Favergeon Loïc

Pijolat Michèle

Soustelle Michel

CIN4 : a multiphysics software for heterogeneous reactions at a reactor scale

Keywords: kinetics, simulation, solid-gas reactions, reactor, aggregates

A rigorous simulation of industrial reactors in the case of solid-gas reacting systems is a complicated task due to several difficulties linked to the kinetic problem at the scale of the solid grains and to the problem of gas and heat transfers within the powder bed. Firstly it requires the knowledge of the kinetic model for the calculation of the kinetic rate of reaction in one part of the reactor and for given conditions of temperature and gas composition, and secondly it necessitates solving the material and heat balance equations for the flow conditions settled in the reactor. The first part of the article is devoted to the presentation of the kinetic models governing a population of grains transformed in the same conditions of temperature and gas composition. Then a short presentation of CIN4, a multiphysics software resulting from the collaboration between ASTEK and EMSE, shows the various scales of the calculations: CIN4 offers the resolution of the thermohydraulic equations combined with the kinetic laws which describe the heterogeneous reactions, including the diffusion aspects in the case of aggregates. Then two examples of application are given in order to illustrate the interest of this multiphysics modeling: the dehydroxylation of kaolinite and the carbonation of CaO by CO 2 .

Introduction

The simulation of heterogeneous reactors has been done in the past using simplifying assumptions as regards to the law expressing the kinetic rate of reaction of a powder which transforms in conditions of constant temperature and gas composition. These assumptions are based on a kinetic law typically of the form:

) ( ) ( α α f P T k dt d = (1)
where k(T) usually follows the Arrhenius law, P is the pressure of the reacting gas and f(α) is a function of the fractional conversion α. The function f(α) most often corresponds to the core-shrinking model applied to various grain geometries. The order respective to the gas is chosen equal to 1. Such assumptions allow using the law of additive reaction times for solving numerically the problem of reacting porous media. However, as a result of about 20 years of experience in kinetic studies of solid-gas reactions in the Centre SPIN of EMSE, it must be recognized that most often, such assumptions are too much restrictive, so that the law of additive reaction times is not valid. Various reasons may be invoked such as a complex dependency of the kinetic rate with the temperature and pressure, several gases are involved in the reaction and influence the kinetics in different manners, the kinetic model equation does not vary according to a f(α) function, etc … In order to overcome the difficulty due to the inadequacy of the law of additive reaction times, we decided to collaborate with ASTEK which previously developed the FLUX-EXPERT software tool applicable to homogeneous reactors, in order to solve the problem of gas-solid reactions in porous media for a large variety of kinetic models including those corresponding to equation (1). ISSN : 1775-335X ;ISBN : 978-2-910239-78-7, Ed. SFGP, Paris, France 104-2

In the following, we present first in section 2 the kinetic models based on a much more general equation such as equation ( 2):

( ) 0 , , , r t S dt d m γ φ φ α = (2)
in which: φ (in units of mol m -2 s -1 ) depends of the thermodynamic conditions only (temperature, partial pressures, etc…) and is called the "areic reactivity of growth", ( )

0 , , , r t S m γ φ
is function of the initial and current morphology of the sample (r 0 represents a characteristic dimension of the initial grains) and therefore depends of time, whereas it also depends of φ, and according to the model, it may depend of the "areic frequency of nucleation" γ (in units of nb. of nuclei m -2 s -1 ).

Then section 3 shortly describes the multiphysics CIN4 software structure, since the mass and heat balance equations and the assumptions settled for solving them were previously detailed by Favergeon et al. (2009). In a last part, section 4, we present examples of application of CIN4 and we discuss the results and the possible improvements to be done in the future before concluding.

The kinetic models at the grains 1 scale

The various scales for modeling

Several geometrical scales characterize all phenomena which occur in a reactor. A rigorous description of our modeling process requires a precise definition of these different scales. We have deliberately set these scales from the objects: mesh, grains, grains population, aggregates and reactor. Four scales will be discussed in this work: grain, grains population, aggregate and reactor. At the grains scale, in the case of solid-gas reactions, it is well admitted that the nuclei appear at the grain surface 1 and grow at the expense of the initial solid phase: the kinetics is governed by the shape of the grains, the type of growth, the localization of the rate determining-step of growth, and the direction of development of the new phase (inwards, outwards). Then a "micro" scale population is a set of grains in the same thermodynamic conditions, i.e. the gradients of temperature and gas partial pressure are extremely small inside the population. At the reactor scale, which corresponds here to the "macro" scale, the chemical reaction is coupled with mass and heat transfers (this case will be illustrated by the kaolinite dehydration reaction). An intermediate scale called "meso" has proven to be useful in the case of powders made of porous aggregates inside which the molecular diffusion of gases may not be neglected (this will be the case of the CaO carbonation reaction).

