Thermodynamic modeling for clathrate hydrates of ozone

J. Chem. Thermodynamics 64 (2013) 193–197

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Thermodynamic modeling for clathrate hydrates of ozone S. Muromachi a,⇑, H.D. Nagashima a, J.-M. Herri b, R. Ohmura a a b

Department of Mechanical Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan Ecole Nationale Supérieure des Mines de St-Etienne, 158 cours Fauriel, 42023 St-Etienne, France

a r t i c l e

i n f o

Article history: Received 7 April 2013 Received in revised form 10 May 2013 Accepted 13 May 2013 Available online 20 May 2013 Keywords: Clathrate hydrate Phase equilibrium Ozone Oxygen Carbon dioxide Thermodynamic model

a b s t r a c t We report a theoretical study to predict the phase-equilibrium properties of ozone-containing clathrate hydrates based on the statistical thermodynamics model developed by van der Waals and Platteeuw. The Patel–Teja–Valderrama equation of state is employed for an accurate estimation of the properties of gas phase ozone. We determined the three parameters of the Kihara intermolecular potential for ozone as a = 6.815  102 nm, r = 2.9909  101 nm, and e  kB1 = 184.00 K. An infinite set of e–r parameters for ozone were determined, reproducing the experimental phase equilibrium pressure–temperature data of the (O3 + O2 + CO2) clathrate hydrate. A unique parameter pair was chosen based on the experimental ozone storage capacity data for the (O3 + O2 + CCl4) hydrate that we reported previously. The prediction with the developed model showed good agreement with the experimental phase equilibrium data within ±2% of the average deviation of the pressure. The Kihara parameters of ozone showed slightly better suitability for the structure-I hydrate than CO2, which was used as a help guest. Our model suggests the possibility of increasing the ozone storage capacity of clathrate hydrates (7% on a mass basis) from the previously reported experimental capacity (1%). Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Ozone (O3) is a strong oxidizing agent and only oxygen (O2) is produced after simple oxidation, such as activated charcoal treatment. This is favorable for many applications, and ozone is currently used in many areas, e.g., water purification, food sterilization, and cleaning. There is difficulty in the preservation of ozone because of the spontaneous decomposition to oxygen, and many potential applications using ozone have not yet been sufficiently developed or explored. However, ozone preservation technology using clathrate hydrates is currently being developed [1–6]. Clathrate hydrates are host–guest crystalline compounds that consist of water molecules (host) and a guest molecule of a substance other than water. In the clathrate hydrate crystal structure, water molecules form the cage-like network by hydrogen bonds and normally one guest molecule is confined in the cage. Since two ozone molecules react with each other to give three oxygen molecules, i.e., 2O3 ? 3O2, when one ozone molecule is confined in the cage it can be isolated from the other ozone molecules and prevent it decaying to oxygen. High density storage is also expected from the molecular level packing. Such hydrate⇑ Corresponding author. Present address: Methane Hydrate Research Centre, National Institute of Advanced Industrial Science and Technology (AIST), 16-1 Onogawa, Tsukuba, Ibaraki 305-8569, Japan. Tel.: +81 29 861 4287; fax: +81 29 861 8706. E-mail address: [email protected] (S. Muromachi). 0021-9614/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jct.2013.05.020

