Jean-Michel Herri

Michel Cournil

Eric Chassefière

THERMODYNAMIC MODELLING OF CLATHRATE HYDRATES IN THE ATMOSPHERE OF MARS

Keywords:

The papers deals with the stability at low temperature and low pressure in the atmospheric conditions of March, that means a gas composition mainly dominated by CO2 plus low concentration of Argon and Nitrogen and traces of gases as methane. The paper is mainly based on the data compilation from Fray et al (2011) for pure gases equilibriums. We fit the classical van der Waals approach to compare deviation. It reveals to be acceptable and allows to simulate a Martian gas of CO 2 dominated (95.3%) plus nitrogen (2.7%) and Argon (2%). The hydrate reveals to be a CO2 based hydrate which is unstable during Mars summer and stable during Mars winter time (at temperature below 154K). The proportion of methane in the hydrate is evaluated and reveals to be one fourth of the composition of the gas phase.

INTRODUCTION

The discovery of methane in the atmosphere of Mars, at first by the spectrometer PFS of the ESA probe Mars -Express (Formisano et al., 2009) then more recently from the telescope of Hawaii [START_REF] Mumma | Strong Release of Methane on Mars in Northern Summer 2003[END_REF], arouse a very big interest in planetology and exobiological communities. The origin of this methane is not known and can be attributed to the serpentinisation during current or past hydrothermal systems, and/or be attributed to the presence of methanogenic bacteria [START_REF] Chassefière | Methane release and the carbon cycle on Mars[END_REF]. A multidisciplinary group of several laboratories from the Universities of Orsay and Versailles Saint-Quentin (IDES, IAS, LATMOS), in association with a team of the École Nationale Supérieure des Mines de Saint-Etienne (LPMG) and the laboratory UTINAM (Besançon) began the study of the clathrates of methane in Martian conditions. This study, financed by the interdisciplinary program EPOV (Planet Environments and Origins of the Life), participates to the preparation of the mission ESA-NASA Exomars Gas Trace Orbiter (2016). It concerns four different but complementary domains of research : i) Question of the origin of the methane and its circulation in the systems sedimentcryosphere-atmosphere, as well as of its reservoirs of storage; ii) the modelling of the domains of stability (Pressure, Temperature) of pure and mixed clathrates in Martian conditions; iii) the research of the sources of methane on the surface via the observation of methane (gaseous and solid) and of tracers of the degassing (radon); iv) the consequences of the existence of methane on our understanding of the climatic evolution and geomorphology of Mars; v) the link between mineral and biological production of methane in the hydrothermal systems. The paper deals with the second topic and presents a modelling of the (Pressure-Temperature) equilibrium curve in the atmosphere of Mars. The paper compiles experimental data from literature [START_REF] Fray | Equilibrium Data of Methane, Carbon Dioxide, and Xenon Clathrate Hydrates below the Freezing Point of Water. Applications to Astrophysical Environments[END_REF]) and fits a model (Herri et al, 2010) based on van der Waals approach. For pure gases and in winter Martian conditions (temperature evaluated in the range of 140-160K), the model reveals to fit experimental data in the range 140-273 K with a mean deviation of 4-20%, depending on the gas. Then, modelling is applied to a gas which composition is that of the atmosphere of Mars. The hydrate reveals to be a

Introduction

The van der Waals and Platteeuw (1959) model describes the equilibrium of hydrate phases by means of an convergence between a statistical thermodynamics approach implementing Kihara parameters and a classical approach implementing reference state parameters. It is the most common model used because of its physical understanding and easy implementation in codes. In a recent publication [START_REF] Herri | Gas Hydrate Equilibria for CO2-N2 and CO2-CH4 gas mixtures-Experimental studies and Thermodynamic Modelling[END_REF], we have optimized the Kihara parameters versus the different sets of reference state parameters which are presented in the literature (Dharmawardhana et al (1980), John et al (1985), Handa and Tse (1986)). The Kihara parameters have been optimized to fit equilibrium data concerning methane, carbon dioxide and nitrogen, taken as pure gases or in mixtures. Equilibrium data was not only classical (Pressure,Temperature) equilibrium curves but also the envelope curves for gases mixtures consisting in (Pressure, Temperature, gas composition and hydrate compotion) equilibrium data. We observed the Handa and Tse (1986) reference state parameters to be the best to fit our data. In the following work, we retain these parameters to simulate hydrate equilibrium. In [START_REF] Herri | Gas Hydrate Equilibria for CO2-N2 and CO2-CH4 gas mixtures-Experimental studies and Thermodynamic Modelling[END_REF], the Kihara parameters have been optimized in (pressure -temperature) conditions on which we can have a maximum of experimental data from literature, that means principally the earth ambient temperature 273-293 K and pressure of the order of magnitude of 1 to some tens of MPa. The ambient Martian conditions are very different. Pressure is around 0.6-0.7 kPa. In winter period, on the southern permanent CO 2 ice polar cap, the temperature can be 148K (temperature of CO 2 anti sublimation at 0.6 kPa). During the summer period, at low altitude and sunny time, temperature can reach 273 K exceptionally. So, the possibility to extrapolate the van der Waals and Platteeuw model from Earth condition to Martian condition need to be questioned. In this paper, we validate the model against available data of pure gas hydrate around the Martian conditions. Data essentially comes from the compilation of [START_REF] Fray | Equilibrium Data of Methane, Carbon Dioxide, and Xenon Clathrate Hydrates below the Freezing Point of Water. Applications to Astrophysical Environments[END_REF], plus rare additional data. The composition of the Martian atmosphere is CO 2 dominated (95.3%) plus nitrogen (2.7%) and Argon (2%) and traces of other compounds : O 2 (1200±100 ppm), CO (700±200 ppm), H 2 O (<1000 ppm with an average at 200 ppm), H 2 (20 ppm ), O 3 (<0.2 ppm) and CH 4 (<50ppb with an average at 15 ppb) [START_REF] Owen | The composition of the atmosphere at the surface of Mars[END_REF][START_REF] Mumma | Strong Release of Methane on Mars in Northern Summer 2003[END_REF] .

Hydrate structure

Three different structures have been identified sI, sII and SH. They differ by their crystallographic structure in which water is organized in a three dimensional network. It liberates internal cavities of different polyhedra cavities called 5 12 , 5 12 6 2 , 5 12 6 4 , 4 3 5 6 6 3 and 5 12 6 8 (e f describes a polyhedra: e is the number of edges of the face, and f is the number of faces with e edge). On Table 1 are described more precisely the SI and SII structure that can be only formed in respect to the nature of the Martian gases.

Table 1 Structure of SI and SII gas Hydrates (1) Sloan (1998, p. 33).

(2) Variation in distance of oxygen atoms from centre of cages (Sloan, 1998, p. 33).

(3) For ethane hydrate, from (Udachin, 2002). (4) For tetrahydrofuran hydrate, from Udachin (2002).

Modelling

In the case of hydrates, in thermodynamic equilibrium, the equality of chemical potentials of water in the ice (or liquid) phase and in the hydrate phase can be written by introducing a reference state which is a hypothetical phase β that corresponds to a hydrate with empty cavities.

        - = ∆ ∑ ∑ - j i j i i β RT θ ν µ 1 ln H w (2)
In eq. ( 2) ν i is the number of cavities of type i per mole of water and i j θ is the occupancy factor

( ] 1 , 0 [ ∈ i j θ
) of the cavities of type i by the gas molecule j. This last parameter is essential to define the thermodynamic equilibrium and to determine the hydrate properties. The occupancy factor is described by a model based on ideas considering the analogy between the gas adsorption in the 3-dimensional hydrate structure and the 2-dimensional Langmuir adsorption. It can be expressed as a function of the fugacity f j of the gas j as:

        - = ∆ ∑ ∑ - j j i j i i β P T f C RT ) , ( 1 ln H w ν µ (3)
Where i j C is the Langmuir constant of component j in the cavity i that describes the interaction potential between the encaged guest molecule and the surrounding water molecules evaluated by assuming a spherically symmetrical cage that can be described by a spherical symmetrical potential:

∫ ∞       - = 0 2 ) ( exp 4 dr r kT r w kT C i j π (4)
Where w is the interaction potential between the cavity and the gas molecule according to the distance r between the guest molecule and the water molecules over the structure. The interaction potential can be determined by different models such as e.g. the van der Waals and Platteeuw model ( 1959), the Parrish and Prausnitz model (1972) or the so-called Kihara model. The latter, being the most precise [START_REF] Mckoy | Theory of dissociation pressures of some gas hydrates[END_REF], can be expressed as:

            + -       + = 5 4 5 6 11 10 11 12 2 ) ( δ δ σ δ δ σ ε R a r R R a r R z r w (5)               - + -       - - = - - N N N R a R r R a R r N 1 1 1 δ (6)
The gas parameters ε, σ and a are the so-called Kihara parameters and can be calculated from experimental data by fitting the model equations to corresponding hydrate equilibrium experimental data. The reference conditions are the temperature T 0 = 273.15 K and the pressure P 0 = 0. The difference of the chemical potential of water between the reference phase (Ice in our case, but it could be liquid vapor or vapour phase) and the (hypothetical) empty hydrate phase β,

β ϕ µ - ∆ w ,
can be written as follows:

P T P P T T T P P T a RT dP v dT T h T T T , L w L w 2 L w 0 , L w L w ln 0 0 0 0 0 - ∆ + ∆ - ∆ = ∆ ∫ ∫ - - - - β β β β µ µ (7)
The activity of water in the ice is 1. If liquid water is present L w a , is given as the product of the mole fraction of water in the liquid phase, w A refinement of the model is given by [START_REF] Sloan | Clathrate hydrates of natural gases[END_REF]Sloan ( , 2008) ) ( ) A minimization algorithm has been implemented in GasHyDyn Sofware (Java language), to determine (P,T, gas composition, hydrate composition) equilibrium from computing by using Kihara parameters and reference state parameters, or inversely, to determine parameters from (P,T, gas composition, hydrate composition) experimental data base (more details in [START_REF] Herri | Gas Hydrate Equilibria for CO2-N2 and CO2-CH4 gas mixtures-Experimental studies and Thermodynamic Modelling[END_REF].

0 L w , , L w , L w , 0 0 0 T T b c c p P T p P p - + ∆ = ∆ - - - β β β (9)

Data Base

The data base of GasHyDyn Sofware has been completed with the data at low temperature compiled in the paper of [START_REF] Fray | Equilibrium Data of Methane, Carbon Dioxide, and Xenon Clathrate Hydrates below the Freezing Point of Water. Applications to Astrophysical Environments[END_REF] for methane and CO2 clathrate below the ice points. Very few data can be found for N2 clathrate below the point, and in a very limited range of temperature of 261.7-270K [START_REF] Mohammadi | Ice-Clathrate Hydrate-Gas Equilibria for Air, Oxygen, Nitrogen, Carbon Monoxide, Methane or Ethane + Water System[END_REF]. For Argon, few data has been found but in a wide range of temperature of 148.8 to 291 K (in Holder et al, 1980, original data from Saito S., andKobayashi,1965, and original data from Barrer and Edge , 1967).

Simulation of Martian Hydrates

The equilibrium is first order governed by the dominating gas (CO2, 95.7%) and secondly affected by second order constituents (N2, 2.7%, and Ar, 2%). It is not affected by very low concentration components such as O2 (1200±100 ppm), CO (700±200 ppm), H2O (<1000 ppm with an average at 200 ppm), H2 (20 ppm ), and O3 (<0.2 ppm)) and CH 4 (<50ppb with an average at 15 ppb). Methane gas is a trace, but we will model it as precisely as possible because we want to understand its repartition between atmosphere and solid hydrate.

Simulation of pure gas equilibrium at Martian condition

Pure Argon hydrates: Figure 2 plots experimental data and simulation from GasHyDyn software. Data compiled from Holder et al (1980) are the results of Saito and Kobayashi (1965) and Barrer et Edge (1967) and covers a range temperature from 283 K down to 150K. The deviation of the model is very good, 4% maximum. At 150K, the experimental equilibrium pressure of Pure Argon hydrate is 0.1 MPa to be compared to the partial pressure on Mars which is 2 % of 0.6 kPa. Pure Argon hydrate can not be stable on Mars. Fray et al, 2010 [START_REF] Yasuda | Phase Equlibrium of Clathrate Hydrates Formed with Methane, Ethane, Propane or Carbon Dioxyde at temperatures below the Freezing Point of Water[END_REF][START_REF] Mohammadi | Equilibrium Data of Nitrous Oxide and Carbon Dioxide Clathrates Hydrates[END_REF]Schmidtt, 1986Fallabela, 1975[START_REF] Adamson | Physical Absorption of Vapor on Ice. IV. Carbon dioxide[END_REF]Miller and Smythe, 1970[START_REF] Larson | Phase Studies of the Two Component Carbon Dioxide-Water System Involving the carbon Dioxide Hydrate[END_REF] GasHyDyn simulation Fray et al, 2010 [START_REF] Yasuda | Phase Equlibrium of Clathrate Hydrates Formed with Methane, Ethane, Propane or Carbon Dioxyde at temperatures below the Freezing Point of Water[END_REF][START_REF] Mohammadi | Equilibrium Data of Nitrous Oxide and Carbon Dioxide Clathrates Hydrates[END_REF]Makogon and Sloan, 1995[START_REF] Falabella | Experimental Determination of Gas Hydrate Equilibrium below the Ice Point[END_REF][START_REF] Delsemme | Physical-Chemical penomena in comets -I.Experimental study of snows in a ccometary environment[END_REF]Deaton and Frost, 1946 Model Figure 4 Pure CH4 gas hydrate at low temperature down to the Martian winter Martian conditions (148K +-10K) Martian hydrates: The composition of the Martian atmosphere is CO 2 dominated (95.3%) plus nitrogen (2.7%) and Argon (2%) and traces of other compounds : O 2 (1200±100 ppm), CO (700±200 ppm), H 2 O (<1000 ppm with an average at 200 ppm), H 2 (20 ppm ), O 3 (<0.2 ppm) and CH 4 (<50ppb with an average at 15 ppb) Pressure is around 0.6 kPa. In winter period, on the CO2 solid crust, the temperature can be 148K (temperature of CO2 anti sublimation at 0.6kPa). During the summer period, at low altitude and sunny time, temperature can rise 273 K exceptionally.

Low concentration compounds are supposed to not affect the hydrate equilibrium. This point will be validated further. So, we do a simulation by assuming only CO 2 , N 2 and A r . Table 4 reports simulation in the range 139-161K. It reports equilibrium pressure of Structure I and Structure II hydrates. We can see that Structure I is the most stable structure to be formed. The table 4 reports also the equilibrium pressure of pure CO2 hydrate. We can see that the mixture on Mars with 2.7% of N2 and 2% of Ar gives a less stable hydrate which shifts the equilibrium temperature at 600 Pa from 155K to 154K, that means a negligible influence. Secondly, if we look at the composition of the hydrate phase, we can observe that the hydrate is practically one hundred percent CO2 hydrate with traces of Nitrogen and Argon. So, the chemical influence of secondary gases such as Nitrogen and Ar is negligible because they do not participate to the structure. So, the shift in the equilibrium temperature of Martian gas hydrate (CO2 dominated but with 5 % of other gases) is only due to a depletion of the partial pressure in CO2. Also, we can affirm that the equilibrium can not be affected by the tertiary gases which are in a two much low concentration such as O2 (1200±100 ppm), CO (700±200 ppm), H2O (<1000 ppm with an average at 200 ppm), H2 (20 ppm ), and O3 (<0.2 ppm)) and CH 4 (<50ppb with an average at 15 ppb). Atmospheric pressure on Mars is about 600 Pa. Temperature at which the hydrates are stable is 155-156K. Because winter temperature can be below (around 148K), it can be said that the hydrate can form during winter. Because summer temperature are much above 160 K in most regions of Mars, except on the permanent south polar cap, most of these hydrates dissociate. Nevertheless, stable hydrates can survive during several seasonal cycles at the top or inside the south CO2 ice polar cap. So, we can postulate cycles of formation and dissociation of hydrates associated with the winter/summer cycles.