The kinetic models for calculation of the rate at grains population scale

As shown by Pijolat et al. (2008Pijolat et al. ( , 2011)), in order to model reactions which involve surface nucleation and growth of the nuclei, it is interesting to use equation ( 2) in which the rate is expressed as the product of two functions, φ and S m , which means that the value of the rate at any time and thermodynamic conditions requires to know the expressions of these two functions. For constant temperature and gas composition, the S m function can take various expressions calculated from assumptions on the growth type (isotropic or anisotropic), the development direction of the new phase (inwards or outwards), the localization of the rate determining step of growth (external interface, internal interface, diffusion) and the grains shape (spheres, cylinders, plates). Soustelle (2006) detailed the method for deriving the laws and the expressions of S m for each model which result from the combination of the previous assumptions, which leads to more than 40 kinetic models. These models can be simulated using the free CIN3 software according to Pijolat et al. (2011). They have been incorporated in the part "micro" of CIN4. Moreover a numerical procedure allows fitting the experimental kinetic data in order to find the best values of φ and γ for any experiment done at constant temperature and gas composition. These models can be divided in two families: those which involve only one process, surface nucleation or growth, and those which involve simultaneously both processes. In the case of two processes family, the models are derived at the grains population scale: both surface nucleation and growth are taken into account, which leads to two kinds of models according to the growth type (isotropic or anisotropic). Two methods are used to calculate the fractional conversion and the rate of reaction: -in the case of isotropic growth of the nuclei, we use a statistical approach based on the works of Johnson and Mehl (1939) and Mampel (1940), then developed by Delmon (1969).

-in the case of anisotropic growth, each grain starts its transformation at various times and the whole microscopic fractional conversion can be obtained from an integral over all the dates of nucleation, considering the nuclei appeared on each grain of the grains population.

For both types of nucleation and growth models, the rate equation involves the ( )

0 , , , r t S m γ φ
which cannot be expressed using a f(α) function. Such a difference in the rate equation between one process and two processes models explains why the classical approach (law of additivity reaction times) may be applied to one-process models while it cannot in the case of two-process models. In this last case, the kinetic rate of reaction depends both on γ and φ which vary independently with temperature and gas composition, so it will be necessary to numerically follow the behavior of all the grains populations vs. time in each part of the reacting bed. This method can be compared to a population balance but here there is no fluxes from a grain size class to another. The calculations corresponding to the kinetic models are available in the part of CIN4 called "micro", which, in addition, enables to account for grain size distribution and successive reactions.

Mathematical modeling

The assumptions of the model

Six assumptions are done for the various scales of the modeling: the production and consummation of gas by the reaction do not disturb the total pressure distribution; the gases are diluted in a carrier gas; the gases flow rates are low; the gases are considered as ideal; the growth process can be written by a succession of elementary step and one of them controls the growth rate (rate-controlling step approximation); the grains population is supposed uniform in shape.

Heat and mass transport at reactor scale

The macroscopic scale is the center of heat and mass exchanges that are described by the thermal and hydrodynamic models, as well as by a mass transfer equation. The various equations corresponding to the thermal model, hydrodynamic and mass transport models can be found in a previous article (Favergeon et al. ( 2009)) so they will not be detailed here.

Coupling between the microscopic population scale and the reactor scale

At microscopic scale, the reaction fractional conversion is followed for a representative population of grains. By finite difference it is possible to evaluate the reaction rate dt dα . This rate allows to calculate heat and mass sources produced by the reaction. Thus the heat source density is equal to:

∑ = ∆ - = g n i i mA A i T H dt d V M Q 1 1 α ε ν (3)
And the mass (partial pressure) source density for a gas i is equal to:

dt d V Q mA i C i α ε ν - = 1 (4)
So, using these sources terms, microscopic reactions have an impact on the spatial and temporal evolution of the thermodynamic processes at the reactor scale. Inversely since it modifies the areic frequency of As previously noted, the CIN4 software also allows to take into account an intermediate scale when grains are structured in aggregates. In this case, aggregates are considered spherical, with the same radius, and isotropic, which means that inside thin crowns of the aggregate, all the grains behave in the same way, with identical temperature and partial pressures at a given time. Furthermore, inside the aggregates, only diffusion accounts for heat and mass transfers (convection is neglected). The coupling is realized by the combination of the flow modeling at the aggregate scale based on the finite difference method, with the kinetics of heterogeneous grains population. A classical way is followed to obtain weak formulations for each state variable: temperature, charge and partial pressures. At the microscopic scale, the reaction fractional conversion is followed for a representative population of grains. It is thus possible to calculate the reaction rate for a given temperature and given partial pressures, and therefore to get heat and mass sources produced by the reaction.

It is worthwhile to notice that CIN4 allows to simulate the behavior of reactions for which the kinetic rate at time t (at the microscopic population scale) depends on the local thermodynamic conditions that were established between t=0 and t. CIN4 highly improves the previous approaches which were limited to kinetic model based at grain scale only, excluding the gas composition effects other than those with one gas with a first order, and the surface nucleation and growth models.

The equations used in CIN4 require numerous physical properties (porosity, viscosity, density, heat capacity, thermal conductivity, diffusivity). CIN4's user should be able to calculate these properties using empirical laws or mathematical models like for example those given by Bird et al. (2002).

4. Numerical modeling

At grain and microscopic population scales, two types of modeling are envisaged according to the growth: anisotropic or isotropic. In the first case, the used method allows to calculate the evolution of the fractional conversion for all grains in each granulometric class. When the growth is isotropic, we used the Mampel's method (1940) for the three grain geometries: spheres, cylinders and plates. At reactor scale, for heat, hydrodynamic and mass transport problems, two finite element formulations were built, depending on the system symmetry (cylindrical or Cartesian coordinates). A classical way was followed to obtain weak formulations for each state variable: temperature, charge and partial pressures.

