurements with model protein systems, J. Phys. Chem. B 115(34) (2011) 10270‒10276. [15] J. Shin, S. Han, S.P. Kang, Investigation of xenon and natural gas hydrate as a storage medium to maintain the enzymatic activity of the model proteins, Korean J. Chem. Eng. 33(4) (2016) 1345‒1351. 17
[16] R.M. Barrer, A.V.J. Edge, Gas hydrates containing argon, krypton and xenon: kinetics and energetics of formation and equilibria, Proc. R. Soc. London, Ser. A 300(1460) (1967) 1‒ 24. [17] G.J. Ewing, L.G. Ionescu, Dissociation pressure and other thermodynamic properties of xenon-water clathrate, J. Chem. Eng. Data 19(4) (1974) 367‒369. [18] Y.A. Dyadin, E.G. Larionov, T.V. Mikina, L.I. Starostina, Clathrate hydrate of xenon at high pressure, Mendeleev Commun. 2(6) (1996) 44‒45. [19] Y.A. Dyadin, E.G. Larionov, D.S. Mirinskij, T.V. Mikina, E.Y. Aladko, L.I. Starostina, Phase diagram of the Xe–H2O system up to 15 kbar, J. Inclusion Phenom. Mol. Recognit. Chem. 28(4) (1997) 271‒285. [20] Y.A. Dyadin, E.G. Larionov, T.V. Mikina, L.I. Starostina, Clathrate formation in KrH2O and Xe-H2O systems under pressures up to 15 kbar, Mendeleev Commun. 7(2) (1997) 74‒76. [21] T.Y. Makogon, A.P. Mehta, E.D. Sloan, Structure H and structure I hydrate equilibrium data for 2, 2-dimethylbutane with methane and xenon, J. Chem. Eng. Data 41(2) (1996) 315‒ 318. [22] K. Ohgaki, T. Sugahara, M. Suzuki, H. Jindai, Phase behavior of xenon hydrate system, Fluid Phase Equilib. 175(1‒2) (2000) 1‒6. [23] K. Sugahara, T. Sugahara, K. Ohgaki, Thermodynamic and Raman spectroscopic studies of Xe and Kr hydrates, J. Chem. Eng. Data 50(1) (2005) 274‒277. [24] L. Aaldijk, Monovariante gashydraatevenwichten in het stelsel xenon-water, Ph.D. Thesis, Delft University of Technology, The Netherlands, 1971. [25] S. Takeya, A. Hachikubo, Structure and density comparison of noble gas hydrates encapsulating xenon, krypton and argon, ChemPhysChem 20(19) (2019) 2518‒2524. [26] A.M. Schilderman, S. Raeissi, C.J. Peters, Solubility of carbon dioxide in the ionic liquid 1-ethyl-3-methylimidazolium bis (trifluoromethylsulfonyl) imide, Fluid Phase Equilib. 260(1) (2007) 19‒22. [27] S. Raeissi, C.J. Peters, Carbon dioxide solubility in the homologous 1-alkyl-3methylimidazolium bis (trifluoromethylsulfonyl) imide family, J. Chem. Eng. Data 54(2) (2008) 382‒386. [28] S. Raeissi, C.J. Peters, Bubble-point pressures of the binary system carbon dioxide + linalool, J. Supercrit. Fluids 20(3) (2001) 221‒228. [29] S. Raeissi, C.J. Peters, High pressure phase behaviour of methane in 1-butyl-3methylimidazolium bis (trifluoromethylsulfonyl) imide, Fluid Phase Equilib. 294(1-2) (2010) 67‒71. [30] P.J. Smits, C.J. Peters, J. de Swaan Arons, High-pressure phase behaviour of {xCHF3 + (1− x) H2O} and {x1CHF3 + x2NaCl + (1− x1− x2) H2O}, J. Chem. Thermodyn. 29(12) (1997) 1517‒1525. [31] P.J. Smits, R.J.A. Smits, C.J. Peters, J. de Swaan Arons, High pressure phase behaviour of {xCF4 + (1− x) H2O}, J. Chem. Thermodyn. 29(1) (1997) 23‒30.
