S Teychené

N Sicre

B Biscans

Is spherical crystallization without additives possible?

Keywords: Spherical crystallization, Interfacial tension, Quasi-emulsion, Agglomeration, Liquid-liquid dispersion, Spherulitic crystal growth

The quasi-emulsion solvent diffusion method of spherical crystallization consists in producing in one step crystallization and agglomeration of small crystals in droplets of an emulsion. Additives are generally used to stabilize the emulsion before crystallization. The aim of this study is to investigate the feasibility of spherical crystallization without surfactant. Experiments were performed in an automated batch laboratory scale crystallization process to study the influence of the process operating conditions on the structure of the particles obtained. The results clearly show that, for the experiments performed two types of particles are formed: primary spherical particles and secondary agglomerates. The pattern of the primary particles, observed under scanning electron microscopy, suggests that these particles results from a spherulitic crystal growth mechanism inside the droplet. The secondary agglomerates results of the agglomeration of the spherical particles. In addition, a set of experiments were performed with carefully selected solvents to study the influence of the crystallization solution/water interfacial tension, at constant hydrodynamic conditions and supersaturation level. The results of these experiments demonstrate that the interfacial tension is not a key parameter for designing such a process.

Introduction

Crystallization from solution is a core technology in pharmaceutical industries. Usually, this process is a part of a wide processing system, including solid-liquid separation, particle design, formulation. As the generation step, the crystallization process determines the powder handling properties (shape, size distribution, purity, etc.). These properties affect the downstream processing. After crystallization, filtration and drying steps, the crystalline product is formulated to obtain tablets, which are the most common drug preparation. However, most of the API powders, obtained by a classical crystallization process, have poor physical properties that prevent direct tableting. To overcome these problems, a wet granulation step is added in the API solid chain production (indirect tableting). Direct tableting of API crystalline powder should enable to reduce cost and delay in the development of a pharmaceutical product. Direct tableting needs powders that excel in bioavailability, bindability, and mechanical strength. Among the methods allowing to improve the physical properties of a crystalline powder, Kawashima et al. (Kawashima andCapes, 1976, 1974;[START_REF] Kawashima | Parameters determining the agglomeration behaviour and the micromeritic properties of spherically agglomerated crystals prepared by the spherical crystallization technique with miscible solvent systems[END_REF][START_REF] Kawashima | Improved flowability and compactibility of spherically agglomerated crystals of ascorbic acid for direct tableting designed by spherical crystallization process[END_REF][START_REF] Kawashima | Development of agglomerates process by using a flocculation phenomenon of particle in liquid and the application to pharmaceutical system[END_REF] suggest to perform size enlargement and crystallization processes in one step. The crystals are generated and confined inside supersaturated droplets of an emulsion to form spherical particles. The possibility of forming large spherical particles directly inside the crystallizer presents many advantages for the powder processing.

The quasi-emulsion spherical crystallization process is one of the most efficient methods for performing crystallization and particles size enlargement in one step. However, this process presents two main difficulties in stirred reactor. The first one is to find the right operating conditions to form and to stabilize spherical and monodispersed droplets. The second one is to control the coexistence of solid particles and droplets to avoid coalescence and agglomeration of particles. In such a process, the structure of the final particles directly depends on the physico-chemistry properties of the compounds used and on hydrodynamics of the crystallizer.

Today the addition of small amounts of surfactants or polymers is the method usually applied to obtain the best spherical particles. Surfactants or polymers can decrease the interfacial tension between fluids, change the viscosity of phases and so influence the droplets sphericity and emulsion stability.

The classical process of quasi-emulsion crystallization consists in three steps: first the creation of an emulsion with a short lifetime, then an increase of supersaturation inside the droplets due to heat and mass transfer and finally, nucleation, growth and agglomeration of crystals inside the droplets. In this process, the continuous phase (generally aqueous phase) is formulated with a polymer (Mowiol, PVA, PEG), which avoids, by steric effect, secondary agglomeration. In literature, the quasi-emulsion spherical crystallization of ketoprofen was performed with mowiol 8-88 (Espitalier et al., 1997a,b,c). For ibuprofen crystallization [START_REF] Kachrimanis | Crystallisation conditions and physicomechanical properties of ibuprofen-eudragit s100 spherical crystal agglomerates prepared by the solvent-change technique[END_REF][START_REF] Kachrimanis | Spherical crystal agglomeration of ibuprofen by the solvent-change technique in presence of methacrylic polymers[END_REF] described crystallization with Eudragit S100 by the solvent-change method.

Because surfactants are sometimes prohibited for toxicological reasons, the objective of this work is to study quasi-emulsion spherical crystallization without additive. When this process is performed without polymer or surfactant, agglomeration of the spherical particles previously formed and droplet coalescence may occur. In this paper, experiments have been designed to study, for a given mean droplet size, the influence of the process parameters (mixing and injected phase volume fraction) and the physico-chemical properties of the liquid phases (interfacial tension) on the morphologies of the obtained agglomerates.

Materials and methods

Materials

Organic solvents ethyl acetate, n-butyl acetate, methylethyl ketone (butan-2-one, MEK) and polyvinylacetate (PVA) were purchased from VWR (purity of 99.5%). The solvents were used without further purification. Double distilled water was processed through a Barnstead EASYpure UVsystem and was used as the continuous phase. The water resistivity was greater than 18 M cm. Ibuprofen (C 13 H 18 O 2 ) or (R)-(S)[2-(4isobutylphenyl)propionic acid] was supplied by BASF.

Ibuprofen crystals belong to the space P21/c (monoclinic system). The melting point of this colorless, crystalline material is commonly quoted as 75-77 • C [START_REF] Macconnel | 2-(4-Isobutylphenyl) propionic acid[END_REF]. Molecular structure of ibuprofen is given in Fig. 1.

The physico-chemical properties of the solvents used in this study are given in Table 1 at 25 • C.

Spherical crystallization experiments

Experimental setup

The experimental device, shown in Fig. 2 is composed of two elements: a pressurized reactor (for the dissolution of the API and the injection of the supersaturated solution) and a 2L crystallizer.

The 500 ml-pressurized reactor is a double-jacketed steel reactor. A marine propeller is used to mix the suspension. The reactor and the pouring device are thermally controlled with a precision of ±0.1 • C. Injection of the saturated solution, from the dissolution vessel to the crystallizer, is performed by pressurization of the reactor. The pressure in the reactor is controlled to ensure a constant flow rate through the injector. The injector was a capillary tube (D ext = 1.6 mm, D int = 1 mm) and was positioned 0.5 cm above the impeller of the crystallizer.

Crystallization was performed in a round-bottomed double-jacketed glass reactor in a Labmax automated system (Mettler Toledo). The crystallizer was equipped with a lid, four baffles and a Pt-100 resistance thermometer. The precision of these thermometers is given to be 1/10 • C. The crystallizer was built according to the classical rules of batch crystallizer design (D reactor = H liquid = 14 cm, l baffles = 1.5 cm). Mixing is ensured by a mixell TT propeller (D p = 0.6D reactor = 8 cm). The stirrer was positioned 5 cm above the bottom of the reactor (h = 0.33D reactor = 5 cm).