For solving heat and mass transfers equations throughout the reactor by the finite elements method, a mesh is available by using Gmsh freeware application (Geuzaine et al. ( 2009)).

Applications

Kaolinite dehydroxylation

The dehydroxylation of kaolinite was followed by means of a SETARAM TAG16 balance. Isothermal and isobaric curves were first obtained by heating the powder samples in an alumina crucible (2 cm height, 1 cm diameter) at different temperatures in the range 420-460°C under water vapor pressure fixed in the range 0.25-2 kPa by means of SETARAM WETSYS. First, experiments were performed under with about 25 mg of powder. During the entire transformation, the sample weight was low enough to prevent any effect of pressure gradients in the powder layer. As expected, the reaction rate increases with temperature and decreases with water vapor pressure.

A second series of experiments was done with various heights of the powder bed. In the same crucible as previously, two distinct TGA experiments were performed at 450°C under a water vapor pressure of 7hPa: the first one with 50 mg of powder corresponding to a powder bed height of 3 mm; the second one with 300 mg corresponding to 10 mm. The kinetic curves and rates curves obtained with both bed heights are presented on Figure 2. One can see that the powder bed height has an important effect on the kinetic curves since the maximum reaction rate obtained for 3 mm is about twice more that obtained with 10 mm (and is about twice less than that obtained with 1 mm).

In order to interpret the experimental curves obtained with thin powder beds, we used a model involving random nucleation followed by anisotropic growth of the nuclei first proposed by Nahdi et al. (2002). Nucleation is supposed to occur at the surface of the individual particles with an areic frequency (γ). Once

Récents Progrès en Génie des Procédés, Numéro 104-2013 ISSN : 1775-335X ; ISBN : 978-2-910239-78-7, Ed. SFGP, Paris, France 104-5 a nucleus has been formed, the anisotropy of growth is such that the particle is instantaneously covered with a very thin layer of solid product. In this case the tangential component of the growth rate is very high compared to its radial component. The particles are considered as flat cylinders, as suggested by SEM characterizations, with a mean radius r 0 equal to 0.15 µm and a thickness h equal to 0.05 µm. The rate-determining step of growth is a diffusion step through the metakaolinite already present in the periphery of the particle. Moreover the growth of the metakaolinite is considered as occurring inwards (shrinking core type). With these assumptions and by considering the volumic expansion factor equal to 1 (i.e. kaolinite and metakaolinite have the same molar volume V mA = 97.3 10 -6 m 3 mol -1 ), the reaction rate dα/dt for the powder can be determined by solving the following equations system:

∫ - - = t m d r h r l V h dt d A ς τ β τ γ π φ γ π α
) ( . . 4 1 1 ) 1 ( ) 1 ( 2 0 0 τ φ β β - = + - - - t r l V Ln A m (6)
where l 0 is equal to the length unity (l 0 = 1 m). The optimization procedure of CIN4 provided the values of γ (in nb nuclei m -2 s -1 ) and φ (in mol m -2 s -1 ) for each experiment so the laws of variation of these quantities with temperature (T in K) and water vapor pressure could be expressed by: Concerning the results obtained with thick powder beds, the "macro" part of CIN4 was used in order take into account the heat and mass transfers between all the various places inside the reactor (sample bed, crucible, gas flow). The first step consisted in defining the domains of calculation. The system being axisymmetric, we used weak forms written in cylindrical coordinates. Inside the granular medium, heat and mass sources were calculated simulating the reaction phenomena at the microscopic level. Figures 1 a andb show the gas flow rate obtained as well as temperature and fractional conversion fields at a given time for bed heights of 3 and 10 mm, respectively. Figure 2 presents a comparison of the whole rate of reaction versus fractional conversion between the experimental data and the simulation for both bed heights. Figure 2 shows that the CIN4 simulation is able to discriminate the height of the powder bed from 3 to 10 mm. It is important to notice that these results at reactor scale are obtained by means of numerical simulation without any adjustment of parameters. Nevertheless the calculated curves are not perfectly superimposed with the experimental ones. Some improvements should be done in order to get better simulations. For example several points could be improved: at the grains scale ("micro" optimization), the particles size distribution could be taken into account to obtain more precise values of γ and φ; at the reactor scale, the change in the physical properties of the granular medium (including porosity) due to the transformation of kaolinite into metakaolinite should be considered using appropriate relationships between the physical properties of the granular medium and fractional conversion.