based molecular storage technologies have applications for the storage of chemically unstable substances [7,8]. The thermodynamic equilibrium conditions of clathrate hydrates, such as pressure and temperature, are the fundamental factors in their industrial production, and mostly depend on the mass, shape and size of the guest molecule [9–12]. From the molecular sizes, the three-phase (liquid water  hydrate  gas) equilibrium conditions for the clathrate hydrate of pure ozone are similar to those of carbon dioxide, i.e., 1.5 MPa at T = 274 K [13,14]. However, gaseous ozone in such pressure and temperature conditions is highly explosive. In practice, ozone is obtained by electric discharge in an oxygen atmosphere and the concentration of ozone in the mixed (O3 + O2) gas is less than 10% of the molar concentration. If this (O3 + O2) gas were used directly for the hydrate formation, extremely low temperature and high pressure conditions would be required, e.g., 10 MPa at T = 274 K. Accordingly, other guest substances that can moderate the equilibrium conditions, so-called ‘‘help guests’’, have been added to the (O3 + O2) gas. McTurk and Waller [15] first reported the formation of the ozone-containing hydrate using carbon tetrachloride as a help guest. We previously reported the phase-equilibrium data and capability of ozone preservation for the {O3 + O2 + (CCl4 or Xe)} hydrates [1,3]. Furthermore, we found that carbon dioxide could also be used as a help guest to form the ozone-containing hydrate [4,5]. The (O3 + O2 + CO2) hydrate is expected to have no lasting effects on the environment, and is therefore suitable for industrial and consumer applications. Recently, we reported the phase

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equilibrium conditions for the (O3 + O2 + CO2) hydrate in the temperature–pressure range relevant to industrial production, i.e., 272 K to 279 K and <3 MPa [2]. To develop the hydrate-based technology, a theoretical method to predict the thermodynamic properties is required. When a hydrate forms from a mixed feed gas, the gas components are trapped in the hydrate with a different composition from the feed gas, since each gas component has a different preference for the hydrate cages that mainly depends on the molecular size. In the continuous formation process of the ozone-containing hydrate, there must be ozone, oxygen and a help guest in the reactor and the composition in the gas, water and hydrate phases are constantly changing. Thus, it should be possible to calculate the properties in the reactor corresponding to an infinite set of the thermodynamic conditions. Among the prediction, although the statistical thermodynamic model is empirical methods it is a good way of extending limited experimental data. The thermodynamic model allows the calculation of not only a specific state point but also long time-scale simulations of a continuous plant operation owing to its low computation cost [16–18]. Furthermore, the thermodynamic model can evaluate the suitability of the guest molecule for the hydrate cages. There is considerable scientific interest in ozone as the guest substance, because it is polar and chemically unstable. In this paper, we report the thermodynamic modeling of the (O3 + O2 + CO2) hydrate. The developed model is based on van der Waals and Platteeuw theory [19]. The parameters of the molecular interaction potential suggested by Kihara [20], which characterizes the capability as a guest substance, were determined for ozone based on previously reported experimental data [2,3]. To understand clearly the effect by ozone on the hydrate phase-equilibrium conditions, the other guest substances, i.e., oxygen and carbon dioxide, were carefully modeled. The present thermodynamic model reproduced the experimentally obtained equilibrium pressure within ±2% of the average deviation. The capability of ozone to form a hydrate is discussed on the basis of the thermodynamic model, and is compared with carbon dioxide. A parametric study about the equilibrium conditions and the corresponding ozone storage capacity is also conducted, which would be basic information for the process design of the ozone-hydrate production plant.

the analogy between the gas adsorption in the hydrate and the Langmuir isothermal adsorption model:

hij ¼

1þ

C ij fj X ; C ij fj

ð3Þ

j

where Cij is the Langmuir constant of a guest component j for cage type i, and fj is the fugacity of a guest component j. The fugacity can be calculated from a thermodynamic model, such as an equation of state. If it is assumed that the cage and guest molecule are both spherical and the guest molecule is located at the centre of the cage, the Langmuir constant is given by

C ij ¼

4p kB T

Z

  wðrÞ 2 r dr; exp  kB T

Ra

0

ð4Þ

where kB is the Boltzmann constant, r is the distance from the centre of the cage, and R is the cage radius when it is assumed to be spherical. w(r) is called the cell potential, which is calculated from the Lennard-Jones Devonshire model. It is originally calculated with the Lennard-Jones intermolecular-potential model [19]. McKoy and Sinanoglu [23] adopted the Kihara potential model to calculate w(r) and obtained better results. Since their report, the Kihara potential has been widely used to calculate the cell potential. The equations are