What about methane ? Methane is a very low concentration gas (15 ppb) and it has not any influence on the equilibrium (Pressure, Temperature curve). At 154 K, the equilibrium pressure 597 kPa without methane, and it is the same value with 15 ppb methane in the gas. Table 6 reports the fractionation (or abundance fraction in the definition of [START_REF] Thomas | Variability of the methane traping in martian subsurface clathrate hydrates[END_REF] on gases between gas and hydrate. At low temperature of March (around 148 K), methane composition in the hydrate structure is between one third to one fourth of the composition of methane in the gas phase. Thomas et al (2011) reports a coefficient of 0.166 in condition similar to us (that means a very low methane concentration). The difference is essentially due to a different choice of the Kihara Parameters, but we can not say more because [START_REF] Thomas | Variability of the methane traping in martian subsurface clathrate hydrates[END_REF] do not present a validation of their Kihara parameters versus a modelling of (Pressure/temperature) diagram of pure gases. But, whatever the value of the abundance fraction (25-33% in our case, 16% for [START_REF] Thomas | Variability of the methane traping in martian subsurface clathrate hydrates[END_REF], we can observe that methane is rather well enclathred in the structure compared to Nitrogen or Argon. As we said before, both of these gases do not participate at all to the structure, firstly because of their low concentration in the gas, but also because of their low affinity to the hydrate structure. So, Nitrogen and Ar are not affected by the formation and dissociation of hydrate between winter and summer. In comparison, methane which is in a very low concentration does not perturb the hydrate equilibrium, but methane is affected by the formation of hydrate because he can participate to the formation. It abundance ratio (table 5) is one order of magnitude higher than for Nitrogen and Argon. So, theoretically, methane in the gas phase is affected by the cycles of formation and dissociation between winter and summer, and its concentration at poles could be different.

Conclusion:

In these work, we have proposed a set of Kihara parameters to model pure gas hydrate equilibria at Martian condition. We validate the model against experimental model and we observe a good correlation of some percent in the better case (Ar) and inferior to 20-25% of deviation in the worst case (CO2 and CH4). N2 has not been validated in reason of the lack of data at very low temperature. Then, we did a simulation of the Martian gas hydrate equilibrium and we found that the hydrate can be considered as stable during the winter, and unstable during the summer. Even if methane does not participate actively at the hydrate equilibrium because its concentration in the gas phase is too much low, inversely the methane gas concentration must be affected by the hydrate equilibrium. The reason is the enclathratation of methane in the structure is not negligible (abundance of methane in the hydrate is about one third or one fourth of the gas composition in the atmosphere). So, variation of methane in the atmosphere could be affected by the cycle of formation and dissociation of hydrates between winter and summer if such cycles are existing.

  means of relations from classical thermodynamics.

.

  x , and the activity coefficient of water in that phase, L w In a good approximation, the aqueous phase can be regarded as ideal and the activity coefficient therefore be set to a fixed value of 1

  Figure 3Pure Nitrogen gas hydrate in a limited range of temperature (273 down to 250 K)

Table 3

 3 Reference state parameters

		Unit	Structure I	Structure II
	,	0
	w

Unstable pure CO2 hydrates Stable pure CO2 hydrates

  Pure N2 hydrate: in the literature, we have one set of data from[START_REF] Mohammadi | Ice-Clathrate Hydrate-Gas Equilibria for Air, Oxygen, Nitrogen, Carbon Monoxide, Methane or Ethane + Water System[END_REF] andKhus et al (2000) at high temperature just under the ice formation conditions, down to 250 K only. We added point at positive temperature to look at the model on the larger range of temperature. The model deviates very rapidly from the data of[START_REF] Kuhs | Chemical Physics of air clathrate hydrates by[END_REF]. So, it is difficult to predict how the model will simulate the Nitrogen equilibrium at Martian conditions around 148K. Like the other gases, we need experimental data at Martian conditions to cross validate model and experiments. But, we will see later that Nitrogen very slightly contributes to the stability of Martian hydrates in reason of a very poor integration in the hydrate structure, and whatever it is not stable as a pure component hydrate at Martian conditions. So, the precision of the model towards pure Nitrogen hydrates is second order.

		well with the data from Fray et al (2011) with a
		deviation of 3-4%, but deviates from the data of
		Falabella and Vanpee (1974) with a deviation of
		42%. It can be said that the data sets from the two
		authors are clearly different and our model fits
		better with experimental corpus of Fray et al
		(2011).
		Intermediate conclusion: from comparison
		between data and model in the temperature range
		around 148K (which is supposed to be a
		temperature close to the winter temperature at
		Mars surface), we can observe the model fits very
		well for rare gas Argon and is acceptable for CO2
		(mean deviation of 17% at 151.5K from the data
		of Miller and Smythe (1970).
		For pure methane hydrate, the model fits will with
		the data from Fray et al (2011) with a mean
		deviation of 3.5% at 145.8K. The model differs
	T=153K Pure CH4 Hydrate: the experimental data covers a	T=152K from the data of Falabella and Vanpee (1974) with a deviation of 42.6% at 148.8K and 39% at
	very wide range of temperature, from 273 K down	159.9K.
	to 80K. Figure 4 plots the data and model.	For N2, we didn't find low temperature data
	As for previous gases (CO2, N2, Ar), model fits	Martian winter enough to validate the Kihara parameters at low
	with data for temperature below the ice point and	temperature.
	deviates as the temperature decreases. In the range PCO2=0.6 kPa temperature close to 148K, the model fits very

Table 4

 4 Equilibirum conditions of pure CO2 and Martian gas hydrates

		Pure CO 2		CO 2 -Ar-N 2 Hydrate
		Hydrate		Gas composition (0.953CO 2 , 0.026Ar, 0.020N 2 )
	T (K)	Equilibrium Pressure (Pa) SI structure SII structure SI structure	θ	Hydrate composition CO2 N2(x10 3 ) Ar(x10 3 )
	139	108	560	114	0.991	0.999	0.271	0.561
	141	137	727	144	0.990	0.999	0.291	0.606
	143	173	938	181	0.989	0.999	0.312	0.654
	145	217	1200	228	0.988	0.999	0.335	0.703
	146	243	1348	254	0.987	0.999	0.346	0.729
	148	297	1715	317	0.986	0.999	0.370	0.782
	149	337	1923	354	0.986	0.999	0.382	0.809
	150	376	2158	393	0.985	0.999	0.394	0.837
	151	417	2415	438	0.985	0.999	0.407	0.865
	153	515	3013	538	0.984	0.999	0.432	0.922
	154	571	3352	597	0.983	0.999	0.446	0.952
	155	631	3735	662	0.982	0.999	0.459	0.982
	156	699	4150	730	0.982	0.999	0.473	1.013
	157	767	4601	809	0.981	0.998	0.487	1.044
	158	841	5102	894	0.981	0.998	0.501	1.076
	160	1036	6234	1086	0.979	0.998	0.530	1.141
	161	1141	6889	1195	0.979	0.998	0.544	1.174

Table 5

 5 Abundance fraction of component in hydrate versus component in gas

	Equilibrium condition
	Gas= 97,4% CO 2 ,		Abundance ration =xi/fi
	2,7%N 2 , 2% Ar, 15 ppb	xi : composition of i in the hydrate
	CH 4		fi : composition of i in the gas
	T (K)	P (kPa)	CO2	N2	Ar	CH 4
	139	113,6	1,048	0,010	0,028	0,270
	143	181,5	1,048	0,012	0,033	0,281
	147	284,5	1,048	0,013	0,038	0,291
	150	393,5	1,048	0,015	0,042	0,299
	151	437,5	1,048	0,015	0,043	0,302
	154	596,8	1,048	0,017	0,048	0,310
	155	661,7	1,048	0,017	0,049	0,313
	157	809,4	1,048	0,018	0,052	0,318
	158	893,6	1,047	0,019	0,054	0,320
	159	985,6	1,047	0,019	0,056	0,323
	160	1086,1	1,047	0,020	0,057	0,326


F L Waelbroeck

E Tassi

A Compressible Hamiltonian Electromagnetic Gyrofluid Model

Keywords: numbers: 52.30.Ex, 52.35.Bj, 94.30.Cq 1

A Lie-Poisson bracket is presented for a four-field gyrofluid model with compressible ions and magnetic field curvature, thereby showing the model to be Hamiltonian. In particular, in addition to commonly adopted magnetic curvature terms present in the continuity equations, analogous terms must be retained also in the momentum equations, in order to have a Lie-Poisson structure.

The corresponding Casimir invariants are presented, and shown to be associated to four Lagrangian invariants, that get advected by appropriate "velocity" fields during the dynamics. This differs from a cold ion limit, in which the Lie-Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants.

I. INTRODUCTION

Fluid models of plasma dynamics play an important role in the investigation of nonlinear phenomena such as turbulence, [1,2] magnetic reconnection, [3][4][5][6] and the propagation of coherent structures. [7][8][9] A highly desirable property for such models is that they not only have a conserved energy, [10] but also that they be Hamiltonian in the ideal limit. [11] By ideal limit, we mean the limit obtained by neglecting dissipative terms such as those related to collisions and anomalous transport terms.