Calcium monoxide carbonation

CIN4 has been used to describe the kinetic behavior of CaO aggregates during the reaction of carbonation into CaCO 3 by CO 2 (5 kPa) at temperatures in the range 450-650°C. The experimental kinetic curves reveal two distinct features (cf. Figure 3a): first there exists an induction time which depends of the thermodynamic conditions (temperature and CO 2 partial pressure); secondly, it systematically appears a huge slowing down of the reaction kinetic rate linked to diffusion limitations in the aggregates (decrease in size of the pores during reaction due to a volumic expansion factor equal to 2.13). A typical shape of the CaO aggregates can be seen in Figure 3b. The kinetic modeling at the "micro" scale could be done due to the initial part of the experimental curves as shown by Rouchon et al. (2013). The model corresponds to a single nucleus appearing at the surface of the CaO grains (nucleation is supposed to be a sluggish process compared to the growth, in agreement with the existence of the induction period) which grows isotropically inside the grain until total consumption of the CaO phase. So all the grains behave in a similar manner, but begin to be transformed at distinct times between 0 to t. Thus in this case, we used for the "micro" kinetic model both functions describing the variations of the induction period τ and the areic growth reactivity φ with temperature and partial pressure as shown in Figure 4. CIN4 was used to describe the behavior of the aggregates, coupling the "micro" and "meso" scales, since was assumed that at the reactor scale, all the aggregates in the powder bed are transformed with the same kinetics. This assumption is justified by the fact that the sample mass was very low, around 10 mg, and also due to a large pore size between aggregates, thus ensuring fast heat and mass transfers between aggregates. The simulation was realized considering the changes in the porous characteristics of the aggregates: -an empirical law of variation of the pore size with the extent of conversion as shown in Figure 5, based on experimental characterizations of the porosity inside the aggregates, -a physical law of variation of the porosity ε vs. the fractional conversion:

ε(α)=1-(1-ε 0 )[1+(z-1)α]
(9) where ε 0 is the initial porosity and z is the volumic expansion factor (z=2.13). A good agreement between the calculated and experimental curves can be observed in Figure 3a. The simulation stops when the diffusion of CO 2 molecules through the aggregates pores becomes impossible, i. e. when the pore size reaches that of the molecules size. Afterwards, when the periphery of the aggregates reaches a densified state, a solid state diffusion regime should be invoked (Rouchon et al. (2013)). However, real improvements could be done in the future for a better description of the changes in porosity, not only inside the aggregates, but also at the scale of the powder bed. A., Mehl R.F., 1939, Trans. Amer. Inst. Ming. Metal. Engrs. 135, 416. Mampel, K., L., 1940, Z. Phys. Chem. A 187, 235. Nahdi K., Perrin S., Pijolat M., Rouquerol F., 2002, Phys. Chem. and Chem. Phys., 4, 1972. Pijolat M., Soustelle M., 2008, Thermochimica Acta, 478, 34. Favergeon L., Morandini J., Jacquier S., Soustelle M., 2009, Récents

Résumé

La compréhension du comportement de réacteurs industriels est assez complexe dans le cas de réactions solide-gaz. En effet la phase solide est un milieu granulaire dans lequel circulent des réactifs et des produits gazeux. Les propriétés d'un tel milieu sont modifiées dans l'espace et le temps en raison des réactions se produisant à une échelle microscopique. Les conditions thermodynamiques sont fixées non seulement par les conditions de fonctionnement du réacteur, mais aussi par les transferts de chaleur et de matière dans le réacteur. Nous proposons de résoudre numériquement les équations régissant ces transferts en les combinant avec les lois cinétiques qui décrivent les réactions hétérogènes. L'avantage majeur de cette approche est la grande variété des modèles cinétiques de transformation de grains disponibles (~ 40) comparé à l'approche habituelle, notamment dans le cas de réactions impliquant la prise en compte de la germination à la surface suivie de leur croissance. Les termes sources de chaleur et de matière entrant dans les bilans à l'échelle macroscopique dépendent de la cinétique à l'échelle microscopique. La résolution numérique de ces équations par la méthode des éléments finis permet d'obtenir la température et les pressions partielles dans le réacteur, ainsi que le suivi de l'avancement local et global de la transformation. Deux exemples d'application sont décrits afin d'illustrer l'intérêt de CIN4 pour l'étude des transformations chimiques de solides divisés : la déshydroxylation de la kaolinite et la carbonatation du monoxyde de calcium.

Figure 1 :

 1 Figure 1: Representation of the temperature field, the fractional conversion and the flowing rate (arrows) at a given time for (a) 3 mm-height bed and (b) 10 mm-height bed. Fractional conversion Temperature Fractional conversion Temperature a) b)

Figure 2 :

 2 Figure 2: Reaction rate vs. fractional conversion from the experiment and from the simulation for 3 and 10 mm powder bed heights.

Figure 3 :

 3 Figure 3: Experimental extent of conversion at 650°C compared to the CIN4 "micro-meso" scale simulation (a); SEM view of a CaO aggregate (b).

7 Figure 4 :

 74 Figure 4: Variations of the induction period (a) and of the areic reactivity of growth (b) vs. temperature and CO 2 partial pressure for the CaO carbonation reaction.

Figure 5 :

 5 Figure 5: Variations of the mean average pore size in the aggregates vs. fractional conversion

  Progrès en Génie des Procédés (Actes du Congrès SFGP, Marseille 2009). Pijolat M., Favergeon L., Soustelle M., 2011, Thermochimica Acta, 525, 93. Reid R.C., Prausnitz J.M., Poling B.E., 1987, « The properties of Gases and Liquids », MCGraw Hill, New-York. Rouchon L., Favergeon L., Pijolat M., 2013, J. Therm. Anal. Calorim. DOI 10.1007/s10973-013-2950-5 Soustelle, M., 2009, « Cinétique Hétérogène 2 », Ed. Lavoisier.

  external interface or diffusion volume). Heat and mass transfers are governed by differential equations and solved by a finite element method. Powder beds consisting in separate individual grains (micro and macro scales) or aggregates (micro-meso-macro scales) may be considered. CIN4 allows simulating not only reactions laboratory scales such as thermobalances, but also industrial reactors. Bird R.B., Stewart W.E., Lightfoot E.N., 2002, « Transport phenomena », Wiley International Edition. Delmon, B., Eds. Technip, 1969, Introduction à la cinétique hétérogène, p. 397. Geuzaine C., Remacle J.F., 2009, Int. J. Numer. Meth. Eng. 79, 1309. Johnson W.