Experimental procedure

For all the crystallization systems studied, the experimental procedure is the same.

First, to ensure the complete dissolution of ibuprofen and to avoid clogging of the nozzle, the reactor and the pouring systems are heated 10 • C above the saturation temperature for 1 h.

The solution is then poured in the crystallizer, initially filled with water at 10 • C, with a constant flow rate during 20 s. The suspension is stirred for 15 min. Whatever the operating conditions are, the first crystals appears between 20 and 30 s after the end of the injection.

At the end of the experiment the obtained particles are recovered by filtration, washed with distilled water and dried in a ventilated drying chamber at 50 • C for 24 h.

Experiments were performed to analyze the influence of stirring rate, volume fraction of dispersed phase, pouring rate and organic phase/water interfacial tension on the morphology and size distribution of the particle obtained. The operating conditions explored in this study are given in Table 2.

Solid phase characterization

The shapes of particles were examined by Scanning Electron Microscopy (LEO 435VP) operating at 5 kV or 10 kV and by reflected light microscopy (Zeiss Axio observer 20×, 50×). Note that for SEM photographs, the dried agglomerates were coated with gold for 3-5 min under an argon atmosphere in a gold coating unit prior observation.

The particle size distribution of the agglomerates was obtained by sieving through four standard sieves (1.18, 1.4, 1.60 and 2 mm). For particles whose diameter is lower than 1.18 mm, a more accurate size distribution is obtained by laser diffraction particle size analyzer (Malvern Mastersizer 2000).

Liquid phase characterization

Interfacial tensions between organic solvents and water were measured by the dynamic Wilhemy plate method using a GBX 3S tensiometer. Time-dependent changes in the interfacial tension were detected automatically. The equilibrium inter- facial tension was automatically determined when a preset of standard deviation (0.1 mN/m) was reached. The temperature of the system was controlled and maintained at 25 • C by a cooling circulating oil bath. The beaker was filled with 70 ml of water saturated with organic solvent. The entire plate was then immersed in the aqueous phase and then 10 ml of organic solvent was carefully deposited on the surface of the aqueous phase. By measuring the applied force according to the immersion depth and the dimension of the substrate perimeter, the contact angle can be calculated using the relationship:

= F L cos (1)
where is the contact angle, F the measured force, L the wetted perimeter (40 mm) and the contact angle.

Results and discussion

In this work, the influence of the process operating conditions (stirring rate, volume fraction of dispersed phase) and the influence of the physico-chemical properties of the liquids (interfacial tension, nature of the crystallization solvents and additives) on the morphologies of the agglomerates have been studied. For the whole set of experiments performed (except for n-butyl acetate), two types of particles were obtained. Representative SEM microphotographs of the particles obtained are given in Fig. 3. The primary particle (Fig. 3a) results from spherical crystallization process within each droplet. The secondary agglomerates (Fig. 3b) results from an agglomeration process of primary particles. Depending on the operating conditions and on the nature of the liquid phases used, different proportions of each type of particles are obtained.

Characterization of the spherical primary agglomerates

Influence of the process operating conditions

The experimental results in terms of mean particle size, measured by laser diffraction, and the mass fraction of primary particles are reported in Table 3. In this table, it can be noticed that when n-butyl acetate is used as crystallization solvent, it was impossible to obtain spherical primary particles. In the vessel the agglomerates are actually spherical but they are soft and sticky and their structure is lost during the downstream processes (filtration and drying). In addition, in the range of operating conditions tested, when no additive is added to the water phase, it was impossible to obtain only individual spherical particles.

The diameter of the spherical particles directly depends on the initial droplets diameter which depends on hydrodynamics of the crystallizer (mixing and pouring rates) and on the physico-chemical properties of the liquids. The sizes of the primary spherical particles, reported in Table 3, are always lower than the internal capillary diameter which is Fig. 4 -Microphotograph of broken primary particles. due to the shrinking of the droplet during crystallization. The initial droplet diameter can be roughly calculated, under the assumption that all ibuprofen crystallizes in the droplet to form a non-porous particle, by the following relation:

d initial = d M / 1/3
ibu . Where ibu = V ibu /V drop is the ibuprofen volume fraction inside the droplet (deduced from the ibuprofen solubility). The obtained hypothetical initial droplet diameters are reported in Table 3.

Morphology of primary spherical particles

In quasi-emulsion crystallization processes, the general explanation of particle enlargement is nucleation of many crystals within each organic drop that agglomerate into dense object. However in our case, the long induction time observed during the experiments (around 30 s) indicates a low nucleation rate which should give a moderate number of crystal within each droplet. This observation is incompatible with the agglomeration process generally claimed for the spherical crystallization process. In addition, whatever the crystallization solvent used, the particles obtained present a radiating pattern, as shown for a broken primary particles in Fig. 4. This pattern is typical of spherulitic crystal growth. In addition, to support this mechanism, two "uncrystallized eyes" characteristic of spherulitic crystal growth can be observed in Fig. 4d.

This mechanism produces space filling spherical polycrystalline objects growing from a central seed by radial non-crystallographic branching [START_REF] Goldenfeld | Theory of spherulitic crystallization[END_REF]. However, considering the possibility of ibuprofen to induce a liquid-liquid phase separation (LLPS) before crystal nucleation (observed under microscope and by [START_REF] He | Generalized phase behavior of small molecules and nanoparticles[END_REF] it is impossible to state that it is droplet (from LLPS) as in the case of l-glutamic acid [START_REF] Roelands | Precipitation mechanism of stable and metastable polymorphs of l-glutamic acid[END_REF] or non-crystallographic Fig. 7 -Organic phase/water interfacial tension as a function of the mass fraction of ibuprofen introduced in the solvent.

branching of acicular crystals [START_REF] Beck | Polycrystalline growth in precipitation of an aromatic amine derivative and l-glutamic acid[END_REF][START_REF] Andreassen | Formation mechanism and morphology in precipitation of vaterite-nano-aggregation or crystal growth?[END_REF] that initiate spherulitic crystal growth of ibuprofen.

Influence of the process operating conditions on the formation of secondary agglomerates

Influence of the stirring rate

The results obtained in terms of particles size distributions (PSD) for a stirring rate ranging from 50 to 450 rpm (experiments 1-4) are given in Fig. 5 for four sieve apertures. The size enlargement process in stirred vessel is a complex mechanism which is a balance between particles agglomeration and breakage. The results, presented in Fig. 5b, show that when the stirring rate is increased, the PSD is narrower.

As shown in Fig. 6, the structure of the agglomerates obtained is the same for all the experiments performed. These particles show a loose structure characteristic of a process limited only by the meeting probability of particles, and thus by the collision frequency of particles (similar to the diffusion limited aggregation mechanism). Indeed ibuprofen is highly hydrophobic which gives rise to strong attractive interaction between particles and in order to reduce the free energy of the system, the hydrophobic particles tend to be in contact with each other.

In addition, the remaining uncrystallized solution (crystallization solvent and dilute ibuprofen) acts as a binding agent which covers the particles, and promotes the agglomeration efficiency. Then, the crystallization of ibuprofen creates solid bridges between particles.