The Hamiltonian property confers several advantages upon the models that possess it.

[12] Krommes and Kolesnikov have shown that the description of the generation of zonal flow and convection cells by short wavelength turbulence depends on the existence of a Poisson bracket. [START_REF] Krommes | [END_REF] Another important property of noncanonical Hamiltonian systems is the existence of families of invariants, called Casimir invariants, that characterize the dynamics. [START_REF] Morrison | Poisson brackets for fluids and plasmas[END_REF] Adding these Casimirs to the Hamiltonian of the system results in a functional whose stationary points describe equilibrium states and coherent structures. [START_REF] Morrison | The free principle, negative energy modes, and stability[END_REF] Note, however, that there may exist additional equilibria that do not correspond to stationary points of the Hamiltonian-Casimir functional. This occurs, in particular, at points where the Poisson bracket changes rank. [START_REF] Morrison | Poisson brackets for fluids and plasmas[END_REF] The existence of non-trivial (e.g. 2D) equilibrium states is essential for physical applications. In the absence of knowledge of the Hamiltonian structure, however, the existence of such equilibrium states can be difficult to ascertain for physically sophisticated models, such as many of the fluid models that are used to describe the dynamics in fusion plasmas (see, for example, [START_REF] Hazeltine | [END_REF][16][17][18][19][20][21]). Even simple models that otherwise appear physically compelling can fail to be Hamiltonian. For example, Ref. [8] shows that a simple modification of the Hasegawa-Mima model [22], aimed at improving the description of zonal flow dynamics, lacks any equilibrium solution with a non-monotonic electrostatic potential. When a Hamiltonian structure can be identified, by contrast, equilibrium states are readily found.

The stability analysis depends on whether the equilibrium of interest is a stationary point of the Hamiltonian-Casimir functional. In the affirmative, convexity arguments may serve to demonstrate nonlinear stability. When convexity cannot be established, the analysis of the second variation of the Hamiltonian-Casimir functional may still provide information about the "energy" stability (which implies linear stability) [23]. For more general equilib-sometimes referred to as "dynamically accessible" perturbations. [23] The "Energy principle" of ideal magnetohydrodynamics is a well-known example of this.

Irrespective of the utility of Hamiltonian formulations, there is a fundamental argument for their necessity. All plasma models aim to describe charged particles interacting with electromagnetic fields, a system that obeys Hamiltonian dynamics. It is thus desirable that any reduced description of this system inherit a Hamiltonian substructure.

An important problem in plasma physics is the description of phenomena such that the effects of finite ion temperature plays a central role. This includes, for example, magnetohydrodynamic instabilities such that the growth rate is comparable to the ion diamagnetic drift frequency and localized modes with wavelengths across the field that are comparable to the ion Larmor radius. The construction of models that account for the effects of ion temperature, have a form compatible with the gyroviscous cancellations and satisfy the Hamiltonian property is difficult. For instance, the four-field model of Ref. [START_REF] Hazeltine | [END_REF] conserves energy, but efforts to identify its Hamiltonian structure have been unsuccessful. Hazeltine et al. [24] have constructed an alternative Hamiltonian model by using a mapping technique to introduce finite ion temperature into the cold ion limit of the model of Ref. [START_REF] Hazeltine | [END_REF], for which a Poisson bracket is known. The resulting system, known as the Hamiltonian four-field finite Larmor radius (FLR) model, describes the evolution of the density, the parallel ion velocity, and the magnetic and electrostatic potentials. Unfortunately, their model involves third-order derivatives of the current and the vorticity, raising questions as to the nature of suitable boundary conditions. Perhaps because of this, it has not been applied or studied further since its inception.

A few years after the formulation of the four-field model, research on turbulent ion thermal transport in fusion experiments led to the discovery of a new class of fluid models obtained by truncating the moment hierarchy of the gyro-averaged kinetic equation, instead of the original kinetic equation. [16][17][18]21] These models, known as gyro-fluid models, rely on nonlocal operators to describe the effect of Larmor gyration on the response of the ions.

As a result, they provide far superior approximations to the kinetic dispersion relation for linear waves. In particular, they describe accurately the propagation of disturbances with wavelengths smaller than the Larmor radius, in contrast with so-called FLR (Finite Larmor Radius) models, such as the four-field model of Ref. [24], that are based on the assumption that the wavelengths are much longer than the Larmor radius.

In the present paper, we construct a noncanonical Hamiltonian formulation for a simple electromagnetic gyrofluid model. Like the model studied in our previous work, [25] the model that we investigate in the present paper is a truncation of the much more complete model proposed by Snyder and Hammett that advances 6 moments for the ions and two moments for the electron dynamics. [21] Two new features in the present work are that (1) we retain the effect of ion compressibility, thereby enabling the description of sound waves and drift Kelvin-Helmholtz instabilities [26] and (2) we account for the role of magnetic curvature, thereby enabling our model to describe geodesic acoustic modes (GAM). [27,28] The present paper is organized as follows. Sec. II introduces the dynamical model that we wish to find a Poisson bracket for, and presents the energy. In Sec. III we construct the Poisson bracket for this model. In Sec. IV we calculate the Casimir invariants, and in Sec. 25 we introduce the normal variables suggested by the Casimirs, and use these normal variables to show that the Poisson bracket satisfies the Jacobi identity. Lastly, we summarize our findings in Sec. VI. Appendix A contains an explicit proof of the Jacobi identity for our Poisson bracket.

II. FORMULATION

We are interested in a model that describes the interaction of kinetic Alfvén waves with drift-acoustic waves. In order to describe both the "inertial" (β e ≪ m e /m i ) and the "kinetic" (β e ≫ m e /m i ) regimes of the Alfvén wave, we leave β e unrestricted. [29] (β e is the ratio of the electron kinetic pressure to the magnetic pressures). For β e ∼ m e /m i (v te ∼ v A ), however, our equations must be modified to account for electron Landau damping. We consider the evolution of the density n i and the parallel velocity of the ion guiding centers

u i = ẑ • v i ,
the electron density n e and parallel velocity u e = ẑ • v e , and the magnetic flux A = ẑ • A where A is the vector potential. The quasi-neutrality condition determines the electrostatic potential φ. We normalize these quantities according to

(n i , n e , A, φ, u i , u e ) = L n ρ s ni n 0 , ne n 0 , Âz ρ s B 0 , e φ T e0 , vi c s , ve c s ,
where the carets denote the dimensional variables. Here n 0 is the background density, ρ s = c s /ω ci where c s = (T e /m i ) 1/2 is the sound speed, ω ci = eB 0 /m i is the ion cyclotron frequency, and L n = n 0 /|∇n| is the density scale-length. We also normalize the independent variables according to

(t, k , k ⊥ ) = ( tc s /L n , k L n , k⊥ ρ s ).
Normalizing the perpendicular lengths to ρ s has the advantage of making the τ = T i /T e → 0 limit transparent.

We obtain our evolution equations from the equations of Ref. [21] by assuming constant temperatures and discarding all but the lowest two parallel moments for the ions and electrons. We also discard all the magnetic pumping terms, proportional to ∇ log B, which involve transfers between the parallel and perpendicular thermal energy. The equations that follow from these choices are

∂n s ∂t + [Φ s , n s ] + 2v d [x, τ s n s ± Φ s ] + ∇ s u s = 0; (1)
µ s ∂u s ∂t + [Φ s , u s ] ∓ E s + τ s ∇ s n s + 2λµ s τ s v d [x, u s ] = 0; (2)
where the upper (lower) sign should be taken for the ions (electrons), s = i, e denotes the species index, τ s = T s /T e , µ s = m s /m i , v d is the curvature drift and λ is a numerical coefficient that is calculated to be 2 in Ref. [21]. We will see below that in order for the two-moment system to be Hamiltonian, only half of this term should be retained: that is, one must choose λ = 1. The same conclusion was reached for the FLR version of the fourfield model in Ref. [24], as can be seen by comparing the curvature term in the Hamiltonian version of the parallel momentum equation to that for the version obtained through an FLR expansion in Ref. [30].