 12   r a  r6  a  wðrÞ ¼ 2ze 11 d10 þ d11 þ 5 d4 þ d5 R R R r R r

ð5Þ

and

dN ¼

  1  r aN  r aN ; 1   1þ  N R R R R

ð6Þ

where z is the coordination number of the cage, a, r and e are the parameters for the Kihara potential model, which is expressed as

( /ðrÞ ¼

for r 6 2a h  i ; r 12  r 6 4e r2a for r > 2a r2a

1

ð7Þ

where /ðrÞ is the intermolecular potential energy. 2.2. Liquid water phase

2. Thermodynamic model 2.1. Hydrate phase The thermodynamic model developed in this study is based on our previous models [21,22]. The model consists of two parts: the chemical potential estimations for water in the hydrate and aqueous (or ice) phases. If the two chemical potentials are equal at a given condition, the condition can be considered as a phase equilibrium condition. The following equation is used in our model: Lb DlHb ¼ Dl W ; W

ð1Þ

where DlHb and DlLb W W denote the difference in chemical potential of water between the hypothetical empty hydrate phase and the hydrate and aqueous phases, respectively. The DlHb can be calcuW lated from the van der Waals and Platteeuw model derived from statistical thermodynamics:

DlHb ¼ R0 T W

X

mi ln 1 

i

!

X hij ;

ð2Þ

j

where R0 is the universal gas constant, mi is the number of cages of type i per mole of water, and hij is the cage occupancy of a guest component j in cage type i, which has a value between 0 and 1. The cage occupancy is calculated by the following expression using

For the chemical potential difference between the aqueous and empty hydrate phases (DlLb W ), the right-hand side of equation (1), we used the following equation simplified by Holder et al. [24] and Menten et al.: [25]

 0 Z T Z P DlLb Dl W Dh W Dv W W ¼  dT þ dP  ln cw xw ; 2 R0 T R0 T T 0 ;P0 T 0 R0 T P0 R0 T

ð8Þ

where Dl0W is the difference in the chemical potential of water between the hypothetical empty hydrate lattice and ice at the temperature T0 = 273.15 K, and the third and fourth terms are the differences related to enthalpy and entropy changes of water between the empty hydrate and pure water phases, respectively. The cw is the activity coefficient, and xw is the mole fraction of water in the liquid water phase. The first term on the right hand side of the equation is the chemical potential difference in the reference conditions: T0 = 273.15 K and P0 = 0. This term becomes

 L  DlLW;0 DlW ¼ : R0 T T 0 ;P0 R0 T 0

ð9Þ

The molar enthalpy difference between the empty hydrate lattice and liquid water is described as follows: 0

Dhw ¼ Dhw þ

Z

T

T0

DC PW dT;

ð10Þ

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S. Muromachi et al. / J. Chem. Thermodynamics 64 (2013) 193–197 0

where Dhw is the molar enthalpy difference between the empty hydrate phase and ice at the ice point and zero pressure, and DCPW is the heat capacity difference between the empty hydrate lattice and the pure liquid water phase, which is given by

DC PW ¼ DC 0PW þ bðT  T 0 Þ;

ð11Þ

where b is the coefficient of the temperature correction. The third term on the right-hand side of equation (8) shows the free energy change caused by the difference in molar volumes of liquid water and the hydrate. Assuming that Dvw is constant, this term becomes

Z

P

P0





Dv W dP ¼ R0 T

Z

P

0

  Dv W Dv W P dP ¼ : R0 T R0 T

ð12Þ

When no highly water soluble substance exists in the system, the cw in equation (8) can be assumed to be unity and xw is corrected with Henry’s law as

xw ¼ 1 

X X xi ¼ 1  i

i

fi  : Pe V1 Hi exp R0 Ti

ð13Þ

Table 1 shows the summary of the macroscopic reference properties for the liquid water phase appearing in equations (9)-(13). 2.3. Gas phase We selected a cubic equation of state (EoS) to estimate the properties in the gas phase. Because ozone is slightly polar [26,27], it should be carefully treated in a cubic EoS. We used a combination of the Patel–Teja–Valderrama (PTV) EoS [28] and the Soave attractive temperature function [29] recommended by Valderrama [30]. These are summarized in the following equations:

P¼

R0 T



aaðT r Þ

;

ð14Þ

aaðT r Þ ¼ 1 þ ð1  T r Þðm þ n=T r Þ:

ð15Þ

v  b v ðv þ bÞ þ cðv  bÞ

The following van der Waals mixing rule was used to apply the PTV EoS to a mixture:

XX 8 a¼ xi xj aij ; aij ¼ ð1  kij Þðai aj Þ0:5 > > < i j XX ; > xi xj bij ; bij ¼ 12 ðbi þ bj Þ > :b ¼ i

ð16Þ

j

where x is the mole fraction and kij is the binary interaction parameter. The parameters m and n in equation (15) have a specific value for each substance, and those for O2 and CO2 are available in the literature [31]. The parameters for O3 were determined in the present study based on the vapor pressure of O3 [32]: m = 0.418, and n = 0.149. The binary interaction parameter kij between O2 and CO2 was also determined to be 0.03 based on the vapor  liquid equilibrium data [33]. The determination processes and results are detailed in the Supplementary Material.

2.4. Determination of the Kihara parameters To complete the model, the Kihara parameters were adjusted to give good agreement between the experimental and predicted equilibrium pressures. We then used the absolute average deviation statistics (AAD), which are defined as

AAD ¼

n



1X

Pi;cal  Pi;exp ;



n i¼1 Pi;exp

ð17Þ

where n is the number of data points, Pi,cal is the phase equilibrium pressure calculated from the model and Pi,exp is the experimental value. Of the three Kihara parameters, only a is determined from the correlation with the acentric factor suggested by Tee et al. [34]. The parameter pair of e and r were adjusted to minimize the AAD value. In general, one optimum value of e was found for each r, and, as a result, an infinite set of e and r values was obtained. Several techniques to choose a unique e–r pair from the infinite set have been suggested in the literature [22,23,35,36]. In the method suggested by Avlonitis et al., the experimental data of cage occupancy should be provided. Each e–r pair gives a different value of the cage occupancy, which results from a change in the Langmuir constant. It is possible to choose one optimum pair of e–r that best reproduces the experimentally obtained cage occupancy or Langmuir constant. The properties required in the technique of Avlonitis are usually difficult to measure experimentally. Fortunately, the experimental cage occupancy data of ozone are available from iodometric measurements of the (O3 + O2 + CCl4) hydrate [3]. However, because of the lack of Kihara parameters for CCl4, the cage occupancies corresponding to these experimental conditions could not be calculated with the present model and directly used for the determination of the e–r pair. Therefore, we reduced the experimental cage occupancy data to the Langmuir constant, which is independent of the other guest substances used for the hydrate formation, i.e., CCl4 and O2 in this case. The Langmuir constant was used to choose the optimum pair of e–r for O3. The procedure for the derivation of the Langmuir constant from the experimental cage occupancy data is detailed in the Supplementary Material. To obtain the infinite e–r set of ozone from the phase equilibrium data for the (O3 + O2 + CO2) hydrates, the Kihara parameters of O2 need to be known. The parameters for O2 have been previously reported [36,37]. However, the macroscopic parameters of the model were different from those used in our study (table 2). Thus, we determined the Kihara parameters for O2 with our present model. There were two reports about the phase equilibrium conditions for the pure O2 hydrate [37–39]. When we used these data to obtain the infinite e–r set for O2, a clear minimum value of AAD was found, and we were able to determine the Kihara parameters for O2. More detailed information is available in the Supplementary Material. 3. Results and discussion 3.1. Determination of Kihara parameters For accurate phase equilibrium modeling of the ozone-containing hydrate, the following parameters were determined: the Kihara