	Récents Progrès en Génie des Procédés, Numéro 104-2013
	ISSN : 1775-335X ; ISBN : 978-2-910239-78-7, Ed. SFGP, Paris, France
	Notations
	α Fractional conversion of the powder β Fractional conversion of a grain S m Space function (m 2 .mol -1 ) ε Porous medium porosity ε 0 Initial porous medium porosity	P i Partial pressure of reactive gas (Pa) Q T Heat sources density (W.m -3 ) Q C Concentration sources density (mol.m -3 .s -1 ) r 0 Initial radius of the grain (m) σ Stefan Boltzmann constant (W.m -2 .K -4 )
	φ Areic reactivity of growth (mol.m -2 .s -1 ) γ Areic frequency of nucleation (nuclei.m-2.s-1) h Cylinder thickness (kaolinite particles) M i Molar mass for constituent i (kg mol -1 )	T Temperature (K) τ (section 3.1) Nucleation date τ (section 3.2) Induction period V mA Molar volume of A (m 3 . mol -1 )
	ν i Stoichiometric coefficient	z expansion factor
	P Total pressure (Pa)
	Literature cited
	104-8

The word "grain" represents an elementary dense particle.

Avrami's laws cannot be used for solid-gas heterogeneous reactions involving powders since they are based on bulk nucleation in an infinite volume.


J Avella Castiblanco
email: jorge.castiblanco@ifsttar.fr

D Seetharamdoo
email: divitha.seetharamdoo@ifsttar.fr

M Berbineau

M Ney

F Gallee

Electromagnetic modeling and definition of antenna specifications and positions for radio system deployment in confined environments

In this paper, the definition of antenna specifications for radio system deployment as well as a methodology to integrate them in confined environments is presented. Modal-based analysis and optimization techniques for mode-weight adjustments are used to determine correct co-excitation of the propagating modes and fulfill the desired specifications.

Introduction

Wireless communication system design in confined environments is complex due to harder electromagnetic and spatial constraints. Indeed the number of on-board wireless systems is increasing and polymers are increasingly used to build modern, lighter railway vehicles. The investigation on novel antennas is of outstanding importance to guarantee reliable communication for systems such as control-command. Appropriate methodologies for antenna design and integration have to be developed to face the challenges on such platforms.

Research work on antenna integration in railway environments by using metamaterials is performed in the framework of the METAPHORT project initiated by the French National Institute of Science and Technology for Transport (IFSTTAR). The aim of this project is to develop metamaterial antennas and to establish a methodology for the antenna integration in urban underground railways. Train-to-Wayside communication systems for control command applications for railways in an urban underground environment based on 802.11 a and 802.11 g WLAN standards are considered. Antenna integration in this environment requires appropriate numerical models of the electromagnetic wave propagation accounting for multi-scale geometries such as tunnels and objects, and in a reasonable computation time. Appropriate methodologies for the antenna integration based on these models will be proposed and validated.

The paper is organized as follows. Section 2 presents a methodology for the correct antenna integration. Results using our methodology is shown in section 3. Finally, conclusions and on-going work are developed.

Methodology for Meta-material antenna design

Metamaterials can be described as artificial media tailored to control wave propagation. They can provide a higher degree of freedom in the design of antennas for wireless systems tailored for specific environments by controlling the EM energy stored or radiated around the antenna. For the development of novel antenna solutions in tunnels different inputs are required as depicted in Fig. 1.

Figure 1. Novel metamaterial antenna design methodology.

Antenna design for tunnels

The specifications and parameters involved in the antenna design for confined environments, such as tunnels, have not yet been established in the scientific or technical literature. A few considerations on the obstruction, separation, installation and maintenance can be found in the U. K. guidance on train rooftop antenna positioning (GK/GN06602) as well as by some manufacturers. However, the problem of antenna operation in tunnel environments is not addressed. Thus, a methodology for antenna design is needed and appropriate models providing physical insight must be considered. Various studies have been done considering several models to analyze radiation characteristics and positioning of antennas in tunnels [START_REF] Huo | Effect of antenna on propagation characteristics of electromagnetic waves in tunnel environments[END_REF] or simply to model the wave-propagation in these environments [START_REF] Didascalou | Ray-optical wave propagation modelling in arbitrarily shaped tunnels[END_REF]. These models are based on asymptotic approaches and they thus fail for antennas which are strongly affected by the surrounding environment. A novel methodology for finding appropriate field specifications and optimum antenna positioning can be obtained by using a modal approach with a full-wave method, such as the Transmission Line Matrix method (TLM) [START_REF] Castiblanco | Wave propagation. Chapter: "Electromagnetic wave propagation modeling for finding antenna specifications and positions in tunnels of arbitrary cross-section[END_REF].