The remaining symbols in the above equations are the parallel gradient, ∇ s ξ = ∂ z ξ + [ξ, A s ] for any field ξ, and the electric field,

E s = - ∂A s ∂t -∇ s Φ s .
The fields Φ s and A s are the gyroaverages of φ and A. We will neglect the gyroradius of the electrons, so that Φ e = φ and A e = A. For the ions, by contrast, we take

Φ i = Γ 1/2 0 φ ≡ Φ; (3)
A i = Γ 1/2 0 A ≡ A. (4)
Here Γ

1/2 0 is an operator introduced by Dorland and Hammett [18] and defined by Γ

1/2 0 ξ = exp( 1
where I 0 is the modified Bessel function of the first kind and τ = τ i = T i /T e . The definition in Eq. ( 5) should be interpreted in terms of its series expansion

Γ 1/2 0 ξ = 1 + ∞ n=1 a n (τ ∇ 2 ⊥ ) n = 1 + (τ /2)∇ 2 + • • • ,
where the a n are real numbers. In collocation-based numerical solutions, the Bessel functions in Eq. ( 5) may be evaluated to required accuracy. For finite difference codes, by contrast, it is generally preferable to use a Padé approximant, such as

Γ 1/2 0 (b) = (1 + b/2) -1 ,
in order to simplify the necessary calculations of Γ 0 , Γ 1/2 0 , and their inverses. Ref. [18] provides a comparison of the accuracy of a selection of approximants of various orders.

The system is completed by the quasi-neutrality equation,

n e = Γ 1/2 0 n i + (Γ 0 -1)φ/τ, (6)
where Γ 0 = (Γ 1/2 0 ) 2 , and by Ampére's law,

∇ 2 A = β e 2 u e -Γ 1/2 0 u i . (7)
To facilitate comparison to earlier models including the cold-ion version of the four-field model [31] and the incompressible (3-field) gyrofluid model, [25] we rewrite the dynamical equations as

∂n i ∂t + [Φ, n i ] + 2v d [x, τ n i + Φ] + ∂ z u i + [u i , A] = 0; ( 8
)
∂M i ∂t + [Φ, M i ] + ∂ z (τ n i + Φ) + τ [n i , A] + 2λτ v d [x, u i ] = 0; ( 9
)
∂n e ∂t + [φ, n e ] -2v d [x, n e -φ] + ∂ z u e + [u e , A] = 0; ( 10
)
∂M e ∂t + [φ, M e ] + ∂ z (n e -φ) + [n e , A] -2λµv d [x, u e ] = 0; ( 11
)
where M i = u i + A and M e = µu e -A, with µ = µ e = m e /m i , are respectively the ion and electron momenta.

The above equations conserve the following energy:

H = 1 2 τ n 2 i + u 2 i + n 2 e + µu 2 e + 2 βe |∇A| 2 + Φ n i -φ n e . ( 12
)
where we have used the quasi-neutrality equation to simplify the result. Here, the angular brackets denote the integral over the volume of interest, g(x, y, z, t) = g(x, y, z, t) dx dy dz, and the boundary conditions have been assumed to be such that the surface integrals vanish. The successive terms represent, respectively, the ion thermal and kinetic energies, the electron thermal and kinetic energies, the magnetic energy, and the electrostatic energy of the ions and electrons. We note that H is conserved regardless of the value of λ.

III. POISSON BRACKET

We adopt n i , M i , n e , and M e as our dynamical variables. In terms of these variables the flux A is given by Ampére's law and the potential φ by the quasi-neutrality condition,

φ = τ (Γ 0 -1) -1 (n e -Γ 1/2 0 n i ), ( 13
)
The variations of the Hamiltonian are

H n i = τ n i + Φ; (14)
H M i = u i ; (15)
H ne = n e -φ; (16)
H Me = u e , (17)
where H ξ j is a shorthand notation for the functional derivative of H[ξ 1 , ξ 2 , ξ 3 , ξ 4 ] with respect to the fields

ξ 1 = n i , ξ 2 = M i , ξ 3 = n e , ξ 4 = M e : H ξ j := δH δξ j , j = 1, 2, 3, 4.
We may write the equations of motion in terms of the variations of H as follows:

∂n i ∂t = -∂ z H M i + [n i -2v d x, H n i ] + [M i , H M i ]; (18)
∂M i ∂t = -∂ z H n i + τ [n i -2λv d x, H M i ] + [M i , H n i ]; (19
)

∂n e ∂t = -∂ z H Me -[n e -2v d x, H ne ] -[M e , H Me ]; (20)
∂M e ∂t = -∂ z H ne -µ [n e -2λv d x, H Me ] -[M e , H ne ], (21)
The corresponding Poisson bracket is

{F, G} = s=i,e ({F, G} s + {F, G} ⊥s ), (22)
where

{F, G} s = -F ns ∂ z G Ms -G ns ∂ z F Ms (23)
and

{F, G} ⊥s = ∓ (n s -2v d x)[F ns , G ns ] + µ s τ s (n s -2λv d x)[F Ms , G Ms ] ∓ M s ([F Ms , G ns ] + [F ns , G Ms ]) , (24)
where the ∓ has the same meaning as in Eq. 2

In Sec. V, we will show that for λ = 1, this Poisson bracket satisfies the Jacobi identity,

J := {F, {G, H}} + {G, {H, F }} + {H, {F, G}} = 0 (25)
for all functionals F , G, H.

IV. INVARIANTS

The Poisson brackets characterize the geometrical properties of the dynamics that are independent from the Hamiltonian. Perhaps the most important of these properties is the existence of Casimir functionals, which are invariants of the motion that specify a foliation of phase space. [23,32] In the presence of an ignorable coordinate, these invariants usually appear in infinite families and become useful to construct variational principles for studying the equilibrium and stability of a system. In this section we assume ∂ z = 0: the generalization to the case of helical symmetry is straightforward and is described in Ref. [32].

We obtain the Casimir functional C[n i , M i , n e , M e ] by expressing the condition that {ξ j , C} = 0 for ξ 1s = n s , ξ 2s = M s . This results in the following system of equations:

[n s -2xv d , C ns ] + [M s , C Ms ] = 0; ( 26
)
µ s τ s [n s -2λxv d , C Ms ] + [M s , C ns ] = 0; (27)
Solving the first equation, Eq. ( 26), yields

C = s G s [n s , M s ] = s g s (n s -2xv d , M s )
where the g s are arbitrary functions. Substituting this result in the second equation, Eq. ( 27), we find that there is no well-behaved solution except if λ = 1. In the Appendix we show that indeed, for λ = 1, the bracket (24) does not satisfy the Jacobi identity and thus is not a Poisson bracket. Considering then the case λ = 1, we obtain

µ s τ s ∂ 2 Ms g s (n s -2xv d , M s ) -∂ 2 ns g s (n s -2xv d , M s ) = 0.
Integrating this using the method of characteristics, we find two pairs of independent solu-

tions G s = G s± [n s , M s ]
where

G s± [n s , M s ] = g s± (M s ± √ µ s τ s (n s -2xv d )) , (28)
and the g s± are two new arbitrary functions. The general form of the Casimir functional for

λ = 1 is thus C[n i , M i , n e , M e ] = s,± g s+ (M s ± √ µ s τ s (n s -2xv d )) . (29)
We will henceforth restrict consideration to the case λ = 1.

V. NORMAL FIELDS AND JACOBI

The form of the Casimirs suggests the introduction of a new set of variables, which we call "normal fields":

V s± = M s ± √ τ s µ s (n s -2xv d ).
In terms of these normal fields, the equations of motion take the form

∂V s± ∂t + [Φ s± , V s± ] = 0; (30)
where

Φ s± = Φ s ∓ A s τ s /µ s (31)
are stream-functions describing the flows that convect the fields V s± . The latter are thus Lagrangian conserved quantities. Grasso et al. [33] have shown that the mixing of such Lagrangian quantities by the convecting flows is an important ingredient in rapid collisionless reconnection. In this respect, it is worth noticing that a limit of this model [31], obtained by neglecting the ion gyroradius, possesses a Poisson bracket of a different type (the sum of direct and semidirect products), leading to only three Lagrangian invariants, two of which undergoing phase mixing.

The Poisson bracket of ( 22) can be written in terms of the normal fields. Dropping the species subscripts for clarity, the functional derivatives take the form

F n = (F V + -F V -) √ µτ ; (32)
F M = F V + + F V -. (33)
Inserting these functional derivatives into the brackets of ( 23) and (24) gives

{F, G} = 2 √ µτ ± ± F V + ∂ z G V + , (34)
and

{F, G} ⊥ = 2 √ µτ ± V ± [F V ± , G V ± ] . (35)
The above sums of independent brackets of the four variables are known as a direct product.