TABLE 1 Macroscopic reference properties of the liquid water phase used in the present thermodynamic model. Units

Structure I

Structure II

Dl0W

J  mol1

1287

1068

Dh0w

J  mol1

931

764

106  m3  mol1 J  K  mol1

4.5959 38.12

4.99644 38.12

J  K2  mol1

0.141

0.141

Dm w

DC 0PW b

TABLE 2 Kihara parameters used in this study. The parameters for CO2 are available from the references, and those for O2 and O3 were determined in this study. Guest

CO2 O2 O3

Kihara parameters

Reference

a/102 nm

r/101 nm

e  kB1/K

6.805 2.676 6.815

2.9830 3.1506 2.9909

171.41 136.15 184.00

22 This work This work

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parameters for oxygen with the presently used macroscopic parameters (shown in table 1), the parameters of ozone for the equation of state, and the binary interaction parameter between oxygen and carbon dioxide (see Supplementary Material). We obtained the infinite e–r set from the phase equilibrium data for the (O3 + O2 + CO2) hydrate. Then, the O3 Langmuir constant for the dodecahedral cage (denoted as the 512 cage) is calculated for each e–r pair. The obtained e–r set and the O3 Langmuir constants are shown in figure 1. The AAD statistics of the equilibrium pressure calculated with each e–r pair was less than 2%. The Langmuir constants calculated with the e–r set at T = 273.25 K are also shown in figure 1, as well as those of CO2 for comparison. Based on the previously reported data for ozone-storage capacity in the (O3 + O2 + CCl4) hydrate [3], we obtained the Langmuir constant 2.22  106 Pa1 at T = 273.25 K for the 512 cage (see Supplementary Material). We chose the unique e–r pair that represents this value. Consequently, the optimum Kihara parameters for ozone were determined as a = 6.815  102 nm, r = 2.9909  101 nm and e  kB1 = 184.00 K. The prediction model showed good agreement with the experimental phase equilibrium data, as shown in figure 2. In comparison with the Kihara parameters for CO2, which has a similar molecular size to ozone, the values of a and r for ozone are similar, but the e value for ozone is slightly larger than that for carbon dioxide. This caused the difference in the calculated Langmuir constant for the large cage of structure-I hydrate (51262 cage), i.e., 1.18  104 Pa1 for O3 and 4.65  105 Pa1 for CO2 at T = 273.25 K. The better suitability of ozone implies structure-I hydrate formation for the pure ozone hydrate, which has not been experimentally confirmed.

FIGURE 2. Prediction results for the phase equilibrium data of the (O3 + O2 + CO2) hydrate. The CO2 mole fraction in the gas phase of the experimental data has approximately three levels. The plot shapes show the average CO2 fraction as follows: circles, 0.7; squares, 0.8; and diamonds, 0.9. We calculated the phase equilibrium conditions with the average gas compositions of each level. The lines show the prediction results.

To design an efficient hydrate-production process, information on ozone-storage capacity for the various thermodynamic conditions should be provided. We performed a parametric study of the ozone-storage capacity in the (O3 + O2 + CO2) hydrate with sev-

eral different compositions of the feed gas. As mentioned in Section 1, the ozone concentration available from ordinary ozone generators is less than 10% on a molar basis. If this (O3 + O2) gas is directly used for the production of the (O3 + O2 + CO2) hydrate, the CO2 fraction may be the key factor of the ozone-storage capacity in the hydrate. Avoiding the risk of explosion of ozone, the partial pressure of the (O3 + O2) gas should be less than 1 MPa [40,41]. Hence, there is an upper limit on the total pressure of the feed gas, corresponding to the CO2 fraction in the gas phase. The equilibrium pressure–temperature conditions predicted with the various CO2 fractions are shown in figure 3. In this figure, the CO2 fraction was varied with a step size of 0.1, and the (O3 + O2) gas with 10% of O3 occupied the residual part, e.g., when the CO2

FIGURE 1. Kihara parameters determined for ozone. The infinite e–r set was obtained from the phase equilibrium data of the (O3 + O2 + CO2) hydrate. Cross, infinite e–r set; square, Langmuir constant of O3 in structure-II 512 cage at T = 273.25 K corresponding to the e–r set; triangle, Langmuir constant of O3 in structure-I 51262 cage at 273.25 K corresponding to the e–r set; solid line, Langmuir constant of O3 in structure-II 512 cage at 273.25 K calculated from the experimental ozone–storage capacity data; chain line, Langmuir constant of CO2 in structure-I 512 cage at 273.25 K calculated from the Kihara parameters; dashed line, Langmuir constant of CO2 in structure-I 51262 cage at 273.25 K calculated from the Kihara parameters.

FIGURE 3. Predicted phase-equilibrium conditions for the (O3 + O2 + CO2) hydrate. The CO2 mole fraction in the gas phase was varied with a 0.1 step, and the residual part was occupied by the (O3 + O2) gas having 10% ozone on a molar basis, which corresponds to the maximum concentration available from an ordinary ozone-gas generator. The values on the plot area are the mole fraction of the CO2 gas. The dashed line indicates the upper limit of the total pressure when the pressure of the (O3 + O2) gas was maintained below 1 MPa for safety reasons.

3.2. Parametric study of ozone-storage capacity in (O3 + O2 + CO2) hydrates

S. Muromachi et al. / J. Chem. Thermodynamics 64 (2013) 193–197

197

biotic, Safe and Secure System Design’’, a Grant-in-Aid for the Japan Society for the Promotion of Science (JSPS) Fellows (23-56572) of the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), and JKA through promotion funds from KEIRIN RACE.

Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jct.2013.05.020. References

FIGURE 4. Predicted ozone–storage capacity in the (O3 + O2 + CO2) hydrate. The mass fraction of ozone in the hydrate is denoted by mO3 =mHyd . The solid lines show the isothermal data at T = (272, 273, 274 and 275) K from left to right, respectively. The CO2 mole fraction in the feed gas is as follows: diamonds, 0.9; squares, 0.8; triangles, 0.7; circles, 0.6; crosses, 0.5; and pluses sign, 0.4.

fraction in the gas phase was 0.9, the fraction of O3 and O2 were 0.01 and 0.09. The upper limit of the total pressure is also shown in the figure corresponding to the CO2 fraction, and there is a specific region in which the (O3 + O2 + CO2) hydrate can form. Thermal decomposition of ozone occurs easily at high temperatures, and thus the formation temperature was also limited. Here we calculated the ozone-storage capacity relevant to the ozone-containing hydrate production over the freezing point of water, which could be depressed to T = 272 K due to the coexistence of CO2, as shown in figure 4. The maximum ozone-storage capacity of 7% on a mass basis was estimated with the feed gas having CO2 fraction of 0.5. The previously reported experimental data for ozone-storage capacity of hydrates are less than 1% [4], and such a large value has not yet been reported. The present parametric study suggests the possibility of further increase in the ozone-storage capacity of hydrates. 4. Conclusions We have reported a thermodynamic model of the (O3 + O2 + CO2) hydrate. For the accurate prediction of the hydrate phase, the fluid-phase model for each guest substance was carefully prepared. The model developed for the (O3 + O2 + CO2) hydrate reproduced well the experimental phase-equilibrium data. The intermolecular potential parameters for ozone showed slightly better suitability to the structure-I hydrate than those for CO2, and structure-I hydrate formation is suggested for the clathrate hydrate of pure ozone. We conducted a parametric study of the equilibrium conditions of the (O3 + O2 + CO2) hydrate for industrial application. The result showed the possibility of forming the hydrate with a higher ozone-storage capacity than the experimental data, even in the limited safe range of the pressure–temperature conditions.

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Acknowledgments This study was supported by the Keio University Global Centre of Excellence Program ‘‘Centre for Education and Research of Sym-

JCT 13-214