Methodology for finding antenna specifications

An efficient modal approach for the analysis of guiding structures is applied to find the fields [START_REF]Surface boundary conditions for lossy dielectrics to model electromagnetic wave propagation in tunnels[END_REF]. Different optimum positions and types of excitation are involved due to multi-modal propagation in these environments. These modes may be combined in some way to produce a desired effect, e.g., the maximization of the power transferred by the modes.The particularity of this approach is that only the cross-section of the tunnel is considered while modeling the field propagation in the longitudinal direction. Tunnels with arbitrary cross section, loaded or unloaded, can be analyzed with this approach.

Description of the Methodology

The proposed methodology is divided into five steps. The first one concerns the calculation of the characteristic mode parameters α, β, E (x, y) and H (x, y). Second, the modes are discriminated by their power carried through the tunnel so that only modes with the highest power are considered. Then, an optimization procedure of the transmitted power on the weight coefficients is carried out. A constrained solution in terms of the maximum or minimum power density that can be transmitted or received, respectively, must be found. In the fourth step, the optimum weight coefficients to maximize the mean power along the propagation direction are obtained. Finally, the optimum fields specifications, type of excitation and source positioning are determined by using the optimum weights. A detailed description of this methodology can be found in [START_REF] Castiblanco | Wave propagation. Chapter: "Electromagnetic wave propagation modeling for finding antenna specifications and positions in tunnels of arbitrary cross-section[END_REF].

Results

The wave propagation in a rectangular tunnel with transverse dimension w = 7m × h = 5m was considered. The tunnel walls were modeled by a lossy dielectric with ϵ r = 12 and σ = 0.02Sm -1 . The power density specifications were defined to be 0 dB Wm -2 at z T x = 0 and -40 dBWm -2 at z Rx = 1, 000 m. This tunnel was simulated using our modeling approach. The attenuation, phase constants, and the field profiles of the E y 11 and E x 11 modes were determined and compared with Marcatilli's approximate theory, results are shown in Fig. 2. The attenuation and dispersion characteristics calculated by our modeling approach are in good agreement with Marcatilli's theory. However, the attenuation constants obtained for low frequencies are different most probably due to the fact that in Marcatilli's approach the fields at the corners are canceled unlike in our numerical method. The optimum electric fields to excite the rectangular tunnel are then calculated by using the above results from the modal approach; optimum weight coefficients are then defined using our methodology. Optimum positions where the mean power along the longitudinal distance is maximum were also calculated. By way of illustration, the three field components of the electric field that satisfy the power density specifications in Tx were calculated at z = 0, as shown in Fig. 3. Intuitively, one could expect that optimum fields are most influenced by the lowest attenuated modes. This result can be confirmed by observing that the points where the x-component of the electric field in Fig. 3 is minimum at corners, and the z-component is minimum at half of the width as it occurs for the E x 11 mode. This observation can also be confirmed as the x-component is maximum at one-third, as well as one-quarter of the width as for both E x 21 and E x 31 modes. Optimum positions to locate the transmitter are those where the mean power along the longitudinal distance is maximum, i.e., for the points close to the white regions in Fig. 3.

Conclusions

A methodology for the definition of antenna specifications for radio system deployment and antenna integration in confined environments has been presented. The methodology relies on a numerical model based on modal analysis. Then, optimization techniques were applied, and mode-weight adjustments are used to determine correct co-excitation of the propagating modes and ensure a good power transfer from the transmitter to the receiver. Results are presented for a tunnel in terms of complex propagation constant and mode profiles. Good agreement with an approximate model has been shown, thus validating our numerical results. The antenna design methodology for positioning and determination of specifications was then applied. It should be noted that this methodology has to be repeated for all the critical scenarios varying for traffic considerations, different tunnel cross sections, multiple tracks and different distances between the transmitter and receiver.

Figure 2 .

 2 Figure 2. Attenuation, dispersion and field profiles for the E y 11 and E x 11 modes.

Acknowledgments

This work was funded by the French National Agency within the ANR-VTT 2009 program in the framework of the METAPHORT project. The authors also acknowledge support from the I-TRANS railway cluster of the Nord Region of France, as well as the Images et Reseaux cluster of the Brittany region.


Févotte Gilles

Ouabbas Yamina

Xingjun Wang

Monitoring Solvent-Mediated Phase Transitions Using Acoustic Emission

Keywords: Cristallisation industrielle, Cristallisation en solution, Contrôle de procédés, Capteur non-intrusif, Emission acoustique

Du fait de son caractère non intrusif et non destructif, l'émission acoustique (EA) est une technique attractive pour le suivi de cristallisations, bien que les rares résultats publiés à ce jour ne soient pas très probants. L'émission acoustique est produite par les processus mécaniques locaux qui résultent de l'apparition ou de la disparition de la phase solide, et de son interaction avec son environnement. Elle se présente sous la forme d'un grand nombre de salves pseudo-périodiques séparées ou continues. Les signaux acoustiques sont acquis par un capteur piézo-électrique externe et traités en temps réel par un système d'acquisition et de traitement qui calcule et enregistre des paramètres globaux caractéristiques des salves (nombre de coups, amplitude, fréquence, énergie …) A partir de la cristallisation de l'acide citrique présentée ici, on montre que les paramètres caractéristiques des salves pendant la cristallisation peuvent être reliés aux différentes étapes élémentaires de la transition de l'anhydre vers le monohydrate. Une analyse en composante principale des variables caractéristiques des salves a été effectuée qui permet de repérer les différentes étapes du procédé.

Introduction

L'émission acoustique est très largement utilisée dans l'industrie des matériaux comme technique de contrôle non destructif (métallurgie, génie civil, mécanique, aéronautique, etc.) Elle a été appliquée aussi avec succès au contrôle de procédés du génie chimique (W. R. Boyd et Varley 2001), dans le cas de la granulation [START_REF] Poutiainen | Predicting granule size distribution of a fluidized bed spray granulation process by regime based PLS modeling of acoustic emission data[END_REF]) [START_REF] Hansuld | Monitoring quality attributes for high-shear wet granulation with audible acoustic emissions[END_REF]), de l'enrobage [START_REF] Naelapää | Acoustic monitoring of a fluidized bed coating process[END_REF]), de la compaction de poudres [START_REF] Serris | Acoustic emission of pharmaceutical powders during compaction[END_REF] (Sawada et al. 1985), etc. Cependant, très peu d'études ont été publiées dans le cas des procédés de cristallisation [START_REF] Lube | In-process monitoring of crystal perfection during melt growth[END_REF] [START_REF] Zlatkin | Estimation of fracture toughness and residual stress in brittle crystals from indentation-induced acoustic emission[END_REF] L'émission acoustique est captée par un transducteur acoustique couplé au matériel où se produit le phénomène d'intérêt (réacteur, lit fluidisé, broyeur, …). Le capteur détecte l'énergie élastique des ondes acoustiques se propageant depuis la source physique de l'émission (les cristaux par exemple.) Les ondes transportent une information sur les processus élémentaires qui les ont générées et, à ce titre, sont susceptibles de délivrer une information indirecte sur les phénomènes dynamiques concerné [START_REF] Grosse | Acoustic Emission Testing[END_REF] Le capteur est connecté à un dispositif d'acquisition et de traitement rapide des données transmises. Les signaux se présentent sous forme de « salves » qui se traduisent par une masse considérable de données à numériser. C'est pourquoi, plutôt que de stocker toutes les salves, le système ne calcule qu'un certain nombre de paramètres descripteurs globaux (paramètres moyens par exemple) dont la sauvegarde requiert beaucoup moins de place mémoire. Grâce à ses caractéristiques non intrusives et non destructives, et grâce à la possibilité de mettre au point des applications « en-ligne », l'émission acoustique est très attractive dans le cas de la cristallisation où il reste très difficile de prélever des échantillons fiables et d'analyser l'état de la suspension en cours de cristallisation.

N° Article-1

En vue de l'utilisation de l'émission acoustique (EA) comme nouvelle technique de mesure, l'objectif de ce travail est de mettre en relation les nombreux paramètres (énergie et fréquence) de l'EA avec les principales étapes d'une cristallisation en solution par refroidissement avec changement de phases.

Dispositif expérimental

La cristallisation par refroidissement de l'acide citrique dans l'eau a été mise en oeuvre dans le cristallisoir de 3L représenté sur la Fig. 1. La température de cristallisation est commandée selon une rampe de pente constante par un bain piloté à l'aide d'un programme Labview®. Une mesure par spectroscopie ATR FTIR in situ permet le suivi de la concentration en soluté et, par conséquent, de la sursaturation. [START_REF] Févotte | New perspectives for the on-line monitoring of pharmaceutical crystallization processes using in situ infrared spectroscopy[END_REF] L'acquisition d'image vidéo in situ permet de suivre qualitativement l'évolution de la suspension de cristaux pendant l'opération discontinue. La cristallisation retenue comme modèle d'étude est celle de l'acide citrique dans l'eau pour laquelle des données cinétiques détaillées sont disponibles [START_REF] Févotte | A population balance model of the solution-mediated phase transition of citric acid[END_REF]. Comme le rappelle la Fig. Les différents paramètres acoustiques mesurés montrent la diversité et la richesse des informations acquises qui, toutes, peuvent être mises en relation avec les différentes étapes de la cristallisation. C'est ce qu'illustre de façon plus précise la Figure 5 suivante.

Discussion

La Figure 5c Plusieurs observations peuvent être faites à partir des résultats de la Figure 5. Comme l'indique la flèche 1 sur la trajectoire des composantes principales, une bifurcation marque le début de la cristallisation avant même qu'elle puisse être observée au niveau de la mesure de concentration. La continuité de la trajectoire entre les marques 1 et 2 semble indiquer une continuité du processus de cristallisation (i.e., croissance de l'anhydre). La trajectoire change et devient plus compliquée après le point 2, mais dès ce point il est clair qu'un nouveau phénomène démarre = la transition de phase avec médiation par le solvant ? La trajectoire suivie entre les marques 2 et 3 montre plusieurs bifurcations que confirment les évolutions de la vitesse de production de solide. Le marqueur * correspond à un rebroussement de la trajectoire PC2=f(PC1) et à un maximum très aigu de la variation de la première composante (Figure 5b.) Compte tenu des tracés 5c et 5d il semble logique d'associer ce pic à la traversée de la courbe de solubilité de l'anhydre (le début de la dissolution de l'anhydre.) A partir de la marque 3, l'évolution des composantes principales redevient monotone, comme lors de la cristallisation de l'anhydre. Cette évolution peut donc raisonnablement être associée à la fin de la cristallisation du monohydrate, sans dissolution concomitante de l'anhydre.