[34] Since the individual brackets satisfy the Jacobi identity, their sums in Eqs. ( 34) and ( 35) also do. By virtue of the coordinate invariance of the Jacobi identity, it follows that the parallel and perpendicular brackets given by Eqs. ( 23) and ( 24) also satisfy the Jacobi identity.

As stated in Sec. I, the identification of a Hamiltonian structure in the presence of both magnetic inhomogeneities and finite ion temperature is difficult, even if a conserved quantity for the model is known. [START_REF] Izacard | Gyromap for a two-dimensional hamiltonian fluid model derived from Braginskii's closure for magnetized plasmas[END_REF] For instance, if we had retained the magnetic curvature terms in the continuity equations ( 1), but neglected them in the momentum equations (2), as done in the partially linearized model of Ref. [19], the functional (12) would still have been a conserved quantity, but the resulting bracket obtained from (22) would have no longer satisfied the Jacobi identity.

VI. SUMMARY

We have presented a Poisson bracket structure for the four-field version of the gyrofluid equations of Snyder and Hammett, [21] Eqs. ( 1)-( 2),that we obtained by keeping only the first two moments of the gyrokinetic equations for both the electrons and the ions. We find that the system is Hamiltonian provided that the coefficient of the curvature term in the parallel momentum equation is halved from the value obtained in the six-moment expansion of Ref. [21]. We note that Hazeltine et al. [24] We see that these terms vanish for all F , G and H only if λ = 1. For λ = 1 then, the form (24) is not a Poisson bracket.

  reached a similar conclusion in the long we find{{F, G} ⊥e , H} ⊥e + {{G, H} ⊥e , F } ⊥e + {{H, F } ⊥e , G} ⊥e = µ(λ -1)2v d x([[F Me , G Me ], H ne ] + [[G Me , H Me ], F ne ]) + [[H Me , F Me ], G ne ] + • • • .

	(A7)

τ ∇ 2 ⊥ )I 1/2 0 (-τ ∇ 2 ⊥ )ξ,(5)

ACKNOWLEDGEMENTS E.T. acknowledges useful discussions with Olivier Izacard. This work was supported by the US DOE under contracts DE-FG03-96ER-54346, and by the European Community under the contracts of Association between EURATOM, CEA, and the French Research Federation for fusion studies. The views and opinions expressed herein do not necessarily reflect those of the European Commission. E.T. acknowledges also the financial support from the Agence Nationale de la Recherche (ANR GYPSI).

wavelength (FLR) limit. One may speculate that the omitted part of the curvature term contributes to the Poisson bracket for a more complete system that includes the contributions from the action of the curvature on the heat fluxes.

With the modification noted above, the bracket admits a Casimir functional, given in Eq. (29), that depends on four so-called normal fields, which are linear combinations of the canonical momentum densities and the guiding center densities. The dynamical equations for the normal fields take the form of simple Lagrangian convection equations, with convecting flows that derive from four different stream functions given by Eq. (31). The existence of a proper Poisson bracket describing the dynamics guarantees that the equilibrium equations obtained by neglecting the time derivatives (possibly in a frame of reference moving at constant velocity) may be integrated once to obtain Grad-Shafranov-like equations that determine the magnetic flux (i.e. the component of the vector-potential along the background field) and the electrostatic potential.

Possible applications of the electromagnetic gyrofluid model presented here include any electromagnetic problems where sound waves, diamagnetic drifts and nonlinear effects are important. This includes, in particular, the problem of the mutual interaction of magnetic islands and resonant magnetic perturbations with plasma turbulence. [36,37] APPENDIX A: VIOLATION OF THE JACOBI IDENTITY FOR λ = 1

In Secs. IV and 25 we showed that, for λ = 1, the bracket ( 24) is a Poisson bracket, because it can be mapped to the direct product form, which is known to satisfy all the properties of a Poisson bracket, and in in particular the Jacobi identity. In the following we show that, if λ = 1, the Jacobi identity is violated, thus leaving λ = 1 as the only admissible choice for obtaining a Hamiltonian structure.

Given that the bracket ( 24) is a linear combination of two independent bilinear forms, one for ions and one for electrons, we consider, only the electron part

although the same arguments can of course be applied to the ion part.

Note now that, in order for (A2) to hold, all the terms containing explicitly, n e , x, M e and second functional derivatives, respectively, have to group together and vanish separately.

Let us focus on the terms depending explicitly on x. We have

where the dots indicate all the terms depending explicitly on n e , M e and on the second functional derivatives. From the last expression one can see that the sum of three terms containing F ne , G ne and H ne vanishes by virtue of the Jacobi identity for the inner bracket [ , ]. Applying the "inner Jacobi" identity to the terms in the last two lines of (A6), however,


Jérôme Douzet

Pedro Brantuas

Jean Michel Herri
email: herri@emse.fr

BUILDING OF A LIFE SIZE TESTING UNIT FOR AIR CONDITIONNING BY USING TBAB HYDRATE SLURRY AS A SECONDARY TWO-PHASE REFRIGERANT

Keywords: gas hydrates, quaternary ammoniums, phase change materials

Building of a life size testing unit for air conditionning by using TBAB hydrate slurry as a secondary two-phase refrigerant

Jérôme Douzet, Pedro Brantuas, Jean-Michel Herri

INTRODUCTION

To answer the modern requirements of comfort, the systems of air conditioning are spreading continuously, from the public room to the private houses, passing through the collective buildings. However, such a development has to take into account environmental and economical stakes. The principle of phase change materials (PCM) is very interesting [START_REF] Oyama | Phase diagram, latent heat, and specific heat of TBAB semiclathrate hydrate crystals[END_REF][START_REF] Obata | Advanced technologies towards the new era of energy industries[END_REF], in terms of environmental stakes: the use of phase change materials allows a reduction of the use of CFC, HCFC and HFC refrigerants, which have a heavy ecological impact. In term of energy, this system harnesses the phenomenon of melting of the solid phase and involves a melting latent heat higher than the sensible heat of a single phase. And finally, the cost of the electrical energy subscription can be smoothed over 24 hours because the slurry is accumulated during the night in an isolated tank. Phase change materials based on ice slurries have been developed by HeatCraft Europe [START_REF] Lakhdar | Facing the challenge to produce ice slurry for freezer applications. 5th Ice slurry workshop[END_REF][START_REF] Lakhdar | Heat transfer with freezing in a scraped surface heat exchanger[END_REF] because of their potential in refrigeration (negative temperature). To that aim, the fluid is composed of water that can crystallize to form ice and the melting point is adjusted by adding appropriate additives, such as alcohols. But, for positive applications, the ice slurry technology is not longer applicable because the melting point cannot be adjusted to positive temperature. In the present work, the PCM is a slurry based on the crystallization of a quaternary ammonium (Tetra-Butyl Ammonium Bromide) that combines to water to form a so-called semi-clathrate, whose melting point can be adjusted up to 12°C. Semiclathrate is material in which the guest is part of the water lattice. We propose here to experiment this solution at industrial scale in order to test the technology and check the energy balances. Also, we are interested in our capacity to handle the slurry, which means to crystallize it, to store it, and to distribute it. This needs to model the system. In a previous work, Darbouret et al. [START_REF] Darbouret | Rheological study of TBAB hydrate slurries as secondary two-phase refrigerants[END_REF] have described and modeled the rheological properties of the slurries. In our work, we focus on the storage tank which is the place of sedimentation. A numerical model is proposed to predict the evolution of the hydrate stock and the available cooling power in function of the tank geometry and the operating conditions. It takes into account hydrate settling, fluid circulation, hydrate formation (in generator) and hydrate melting (in utilization loop and because of heat losses). It can be used to optimize the operating conditions and the geometrical parameters of the hydrate slurry storage tank.

METHODS AND MATERIALS

Principal of the TBAB hydrate slurry airconditioning system

The thermodynamic properties of an aqueous solution of TBAB are well adapted for an airconditioning application. Indeed, used as a secondary refrigerant, it can crystallize to form a slurry in which the phase change material is a semi-clathrate compound. With Tetra-n-butyl ammonium bromide, the crystallization temperature can be raised up to 12°C. On the figure 1 we can see the crystallization temperature in function of TBAB mass fraction. The difference between A and B types hydrates consists in the value of hydration number N (respectively 26 and 38). The hydration number is the number of water molecules associated with one molecule of TBAB to form one hydrate.