Influence of the dispersed phase volume fraction

To increase the yield of the crystallization process it seems reasonable to increase the volume fraction of the introduced dispersed phase up to the miscibility limit of the organic phase in the aqueous phase. The experiments nos. 5 and 6 have been performed by injecting 5% of the organic phase. In the range of the operating conditions tested, when the concentration of the organic phase is close to the saturation of the liquid phase, the particles are soft (gel like structure) and sticky. As a consequence, during the process, the particles stick to the impeller and to the crystallizer wall. The gel like structure is obtained when the mass transfer from the organic phase to the aqueous phase is too slow [START_REF] Kawashima | Parameters determining the agglomeration behaviour and the micromeritic properties of spherically agglomerated crystals prepared by the spherical crystallization technique with miscible solvent systems[END_REF][START_REF] Nocent | Definition of a solvent system for spherical crystallization of salbutamol sulfate by quasi-emulsion solvent diffusion (qesd) method[END_REF]. The lifetime of the emulsion is then much higher which leads to an increase of the coalescence frequency of the droplets.

Influence of the crystallization solvent

As the agglomeration process is a complex mechanism, in order to study the influence of the interfacial tension between the crystallization solvent and water, the crystallization solvents were carefully chosen. The selected solvents have a significant difference in interfacial tension with water and have roughly the same physico-chemical properties (density, viscosity, ibuprofen solubility and miscibility with water).

Interfacial tension measurements.

The measured interfacial tension at the pure organic solvent/water interface ranged from 0.5 ± 0.1 for butan-2-one to 13.0 ± 0.9 mN/m for n-butyl acetate. When the measurements were performed with organic phase containing ibuprofen, the solvent/water interfacial tension varied with the mass fraction of ibuprofen introduced. To be sure that the scale of interfacial tension values is not modified by the addition of the solute, measurements of interfacial tension between water and organic solvent with different concentrations (up to the solubility) of ibuprofen were performed. The equilibrium interfacial tensions values are plotted against the mass fraction of ibuprofen introduced in Fig. 7.

The results show that for ethyl acetate and n-butyl acetate, the interfacial tension decreases with increasing the amount of ibuprofen. On the contrary, for butan-2-one the interfacial tension increases when increasing the amount of ibuprofen. The ibuprofen and the organic solvent molecule will compete for the interface based on their interfacial activity. The contribution of each component of the organic phase in the interfacial energy will affect the stability of the system. Therefore, the introduction of ibuprofen in the solvent stabilizes the organic phase/water system. In the case of MEK, the reverse is observed. Nevertheless, the interfacial tension scale is maintained when ibuprofen is added to the organic solvent, i.e.:

MEK/water < Acet/water < n-but/water .

Influence of the organic solvent-water interfacial tension of secondary agglomeration.

The reduction in the interfacial free energy of an emulsion system contributes to the emulsion stability. This means that reducing the interfacial energy between the organic solvent and the water would reduce droplets coalescence and secondary agglomeration. A set of experiments was performed with different organic solvents for the same hydrodynamic conditions (N = 200 rpm) and at constant volume fraction of dispersed phase ( = 2%). The results obtained in terms of particles size distribution are presented in Fig. 8. As already mentioned, when n-butyl acetate is used as the crystallization solvent (experiment no. 9), it was impossible to obtain spherical primary particles. In fact, due to the high interfacial tension between n-butyl acetate solution and water, the generated droplets are large and tend to coalesce to form larger droplet, which is incompatible with the spherical crystallization process. On the contrary, when butan-2-one is used as the crystallization solvent (experiment no. 8), the proportion of fully spherical agglomerates obtained is increased by 200% in comparison to the process using ethyl acetate (cf. Table 3). As expected, the rate of secondary agglomeration process is greatly decreased.

In addition, to show that interfacial energy is not the key parameter for designing such a process, classical spherical crystallization experiments were performed by adding a polymer to the aqueous phase (experiment no. 7). The water phase was formulated in a way that the interfacial tension of the system ethyl acetate/water-PVA (0.1%) was the same as for the butan-2-one/system. The SEM photographs of the particles obtained are shown in Fig. 9. This figure and the results of the mass fraction of primary particles obtained given in Table 3 clearly show that, when a polymer is added to water (i.e. for the same interfacial tension energy and a slightly increase in the viscosity of the aqueous phase), the secondary agglomeration process is totally suppressed by steric effects.

Conclusion

The experiments performed in this study allow to reveal the different mechanisms during a quasi-emulsion spherical crystallization process without additives. Indeed, whatever the operating conditions are, two types of agglomerates are obtained: primary spherical agglomerates whose size is determined by the capillary diameter, and secondary agglomerates resulting of the agglomeration of the spherical primary particles.

On the contrary of what have been observed for other spherical quasi-emulsion crystallization processes, the results obtained in this study suggest that the formation of the primary spherical particles results from spherulitic crystal growth mechanism and not from a agglomeration of fine crystals in the droplet.

In addition, even if the agglomeration process is a very complex mechanism, thanks to the experiments performed with different solvents and with polymer, it can be stated that the interfacial tension is not the key parameter to consider for designing such a process.

So, the answer to the question: "is spherical crystallization without additive possible?" is no for ibuprofen (and probably for other hydrophobic compounds) in the range of operating conditions tested. It is shown, in this study, that for totally avoiding agglomeration of primary particles, the aqueous phase has to be properly formulated. pouring rate (ml s -1 ) V droplet volume (ml) W pp primary spherical particles mass fraction

Fig. 1 -

 1 Fig. 1 -Ibuprofen molecular structure.

Fig. 2 -

 2 Fig. 2 -Spherical crystallization process experimental setup.

Fig. 3 -

 3 Fig. 3 -Microphotograph particles obtained in ethyl acetate/water system. (a) Spherical primary particles; (b) secondary agglomerates.

Fig. 5 -

 5 Fig. 5 -Particle sizes distribution for different stirring rates in ethyl acetate/water system. (a) Experimental size distribution; (b) interpolated sizes distribution.

Fig. 6 -

 6 Fig. 6 -Microphotograph of secondary agglomerates. (a) 100 rpm; (b) 200 rpm; (c) 450 rpm.

Fig. 8 -

 8 Fig. 8 -Particle sizes distribution for different crystallization solvents. (a) Experimental size distribution; (b) interpolated sizes distribution.

Fig. 9 -

 9 Fig. 9 -Microphotograph of particles obtained in water-PVA systems.

Table 1 -

 1 Physico-chemical properties of the used solvents at 25 • C.

	Solvent	(kg m -3 )	(mPa s)	(mN m -1 )		Solubility, w a
					In water	Ibuprofen b
	Butan-2-one	805	0.42	1	0.080	0.401 (0.486)
	Ethyl acetate	905	0.51	6.8	0.082	0.361 (0.520)
	N-butyl acetate	835	0.83	14.5	0.070	0.391 (0.531)
	Water	999	1	-		1.16 × 10 -6

a Mass fraction solubility from

[START_REF] Cano | Experimental study on the ibuprofen crystal growth morphology in solution[END_REF]

,

[START_REF] Garzón | Temperature dependence of solubility for ibuprofen in some organic and aqueous solvents[END_REF]

, and

[START_REF] Gracin | Solubility of phenylacetic acid, p-hydroxyphenylacetic acid, p-aminophenylacetic acid, p-hydroxybenzoic acid, and ibuprofen in pure solvents[END_REF]

. b Solubilities of ibuprofen at 40 • C are given in parenthesis.

Table 2 -

 2 Process operating conditions of spherical crystallization.

	Exp no.	Q

C (ml s -1 ) N (rpm)

Pe a (mW kg -1 ) M i (g) a Pe = wNeTN 3 D 5 p .

Table 3 -

 3 Mean sizes and mass fractions of spherical primary particles obtained.

	Exp no.	d M (m)	d initial (m)	W pp
	1	775 (±50)	953	0.27
	2	940 (±95)	1150	0.15
	3	880 (±90)	1080	0.19
	4	719 (±60)	885	0.17
	5	-	-	0
	6	-	-	0
	7	715 (±40)	865	0.39
	8	712 (±20)	861	1
	9	-	-	0

Acknowledgments

The authors would like to thank the French National Research Agency (ANR) for its financial support (ANR-06-BLAN-0355) and Sophie Cerdan for her technical help.


M Rebai

M Prat
email: prat@imft.fr

M Meireles

P Schmitz

R Baclet

Clogging modeling in pleated filters for gas filtration

Keywords: Gas filtration, Fibrous materials, Pleated Filters, Modeling, Clogging

A model of clogging of a pleated filter in gas filtration is presented. The model is obtained by combining a semianalytical model of the flow in a pleated filter and an empirical model of clogging of the planar filter medium applied locally along the pleated filter channels. The model takes into account the formation of a filtration cake of variable thickness at the porous wall of the pleated filter entrance channels and the resulting evolution of the entrance channels aperture distribution during the filtration/clogging process. Based on the numerical predictions, two main clogging scenarios are identified and analyzed. The optimum pleat density, defined as the pleat density maximizing the filter capacity, is determined and shown to be greater than the pleat density minimizing the pressure drop for a clean filter. Predictions of the evolution of overall pressure drop across the pleated filter due to clogging are compared with experimental data.

Introduction

Air filtration pleated filters are commonly encountered in various applications related notably to the nuclear or automotive industries. Our general objective in this context is to develop a numerical tool for simulating the process of gas filtration through pleated fibrous filters. This tool is developed in order to reduce the design time, the number of trials as well as to optimize the filter geometry (transport properties, pleat length and density, impact of filter supply and exit pipes position and sizes, etc.). It should be able to perform accurate predictions of the filter global pressure drop before clogging, as well as its variations during the clogging phase. As illustrated schematically in Fig. 1, four main scales can be distinguished. The first scale is the fiber scale. The second scale is that of the filter medium, a fibrous material in our case. The third scale is the pleat scale. At this scale, the main objective is to compute the evolution of the overall pressure drop across a pleated filter element during clogging for a given flow rate. The fourth scale is the filter scale, i.e. the scale of the cartridge or the filter panel placed within the engine air supply line. In the present work, we concentrate on the third scale, i.e. the pleat scale.

The flow configuration as well as the pleated filter element of interest to this study, is sketched in Figs. 2 and 3 flow . As shown in Fig. 2, the flow average direction is perpendicular to the filter element. The fluid (gas in our case) enters the filter element through the pleat entrance channel, flows through the fibrous materials separating the entrance channel from the pleat exit channel and leaves the filter element through the pleat exit channel. Owing to the symmetries of this system, the study can be performed over the reduced domain shown in Fig. 2. The main objective is to predict the evolution of overall pressure drop across a filter panel as a function of the aerosol properties, filter medium properties, average flow the Navier-Stokes equations directly is possible, at least in the absence of particles, but computational time consuming. Simulation of particles transport and formation of filtration cake in the pleated filter elements shown in Fig. 2 using direct CFD simulations is significantly more involved and the computational times may become prohibitive for a practical use, not mentioning the remeshing problem associated with the cake formation within the pleated filter entrance channel. For this reason, approaches based on direct CFD simulations are not very efficient for performing optimization studies or sensitivity analysis.

It is therefore attractive to develop simplified numerical approaches so as to increase the efficiency of the numerical simulations considerably in this context. In model of the flow (in the absence of particles) can be derived by channel cross-sectional averaging of the Navier-Stokes (NS) equations and by using similarity solutions of the NS equations locally. As discussed in Rebai et al. (submitted for publication), there exist two variants to this approach depending on the average filtration velocity range of interest: one for the low to moderate filtration velocities, i.e. [START_REF] Oxarango | Laminar flow in channels with wall suction or injection: a new model to study multi-channel filtration systems[END_REF], [START_REF] Benmachou | Dynamical clogging of a pleated filter: experimental and theoretical approaches for simulation[END_REF], [START_REF] Benmachou | Etude et modélisation du colmatage d'un filtre plissé[END_REF], the other for the "large" filtration velocities, Rebai et al. (submitted for publication). These variants differ in the similarity solutions to the NS equations used in the derivation of the simplified model, see Rebai et al. (submitted for publication) for more details. As in Rebai et al. (submitted for publication), we are interested in "large" filtration velocities since this is the situation encountered in the application (air filtration in automobile filters) motivating our study. For the sake of completeness, our large filtration velocity simplified flow model is recalled here and reads,

d u 1 dx + 1 h 1 dh 1 dx u 1 + 1 h 1 Re 0 k p 1 -p 2 e = 0 (1) 1 2 1 - 1 2h 1 1 + dh 1 dx 2 d u 1 2 dx + 1 h 1 dh 1 dx × 3 2 - 1 h 1 1 + dh 1 dx 2 u 1 2 - 1 h 2 1 1 Re 0 × h 1 + 1 + dh 1 dx 2 + h 1 d 2 h 1 dx 2 u 1 - 1 h 1 2 Re 0 dh 1 dx d u 1 dx - 1 Re 0 d 2 u 1 dx 2 + d p 1 dx = 0 (2) d u 2 dx + 1 h 2 dh 2 dx u 2 - 1 h 2 Re 0 k p 1 -p 2 e = 0 (3) 8 d u 2 2 dx - 1 Re 0 d 2 u 2 2 dx 2 + 3 2 16 1 h 2 dh 2 dx u 2 2 - 2 Re 0 1 h 2 dh 2 dx d u 2 dx + 1 h 2 2 1 Re 0 2 4 1 + dh 2 dx 2 -h 2 d 2 h 2 dx 2 - 2 h 2 u 2 + d p 2 dx = 0 (4)
where is a numerical constant, ≈ 0.8251. The notations are the same as in Rebai et al. (submitted for publication). Lower case notations are for dimensionless variables whereas upper case letters are used for the dimensional variables. The notation is for the spatial average of variable over the channel half-width. In the above equations, Re 0 is the channel Reynolds number defined as Re 0 = ( U 0 H 0 )/, where is the fluid dynamic viscosity, U 0 is the averaged velocity at the inlet of the entrance channel and H 0 is the pleat entrance half-width at the inlet (see Fig. 2); u and p are the spatially averaged longitudinal velocity component and pressure, respectively; h is the half-width channel, e and k are the thickness and the porous medium permeability respectively. H 0 and U 0 are used as the reference length and velocity respectively. Subscript 1 refers to entrance channel (channel 1), and subscript 2 to the exit channel (channel 2).

As pointed out in Rebai et al. (submitted for publication), the particular properties of the pleat bottoms (see Fig. 2) have a significant impact on the overall pressure drop when the filter is clean. These properties (permeability, thickness) are affected by the compression of the medium in the pleats when the pleated panel is formed from a planar filter medium, but are generally unknown. As in Rebai et al. (submitted for publication), we will therefore consider two "extreme" cases: non-porous pleat bottoms, and pleat bottoms with the same permeability and thickness as for the flat fibrous material. These two cases will be referred to as the case with impervious pleat bottoms and the one with fully porous pleat bottoms, respectively. It is expected that the system actual behavior lies between the behaviors corresponding to these two "extreme" cases.

The above equations have to be solved in conjunction with the following boundary conditions:

u 1 = 1, u 2 = 0 at x = 0; u 1 = 0, p 2 = p e at x = L/H 0 (5)
when the pleat bottoms are considered as impervious or

u 1 + u 2 = 1, u 2 = Re 0 k e (p 1 -p 2 ) at x = 0 (6) u 1 + u 2 = 1, u 1 = Re 0 k e ( p 1 -p 2 ), p 2 = p e at x = L/H 0 (7)
when the pleat bottoms are porous.

From the numerical solution of Eqs. ( 1)-( 5) or ( 1)-( 4) and ( 6)-( 7), see Rebai et al. (submitted for publication) for the details on the numerical procedure, one can determine the velocity field in the pleat channels. For the entrance channel, which is the channel of interest regarding the particle transport problem, the local velocity field is given by,

u(x, y) = u(x)f ′ y h(x) (8) v(x, y) = Re w Re 0 f y h(x) (9)
where Re w = (V w H 1 )/ is the local filtration Reynolds number. The function f is given by [START_REF] Terrill | Laminar flow in a uniformly Porous channel[END_REF] and is given by

f = 1 + 1 Re w -1 f 1 + 1 (Re w -1) 2 f 2 + 1 (Re w -1) 3 f 3 (10) z = 1 - y h(x) (11) f 1 (z) = 1 -e -Rewz -Re w z (12) f 2 (z) = - 1 2 Re 2 w z 2 -3Re w z e -Rewz -Re w z + 3Re w (1 -z -e -Rewz ) Re w -1 (13) f 3 (z) = - 1 8 Re 4 w z 4 - 3 2 Re 3 w z 3 -8Re 2 w z 2 -22Re w z e -Rewz + 1 4 e -2Rew + 11Re w z Re w z -1 1 -z 2 -e -Rewz (14)
In what follows the velocity field given by Eqs. ( 8) and ( 9) is used for computing the particle transport within the pleat entrance channel. Hence the clogging model relies on the assumption that the entrance channel similarity solution is still valid during clogging. This assumption is supported by the fact that the similarity solution is a good approximation in a channel of variable width, as discussed in Rebai et al. (submitted for publication), and the fact that the flow can be considered as quasi-steady in a channel of variable width during clogging, see Section 1. In passing we also note that [START_REF] Dauenhauer | Exact self-similarity solution of the Navier-Stokes equations for a porous channel with orthogonally moving walls[END_REF] have studied the flow inside a porous channel with moving wall with suction or aspiration. For a high filtration Reynolds number with suction in a contracting channel, the similarity of the flow was found to be the same as for a fixed wall for wall velocities and filtration Reynolds numbers corresponding to our case.

The paper is finally organized as follows. In Section 1, the clogging model of a pleated filter is presented. Clogging scenarios are studied in Section 2. In Section 3, the model is used for studying the pleat density maximizing the filter capacity. Some comparisons with experimental results are presented in Section 4.

Pleated filter clogging model

The development of clogging model begins with the consideration of clogging of a planar filter medium. The analysis and the modeling of clogging of a plane fibrous medium has been studied in many works, e.g. [START_REF] Brown | Air Filtration[END_REF], [START_REF] Thomas | Modelling pressure drop in hepa filters during dynamic filtration[END_REF][START_REF] Thomas | Clogging of fibrous filters by solid aerosol particles experimental and modelling study[END_REF], see also Chapter 4 and the references therein in [START_REF] Bejan | Porous and Complex Flow Structures in Modern Technologies[END_REF]. However, the filtration velocities considered are often significantly lower than the ones of interest in our study. Also, in many previous modeling studies, the aerosol particle size distribution is much narrower than the one that used in the application motivating the present study. For these reasons, we have used an empirical model of clogging at the planar filter medium scale based on experimental results, preferring to concentrate our effort on the clogging modeling at the pleated filter element scale. It would be easy, however, to incorporate any better model at the plane filter medium scale in our approach. Fig. 4 shows the evolution of pressure drop across a plane synthetic two-layer fibrous medium as a function of collected mass during experiments of frontal filtration using the ISO Fig. 4 -Evolution of the pressure drop P across a plane synthetic fibrous medium as a function of the collected particles mass per unit area M during the experiment of frontal filtration; P 0 is the pressure drop for the clean medium.

12103 -1 A2 fine Arizona test dust for different mean filtration velocities ranging between 0.77 and 1.42 m/s. The permeability and porosity of the fibrous medium (when clean) are 1.66 × 10 -10 m 2 and 0.9, respectively. The fiber average diameter is 13 m in the first layer and 14 m in the second layer. The total thickness of the planar medium is 2.22 mm. The Arizona test dust is the reference aerosol for testing filters in the automobile industry. This is a polydisperse distribution of roughly spherical grains in the range [0 m, 80 m] with a volume mean diameter of 9.15 m and a standard deviation of 19.6 m whereas the median diameter based on particle number is 0.99 m. One can refer to [START_REF] Rebai | Analyse et modélisation du colmatage d'un filtre plissé au sein d'un filtre à air plissé[END_REF] for more details on this aerosol, as well as for a description of the experimental facility used to perform the experiments.

As can be seen from Fig. 4, the pressure drop evolutions for the various filtration velocities collapse reasonably well onto a single curve, at least for the range of filtration velocity considered here. Also, the results shown in Fig. 4 are consistent with a classical evolution in two main periods, e.g. [START_REF] Brown | Air Filtration[END_REF], namely a period of depth filtration where most of the collected mass corresponds to particles trapped within the filter medium followed by a period of surface filtration with the formation of a filtration cake at the filter medium surface. Neglecting the transition period between these two periods, we assume in what follows that the change from the depth filtration period to the surface filtration period occurs suddenly for a "critical" collected mass per unit of area M cr . Noting that the evolution of the pressure drop is quasi-linear in each period, the simple representation of the pressure drop evolution depicted in Fig. 4 can be expressed as,

P = E K(M) + E c K c V w ( 15
)
where E, E c , K and K c are the filter medium thickness, the filtration cake thickness, the filter medium permeability (which depends on collected mass M) and cake permeability, respectively. The evolution of K as a function of M can be fitted from the experimental data. For the data shown in Fig. 4, this yields an expression of the form (with ˛= 6.72),

K(M) = K 0 ˛M + 1 (16)
as long as M ≤ M cr whereas K(M) = K(M cr ) for M ≥ M cr (M cr ∼ 0.59 kg/m 2 for the data shown in Fig. 4).

Assuming that the cake is a porous medium of uniform porosity ε c , the cake thickness can be expressed as a function of collected mass as.

E c (M) = 0 for M ≤ M cr ; E c (M) = M -M cr p (1 -ε c ) for M ≥ M cr ( 17
)
where p is the particle density ( p = 2650 kg/m 3 ). Combining Eqs. ( 15) and ( 17) yields

P = E K(M) + M -M cr p (1 -ε c )K c V w (18)
Hence in this approach, the cake is characterized by two parameters: its porosity and its permeability. However, from the experimental data (slope of pressure drop evolution in the second period) only the product (1 -ε c )K c can be determined. For example, this yields (1 -ε c )K c = K 0 /(17.7 p e 0 ) from the data shown in Fig. 4. Note that the dimensions of the numerical constant are m 2 /kg. We will return to this problem below when we present the clogging model at the pleated filter element scale. Note that it is not easy to use here a standard expression of permeability as a function of porosity and grain diameter such as the Kozeny-Carman relationship, e.g. [START_REF] Dullien | Porous media[END_REF], owing to the large distribution in the size of particles forming the cake.

In order to model clogging at the pleated filter element scale the next step consists in combining the clogging model for the plane filter medium with the flow model given by Eqs.

(1)-( 5) (impervious pleat bottoms) or Eqs. ( 1)-( 4) and ( 6)-( 7) (full porous pleat bottoms). Before reaching the filter medium or the filtration cake, the particles are transported by the gas flow within the pleated filter entrance channels. As it is well known, e.g. [START_REF] Elimelech | Particle Deposition and Aggregation[END_REF], the trajectory of a particle depends on the various forces acting on it during its travel in the entrance channel. In the present work, it is assumed that the particles follow the streamlines. This assumption is supported by the following estimates of the particle Stokes, Peclet and Froude numbers. Taking as representative values a pleated filter frontal surface of 220 mm. 89 mm, a pleat height of 55 mm, a pleat density of 13 pleats per 100 mm, a gas flow rate of 389 m 3 /h, and as particle diameter the median diameter of the Arizona Fine dust (0.99 m), we obtain for the particle Stokes, Peclet and Froude numbers, St ≈ 0.2, Pe ≈ 10 5 , Fr ≈ 50, respectively. These values indicate that the Brownian diffusion and gravity effects are negligible on the motion of particles. The assumption of neglecting the inertia effects is more questionable since the Stokes number is not very small compared with unity and has been estimated considering only the median diameter of a relatively broad distribution in particle size. Hence the motion of the biggest particles in the distribution is likely to be affected by inertia effects. This suggests that the influence of inertia effects on the particle transport would deserve to be considered more carefully in future works. Nonetheless, on the basis of the above estimates, the assumption of particles following streamlines appears to be a fair first approximation. Under these circumstances, the particles deposition rate is simply proportional to the local filtration velocity. As a result of the particles capture, a filtration cake eventually forms within the pleated filter element entrance channel. This induces a modification of the free fluid domain within the entrance channel since the local aperture decreases as the cake grows. This results in turn in the evolution of the fluid flow within the entrance channel due to the modification of both the fluid domain geometry and filtration velocity distribution along the pleat channel. These effects are taken into our model under the assumption of a quasi-steady evolution of the entrance channel flow. This assumption is supported by the fact that the travel time of a particle in the entrance channel is much smaller than the time needed for observing a significant change in the filtration cake local thickness.

To compute the evolution of filter medium properties (depth filtration period) and cake local thickness (surface filtration period), the pleat entrance channel is discretized along the x-axis into small elements (grid spacing) of uniform size dX. The mass collected during the time dT over the element located at the position x is expressed as

dM s = C u V w (x)ϕℓdX dT ( 19
)
where C u and ℓ are the particle concentration in the aerosol (kg/m 3 ) and extension of filter element in the z direction (direction perpendicular to the figure in Fig. 3), respectively. In Eq. ( 11), ϕ is the porous medium particles capture efficiency. The efficiency evolves during the clogging stage [START_REF] Brown | Air Filtration[END_REF] and this evolution has to be taken into account to determine the evolution of the mass of particles collected by the filter as a function of time. Here we are primarily interested in the evolution of pleated filter properties as a function of the overall collected mass assuming the porous medium properties local evolution is known as a function of the mass collected locally. Hence we can set arbitrarily the efficiency ϕ to 1 in the simulations. The deposit kinetic at the filter scale is clearly affected by this assumption but not the evolution of pleated filter properties as a function of overall collected mass.

As long as the local collected mass M(x) = M s (X)/dX/ℓ is lower than M cr , the evolution of the local permeability of the filter medium is computed according to equation ( 16). For M(X) ≥ M cr , we apply equation. ( 17) so that the evolution of the channel local half-width is given by

dH 1 (x) = - C u V w (x) p (1 -ε c ) dT ( 20
)
whereas the local filtration velocity is computed using Darcy's law, Eq. ( 18), applied locally,

P 1 (x) -P 2 (x) = E K(M(X)) + H 1 (X, 0) -H 1 (X, T) K c V w (X) (21)
where the pressure distributions P 1 (X) and P 2 (X) are obtained by solving the system of Eqs. ( 1)-( 5) or Eqs. ( 1)-( 4) and ( 6)-( 7).

To be consistent with the system (1)-( 5) or ( 1)-( 4)-( 6)-( 7), the clogging model is expressed in dimensionless forms using, as for the flow model, H 0 as characteristic length and U 0 as 16)-( 18).

characteristic velocity. This leads to

h 1 (x, t + dt) = h 1 (x) -Re 0 k(x)c p 1 (x) -p 2 (x) (1 -ε c )e(x, t) dt (22) e(x, t + dt) = e(x) + Re 0 k(x)c p 1 (x) -p 2 (x) (1 -ε c )e(x, t) dt (23) dm = k(x)Re 0 c p 1 (x) -p 2 (x) e(x, t) dt ( 24
)
where c = C u / p , m is the amount of particles per unit area deposited on fibrous media scaled by 1/ p H 0 ; dm is the amount of particles deposited during a time step dt.

Hence the simulation of clogging is obtained by solving numerically Eqs. ( 1)-( 5) or Eqs. ( 1)-( 4) and ( 6)-( 7) together with Eqs. ( 16)-( 17) and ( 22)-( 24). One can refer to Rebai et al. (submitted for publication) for details on the numerical procedures used to solve equations ( 1)-( 5) or Eqs. ( 1)-( 4) and ( 6)-( 7). Since the flow is considered as quasi-steady, the clogging model does not bring any significant additional numerical difficulties, see [START_REF] Rebai | Analyse et modélisation du colmatage d'un filtre plissé au sein d'un filtre à air plissé[END_REF] for the details. The only one worth mentioning is the fact that the length of the free fluid domain in the entrance channel can decrease as a result of cake growth in the pleat bottom. This can be obtained when the cake lateral growth (growth in the y direction) is sufficient for blocking entirely the channel or when there is a sufficient upward longitudinal growth from the pleat bottom wall (this second scenario is possible only with a porous pleat bottom). To deal with the modification of the free fluid domain in the longitudinal direction, we have used a rather crude but simple procedure: when the cake growth is sufficient for blocking entirely the last grid mesh, this mesh is removed from the free fluid domain. The corresponding region is then considered as porous (or impervious) when the pleat bottom is porous (or impervious). This procedure amounts to considering a discrete evolution of the entrance channel free fluid domain length. Although it would be probably interesting to Fig. 6 -Evolution of filtration velocity v w scaled by the averaged filtration velocity, and pleat half-width H 1 along the pleat at different stages of clogging in a pleated filter with impervious pleat bottoms. The frontal surface of pleated filters is 220 mm. 89 mm, the pleat height is 55 mm and the pleat density is 31 pleats/100 mm. The fibrous material is a two-layer medium with an overall thickness and a permeability of 2.84 mm and 2.28 × 10 -10 m 2 respectively. The flow rate is 386 m 3 /h. consider a more refined technique involving for example frequent remeshing and the associated interpolation techniques, the simple procedure used in the present effort is acceptable for small grid spacing. After some tests, it was found that a grid of 300 regularly spaced points was sufficient to simulate the pleat clogging.

Clogging scenarios

For the 'large' average filtration velocities considered in this paper, the filtration velocity is not uniform along the pleat channels and generally increases with x, at least when the filter is clean, see Rebai et al. (submitted for publication).

For this reason, the cake tends to grow more rapidly in the bottom of the entrance channel initially. This increases the hydraulic resistance in the entrance channel bottom. As a result, the pleat bottom properties (=hydraulic resistance of filter medium forming the pleat bottom) are parameters much less sensitive than for the prediction of pressure drop for a clean filter, Rebai et al. (submitted for publication). This is illustrated in Fig. 5 which shows that the evolution of pressure drop is identical during clogging, except at the very beginning, regardless of the porous or impervious nature of pleat bottom. In the remainder of this paper, all the results presented are obtained for a gas flow rate of 389 m 3 /h, a frontal surface of the filter panel = 89 mm. 220 mm, a pleat length L = 55 mm and the fine Arizona dust as particles. Hence the details of the pleat bottom properties, which are generally unknown, are important for the prediction of pressure drop across a clean filter but are not important for the evolution of pressure drop during clogging.

Our model can be used for analyzing the pleated filter clogging scenarios, as well as the influence of the problem various parameters on clogging. This is illustrated in what follows with the consideration of the impact of the pleat density on clogging scenarios. We begin with the study of clogging in a pleated filter channel of initially uniform width. A first clogging scenario is shown in Fig. 6 and can be termed as "cake lateral growth clogging". As shown in Fig. 6, this case is characterized by the cake growth from the channel lateral walls and a significant evolution of the filtration velocity distribution during clogging. Initially, the filtration velocity monotically increases along the channel and is maximum at the channel bottom. This induces a preferential cake growth in the channel bottom region, which induces in turn a displacement of the maximum filtration velocity in the upstream direction. As a result, the filtration velocity distribution becomes more uniform and eventually can even decrease along the channels. This flow reorganization is accompanied by the progressive growth of the cake all along the channel lateral walls.

As illustrated in Fig. 7, the clogging scenario is completely different for the pleat density of 14 pleats/100 mm (for which Re 0 = 1110 whereas Re 0 = 1973 for the pleat density of 8 pleats/100 mm, noting that the greater Re 0 , the greater the filtration velocity in the pleat bottom for a given porous medium permeability, a given porous medium thickness and a given pleat length). This case can be referred to as the "from the bottom cake growth clogging". In this case, the filtration velocity is always a maximum at the channel bottom and the filtration cake essentially growths from the channel bottom. Intermediate scenarios with both types of cake growth, i.e. lateral and from the bottom, are obtained for intermediate pleat densities, [START_REF] Rebai | Analyse et modélisation du colmatage d'un filtre plissé au sein d'un filtre à air plissé[END_REF].

As shown in [START_REF] Rebai | Analyse et modélisation du colmatage d'un filtre plissé au sein d'un filtre à air plissé[END_REF], simulations performed for pleated filter channels of variable width indicate that the pleated filter element shape can also have an influence on the clogging scenario. For example, in a channel of decreasing width, Fig. 3b, the maximum filtration velocity is not necessarily located at the channel bottom initially (clean filter) but significantly upstream from the bottom. For the highest pleat densities, this may lead to the formation of a filtration cake bridge upstream from the pleat channel bottom. In this case, the part of the filter located downstream from the cake bridge becomes useless for filtration. One can refer again to [START_REF] Rebai | Analyse et modélisation du colmatage d'un filtre plissé au sein d'un filtre à air plissé[END_REF] for more details.

Optimal pleat density

Consider a filter panel of given frontal area and volume. We want to determine the pleat density (=number of pleats per 100 mm) that maximizes the filter capacity. The filter capacity is classically used to characterize the performance of a filter. For a specified flow rate, the filtration capacity FC can be defined as the collected mass for a given overall pressure drop increase ıP = P -P 0 from the pressure drop P 0 across the clean filter. For the same conditions (flow rate, pleat optimal pleat density decreases as the flow rates is increased whereas an inverse evolution is found for the capacity optimal pleat density, as shown in 10. Also, Fig. 10 shows that the optimal pleat density range becomes larger when the flow rate is reduced and tends to include the optimal range obtained for higher flow rate. Interestingly, the optimal capacity is not very sensitive to the flow rate.

As a last illustration of the results that can be obtained using the model proposed in this paper, Fig. 11 shows the influence of pleat length. As can be seen from Fig. 11 for a pleat of uniform width the capacity increases with the pleat length and the optimal pleat density decreases with the pleat length. As shown in Fig. 12, however, increasing the pleat length does not implies systematically an increase of capacity when the pleat aperture is non-uniform.

Comparison with experimental data

In order to evaluate the performance of the clogging model, comparisons are made in this section with some experimental data for the overall pressure drop evolution during clogging. However, as explained below, we have not been in a position to perform this comparison as well as it would have been desirable for a full validation. Nonetheless the partial validation presented in this section together with the results presented in Section 3 is quite encouraging. The comparisons are made for the following conditions: air flow rate of 389 m 3 /h, frontal surface of the test panel = 89 mm. 220 mm, pleat length L = 55 mm and the fine Arizona dust as particles. Details on the experimental facility can be found in [START_REF] Rebai | Analyse et modélisation du colmatage d'un filtre plissé au sein d'un filtre à air plissé[END_REF]. Results of the experiments are shown in Fig. 13 for two pleat densities, namely 8 pleats/100 mm (which corresponds to a total number of 17 pleats) and 12 pleats/100 mm (total number of pleats = 26). In the simulation, we have considered a pleated filter element of variable aperture with H s /H 0 = 0.2 (see Fig. 3 for details on the notations) and with impervious pleat bottoms. The first simulation attempts have been performed using the parameters of plane filter medium clogging model given in Section 1. This led to poor results, [START_REF] Rebai | Analyse et modélisation du colmatage d'un filtre plissé au sein d'un filtre à air plissé[END_REF]. For technical reasons linked to the applied context in which this work has been developed, it must, however, be pointed out that the filter medium used in the pleated filter panel experiments is unfortunately not the same as the one that used in the plane filter medium experiments (see Fig. 4). The thickness and permeability of the filter medium used in the clogging plane medium experiments are 2.22 mm and 166 m 2 , respectively, whereas the corresponding values for the filter medium used in the respectively. Thus, one can surmise that this affect the clogging curve at the plane filter medium scale. For this reason, we have decided to adjust two parameters so as to obtain a good agreement with the experimental data for the pleat density of 8 pleats/100 mm (17 pleats). The two parameters are the cake porosity ε c and the critical collected mass M cr marking the transition between the depth filtration and the surface filtration. This leads to M cr = 0.118 kg/m 2 (as opposed to 0.59 kg/m 2 for the plane medium experiments presented earlier) and ε c ∼ 0.65-0.7. As can be seen from Fig. 13, a good agreement with the experimental results is indeed obtained with these values for the pleat density of 8 pleats/100 mm. Note that the parameter ˛of the evolution of filter medium permeability during the depth filtration period (see Eq. ( 1)) and the cake permeability are not adjusted but deduced from the plane filter medium experiments. More interestingly, Fig. 13 shows that a good agreement is also obtained for 12 pleats/100 mm with the adjusted values for 8 pleats/100 mm. Thus we believe that our model captures the essential features of clogging in a pleated filter element and can be used with some confidence for analyzing clogging. This conclusion is also supported by the study of the optimum pleat density presented in Section 3, which is also consistent with the available experimental data and the fitting of M cr used in this section. The results reported in Fig. 13 also indicate that the cake properties are sensitive parameters. A change in cake porosity directly affects the growth rate of the cake and this has a direct impact on the pressure drop generated in the pleat entrance channel.

Conclusion

A numerical model was proposed to simulate the clogging of a pleated filter element for the high filtration velocities representative of gas filtration in automobile applications. The model is obtained by combining a semi-analytical model of flow in a pleated filter and an empirical model of clogging of the plane filter medium applied locally along the pleated filter channels. The model takes into account the formation of a filtration cake of variable thickness at the porous wall of the pleated filter entrance channels and the resulting evolution of the entrance channels aperture distribution during the filtration/clogging process.

The results of simulation are consistent with the available experimental data. Nevertheless, more comparisons with experimental data are certainly needed. In particular, it would be interesting to test the model for simpler conditions than those considered here, for example from experiments with much narrower aerosol particle size distribution. This would also permit estimation of the porosity and permeability of the filtration cake using conventional models, e.g. for example the Kozeny-Carman relationship, or from a specific experiment. This would limit the number of unknown parameters in the model.

The model is much easier to use and much less computationally time demanding than a CFD code. For this reason, it greatly facilitates the performing of sensitivity studies. This has notably allowed us to identify two main clogging scenarios; clogging by filtration cake lateral growth and clogging by upstream growth of the cake from the pleat channel bottom. Experimental confirmation of this finding would be of great interest.

From a practical standpoint, the proposed model can be used as a design-sizing tool to limit for example the number of trials in the development of a filter. This has been illustrated in the present work by determining the pleat density maximizing the filter capacity.

Fig. 2 -

 2 Fig. 2 -Sketch of flow configuration and pleated filter element. The regions in grey correspond to the porous medium (in light grey for the pleat bottoms and in darker grey elsewhere along the pleat). The red arrows schematize the flow through the porous medium whereas the blue arrows are for the flow in the entrance and exit channels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

  Fig. 1 -Four scales of interest for the analysis of filtration in a filter. In this work, we concentrate on the pleated filter element scale (box with the dashed frame).

Fig. 5 -

 5 Fig.5-Evolution of pressure drop P as a function of mass M T of deposited particles for a pleated filter of uniform pleat aperture with impervious pleat bottoms or fully porous pleat bottoms. The frontal area is 220 mm. 89 mm and the pleat height is 55 mm. The total number of pleats is 26. The fibrous medium is a two-layer medium with an overall thickness and permeability of 2.84 mm and 2.28 × 10 -10 m 2 , respectively. The flow rate is 389 m 3 /h. Note that Kozeny-Carman model has been used for this simulation instead of the empirical model described by Eqs. (16)-(18).

Fig. 7 -

 7 Fig.7-Evolution of the filtration velocity v w (scaled by the averaged filtration velocity) and dimensionless pleat half-width H 1 along the pleat at different stages of clogging in a pleated filter with impervious pleat bottoms. The frontal surface of pleated filters is 220 mm. 89 mm, the pleat height is 55 mm and the pleat density 31 pleats/100 mm. The fibrous material is a two-layer medium with an overall thickness and a permeability of 2.84 mm and 2.28 × 10 -10 m 2 respectively. The flow rate is 386 m 3 /h. Simulation has been performed with impervious pleat bottoms.

Fig. 8 -

 8 Fig. 8 -Optimal pleat density. (i) without clogging (clean filter): variations of net pressure drop P (Pa) as a function of pleat density N (pleats number/100 mm), (ii) with clogging: variations of pleated filter capacity FC as a function of pleat density N (pleats number/100 mm) for an extra pressure drop of ıP = 15 mbar.

Fig. 10 -

 10 Fig. 10 -Evolution of pleated filter capacity as a function of pleat density N (number of pleats number/100 mm) for different flow rates. The pleated filter frontal surface is 220 mm. 89 mm and the pleat height is 55 mm. The pleat bottoms are impervious. The fibrous material is a two-layer medium with an overall thickness and permeability of 2.84 mm and 2.28 × 10 -10 m 2 , respectively.

Fig. 11 -

 11 Fig. 11 -Evolution of the filter capacity as a function of pleat density N for different pleats heights. The pleat aperture is uniform and the pleat bottoms are impervious.The flow rate is 389 m 3 /h. The frontal surface of pleated filter is 220 mm. 89 mm. The fibrous material is a two-layer medium with an overall thickness and permeability of 2.84 mm and 2.28 × 10 -10 m 2 , respectively.

Fig. 12 -

 12 Fig. 12 -Evolution of filter capacity as a function of pleat density N for two pleat heights 55 and 75 mm with uniform (ı = 1) and non-uniform (ı = 0.2) pleat aperture. The flow rate is 389 m 3 /h. The frontal surface of pleated filter is 220 mm. 89 mm. The pleat bottoms are impervious The fibrous material is a two-layer medium with an overall thickness and permeability of 2.84 mm and 2.28 × 10 -10 m 2 , respectively.

Fig. 13 -

 13 Fig.13-Evolution of overall pressure drop P (mbar) of pleated filter as a function of mass of deposited particle mass. The frontal surface of pleated filter is 220 mm. 89 mm and pleat height is 55 mm. The fibrous material is a two-layer medium with an overall thickness and permeability of 2.84 mm and 2.28 × 10 -10 m 2 , respectively. The flow rate is 389 m 3 /h.