Un autre point de vue sur les différentes étapes du procédé est obtenu en calculant, sur différentes périodes de temps données, les histogrammes de distribution cumulés en nombre des salves placées sur une grille dont les coordonnées sont définies pas deux descripteurs (X,Y). La Figure 6 donne un exemple d'une telle distribution dans le cas où les coordonnées sont l'amplitude et le nombre de compte par salve.

N° Article-5

Ici encore, on observe que les distributions des paramètres de l'émission acoustiques présentent des profils qui varient significativement selon les différentes étapes du processus de cristallisation avec transition de phase. Ils fournissent donc le moyen de contrôler « en temps réel » le déroulement du procédé et, ainsi, de corriger d'éventuelles dérives et/ou de diagnostiquer des problèmes spécifique de la cristallisation, une nucléation retardée ou l'absence éventuelle du déclenchement de la transition de phase par exemple.

Conclusions

Les processus de cristallisation par refroidissement s'accompagnent d'une émission acoustique spontanée constituée de la succession d'un très grand nombre de salves. Ces salves sont acquises par un capteur non intrusif et transmises à un analyseur de signaux de haute fréquence qui les enregistre et les analyse. Des paramètres descriptifs des salves sont ainsi calculés et l'analyse de leur évolution temporelle permet d'identifier différentes phases, évènements ou perturbations du procédé de cristallisation. L'application de l'émission acoustique au suivi d'une cristallisation en solution de l'acide citrique a permis de montrer que la cristallisation génère un très grand nombre de signaux dont les évolutions moyennes continues sont représentatives des phénomènes élémentaires du processus. On a pu ainsi observer, à l'aide de différentes techniques statistiques (ACP= Analyse en Composantes Principales, calculs d'histogrammes 3D) que les étapes importantes de la cristallisation de l'acide citrique avec transition de phase (Anhydre → Monohydrate) peuvent être identifiées et, par conséquent, permettre un diagnostic de l'opération. L'émission acoustique devrait également permettre de concevoir de nouvelles stratégies de conduite des cristallisoirs en vue d'un contrôle amélioré du procédé.

N° Article-6

Figure 1 .

 1 Figure 1. Schéma simplifié du dispositif expérimental. 1. Réacteur double enveloppé, 2. Bain thermostaté, 3. Sonde infra-rouge in situ ATR, 4. Sonde acoustique, 5. Sonde de température, 6. Spectromètre infrarouge, 7. Système d'acquisition et de traitement des données acoustiques, 8. Sonde de suivi vidéo in situ (EZProbe.) La figure 2 montre quelques-unes des caractéristiques essentielles des salves. Les paramètres indiqués (les "descripteurs") sont supposés donner une information quantitative sur chaque train d'ondes émis lors de transformations mécaniques se produisant dans le cristallisoir. La forme des salves dépend de la nature des processus élémentaires impliqués, on doit pouvoir analyser les évolution temporelles des descripteurs pour en extraire une information sur les déroulement des processus de cristallisation en cours. Lorsque les évènements acoustiques sont trop nombreux, le signal acquis ne retombe jamais en dessous du seuil indiqué en Figure 2 et les salves sont alors considérées continues.

Figure 2 .

 2 Figure 2. Représentation d'une salve acoustique et des principales caractéristiques enregistrées par le système d'acquisition.

Figure 3 .

 3 Figure 3. Courbes de solubilité dans l'eau de l'acide citrique (Anhydre/monohydrate.) Les données ont été mesurées par spectroscopie ATR FTIR et confrontées aux données publiées par Apelblat et Manzurola (1987)

  montre l'élévation de température consécutive aux phénomènes de nucléation et croissance induits par l'apparition de la phase solide. La transition de l'anhydre en monohydrate se produit vers 30°C où un deuxième pic exotherme est observé. La figure 5d montre que la mesure de concentration en soluté est cohérente avec ces observations. Les vitesses de production de solide ont été calculées par dérivation de la concentration mesurée en fonction du temps. Deux pics successifs de production de solide sont clairement mis en évidence par ce tracé, les marqueurs  et ♦ indiquent les maximas de production de solide sur la Figure5d, et indiquent également les instants correspondants sur la courbe indiquant la trajectoire de température T=f(t).N° Article-4Une analyse en composante principale des descripteurs a été effectuée en réduisant le nombre de variables prises en compte. En effet, après analyse, plusieurs variables de la Figure4s'avèrent très fortement corrélées, et la conservation de l'ensemble des données ne ferait qu'alourdir inutilement l'analyse. On retient donc finalement le vecteur suivant : [Eabs f f ~n ρ ]; où ρ est un rapport d'énergie cumulée dans deux bandes de fréquence différentes choisies par "essais et erreurs".

Figure 5 .

 5 Figure 5. (a) Tracé des deux premières composantes principales (PC1 et PC2) des descripteurs mesurés et calculés pendant la cristallisation de l'acide citrique. Les marqueurs permettent de repérer les différentes étapes du procédé. (b) Evolution temporelle de la première composante (c) Profil de température suivi (d) Evolution de la vitesse globale de production de solide calculée d'après les mesures ATR FTIR de concentration du soluté.

Figure 6 .

 6 Figure 6.Tracé des évolutions des histogrammes 3D donnant la fraction en nombre de salves caractérisées par une classe de nombre de coup en fonction de l'amplitude des salves enregistrées.