Figure 1: Phase diagram of TBAB hydrates [START_REF] Darbouret | Rheological study of TBAB hydrate slurries as secondary two-phase refrigerants[END_REF].

We built a real size prototype, which stands both for a testing stand and for a life-size application. A compression cycle refrigeration unit (with a condenser installed outside the buildings) allowed via a scraped surface heat exchanger (cooled down by the evaporator) to create TBAB hydrates slurries which were stored in a tank during the night. In the daytime, a pump supplies the slurry to the air conditioning heat exchangers. The capacity of our installation corresponds to the airconditioning of 4 rooms situated on 3 different floors. Since 2005 the JFE Engineering Corporation (Japan) has shown the feasibility of large-scale utilization of clathrate hydrate slurry for air conditioning [START_REF] Mizukami | Thermal Energy Storage System with Clathrate Hydrate Slurry[END_REF]. Our research aims to adapt a refrigerating technology for air conditioning in order to give us the know-how and an expertise at an industrial scale. The principle of our technology consists in using different loops.

The primary loop:

The compression cycle refrigeration unit is connected to the hydrates generator via a refrigerant fluid (R-407C). The hydrates generator crystallizes the aqueous solution of TBAB by recovering the R407C vaporization enthalpy. This compression cycle refrigeration unit is a standard model from LENNOX ®. It is located outside the building. Its cooling capacity is 29,3kW (Electricpower: 13,8kW). It is classically composed of a scroll compressor and a condensing unit, i.e. a heat exchanger that condenses the R407C fluid at ambient temperature. Then the liquefied R407C fluid is directed to the hydrate generator after passing through an expansion valve.

The generating loop:

The aqueous solution of TBAB replaces traditional secondary refrigerant fluids. The hydrates generator transforms the aqueous solution of TBAB into slurry. This hydrates generator is a scraped surface heat exchanger with a capacity of approximately 200L. It was supplied to us by the company HeatCraft (France, 69 Genas) and is usually used to produce ice slurries in the refrigeration field (Ben [START_REF] Lakhdar | Facing the challenge to produce ice slurry for freezer applications. 5th Ice slurry workshop[END_REF][START_REF] Lakhdar | Heat transfer with freezing in a scraped surface heat exchanger[END_REF]. The slurry is stored in an intermediate tank during the night (when electricity is cheaper). The storage tank communicates with the hydrates generator. The liquid solution is sucked at the top of the storage tank where only liquid is present. In fact, the hydrate particles have a density higher than the liquid, and they settle into the tank. Liquid remains at the top of the tank whereas solids accumulate at the bottom. This tank is approximately 2,5m height and its volume is of about 2m3. It is entirely made of stainless steel.

The distribution loop:

During the day, the slurry is pumped from the storage tank to the secondary loop (by pumps P1 and P2 on the figure 2) and it is melt in heat exchangers located in the room to condition. The secondary flow loop constitutes the distribution network of the slurry. The slurry is sucked from two different levels of the storage tank. The pump P1(lobes pump) imposes the flow rate. Because it sucks the slurry at the bottom of the storage tank, the solid concentration can be important, up to 40-50% in volume. In order to dilute it, a secondary pump P2 sucks the liquid at the top of the storage tank and injects it just before the pump P1. This allows controlling the volume fraction of solid particles in the slurry of the distribution loop. A volume fraction of 20% was chosen so that viscosity is moderate. The length of this loop is approximately 100m. A flow meter (Coriolis flow-meter from Micro Motion © F) is installed on this loop. The heat exchangers are standard models from LENNOX ®. They are connected on the secondary flow loop with brass pipes, using a little pump installed on each of them. The heat exchangers cool down the air of the room by melting the slurry. 3 floors are conditioned, by the use of 7 heat exchangers (3 on the 1st floor, 2 on the 2nd floor and 2 on the 3rd floor). One of these exchangers is instrumented in order to take measures and to model heat exchanges.

Figure 3: Scheme of the full size installation and location of sensors.

Compr ession cycle

The sensors:

On the figure 3 we can see the localization of some sensors.

Temperature sensors are Pt 100 type. Differential pressure sensors allow measuring the pressure drop during the slurry's circulation this allows to estimate the hydrate fraction in the slurry [START_REF] Darbouret | Rheological study of TBAB hydrate slurries as secondary two-phase refrigerants[END_REF].

The absolute pressure sensor located near the storage tank is used to verify the slurry height.

System control

For practical reasons, the generation of the hydrate slurry is controlled manually for the moment.

The distribution of the slurry in the rooms is controlled by end-users. In each room, when an end-user activates a heat exchanger, it activates simultaneously the two circulation pumps (P1 and P2).

MODELLING

The experimental study is very time consuming so only a very limited number of configurations (TBAB concentration, operating conditions, geometry of the tank, location of the tank inlets and outlets …) can be investigated. In order to optimize the system a modeling approach was developed to study a large number of configurations by numerical simulation. One of the crucial phenomena is the settling of the hydrate particles inside the storage tank. Because the hydrate density is higher than that of the solution, hydrate tends to be more concentrated at the bottom of the tank. This can be an advantage because by adjusting the flow rates coming from the bottom and the top of the tank (via the pump P1 and P2), it is possible to adjust the hydrate concentration of the slurry which is supplied to the air-conditioning heat exchangers. The constitutive equations are concerning the mean velocities of the two phases at a given height, the balance of hydrate particles, the enthalpy balance, the equilibrium curve (figure 1) and the conservation equation of TBAB. The inlets and outlets of the tank correspond to source and sink terms for the conservation equations, which are calculated from the hydrates generation and utilization operating conditions. These constitutive equations were discritised by a semi-implicit finite volume method. Numerical resolution was performed with Matlab ®. The model will be compared with experimental results for validation before its use for system optimization.

PRELIMINARY RESULTS

Handling of the hydrate slurry

Conforming to the results of Darbouret et al [START_REF] Darbouret | Rheological study of TBAB hydrate slurries as secondary two-phase refrigerants[END_REF], the distribution of the TBAB hydrate slurry within the secondary loop has not generated difficulties. The volumetric pump (lobes pump) works with the slurry without any difficulty. Concerning the centrifugal pumps, we observed that they work efficiently up to a solid content of 20% vol.

Hydrate generator

The scraped surface heat exchanger was able to produce hydrate slurries with a solid content up to approximately 20 % vol. Figure 4 presents the temperature evolution at inlet and outlet of the hydrate generator during a typical run where the TBAB solution circulates only in the closed loop between the storage tank and the generator, the solution is at eutectic concentration and the initial temperature is 21°C in the tank. During about one hour, the outlet temperature decreases but remains higher than the equilibrium change phase temperature (12.3°C). During this step, the solution remains totally liquid and it is possible to estimate the transmitted power from the temperature and flow rate measurements as follow:

( )

outlet inlet solution TBAB solution TBAB solution TBAB T T Cp Q P - ⋅ ⋅ = (1)
where Q TBAB solution is the TBAB solution's flow rate and Cp TBAB solution is heat capacity. The transmitted power decreased from 20 to 7 kW during this first hour.

During a second step, the outlet temperature decreased significantly below 12.3°C, which means that the solution was of thermodynamic equilibrium. For about 20 min, the temperature decreased (from 12.3°C to 11.5°C) in a similar way as previously and without hydrate generation. Then crystallization began, and the outlet temperature increased progressively up to 12.3 °C.

After about 2 hours, the inlet temperature reached also the phase change temperature. That means that the hydrate generator is fed with a fluid containing already hydrate particles. From this point, the hydrate concentration in the secondary loop (tank, hydrate generator) increases progressively. Consequently, the apparent viscosity and the mechanical friction increases inside the scraped surface heat exchanger. Finally after 5 hours, the rotation of scrapers in the crystallizer is blocked and hydrate generation is stopped. This phenomenon could be explained by the formation of a compact hydrate layer on the heat exchanger plates. It becomes to much thick and difficult to scrape. To prevent the blockage, a solution could be to reduce the temperature difference between the primary refrigerant evaporating temperature (-30°C in our experiment) and the hydrate formation temperature (12,3°C). Another possible solution is to install a system which periodically warms up the hydrate generator in order to eliminate the hydrates accumulated on the plates.

Hydrate storage in the tank

We didn't encounter any problem in the storage of the hydrate slurry. The slurry is maintained in its original state during the storage and it does not agglomerate to form a plug. It is interesting to note that during the winter time and because the ambient temperature decreases at a value lower than 12.3°C, the crystallization can occur spontaneously in the quasi totality of the tank without damaging it. But we observed that pipes can be damaged during melting (molar volume of liquid is superior to solid) Settling explains that hydrate concentration is higher at the bottom of the tank and tends to the maximum value of 50% which is a modeling parameter (volume fraction of packed solid particles). The simulation shows that the volume fraction of particles at the top of the tank is in the same order of magnitude than near the bottom. This can be explained by the upward flow in the tank (from h inlet =0.4 m to h outlet = 2m) which compensate the settling phenomena. Indeed, for this concentration of TBAB (35%) the difference of density between the liquid and solid phase is only about 4% so that the settling velocity is low.

Prediction of the hydrate concentration in the tank

A practical consequence is that particles are always present at the top of the tank, in concentration which is never negligible. So, the flow which is sucked at the top of the tank and is directed to the crystallizer contains particles, with a volume fraction which increases with time. From an experimental point of view, we have observed this tendency: by sampling the flow before entering the crystallizer we observed an increase of the volume fraction of particles. As a consequence, the viscosity of the slurry increases and the flow rate of the centrifugal pump decreases down to zero. This is an important phenomenon for the design of the pump P3. We have changed it to the profit of a volumetric pump for which the flow rate is less sensitive to the fluid viscosity in the loop. We are now in the process of building a new installation for the production of TBAB hydrates. This pilot consists in both capturing CO 2 by hydrate formation and using the formed hydrates for cooling. This is based on the existing setup (Figure 6) where we add a new reactor that is located directly above the hydrates storage. The apparatus consists of a bubble column reactor where CO 2 and the TBAB solution comes into contact and forms mixed hydrates of CO 2 + TBAB. The hydrates is then stored until being re-used for the cooling application. The gas is compressed and returned to the bubble column to continue the production of mixed hydrates. This pilot was built with the objective of producing 50 kg/h of mixed hydrates with a gas flow of 2.5 m 3 /h and a liquid flow of 1 m 3 /h. The reactor works at about 20 bar and 15°C.

NEW CO 2 CAPTURE PROCESS

Equipments

Figure 7 is the picture of our current pilot located above our other prototype. In left we have our gas compressor/circulator system which consists in 2 ballasts with a volume of about 50 liters and a gear pump in beetween (Figure 8). The CO 2 is provided by 2 bottles situated beneath the gas circulating system. By sucked water from first ballast to the other one, we can push gas to the reactor. We have pressure controllers coupled with automated valves which allows us to have a continuous pumping of gas towards the reactor and extraction from it.

Figure 8: View from the top of the gas circulating device.

In the bottom right we shows our reactor which volume is 50 liters: The reactor has a body made of PMMA to be transparent. It is 2000mm height and diameter of 188mm (Figure 9). The reactor is transparent in order to have a better assessment of gas dispersion phenomena inside the bubble column. After optimization of the gas dispersion system, it will be replaced by a body in 304l steel for having the adequate resistance to the pressure required for the hydrate formation conditions. There is a volumetric pump designed for pumping the hydrates formed between the reactor and the hydrate storage. This pump is placed next to the hydrate storage in a vertical position (Figure 9).

At right of the photo(figure 10) we present the heat exchanger which has 4 meters long and consists of 2 concentric tubes with a counter current flow connected to a cooling device with a power of 3 kW (Figure 10).

G AB L L L L T L L C C AB L C G L L C L C G G ε D ρ µ µ g ρ D σ g ρ D D D a k gD V µ ρ gD σ g ρ D α ε ε                         =                 = - [7]
Hikita et al

AB L L G L G L I L G L G L L L G G D V g a k f l ion g I g V f - - - = = ⇒ ⋅ < <                               = σ µ µ ρ µ µ ρ ρ σ ρ µ σ µ ε
- - - - -                                 × + × ×                         =                 - +               = - L L G C L L C C t S S G L L L AB L L L AB L L L G C L L S S L L L G G G V D g D g D V C g D g D a k V D g V k µ ρ σ ρ ρ ε σ ρ µ ρ µ σ ρ ρ ρ ρ ρ ϕ σ ρ µ σ µ ε ε [9] Sauer & Hempel ( ) ( ) ( ) ( )
+ + + = =                         =         =                         = - - [10] Chisti ( ) ( ) G L G G L L b L S L L AB L b L a k a k d k C D g d k ε ε ε µ σ ρ 6
        + × = = ×         = × = - - - × ×         +         -         × = - - C C AB S G G G L L L L α d W S SM S S C C v T T G L L G L G D D T D d V ε ρ σ µ ρ x a k e d N K X d ρ D D P P P V σ µ ρ ρ ε [12]
Table 1: List of equations from literature for gas holdup and k L a for three phase systems in bubble column reactors.

Objectives

Figure 11 shows our gas dispersion tube and also our transparent column with a water solution and air bubbling during our test phase. The presence of three different phases inside the column: liquid (TBAB solution), gas (CO 2 ) and solid (mixed hydrates) increases the complexity. Especially, bubble columns are equipments which are difficult to design and scale-up due to their complex hydrodynamic behavior. [11, 12, 13 and 14]. A review of the literature concerning three phase system in bubble columns was done in order to find what parameters are determinant in this type of process. Two important parameters are gas holdup (ε G ) and the mass transfer coefficient between the liquid and gas phase (k L a). The gas holdup is the fraction of gas in the liquid phasethe gas bubbles. By analyzing some of the equations from the literature concerning three phase systems in bubble column reactors we have found out that the physical properties of the liquid and solid phase have a strong influence in the gas holdup as well as the gas injection speed into the column. As for the k L a is the gas holdup that shows a bigger influence followed by the gas injection speed and the physical properties of the liquid phase (viscosity, density and surface tension).

CONCLUSIONS

Our experiments demonstrate the feasibility of an air-conditioning system using TBAB hydrate slurry as secondary refrigerant. Some technological problems have still to be solved but different solutions are proposed and will be now tested. A model was developed in order to predict the amount of hydrate and its repartition in the tank. A comparison between numerical and experimental results will be carried out to validate the model.

Then the model will be used to simulate new alternatives (other TBAB mass fractions, other inlet and outlet positions, etc.). The best predicted configuration can finally be tested experimentally.

Figure 2 :

 2 Figure 2: Basic scheme of the full size installation.

Figure 4 :

 4 Figure 4: Temperature of slurry at inlet and outlet of the hydrates generator.

Figure 5 :

 5 Figure 5: Prediction of hydrate concentration in the tank.

Figure 5

 5 Figure 5 presents an example of the predicted evolution of hydrate volume fraction in function of the height position in the tank for different instants during the generation time (night) for a TBAB concentration of 35%. It can be seen that hydrate concentration increases progressively in the tank. Settling explains that hydrate concentration is

Figure 6 :

 6 Figure 6: Our real size prototype (generating loop).

Figure 7 :

 7 Figure 7: The CO 2 capture pilot.

Figure 8 :

 8 Figure 8: Bubble column reactor.

Figure 9 :

 9 Figure 9: Liquid solution pump.

Figure 10 :

 10 Figure 10: Heat exchanger.

Figure 11 :

 11 Figure 11: Flow of air in a water solution.

  geometry and the operating conditions (generator and utilization power and flow rate). This needs in fact to predict the evolution of the hydrate slurry state (hydrate fraction, temperature, composition of the residual solution, mean diameter of the hydrate particles) at each height in the tank in function of time. The main hypotheses are the following: The solution flow and the hydrate settling are vertical. The slurry is homogeneous in a given horizontal section. The volume variation related to hydrate melting is not taken into account and the total height of slurry in the tank is supposed constant (the density difference between the liquid and the solid phase is approximately 40 kg/m3). Nevertheless, the buoyancy forces, depending on density difference, are taken into account in the settling law (Stokes law). The volume of slurry in loops is neglected in comparison with the volume of the tank.

	Our first
	experiments were conducted with an eutectic
	mixture (molar fraction of water corresponding to
	the hydration number) this is convenient because
	melting of the hydrate does not change the TBAB

concentration in the solution, but melting occurs at the given eutectic temperature of 12.3°C. However the model was developed to predict variation of TBAB concentration, settling and melting temperature. More generally, it was developed to predict the evolution of the hydrate stock and the available cooling power in function of the tank

ACKNOWLEDGEMENTS

This study benefits of the financial support of the Rhône Alpes region in France.