18
[32] P.J. Smits, R.J.A. Smits, C.J. Peters, J. de Swaan Arons, High pressure phase behaviour of {x1CF4 + x2NaCl + (1− x1− x2) H2O)}. J. Chem. Thermodyn. 29(4) (1997) 385‒393. [33] J.H. Van der Waals, J.C. Platteeuw, Clathrate solutions. Adv. Chem. Phys, 2 (1958) 1‒ 57. [34] W.R. Parrish, J.M. Prausnitz, Dissociation pressures of gas hydrates formed by gas mixtures, Ind. Eng. Chem. Process Des. Dev. 11(1) (1972) 26‒35. [35] A. Rasoolzadeh, A. Shariati, Considering double occupancy of large cages in nitrogen and oxygen hydrates at high pressures, Fluid Phase Equilib. 434 (2017) 107‒116. [36] G.D. Holder, S.P. Zetts, N. Pradhan, Phase behavior in systems containing clathrate hydrates: a review, Rev. Chem. Eng. 5 (1988) 1‒70. [37] A. Michels, T. Wassenaar, Vapour pressure of liquid xenon, Physica 16(3) (1950) 253‒ 256. [38] B.E. Poling, G.H. Thomson, D.G. Friend, R.L. Rowley, W.V. Wilding, in: R.H. Perry, D.W. Green (Eds.), Chemical Engineers Handbook, 2008. [39] D.Y. Peng, D.B. Robinson, A new two-constant equation of state, Ind. Eng. Chem. Fundam. 15(1) (1976) 59‒64. [40] T. Kihara, The second virial coefficient of non-spherical molecules, J. Phys. Soc. Japan 6(5) (1951) 289‒296. [41] V. McKoy, O. Sinanoglu, Theory of dissociation pressures of some gas hydrates, J. Chem. Phys. 38(12) (1963) 2946‒2956. [42] I.R. Krichevsky, J.S. Kasarnovsky, Thermodynamical calculations of solubilities of nitrogen and hydrogen in water at high pressures, J. Am. Chem. Soc. 57(11) (1935) 2168‒ 2171. [43] J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamics of Fluid Phase Equilibria, 3rd Ed., Prentice Hall, 1998. [44] National Institute of Standards and Technology (NIST). Chemistry Webbook, SRD 69, 2018. [45] B.B. Benson, D. Krause Jr, Empirical laws for dilute aqueous solutions of nonpolar gases, J. Chem. Phys. 64(2) (1976) 689‒709. [46] R.A. Pierotti, Aqueous solutions of nonpolar gases, J. Phys. Chem. 69(1) (1965) 281‒ 288. [47] A.S. Kertes, H.L. Clever, Solubility Data Series. 2: Krypton, Xenon and Radon: Gas Solubilities, International Union of Pure and Applied Chemistry (IUPAC). Analytical Chemistry Division, Commission on Equilibrium Data, Subcommittee on Solubility Data, Pergamon, 1979.
19
Table 1 The specifications of materials used in this work Material
Supplier
Molar.mass (g.mol-1)
Purity (mole fraction)
xenon
Philips
131.293
0.9999
water
TU Delft
18.053
distilled
20
Table 2 The thermodynamic reference properties and the geometry of cages for the sI hydrate [1, 34] Property
Value
∆
1263.60
∆ℎ ∆ℎ
>
(J.mol-1)
>
(J.mol-1)
-4858.90
>
(J.mol-1)
1150.60
∆R ∆R
(cm3.mol-1)
4.60
(cm3.mol-1)
3.00
Structure
I
Cavity description
512
51262
Cavity
Small
Large
Number of cavities per unit cell
2
6
Average cavity radius (Å)
3.95
4.33
Coordination number
20
24
21
Table 3 Critical properties, acentric factor, and Kihara parameters for xenon [1, 44] Component
xenon
€
Tc (K)
Pc (MPa)
[44]
[44]
[44]
289.73
5.842
0.0036
22
a (Å) [1]
E ∗ (Å) [1]
ε/k (K) [1]
0.2357 3.32968 193.708
Table 4 The experimentally measured P-T data points of xenon hydrate equilibrium conditions T/K 273.15 a 274.43 a 276.67 a 280.62 a 284.66 a 284.05 b 291.25 b 294.73 b 297.11 b 299.13 b 302.88 b 307.00 b 310.85 c 313.96 c 318.34 c 321.46 c 323.41 c 325.85 c 328.58 c 331.65 c 337.11 c 341.45 c
P / MPa 0.15300 a 0.17468 a 0.21805 a 0.32839 a a
0.49426 b 0.47 b 0.97 1.37 b 1.75 b 2.17 b 3.29 b 5.30 b 9.91 c 26.58 c 55.02 c 78.56 c 94.25 c 113.86 c 144.54 c 180.14 c 257.36 c 334.42 c
263.61 a 0.10203 a 266.71 a 0.11713 a 268.67 a 0.12605 a 270.04 a 0.13294 a 270.67 a 0.13821 a 271.15 a 0.14044 a 271.87 a 0.14490 a 272.43 a 0.14854 a Standard uncertainties: a
T/K 273.49 a 274.75 a 277.66 a 281.66 a 285.66 a 285.17 b 292.24 b 295.11 b 297.99 b 300.05 b 303.77 b 308.15 b 311.21 c 314.83 c 319.44 c 321.72 c 324.45 c 326.69 c 329.48 c 333.43 c 338.18 c 342.74 c 264.79 a 266.79 a 268.76 a 270.17 a 270.76 a 271.15 a 271.87 a 272.60 a
Lw-H-V P / MPa T/K a 0.15949 273.56 a a 0.18016 274.83 a 0.24075 a 278.67 a 0.36416 a 282.66 a 0.55253 a 282.18 b 0.52 b 286.06 b b 1.06 293.33 b 1.42 b 296.06 b b 1.93 298.36 b 2.37 b 300.89 b b 3.61 304.93 b 6.37 b 309.35 b c 11.87 311.99 c c 31.48 316.13 c 62.86 c 319.80 c c 79.34 322.48 c 103.13 c 324.88 c c 121.71 327.40 c 153.09 c 330.17 c c 203.87 334.66 c 276.06 c 339.29 c c 358.24 343.75 c I-H-V 0.10670 a 265.69 a 0.11602 a 267.68 a 0.12605 a 269.61 a 0.13507 a 270.65 a 0.13882 a 270.77 a 0.14044 a 271.69 a 0.14540 a 272.08 a 0.15037 a 272.72 a
For tensimeter: u(T) = is 0.01 K and u(P)= 0.025 kPa. 23
P / MPa 0.15979 a 0.18178 a 0.26851 a 0.40419 a 0.39 b 0.59 b 1.19 b 1.58 b 1.99 b 2.63 b 4.24 b 7.82 b 14.81 c 39.33 c 63.97 c 86.40 c 106.02 c 129.55 c 162.41 c 221.75 c 292.96 c 376.01 c
T/K 273.82 a 275.74 a 279.56 a 283.62 a 283.28 b 289.07 b 294.15 b 296.11 b 298.86 b 302.08 b 305.77 b 310.15 b 312.51 c 317.28 c 320.49 c 323.02 c 325.08 c 328.06 c 330.88 c 335.93 c 340.30 c
P / MPa 0.16405 a 0.19789 a 0.29526 a 0.44654 a 0.43 b 0.78 b 1.29 b 1.57 b 2.09 b 3.00 b 4.68 b 8.82 b 18.73 c 47.18 c 70.71 c 91.17 c 109.05 c 137.49 c 168.78 c 239.48 c 310.81 c
0.11065 a 0.12088 a 0.13030 a 0.13740 a 0.13871 a 0.14419 a 0.14642 a 0.15077 a
265.70 a 267.73 a 269.73 a 270.66 a 271.09 a 271.77 a 272.30 a
0.11095 a 0.12149 a 0.13253 a 0.13831 a 0.14104 a 0.14378 a 0.14814 a
b
For Cailletet: u(T) = is 0.02 K and u(P)= 0.01 MPa.
c
For autoclave: u(T) = is 0.01 K and u(P)= 0.05 MPa.
Table 5 AAD and AARD% of the thermodynamic model for the different types of xenon hydrate equilibria Experimental data
NP*
AAD (K)
AARD%
(Lw-H-V)
87
0.606
0.199
(I-H-V)
31
0.416
0.153
Total
118
0.556
0.187
*NP is the number of data points
24
List of Figures Fig. 1. Comparison between the experimental vapor pressures of xenon and water [37, 38] Fig. 2. Comparison between the experimental xenon hydrate stability conditions of this work to the data available in the literature. (a): Lw-H-V, (b): I-H-V Fig. 3. Comparison between the experimental data points on the solubility of xenon in water to the modeling results at atmospheric pressure [47] Fig. 4. The Boltzmann probability factor as a function of the xenon distance from the cavity center at 300 K Fig. 5. Comparison between the experimental data of xenon hydrate and the modeling results
25
10
1 Xenon Water Psat / MPa
0.1
0.01
0.001
0.0001 270
275
280
285
290
T/K Fig. 1. Comparison between the experimental vapor pressures of xenon and water [37, 38]
26
1000
Lw-H-V (This work) Lw-H-V [17] Lw-H-V [18-20] Lw-H-V [21] Lw-H-V [22] Lw-H-V [23] Lw-H-V [14]
P / MPa
100
10
1
0.1 270
280
290
300
310
320
330
340
350
360
T/K (a)
1 I-H-V (This work) I-H-V [16]
P / MPa
I-H-V [21]
0.1
0.01 210
220
230
240
250
260
270
280
T/K (b) Fig. 2. Comparison between the experimental xenon hydrate stability conditions of this work to the data available in the literature. (a): Lw-H-V, (b): I-H-V 27
390
Model Experimental data [47]
370
T/K
350 330 310 290 270 250 0
0.00002 0.00004 0.00006 0.00008
0.0001
0.00012 0.00014
xenon mole fraction in water Fig. 3. Comparison between the experimental data points on the solubility of xenon in water to the modeling results at atmospheric pressure [47]
28
6.E-16 Small cavity Large cavity
Boltzmann Probability Factor
5.E-16
4.E-16
3.E-16
2.E-16
1.E-16
0.E+00 0.0
0.5
1.0
1.5
2.0
2.5
Distance from the center of the cavity (Å) Fig. 4. The Boltzmann probability factor as a function of the xenon distance from the cavity center at 300 K
29
1000 Lw-H-V experimental data (This work) I-H-V experimental data (This work) Lw-H-V model I-H-V model
P / MPa
100
10
1
0.1 250
270
290
310
330
350
T/K
Fig. 5. Comparison between the experimental data of xenon hydrate and the modeling results
30
Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: