Audrey Juban

Stéphanie Briançon

François Puel
email: francois.puel@centralesupelec.fr

Stéphanie Briancon

Prof F Puel

Processing-induced-transformations (PITs) during direct compression: Impact of tablet composition and compression load on phase transition of caffeine

Keywords: Polymorphic transformation Processing induced transition (PITs) Tabletting Direct compression DSC 2519), Microcrystalline cellulose (PubChem CID: 14055602), Anhydrous dicalcium phosphate

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Processing-induced-transformations (PITs) during direct compression: Impact of tablet composition and compression load on phase transition of caffeine

Introduction

Solid-state transitions during manufacturing of pharmaceuticals are of great importance for pharmaceutical industry. For several years, processing-induced-transformations (PITs) in pharmaceutical manufacturing have been reported. Many authors [START_REF] Morris | Theoritical approaches to physical transformations of active pharmaceutical ingredients during manufacture processes[END_REF][START_REF] Brittain | Effects of mechanical processing on phase composition[END_REF][START_REF] Zhang | Phase transformation considerations during process development and manufacture of solid oral dosage forms[END_REF][START_REF] Descamps | Transformation of pharmaceutical compounds upon milling and comilling: the role of Tg[END_REF] have started to set up theoretical and experimental approaches of these transformations. The PITs includes all the physical or chemical transformations of molecular crystals caused by one or more steps of preparation and shaping processes. In the pharmaceutical field, these transformations have both an economic and health impact. These changes can affect the quality of products produced and it is possible that non-compliant consignments are not released on the market for this reason [START_REF] Bauer | Ritonavir: an extraordinary example of conformational polymorphism[END_REF]. Today, manufacturers have no control over these phenomena. The theoretical approach of the PITs [START_REF] Morris | Theoritical approaches to physical transformations of active pharmaceutical ingredients during manufacture processes[END_REF][START_REF] Petit | Transformations physiques de cristaux moléculaires induites par les procédés industriels de fabrication : approche théorique, conséquences et exemples. Cristal 3, conférence sur la cristallisation et la précipitation industrielles[END_REF] is recent and proposes kinetics and thermodynamics consideration of these transformations. This theory assumes that each step of a process can generate a stress (thermal, mechanical, or chemical) for the product which cause a shift in the thermodynamic equilibrium of the system. In the literature, the more documented PITs are the ones observed during milling, granulation and compression.

Tablets are the most common dosage form for oral administration in the world [START_REF] Wu | A comparative study of compaction properties of binary and bilayer tablets[END_REF].

The use of direct compression has rapidly increased [START_REF] Zhang | Physical properties and compact analysis of commonly used direct compression binders[END_REF] in the past few years because of its economic interest and its process which removes the steps of wet granulation and drying processes. The basis of the direct compression is to add a key component (generally a diluent) in the formulation which possesses the necessary degree of fluidity and compressibility. If it necessary, other ingredients could be included, such lubricant, a glidant and/or a disintegrating agent [START_REF] Armstrong | Tablet manufacture[END_REF]. However, active pharmaceutical ingredients (APIs) and excipients may exist under different crystalline forms or amorphous phase depending on the nature of the treatment that they received.

Many parameters in the process itself could have an impact on the average transformation in final tablets like pressure, temperature and mechanical stress. It is not uncommon to observe phase transition as polymorphic transformation during the compression process [START_REF] Chan | Polymorphic transformation of some drugs under compression[END_REF][START_REF] Lefebvre | Polymorphic transitions of carbamazepine under grinding and compression[END_REF][START_REF] Boldyreva | Effect of pressure up to 5.5 GPa on dry powder samples of chlorpropamide form-A[END_REF][START_REF] Otsuka | Nano-and macro-geometrical structural change of caffeine and theophylline anhydrate tablets during hydration process by using X-ray computed tomography[END_REF][START_REF] Hubert | Process induced transformations during tablet manufacturing: phase transition analysis of caffeine using DSC and low frequency micro-Raman spectroscopy[END_REF]. The consequences of this phenomenon are an unpredictable solid phase composition and may induce a modification of bioavailability (Doelker, 2002, Ticehurst and[START_REF] Ticehurst | Integration of active pharmaceutical ingredient solid form selection and particle engineering into drug product design[END_REF].

In this study, caffeine (C8H10N4O2) was used as an API model. Caffeine is one of the most widely used psychoactive drug in the world and it is cheap and readily available [START_REF] Sahoo | Molecular recognition of caffeine in solution and solid state[END_REF]. This common molecule is well known for polymorphism studies (Pirttimäki et al., 1993;[START_REF] Lehto | A kinetic study of polymorphic transition of anhydrous caffeine with microcalorimeter[END_REF][START_REF] Manduva | Calorimetric and spatial characterization of polymorphic transitions in caffeine using quasi-isothermal MTDSC and localiezd thermomechanical analysis[END_REF][START_REF] Mazel | Polymorphic transformation of anhydrous caffeine under compression and grinding: a re-evaluation[END_REF] which presents a simple enantiotropic behaviour.

Two polymorphic forms of caffeine is known, the Form II is the stable polymorph at room temperature until 145°C and the Form I is stable at ambient pressure from the transition temperature (145°C) to its melting point 236°C [START_REF] Pinto | Thermochemical study of two anhydrous polymorphs of caffeine[END_REF]. The kinetics of the phase transformation of caffeine form I into form II at solid state was studied at 363 K using time-resolved dielectric relaxation spectroscopy [START_REF] Decroix | Time-resolved dielectric investigation of relaxation kinetics in metastable caffeine[END_REF] and law frequency raman spectroscopy (Hedoux et al., 2011b). [START_REF] Hédoux | Polymorphic transformation of anhydrous caffeine upon grinding and hydrostatic pressuring analyzed by low-frequency raman spectroscopy[END_REF] have thoroughly investigated the transition of the caffeine form I to form II by hydrostatic pressurizing and grinding. They have shown that an intermediate metastable form may exist under pressure or stabilize during the grinding, but disappear as soon as the application of pressure or the milling process is stopped.

The purpose of this study was to improve the understanding of the PITs in a direct compression process. This work is focused on two formulation parameters: nature of the diluent and the impact of the caffeine dilution, and one process parameter: the compression pressure that may impact the phase transition of the caffeine form I. Tablets were made at three different compression pressures. Two diluents exhibiting different behaviours under compression (plastic: microcrystalline cellulose and fragmentary: anhydrous dicalcium phosphate [START_REF] Ilkka | Prediction of the compression behaviour of powder mixtures by the Heckel equation[END_REF]) and their mixtures with caffeine form I were studied. The determination of the transition degree of caffeine was performed by thermal analysis.

Materials and methods.

Materials

The material used in this study is caffeine Form I (CFI). It was obtained by using the same method than [START_REF] Hubert | Process induced transformations during tablet manufacturing: phase transition analysis of caffeine using DSC and low frequency micro-Raman spectroscopy[END_REF] which based on these suggested by [START_REF] Derollez | Ab initio structure determination of the high-temperature phase of anhydrous caffeine by X-ray powder diffraction[END_REF] and [START_REF] Griesser | Vapor pressure and heat of sublimation of crystal polymorphs[END_REF] as follows. Commercial anhydrous caffeine Form II (CFII), purchased from Cooper, was heated to 170°C in an oven for 24 hours in order to anneal the Form II. Freshly Form I was quickly cooled in liquid nitrogen until room temperature. On each batch a Differential Scanning Calorimetry (DSC) analysis was performed at a heating rate of 10°C/min from 20°C to 270°C and no trace of the transition III was found (Fig. 1). Microcrystalline cellulose (MCC), Avicel® PH-102, was obtained from FMC Biopolymer and anhydrous dicalcium phosphate (DCPA) from JRS Pharma. The particle size distribution for these two materials was measured with a Mastersizer 3000 using an AeroS dry dispersion unit (Malvern Instruments, U.K.). Three measurements of the particle size distribution were performed on each powder. The Fraunhofer model was selected in order to transform the dynamic light scattering measurements in a particle size distribution. The resulting volume density function distribution was averaged. The calculation of the different standard percentile (Dv10, Dv50 and Dv90) is performed following the British standard BS2955.1993. Their Their values with their standard deviations are listed in Table 1.

Tablet preparation

Cylindrical tablets were prepared with a binary mixture of anhydrous caffeine Form I (a day after its manufacture) and MCC or DCPA as a binder/diluent. Each diluent was mixed in various proportions with caffeine content of 60, 78, 90 wt% of CFI for MCC, and 40, 60 and 80 wt% of CFI for DCPA. The blending was performed with a tridimensional mixer (Turbula® T2F) at a rotational rate of 49 rpm (for 10 minutes). A series of pure CFI tablets (100 wt%) was also made. All compacts were made at constant mass (about 300 mg) by using a compaction simulator Styl'One Classic (Medelpharm, Beynost, France) and its data acquisition software (Analis, 2.03 versions, Medelpharm). This tableting press is a single station press. Compression was made by the lower punch. The compression forces were measured with strain gauges located on the upper and lower punch holders with an accuracy of 0.5% of full scale, and the displacements of the punches were monitored using Linear Variable Differential Transformers (LVDTs) with an accuracy of 20 µm for upper punch and 50 µm for lower punch. Standard Euro D tools with flat-punches 11.28 mm of diameter (corresponding to an area of about 1 cm²) were fitted on the simulator.

The deformation of the machine (including punch deformation) was taken into consideration and measured before experiments to correct the values of the displacement. In this work, tablets were produced (without lubricant) with one main compression driven by force under three different compression forces (5 kN, 10 kN, 20 kN) corresponding at 50 MPa, 100 MPa and 200 MPa in compression pressure, respectively. The tablet weights and thicknesses (shown in Table 2) were measured just after compression according to the European Pharmacopoeia method section 2.09.05 (European Pharmacopoeia, 2014). The storage was made at room temperature during six months (over one year the temperature may vary in a range of from 18 to 30°C).

Thermal analysis

The Differential Scanning Calorimetry (DSC) apparatus used for the calorimetric measurements was a Q200 ® from TA Instruments. For all measurements, aA nitrogen purge of 50 mL/min was employed.

Aluminium crucibles with perforated caps were used and the heating rate was 10°C.min -1 from 20°C to 270°C. It was not possible to increase the MCC content in the powder mixture since the endotherm of evaporation hid the transition endotherm from CFII to CFI. Analysis were made in triplicate (i.e., for one batch of caffeine form I, and thus for one composition studied, three tablets were analysed in the same conditions (storage, compression pressure) in order to ensure the reliability of data and the reproducibility of the measurements). DSC takes into account a macroscopic fraction (about 10 mg) of the tablet and was applied in order to estimate transition degree in tablet parts (core and border) as

shown in Figure 2. For the studies of CFI transition degree, a kinetic analysis has been performed for each mixture, at the following specific times: 2, 15, 30, 45, 60, 90 and 180 days after tablets fabrication.

Validation of the analytical DSC method to quantify caffeine in tablets

In this work, the transition temperature II→I of caffeine was measured in triplicate for the commercial anhydrous caffeine Form II. The transition temperature considered Ttrans(II→I) was 140.63 ±0.22 •C and the associated enthalpy of transition ∆transH was 17.42 ± 0.76 J/g. These data are in accordance with the ones find in the literature (collected by [START_REF] Hubert | Process induced transformations during tablet manufacturing: phase transition analysis of caffeine using DSC and low frequency micro-Raman spectroscopy[END_REF]. In order to quantify caffeine in tablets, the three following equations, described by Hubert et al., 2011, are needed. Equation 1 allows correlating between the amount of caffeine Form II in a sample and the enthalpy of transition (∆transH)

of pure Form II:

% 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐𝑜𝑜𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝐹𝐹𝑜𝑜𝐹𝐹𝐹𝐹 𝐼𝐼𝐼𝐼 = ∆ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐻𝐻 (𝐼𝐼𝐼𝐼→𝐼𝐼) 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 ∆ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐻𝐻 (𝐼𝐼𝐼𝐼→𝐼𝐼) 𝑜𝑜𝑜𝑜 𝑠𝑠𝑝𝑝𝑝𝑝𝑒𝑒 𝑜𝑜𝑜𝑜𝑝𝑝𝑠𝑠 𝐼𝐼𝐼𝐼 × 100 (1)
For tablets made from binary mixtures of CFI and MCC, the dilution impact of the excipient must be taken into account for the calculation of the transition degree of Form I. The total amount of caffeine in the sample is given below by the equation 2:

% 𝑜𝑜𝑜𝑜 𝑐𝑐𝑐𝑐ℎ𝑦𝑦𝑦𝑦𝐹𝐹𝑜𝑜𝑦𝑦𝑦𝑦 𝑐𝑐𝑐𝑐𝑜𝑜𝑜𝑜𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = ∆ 𝑓𝑓𝑓𝑓𝑡𝑡 𝐻𝐻 (𝐼𝐼) 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 ∆ 𝑓𝑓𝑓𝑓𝑡𝑡 𝐻𝐻 (𝐼𝐼) 𝑜𝑜𝑜𝑜 𝑠𝑠𝑝𝑝𝑝𝑝𝑒𝑒 𝑜𝑜𝑜𝑜𝑝𝑝𝑠𝑠 𝐼𝐼 × 100 (2)
The relative amount of caffeine Form II in the tablet corresponds to the transition degree of caffeine Form I into Form II during the compression process. By combining expressions (1) and ( 2), the transition degree (τ) after compaction was obtained with equation 3:

𝜏𝜏 = ∆ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐻𝐻 (𝐼𝐼𝐼𝐼→𝐼𝐼) 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 ∆ 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝐻𝐻 (𝐼𝐼𝐼𝐼→𝐼𝐼) 𝑜𝑜𝑜𝑜 𝑠𝑠𝑝𝑝𝑝𝑝𝑒𝑒 𝑜𝑜𝑜𝑜𝑝𝑝𝑠𝑠 𝐼𝐼𝐼𝐼 • ∆ 𝑓𝑓𝑓𝑓𝑡𝑡 𝐻𝐻 (𝐼𝐼) 𝑜𝑜𝑜𝑜 𝑠𝑠𝑝𝑝𝑝𝑝𝑒𝑒 𝑜𝑜𝑜𝑜𝑝𝑝𝑠𝑠 𝐼𝐼 ∆ 𝑓𝑓𝑓𝑓𝑡𝑡 𝐻𝐻 (𝐼𝐼) 𝑜𝑜𝑜𝑜 𝑡𝑡ℎ𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒 (3)
The experimental validation of this method (shown in Fig. 3 and Fig. 4) was performed, for each composition studied, in order to confirm the expression (3). According to the tablet formulations, samples of 5g were prepared with different ratios of caffeine Form I and Form II. The ratios were 0 wt%, 20 wt%, 40 wt%, 60 wt%, 80 wt% and 100 wt% of caffeine Form II compared to the Form I. Three samples were analysed for each ratio. Whatever the diluent used, the maximal relative deviation increased with the caffeine dilution from 3% to 9%. In the case of 40 wt% of caffeine with DCPA, this relative deviation goes up to 15%. This explains why a higher dilution of caffeine in the blend containing DCPA as diluent was not studied.

Statistical analysis

Statistical analyses were conducted using the R software (The R foundation) with a threshold of 5% corresponding to the alpha risk. As all data were not normally distributed and variances were not equal, consequently, data could not be normalized, a non-parametric Kruskal-Wallis test was performed. In order to determine significant differences between two parameters, a multiple comparison post hoc test (kruskalmc) was conducted.

Results

Pure CFI and binary mixtures with MCC as diluent

Evolution of the CFI transition degree (τ) in uncompressed powders

In uncompressed powders, a kinetic study was conducted during six months in order to observe the evolution over time of the CFI transition degree (see Fig. 5). Each curve represents one of the four compositions studied: pure CFI and mixtures CFI/MCC: 90, 78 and 60 wt% of CFI.

Whatever the composition, the transition profile showed a first rapid growth stage over the first 50 days until reaching a plateau from the fiftieth day. For the transition degree values, very small differences appeared between the compositions over the first 50 days, after that, no significant difference was observed. In uncompressed powders, the transition of the caffeine form I into form II seems not be influenced by the dilution of the caffeine with MCC. The evolution of the CFI transition degree in tablets two days after their fabrication is given in Fig. 6. In order to observe the effects of direct compression process on τ, pure caffeine and the three binary mixtures of CFI/MCC were analysed. Results observed for tablets were compared to those obtained for uncompressed powders. Analyses were made in triplicate (n=3). Three parameters that may influence τ were analysed on Fig. 6: compression pressure (symbol type), dilution/composition (x-axis)

and tablet parts (full or empty symbol corresponding to the borders and the core of tablets, respectively).

Two days after tablet fabrication, the transition of metastable caffeine into its stable form was observed both in tablets and in uncompressed powders. Furthermore, τ values were higher in tablets than in uncompressed powders. This indicated that CFI transition degree was more important after compression.

For all the compositions, no significant difference on the transition degree was observed between the three compression loads. CFI transition degree was also the same between the core and the border of tablets. The transformation of the caffeine form I into form II could be considered uniform throughout the volume of the tablet. Consequently, for the rest of our study, these results will be combined.

Nonetheless in tablets, significant differences of the transition degree were observed with the compositions. This indicates that unlike uncompressed powders, after compression the transition of the caffeine form I into form II seems to be influenced by the dilution of the caffeine with MCC.

Influence of the compression pressure

Like previously for uncompressed powders, a kinetic study of the CFI transition degree occurring in tablets was also conducted during six months. For each composition, results obtained for tablets were compared to the ones obtained in uncompressed powders. For clarity reasons, Fig. 7 presents the kinetic study of the CFI transition degree for one composition (78 wt% of CFI). However, the same results were observed for the other compositions.

Uncompressed powder and tablets made at three different compression loads (50,100 and 200 MPa) were represented by the black and colour symbols, respectively. In tablets, the τ profile was similar to the one of uncompressed powder: a first rapid growth stage over the first 50 days until reaching a plateau. All along the kinetic study of the CFI transition degree for pure caffeine and binary mixtures of CFI/MCC, a constant gap of around 20% was observed between uncompressed powders and the tablets. The plateau value was more important in tablets than in uncompressed powders. Whatever the composition, no significant difference was observed between the three compression loads. These results mean that the compression pressure doesn't alter the transition of the caffeine form I into form II after compression, but the content of form II was initially more important in tablets than in uncompressed powder.

Influence of the dilution

In order to observe the dilution impact on τ, Fig. 8 presents the evolution over time of the CFI transition degree in tablets for pure caffeine and the three binary mixtures of CFI/MCC. As seen before, compression pressure had not an influence on τ, consequently results obtained for tablets made at different compression pressures were regrouped. This means that each dot on figure 8 corresponds to the mean value of eighteen DSC analyses made on tablets. Significant differences between the compositions were observed over time for the τ values. These results indicate that in tablets, the dilution of caffeine with MCC had an impact on the CFI transition degree: τ increased with the caffeine content in the blend. These results match up well with the ones obtained two days after tablet fabrication.

Furthermore, the plateau value of the transition degree increased with caffeine content in the blend.

In addition, the higher caffeine content in the blend, the faster the plateau value was reached. The plateau values for τ in tablets and in uncompressed powders are given in Table 3. Whatever the composition, the plateau value was higher in tablets than in uncompressed powders. Moreover, the plateau was more quickly reached for tablets than for uncompressed powders.

Binary mixtures with DCPA as diluent

As indicate in "Tablet preparation" (section 2.2), the storage of uncompressed powders and tablets was made at room temperature and during one year the temperature varies in a range from 18 to 30°C. Unfortunately, the study of the CFI transition degree of pure CFI, CFI/MCC and CFI/DCPA were not carried out at the same time. Consequently, for tablets it is not possible to compare the pure caffeine with the binary mixtures of CFI/DCPA, neither compare CFI/MCC with CFI/DCPA. Nevertheless, it's still interesting to observe the evolution of CFI transition degree for mixtures containing DCPA as diluent.

Evolution of the CFI transition degree (τ) in uncompressed powders

As well as MCC, a kinetic study was performed during six months in order to observe the evolution over time of the CFI transition degree for binary mixtures composed of CFI and DCPA. In uncompressed powders, evolution over time of the transition of caffeine form I into form II is given in Fig. 9. Analyses were made in triplicate. All along the kinetic study, no significant difference was observed whatever the compositions. These profiles are similar with the ones obtained for uncompressed binary mixture of CFI/MCC. Furthermore, for all the compositions, no significant difference on the transition degree was observed between the three compression loads. The transformation of the caffeine form I into form II uniformly occurred throughout the different parts of the tablet (core and border). Therefore, results obtained for the core and the border of tablets will be combined. Contrary to CFI/MCC, a small impact of the composition was observed in tablets.

Influence of the compression pressure

For the three binary mixtures containing DCPA as diluent, tablets made from three compression loads (50,100 and 200 MPa) were studied in order to evaluate the influence of this parameter on τ. For each composition, a comparison between uncompressed powder and tablets was made. For clarity purposes, Figure 11 presents only the kinetic study of the CFI transition degree for one composition: 80 wt% of CFI (the same results were observed for the other ones). Uncompressed powder and tablets made at three different compression loads were represented by the black and colour symbols, respectively. The transition degree profile was similar both in uncompressed powder and in tablets. All along the kinetic study of the CFI transition degree for binary mixtures of CFI/DCPA, a constant gap of around 30% was observed between uncompressed powder and tablets. The transition degree of the caffeine form I into form II continued to increase with time, no threshold level was observed for tau. Different values of the transition degree appeared according to the compression pressure applied over the first 50 days, after that, no significant difference was observed. For binary mixtures of CFI/DCPA, the transition of the caffeine form I into form II did not seem to be influenced by the compression loads.

Influence of the dilution

The time evolution of τ in tablets for the three compositions (CFI/DCPA) is given in figure 12.

Considering that compression pressure had not an impact on the CFI transition degree, the results obtained for tablets made at different compression loads were regrouped. Little differences on the transition ratio appeared according to the compositions over the first 50 days, after that, no significant difference was observed. The transition degree of the caffeine form I continued to increase with time.

In tablets, the dilution of caffeine with DCPA did not influence the CFI transition degree. These results are similar with the ones obtained with the uncompressed powders.

Discussion

Evolution of the CFI transition degree (τ) in uncompressed powders

Whatever the diluent used, the transition of pure caffeine form I into form II was rapidly observed in uncompressed powders (two days after the batch production of CFI). These results are consistent with those find in the literature by [START_REF] Mazel | Polymorphic transformation of anhydrous caffeine under compression and grinding: a re-evaluation[END_REF] and Hédoux et al. (2011) and confirm that the metastable form I of the caffeine tends spontaneously to move towards its stable form II. At long time storage, the transition kinetic rate is slowed down for two possible reasons: (i) the converted parts in the particle stop their expansion when they are in contact each other (ii) the inner volume fraction of form I available for nucleation of form II decreases.

However, the τ profiles of mixtures containing MCC shows a first rapid growth stage over the first 50 days until reaching a plateau from the fiftieth day. By contrast, for mixtures containing DCPA, none plateau was reached, CFI transition degree profile presents a first rapid growth stage only over the first 15 days, after that, τ continue to increase progressively over the time. This difference cannot be explained by the nature of the diluent, but may be due to the batch of caffeine form I and/or to the storage temperature changes (as mentioned previously).

Evolution of the CFI transition degree in tablets

These kinetic studies help to highlight the influence of the diluent on the CFI transition degree.

Whatever the diluent used, CFI transition degree was higher in tablets than in uncompressed powders.

This means that a part of the transformation was induced by the direct compression process. These results are in accordance with those reported in the literature in compression (Pirttmäki et al., 1993[START_REF] Mazel | Polymorphic transformation of anhydrous caffeine under compression and grinding: a re-evaluation[END_REF] and in hydrostatic pressurizing [START_REF] Hédoux | Polymorphic transformation of anhydrous caffeine upon grinding and hydrostatic pressuring analyzed by low-frequency raman spectroscopy[END_REF]. Moreover, the compression pressures did not have an impact on τ. In accordance with [START_REF] Hubert | Process induced transformations during tablet manufacturing: phase transition analysis of caffeine using DSC and low frequency micro-Raman spectroscopy[END_REF], the compression load did not alter the transition degree of the caffeine form I into form II. For CFI, whatever the values of the compression pressure applied in this study, the transition degrees just after compression were similar. One could assume that the compression pressure has just a triggering effect when a sufficient compression pressure is applied in tableting.

It is known that the stress applied during tableting is not uniformly distributed in compact, but the stress and density in compressed powders are localized [START_REF] Moe | Non-destructive viscoelastic analysis of anisotropy in compressed tablets[END_REF]. Nonetheless, in agreement with observations made by [START_REF] Hubert | Process induced transformations during tablet manufacturing: phase transition analysis of caffeine using DSC and low frequency micro-Raman spectroscopy[END_REF], no difference on the transition of the caffeine form I into form II was observed between regions considered (core and border). These observations mean that even if the stress was not uniformly distributed in all parts of the tablet, the transition of CFI is occurring throughout the tablet volume. This is in accordance with previous results which showed that transition degree of the caffeine form I into form II was not affected by the compression level.

The nature of the diluent has an impact on the transition degree occurring during the compression process. Concerning the impact of the API dilution, two situations were observed in function of the diluent used. With MCC, the API dilution had an impact on τ: CFI transition degree increased with the caffeine content in the blend. Conversely, the effect of the diluent content was not significant for CFI/DCPA mixtures. These differences observed between mixtures containing MCC and those with DCPA could be due to the different behaviour of the diluent under compression. In addition to their different mechanical properties, MCC exhibits a plastic deformation and DCPA a fragmentary behaviour under compression [START_REF] Ilkka | Prediction of the compression behaviour of powder mixtures by the Heckel equation[END_REF]. Indeed, diluents are well known to be able to affect the solid-state transition kinetics of APIs during various unit operations (Allesø et al., 2009). [START_REF] Schmidt | Potential of carrageenans to protect drugs from polymorphic transformation[END_REF][START_REF] Schmidt | Polyethylene oxides: protection potential against polymorphic transitions of drugs?[END_REF] have studied the potential of some diluents for minimizing polymorphic or pseudopolymorphic transitions of API, which are induced by the tableting process. They have indicated that some diluent could have the ability to partially prevent the transformation of the drug. This difference can be attributed to the different behaviour of the diluent under compression.

The τ profile in tablet is similar than in uncompressed powder: a first rapid growth step followed by a plateau or a small increasing rate. A constant gap between the two profiles was observed for both diluent. The values of these gaps were not dependent on the caffeine content in mixtures, but differ according to the diluent: a 20% gap for MCC mixtures and 30% gap for DCPA mixtures. It means that the transition mechanism is the same, the only difference is due to the higher initial transition value in tablet due to the initial compression load. However, the plateau value was more quickly reached in tablets than with uncompressed powders. Consequently the compression has not only triggered the transition, but the transition mechanism after compression is afterwards more intense.

Analysis of the transformation kinetic rate and mechanism of the caffeine form I into form II

The solid-state transformations are usually controlled by the nucleation and growth processes of the new phase, which is described with a first-order transition kinetic rate. The Johnson-Mehl Avrami (JMA) approach is widely used in studies dedicated to transformation kinetics of powders at solid state [START_REF] Duan | Reaction kinetics for the solid state synthesis of the AlH3/MgCl2 nano-composite by mechanical milling[END_REF][START_REF] Jiang | Solid-state phase transformation kinetics in the nearequilibrium regime[END_REF]. The JMA equation is expressed as follows:

𝑋𝑋(𝑡𝑡) = 1 -𝑐𝑐 �-� 𝑡𝑡 𝜏𝜏 𝐴𝐴 � 𝑡𝑡 𝐴𝐴 � (4)
Where X is the volume fraction of the transformed phase (caffeine form II in this study), τA is the overall rate constant that generally depends on the germination rate and its type, as well as growth rate, nA is usually known as Avrami's exponent which is characteristic of the growth mode and t represents time.

An initial induction time is a general feature of a first order transition, and this induction time increases with the value of n (n from 1 to 4). Consequently the transition profile is sigmoidal. This sigmoidal Avrami law becomes questionable when the transformation occurs in constrained geometry [START_REF] Delcourt | Size effect in a nucleation and growth transformation[END_REF] like the profile measured in this study. The particle volume could be infinitely subdivided into crystallites so small enough that they are immediately converted once nucleated. The transformed fraction X(t) coincides with the nucleation fraction [START_REF] Decroix | Time-resolved dielectric investigation of relaxation kinetics in metastable caffeine[END_REF]. When the transformation proceeds only by nucleation, without growth process, a stretched exponential law is then used:

𝑋𝑋(𝑡𝑡) = 1 -𝑐𝑐 �-� 𝑡𝑡 𝜏𝜏 𝑁𝑁 � 𝑡𝑡 � (5)
Where τN, the characteristic kinetic time, is controlled by nucleation only and becomes highly dependent on the cell size. The exponent n is then equal or lower than 1 (n = 1 indicate that the crystallites exhibit the same size) By contrast, when crystallites are distributed in size, n is lower than 1, meaning that nuclei will appear in the largest ones and were converted first, and therefore, the smallest ones are transformed later (or hardly ever) since the nucleation probability is infinitesimal.

Modelling the transition kinetic rate of uncompressed caffeine form I into form II followed at different temperature between 45 to 90°C with several techniques like X-Ray diffraction and microcalorimetry (Letho and Laine, 1998), dielectric relaxation [START_REF] Decroix | Time-resolved dielectric investigation of relaxation kinetics in metastable caffeine[END_REF][START_REF] Decroix | Stabilité physique et désordre orientationnel dans un cristal à usage thérapeutique: la caféine[END_REF], Raman spectroscopy (Hedoux et al., 2011b) has been already published in the literature. The stretched exponential law is in good agreement with the experimental data, the value of n seems to depend on the technique used.

With X-Ray diffraction measurement, the reported values of n are 0.24, 0.33 and 0.37 at 45°C, 60°C and 80°C, respectively. With microcalorimetry, n varies between 0.37 and 0.47 and with the dielectric relaxation technique n is about 0.5.

For this study on uncompressed powders and tablets, the values of n and τN were determined according to the expression (5). Results are presented in Table 4. The calculated profiles are shown in Fig. 13 and compared with the experimental points. The mean relative difference between the experimental and the calculated volume fractions is 6 % and 3% for uncompressed powders and tablets, respectively. This indicates that the transition kinetic rate measured in this study could be represented correctly with this stretched exponential law. The values of n are in a close range from 0.22 to 0.31 with DCPA as the diluent, and more scattered from 0.19 up to 0.50 with the MCC mixture.

These values are in accordance with the ones previously reported in the literature. The presence of the diluent has not a decisive impact on the transformation mechanism, and the exponent value lower than 1 confirms the presence of a distribution of small crystallites, probably a majority of them exhibiting a lower size dimension than the natural length scale [START_REF] Decroix | Time-resolved dielectric investigation of relaxation kinetics in metastable caffeine[END_REF].

A comparison of the values of n and τN obtained between tablets and uncompressed powders bring two indications. First, the values of n are close. Considering the assumptions of this model, the compression load during tableting had neither reduced the size of the crystallites nor their size distribution. This means that for both situations, the transition mechanism is similar, i.e controlled by the nucleation rate, until a plateau value for which the nucleation frequency is negligible. Secondly, the value of τN is several times lower for tablets than for uncompressed powders. The reduction of τN is the result of the triggering of the transition during the tableting. At least these considerations are the same whatever the diluent and the caffeine content.

Conclusion

In this work, the polymorphic transitions of the caffeine form I into form II that occur during the direct compression process were studied in order to improve the understanding of the PITs. Three parameters that could impact the phase transition of the caffeine form I were analysed: nature of the diluent, dilution of the API and compression pressures. Results shown that the transition degree of the caffeine form I into form II, was higher in tablets than in uncompressed powders. This means that direct compression process has a triggering effect on the polymorphic transition of the caffeine form I  II. Like for uncompressed powders, the transition of the caffeine form I into form II in tablets continue after compression, but starting at a higher value. Moreover, the CFI transition degree in tablets depends on the formulation parameters (diluents used and API dilution). The dilution impact is only observed for mixtures containing MCC. Nevertheless, the processing parameter (compression load) has not an impact on the CFI transition degree.

This study allowed improving the understanding of the transformation of the metastable form of the caffeine (form I) into its stable form (form II) during the direct compression process. A stretched exponential law of the Johnson-Mehl-Avrami model well describes the kinetics transition of the caffeine form I into form II at room temperature in both uncompressed powders and tablets. The tabletting accelerates the polymorphic transition without modifying its mechanism (controlled by nucleation only). Certainly, the protocols reported here should provide referenced methods for investigating solid phase transformations occurred in other oral solid dosage forms such as threedimensional printing tablets [START_REF] Yu | Novel drug delivery devices for providing linear release profiles fabricated by 3DP[END_REF] and electrospun medicated nanofibers [START_REF] Yu | Nanofibers fabricated using triaxial electrospinning as zero order drug delivery systems[END_REF][START_REF] Yu | Medicated janus fibers fabricated using a Teflon-coated side-by-side spinneret[END_REF].

Fig. 1 .

 1 Fig.1. (a) DSC curve of caffeine Form I. (b) DSC curve of caffeine Form II with a Y-axis offset for comparison.

Fig. 2 .

 2 Fig.2. Core and border sample fraction (coloured parts) of tablet for DSC Analysis

Fig. 3 .

 3 Fig.3. Validation of the quantification method of caffeine Form II in the powder mixture for the pure caffeine and the three blends containing MCC as diluent.

Fig. 5 :

 5 Fig.5: Evolution over time of the CFI transition degree in uncompressed powders for the four binary mixtures (CFI/MCC), n=3.

Fig. 6 :

 6 Fig.6: Evolution of the CFI transition degree in tablets two days after their fabrication, n=3.

Fig. 7 :

 7 Fig.7: Evolution over time of the CFI transition degree for the composition 78 wt% of CFI. (powders: n=3; tablets: n=6)

Fig. 8 :

 8 Fig.8: Evolution over time of the CFI transition degree in tablets (mean value of the three compression loads) for the four binary mixtures of CFI/MCC (n=18).

Fig. 9 :

 9 Fig.9: Evolution over time of the CFI transition degree in uncompressed powders for the four binary mixtures (CFI/DCPA), n=3.

Fig. 10 :

 10 Fig.10: Evolution of the CFI transition degree two days after compression versus the caffeine content in tablets (CFI/DCPA) Two days after tablets fabrication, the transition of the caffeine form I into form II was observed both in tablets and in uncompressed powders. The τ values in tablets were more important than those observed in uncompressed powders, these mean that CFI transition degree was more important after compression.

Fig. 11 :

 11 Fig.11: Evolution over time of the CFI transition degree for the composition 78 wt% of CFI. (powders: n=3; tablets: n=6)

Fig. 12 :

 12 Fig.12: Evolution over time of the CFI transition degree in tablets (mean value of the three compression loads) for the three binary mixtures of CFI/DCPA (n=18).

Fig. 13 :

 13 Fig.13: Evolution over time of the volume fraction of the transformed phase. Comparison between experimental data and calculated values obtained from the kinetic laws (5) for uncompressed powder and tablets, respectively.

Table 1 : Particle size distribution parameters of CFI, MCC and DCPA (n=3)

 1

Table.2: Tablet characteristics

  Calculated from true density of pure MCC or pure DCPA and CFI by using a mixing rules (ρ MCC = 1.561 g.cm -3 and ρ DCPA = 2.890 g.cm -3 )

	a Average calculated from 3 measurements a Measured by helium pycnometry Hubert (2012)	b Average calculated from 20 tablets
	Powders	D v 10 a (µm)		D v 50 (µm) a	a (µm) D v 90
	CFI MCC DCPA Diluent	19.7 ± 0.7 28.3 ± 0.1 54.8 ± 5.0 Composition	67.8 ± 2.1 119.0 ± 0.0 182.7 ± 11.0 Compression load (P) Tablet weight b (w) 160.5 ± 3.5 268.7 ± 1.2 318.7 ± 14.0 Tablet thickness b (t)	Particle density (ρ blend )	Tablet Density b (d)
	(-)	(wt% CFI)	(MPa)	(mg)	(mm)	(g.cm -3 )	(g.cm -3 )
			200	301.8 ± 1.7 2.27 ± 0.02		1.333 ± 0.004
		100	100	298.6 ± 2.5 2.38 ± 0.02	1.446 a	1.253 ± 0.003
			50	303.3 ± 2.3 2.64 ± 0.02		1.147 ± 0.005
			200	302.5 ± 1.2 2.25 ± 0.01		1.343 ± 0.006
		90	100	301.4 ± 2.2 2.41 ± 0.01	1.457 c	1.253 ± 0.004
			50	302.4 ± 2.5 2.63 ± 0.00		1.149 ± 0.003
			200	293.2 ± 4.5 2.18 ± 0.03		1.344 ± 0.003
	MCC	78	100	294.0 ± 4.5 2.40 ± 0.03	1.470 c	1.227 ± 0.007
			50	297.0 ± 2.9 2.68 ± 0.02		1.108 ± 0.003
			200	301.2 ± 2.3 2.22 ± 0.02		1.361 ± 0.002
		60	100	300.4 ± 2.3 2.42 ± 0.02	1.490 c	1.240 ± 0.004
			50	302.4 ± 1.8 2.75 ± 0.01		1.099 ± 0.003
			200	300.3 ± 3.0 2.12 ± 0.02		1.397 ± 0.005
		80	100	301.4 ± 2.0 2.27 ± 0.02	1.607 c	1.311 ± 0.003
			50	298.5 ± 3.2 2.44 ± 0.04		1.212 ± 0.006
			200	301.9 ± 2.7 2.02 ± 0.02		1.469 ± 0.007
	DCPA	60	100	300.7 ± 3.2 2.18 ± 0.02	1.807 c	1.365 ± 0.005
			50	299.6 ± 2.5 2.34 ± 0.02		1.264 ± 0.004
			200	301.4 ± 2.0 1.90 ± 0.01		1.561 ± 0.006
		40	100	301.7 ± 2.4 2.07 ± 0.02	2.065 c	1.435 ± 0.005
			50	303.9 ± 1.5 2.27 ± 0.01		1.326 ± 0.004

c

Table . 3: Transition degree τ at long storage (plateau value) in the cases of uncompressed powder and tablet for several caffeine content blends. Tp: Time plateau.

 .

		Uncompressed powders	Tablets
	wt% CFI τ (%)	Tp (days)	τ (%)	Tp (days)
	100	62	90	80	60
	90	57	90	80	60
	78	55	90	75	90
	60	55	110	70	90

Table . 4: Values of τN and n calculated with the kinetic laws (5) for uncompressed powder and tablets, respectively.

 .

	Diluent wt% CFI	n Uncompressed powders τ N	n	Tablets	τ N
		100	0.25	934	-		-
	DCPA	80 60	0.31 0.30	776 1321	0.26 0.26		40 45
		40	0.28	1796	0.22		36
		100	0.33	114	0.19		8
	MCC	90 78	0.30 0.20	147 398	0.25 0.19		13 23
		60	0.50	174	0.32		84

Acknowledgements

The support of the Ministère de l'Enseignement Supérieur et de la Recherche is gratefully acknowledged. We would like to thank Medelpharm especially T. Ménard, G. Tardy and B. Villa as well as the technician G. Conod-Nardi for her technical support. The authors are also grateful to Pr. Alain Hédoux for stimulating discussions.


D Bourcier
email: damien.bourcier@cpe.fr

J P Féraud

D Colson

Influence of particle size and shape properties on cake resistance and compressibility during pressure filtration

Keywords: Pressure filtration, cake resistance, Compressibility, Particle properties, Experimental, Modelling

The aim of this paper was to study cake filterability and compressibility as a function of the particle shape and particle size distribution (PSD). Different shapes and PSD of calcium carbonate and uranium oxalate particles covering the classical types encountered in industry (sphere, cube, needle and platelet) were obtained by precipitation. The size and shape factor distributions were measured using an image analysis system on SEM pictures. The cake filtration properties were measured in ideal monitored operating conditions, because of a miniaturized filtration cell set-up. The impact of the PSD and the shape were quantitatively assessed. These two solid features have an impact on cake resistance and compressibility, but not in the same way. The PSD has the strongest effect on cake resistance and compressibility. The particle shape is a decisive parameter for cake compressibility when the shape is far from the sphere. Both parameters need to be considered when working on the development of a filtration operation. Here, a practical model built following the Darcy law coupled with a new correlation for compressibility factor assessment is proposed. It gave satisfactory estimates of cake filterability and compressibility for the four shapes studied.

Introduction

Solid-liquid separation technologies are needed in many industries: biotechnology, chemistry, oil and pharmaceutics, for instance. Their performance is strongly dependent on particle properties such as size distribution and shape. Solid-liquid separation involves the separation of a solid and a liquid phase from a slurry. Cake filtration performed at constant pressure is one of the most popular techniques, and is considered in this study.

At plant scale, the filtration operation is often very time consuming, since cake filterability is low, especially with small particles (< 100 µm) exhibiting quite a wide size distribution and/or asymmetrical shapes. Increasing the pressure in order to accelerate the process may be counterbalanced by an increase in cake compressibility. Determining filterability and compressibility is essential when the appropriate solid-liquid separation technologies need to be defined.

At laboratory scale, when a toxic, harmful solid material involved, as is especially the case for some active pharmaceutical ingredients or for nuclear fuel, filterability measurements are quite difficult and must be limited in number and volume. Being able to estimate solid cake behaviour when only a few solid physical properties are known may help to overcome such difficulties.

The influence of the conditions of the particle production on filterability has been described in several studies reported in the literature (e.g. [START_REF] Hakkinen | The influence of cryztallisation conditions one the filtrationcharacteristics of sulphathiazolesuspensions Lappeenranta University of Technologie[END_REF]. Usually the crystallization conditions (temperature profile, supersaturation, and seeding procedure) were varied in order to generate different particle size distributions (PSD). A recent study by [START_REF] Beck | The influence of crystallization conditions on the onset of dendritic growth of calcium carbonate[END_REF] went further, since the impact of different particle shapes on filterability was investigated.

Nevertheless, only the cake resistance was examined, without considering other cake properties such as its compressibility. These interesting results are not sufficient to provide guidelines when designing a filtration set-up for industrial materials.

The objective of this work was twofold. First, an experimental study was performed in order to assess changes in filterability and compressibility compared to the shape factor and particle size distributions. The second objective was to develop a model of the filtration in question using the experimental data collected. Such a model was expected to be able to estimate cake filtration properties (i.e. cake resistance and compressibility) using particle physical features and cake properties, such as porosity. A correlation based on physical parameters is proposed for the compressibility coefficient. It will be presented in Part 6 of this paper.

A model system which is easy to synthesize and does not involve hazards, and which exhibits different shape and size distributions, was chosen. The Beck and Andreassen studies [START_REF] Beck | The influence of crystallization conditions on the onset of dendritic growth of calcium carbonate[END_REF] have shown that calcium carbonate could be used to meet this requirement, since its polymorphic solid forms are well known, easy to handle and have a wide range of shapes. The particles were generated in very soft conditions (from 10 to 90°C). Controlling the supersaturation and temperature levels coupled with the suitable choice of the chemical reagents involved in the precipitation enabled us to produce several sphere, cube, plate and needle shapes. Once the different shapes had been obtained, they were sent to a metallographic laboratory for SEM expertise. An imaging software analysed the pictures and provided the particle size and volume and a volume shape factor for each particle. Filterability tests were carried out at pressures ranging from 1 to 5 bar using the mini-test set-up already developed and validated in a previous study [START_REF] Feraud | New filterability and compressibility test cell design for nuclear products[END_REF].

These tests gave the cake specific resistance and compressibility for each shape. The relationships between the particle features and the cake filterability behaviours could then be obtained. Relying on these experimental observations, a model based on Darcy's law was developed. Experimental and simulation results were compared, in order to assess the reliability of the modelling.

Materials, experimental setup and operating conditions

Model substance, raw materials

The model substance used here was calcium carbonate with an intrinsic relative density close to 2.7g.cm -3 . The different polymorphic forms were obtained by precipitation in water (see next section) by combining pure reagents (> 99%) supplied by Sigma Aldrich. The ethanol used was >95% pure and also supplied by Sigma-Aldrich. All runs were performed at a solid concentration of 30g.L -1 .

Precipitation set-up and operating conditions

The operating conditions for the precipitation of four differently-shaped forms of calcium carbonate particles corresponding to three polymorphic forms in water solution are presented in Table 1. Once the precipitation was complete, the suspension was quickly filtered on a Büchner. The cake was washed with ethanol and dried. The dry solid was stored in closed bottles before filtration.

Run

An additional solid (Uranium Oxalate) was used to validate the methodology. They were precipitated from uranium nitrate and oxalic acid in a vortex reactor, as presented in [START_REF] Bertrand | Methods for preparing an actinide oxalate and for preparing an actinide compound[END_REF].

Filtration set-up and operating conditions

Cake resistance experiments were carried out with the mini-test cell which was fully described in Feraud and al [START_REF] Feraud | New filterability and compressibility test cell design for nuclear products[END_REF]. This minitest allowed using small amount of slurry (about 100 mL). In this previous study, the authors have validated that the reduction of the cell diameter down to 1.8cm has no impact on the cake specific resistance measurement. The test cell diameter used was 1cm. The absence of wall effect on the filterability was again estimated for this smaller cell and is presented in section 5.1. The filterability test was based on experiments described by Roorda and van der Graaf [START_REF] Roorda | SUR test used for optimisation of membrane filtration plants treating wastewater effluents[END_REF] and by [START_REF] Couturier | Enhanced expression of filter cakes using a local thermal supply[END_REF]).

The slurry containing precipitated particles at 30 g/L was filtered at three different pressures in ethanol, a solvent which prevents particle redissolution.

Particle size distribution measurement

An in-house method for characterising precipitated particles coupled a scanning electronic microscope (SEM) for image recording and an image analysis software INCA future provided by Oxford Instruments for the determination of specific particle dimension and base area of a prism. This procedure is described in [START_REF] Brackx | Method for characterising particles by image analysis[END_REF]. The dry particles were deposited on conductive pads before placing them in the SEM.

Several tens of grayscale SEM images were recorded by the software. The image processing started by a particle detection step in the image by thresholding the intensity of its gray level. Multiple levels of thresholds may be used to improve the efficiency of detection. Once the particles were detected, some image analysis procedure including erosion, dilatation were conducted to separate the particles one another. Only the separate particles were considered, since contiguous particles were rejected. Since a few thousands of particles were kept, the rejection of particles did not bring any segregation. For each separated particle, the software measured the maximum length of Féret and the minimum width of Féret. The operator indicated to the software a geometric model representative of the particle analyzed which appeared on the SEM image. This indication was made once, since all the particles belonging to the sample exhibited the same shape. Considering the geometric model and the minimum width of Féret, the base area of a prism of the particle Ap was calculated by the appropriate expression.

Here, several thousand particles were taken into consideration for shape and size analysis, which is far beyond the minimal requirement of about a few hundred [START_REF] Mullin | Crystallization. 4th edition : Chapter 2 Physical and thermal properties[END_REF]. This gave the study consistent quantitative data.

An equivalent circular diameter corresponding to the diameter of the circle having the same base area of a prism of the particle is calculated by the expression (1):

𝐸𝐸𝐸𝐸𝐸𝐸 = � 4𝐴𝐴 𝑝𝑝 𝜋𝜋 (1)
where Ap is the base area of a prism of the particle. The Figure 2 shows the equivalent circle for a needle and a platelet.

The volume of the particle was obtained as the product of the maximum length of Féret and the base area of a prism. This calculation which has already been applied for spheres and cubes was extended for needles and plates, even though the orientation of the particles in the imaging plane would lead to a deviation. In the case of platelets, we don't apply this procedure and we prefer to estimate on few images with the naked eye the mean particles thickness. This replaces the Féret diameter in the calculation of the volume.

Since the volume and the ECD are known for a population of several thousands of particles, it is possible to calculate the volume particle size distribution of the particles and its main characteristics, as its mean size and its standard deviation. The variation coefficient was defined as the standard deviation to the mean size ratio.

Recently, the literature presents an estimation that serves to reconstruct the three-Dimensional crystal shape from the two-Dimensional crystal projection [START_REF] Borchert | Image-Based in Situ Identification of Face Specific Crystal Growth Rates from Crystal Populations[END_REF]. Also, an alternative to this method is to use stereoscopic images [START_REF] Kempkes | Measurement of 3D particle size by spectroscopic imaging[END_REF].

Volume shape factor determination

Many shape factors can be calculated: volume, surface area, circularity factor… The volume shape factor is particularly interesting for this study because of its simplicity and of its easy access by the software. A volume shape factor is defined as the ratio of the surface area of the sphere having the same volume as the particle to the apparent estimated surface area of the particle [START_REF] Garside | Measurement of crystal growth and nucleation rates : second edition[END_REF] as follows:

𝜑𝜑 𝑣𝑣 = 𝜋𝜋𝑑𝑑 𝑣𝑣 2 𝐴𝐴 (2)
with dv the diameter of a sphere having the same volume of the particle.

This factor allows a comparison between the real particle shape and an equivalent sphere.

The table 2 exhibits few theoretical values of volume shape factor for known shapes [START_REF] Allen | A study of filtration : formulation as a Stefan problem[END_REF] (Table 2): (1) : Length = 10µm, Width = 1µm, Thickness = 0,77µm (2) :

𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿ℎ 𝑊𝑊𝑊𝑊𝑊𝑊𝐿𝐿ℎ = 10 𝑡𝑡𝑡𝑡 30
Once the volume of the particle was obtained, the volume shape factor ϕv can be calculated as the ratio of the area of the sphere having the same volume divided by the particle area.

A number distribution of volume shape factor was then calculated.

Porosity determination

For each shape, cake porosity was determined by a mass balance. The wet cake was weighed after draining, carried out using regulated compressed air. The cake was dried 24h at 80°C and weighed again. The difference between the two weights gave the amount of moisture. The volume of ethanol in the cake before and after deliquoring added to the volume recovered during draining gave the total volume of ethanol in the cake, thus giving the porous volume.

Theoretical considerations

Standard Filterability Laws

Considering the cake as a solid porous block [START_REF] Wakeman | m.s -1 ) V: filtrate volume (m 3 ) Vmes+ : upper limit of the uncertainty domain for the filtrate volume (m 3 ) Vmes-: lower limit of the uncertainty[END_REF] traversed by circular capillaries of the same diameter, d, the pressure drop through a filtering media is given by Poiseuille's equation:

2 32 U l P d µ ⋅ ⋅ ∆ = (3)
This law comes from the Navier-Stokes equation, with several assumptions: laminar and stationary flow, and a non-compressible Newtonian fluid.

The Kozeny-Carman model can determine the value of this resistance based on experimental data [START_REF] Léger | Etude de l'essorage en vue d'assister thermiquement le procédé[END_REF]. It considers the cake as incompressible, and consisting of spherical particles. The resistance value comes from an identification of Darcy's law for filtration [START_REF] Darcy | Les fontaines publiques de la ville de Dijon[END_REF][START_REF] Carman | Slip Flow in Granular Beds[END_REF]:

𝑈𝑈 0 = -1 𝜇𝜇𝜇𝜇 𝑊𝑊𝑝𝑝 𝑊𝑊𝑑𝑑 (4)
Equation 3 is also known as the modified Darcy's Law.

Combining Equations ( 3) and ( 4), and using the assumption of the Kozeny-Carman model [START_REF] Carman | Fluid flow through granular beds[END_REF]), the specific cake resistance "α" can be given by:

(5)

The parameter α is dependent only on particle and cake properties, which can be measured. The factor "180" was proposed by Leva considering the tortuosity and the external surface of the pores (see details in [START_REF] Tiller | Compressibility of Particulate Structures in Relation to Thickening, Filtration, and Expression -a Review[END_REF]).

This relation is typical of the solid properties. In a constant-pressure filtration process, Equation ( 5) leads to [START_REF] Ergun | Fluid Flow through Randomly Packed Columns and Fluidized Beds[END_REF]):

2 2 s T T t w R V V P P α µ µ = ⋅ + ⋅ ∆ Ω ∆ Ω (6)
Equation ( 6) is the basis of the entire filterability test. All the parameter values can be determined experimentally, and the specific cake resistance "α" can then be extracted. The pressure effect on filter cake can decrease porosity and thus drastically increase the cake resistance. This behaviour is given by many empirical equations describing the variation of α with pressure, such as the power law (see Equation 7) [START_REF] Tarleton | The Role of Porosity in Filtration -Numerical Methods for Constant Rate and Constant Pressure Filtration Based on Kozeny Law[END_REF]:

𝛼𝛼 = 𝛼𝛼 0 ( ∆𝑃𝑃 ∆𝑃𝑃 0 ) 𝐿𝐿 (7)
The pressure ∆p0 is a reference pressure used to give a dimensionless pressure drop in Equation ( 7). The relation ( 7) can be linearized by a logarithmic approach. In this way, the slope easily gives the compressibility coefficient, n, which is decisive information for the optimization of solid/liquid separation conditions and for a suitably-adapted filter technology.

Filtration modelling strategy

In pressure filtration, particle characteristics must be taken into account in order to model cake resistance and compressibility. The first models for compressible cake [START_REF] Smiles | A Theory of Constant Pressure Filtration[END_REF]) [START_REF] Philip | Diffusion by Continuous Movements[END_REF]) built during filtration were based on the differential equations of continuity and momentum, through a Lagrangian approach. One of the major difficulties was to take into account the boundary condition at the cake-slurry interface. Atsumi and Akiyama (Atsumi and Akiyama 1975) and Tosun [START_REF] Tosun | Formulation of Cake Filtration[END_REF] worked on a moving boundary condition at the growing cake surface. Generally, interparticle interactions are ignored in these models.

More recently, Auzerais et al [START_REF] Auzerais | The Resolution of Shocks and the Effects of Compressible Sediments in Transient Settling[END_REF]), Buscall and White [START_REF] Buscall | The Consolidation of Concentrated Suspensions .1. The Theory of Sedimentation[END_REF] and Landman et al [START_REF] Landman | Dewatering of Flocculated Suspensions by Pressure Filtration[END_REF]) used the particle network concept, closer to a sedimentation approach. This leads to the development of an empirical power law, with many empirical constants to determine.

Both approaches rely on measurements of the local solid compressive pressures to determine cake resistance and porosity, using a power law which can be written as:

𝛼𝛼 = 𝛼𝛼 0 (𝑎𝑎 1 + 𝑏𝑏 1 𝑝𝑝 𝑠𝑠 ) 𝐿𝐿 1 (8)
where α0, a1, b1, and n1 are empirical constants determined through experiments. The same power law can be written for porosity.

The model presented in this study is based on the Kozeny-Carman model, since it is the simplest and its application is consistent with our assumptions. This model requires a porosity lower than 0.8 [START_REF] Tarleton | The Role of Porosity in Filtration -Numerical Methods for Constant Rate and Constant Pressure Filtration Based on Kozeny Law[END_REF]).

The modelling approach proposed here consists of two parts. The first enables calculation of the cake specific resistance of a polydisperse population of particles. This population could be considered as a sum of monodisperse populations, each having a specific resistance which could be calculated with the well-known Kozeny-Carman [START_REF] Carman | Slip Flow in Granular Beds[END_REF]) equation and the associated Léva [START_REF] Ergun | Fluid Flow through Randomly Packed Columns and Fluidized Beds[END_REF] equation. In this study, the cake resistance was then obtained considering the different specific resistances and the relative importance of each particle population present in the whole cake (see section 3.4). The second part focuses on the estimation of cake compressibility. Knowing local particle interactions with pressure was not in our scope, since the goal was to obtain a practical modeling easily transferable to industry. Thus an empirical power law was developed taking into account the porosity and the extent of the particle size distribution compared to the average size. This data is relatively easy to obtain experimentally. The shape incidence on the compressibility was taken into account through the exponents of this law (see section 3.6).

The innovations in this modeling lie in the fact that the polydisperse feature of a particle population is taken into account, and that it gives a relation for compressibility based on a physical approach.

Filtration resistance modelling and cake formation calculation

The extent of the distribution was taken into account by a discretization of the probability density function. For each class of particle diameter dP, an associated specific resistance α0 was calculated using the Léva equation ( 5).

(9)

The particle shape was taken into account via the volume shape factor ϕv (see definition in section 3.2.).

A resistance additivity hypothesis was proposed in order to take into account the specific resistance generated by each particle fraction of size dp in the total cake specific resistance α0. This was made possible by considering the probability of the particle size fraction in the population volume density function. This value α0m represents a mean value on the cake. It was then used in the governing equations of liquid flow in porous media. The value of the cake resistance was finally calculated with the classical equation ( 5) describing the effect of pressure drop on cake resistance.

𝛼𝛼 0𝑚𝑚 = ∑ 𝑝𝑝�𝑑𝑑 𝑝𝑝 �. 𝛼𝛼 0 (𝑑𝑑 𝑝𝑝 ) (10)
The calculation of the compressibility coefficient was possible through a power law based on physical considerations. The establishment of this power law is described in section 3.5.

The model also calculated the profiles with time of the filtrate volume, the liquid flow rate and the cake height by mass balance carried out on the solid with the following expressions:

𝑉𝑉(𝑡𝑡) = Ω([� 𝑅𝑅 𝑠𝑠 𝜇𝜇.𝑤𝑤 � 2 + 2∆𝑃𝑃.𝐿𝐿 𝜇𝜇𝜇𝜇𝑤𝑤 ] 0.5 - 𝑅𝑅 𝑠𝑠 𝜇𝜇𝑤𝑤 ) (11)
𝑄𝑄 𝑙𝑙 (𝑡𝑡) = 𝑑𝑑𝑑𝑑(𝑡𝑡) 𝑑𝑑𝑡𝑡 (12)

ℎ(𝑡𝑡) = 𝑉𝑉(𝐿𝐿).𝑤𝑤 (1-𝜀𝜀)Ω𝜌𝜌 𝑆𝑆𝑑𝑑 (13)
Finally, the model considers that the local porosity and the specific resistance are uniformly distributed in the cake volume.

Filter media resistance

The dimension of cake resistance is expressed in m.kg -1 whereas that of the filtering media resistance is m -1 . To compare them and to integrate the filtering media resistance in the model, the two resistances must have the same dimension. The filtering media resistance can be compared to the cake resistance via the surface mass of the media [START_REF] Tarleton | The Role of Porosity in Filtration -Numerical Methods for Constant Rate and Constant Pressure Filtration Based on Kozeny Law[END_REF]. This resistance is considered as the cake resistance at t=0.

𝛼𝛼(𝑡𝑡

= 0) = 𝛼𝛼 𝑚𝑚𝐿𝐿𝑊𝑊𝑊𝑊𝑚𝑚 = 𝑅𝑅 𝑠𝑠 𝑚𝑚 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 (14)
Possible filtering media clogging was ignored. For 10 µm REPS metallic media from Gantois, Rs was set at 10 10 m -1 .

The parameter msurf represents the surface mass of the filtering media calculated as the ratio of the filtering media mass and the filtering media surface.

Compressibility law

As part of the experimental plan, a preliminary study showed that three main parameters had an influence on the compressibility factor: cake porosity, mean particle size and the extent of the size distribution. These observations were in agreement with Tarleton [START_REF] Tarleton | The Role of Porosity in Filtration -Numerical Methods for Constant Rate and Constant Pressure Filtration Based on Kozeny Law[END_REF].

In order to define a practical compressibility model, a phenomenological law was built considering the effect of both the porosity and the particle size distribution. When taking into account the cake porosity, attention needs to be paid to two extreme conditions. When the porosity tends to 0, compressibility becomes negligible. On the contrary, when porosity is close to 1, the cake is fully compressible. This is why the first part of the law includes a ratio with the porosity ε and the dryness

(1-ε). In order to consider the extent of the size distribution compared to the mean size, it was decided to consider the variation coefficient (VC), which is the ratio of the standard deviation to the mean size of the size distribution. The compressibility law is then defined as follows:

𝑛𝑛 = ( 𝜀𝜀 1-𝜀𝜀 ) 𝛽𝛽 (𝑉𝑉𝐸𝐸) 𝛾𝛾 (15)
It was not necessary to use a proportional coefficient in expression (15), since the compressibility law is built with two dimensionless ratios. This expression will be used later to characterize cake resistance evolution due to pressure drop. The exponents β and γ were determined for each particle shape (see section 5.2).

Simulation procedure

Figure 3 summarizes the working flowchart of the simulation: With all these parameters, equations ( 4) and ( 5) gave the specific cake resistance at 1 bar. The compressibility calculated with equation ( 15) combined with equation ( 7) led to the calculation of the cake resistance at every pressure drop. The use of equations ( 6), ( 11), ( 12),and (13) enabled the calculations and then profile plotting with time for the volume of filtrate collected, the liquid flow rate, and the cake height as a function of filtration time. These three profiles were also measured.

Characterization of synthetized particles

Spheres

The particles precipitated from potassium carbonate and calcium acetate presented a spherical shape with considerable surface roughness, as shown in Fig. 4. The particles were not agglomerated and were well dispersed in the slurry.

Figure 4: SEM image of dried spherical particles

The ECD of the particles were in a narrow range from a few µm up to 10µm (see Fig. 5). The PSD fitting a normal distribution was centred on 5.7 µm, with a standard deviation of 1.3 µm.

Cubes

The particles precipitated from sodium carbonate and calcium chloride had a cubic shape with irregular surfaces, as shown in Fig. 7. The ridge length was about 10µm. The particles were not agglomerated and were well dispersed in the slurry.

The ECD of the particles laid in the range from a few µm up to 15µm (see Fig. 8). The PSD fitting a Normal distribution was centred on 9.5µm, with a standard deviation of 1.6µm.

Number frequency (%)

Volume shape factor

Normal law

Platelets

The platelets of calcium carbonate obtained were very thin, small and strongly agglomerated. Their characteristic length was about 3 µm. The particles were so thin that they could be broken by an ultrasonic bath if it were used to separate them.

Figure 10: SEM image of dried platelets

These particles were used to determine the β and γ coefficients, as the volume shape factor was not needed to calculate them. Nevertheless, the volume shape factor was needed to model the cake 0 5 0,2 0,25 0,3 0,35 0,4 0,45 0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9 0,95 1

Number frequency (%)

Volume shape factor specific resistance and compressibility. The calcium carbonate particles were therefore replaced by uranium oxalate particles. They were precipitated from uranium nitrate and oxalic acid in a vortex reactor, as presented in [START_REF] Bertrand | Methods for preparing an actinide oxalate and for preparing an actinide compound[END_REF]). These particles are shown in Figure 11: The ECD of the particles laid in the range from a few µm up to 15µm (see Fig. 12). The PSD fitting a lognormal distribution is centred on 14.5µm, with a standard deviation of 6.6µm. The distribution correctly fits a log-Normal law.

Needles

The needles precipitated from sodium carbonate and calcium chloride are presented in Figure 14.

Figure 14: SEM image of needles

The ECD of the particles was in a wider range from a few µm up to 40µm (see Fig. 15). The PSD fitting a log-normal distribution was centred on 17µm, with a standard deviation of 10µm. Volume distribution (%)

Equivalent Circular Diameter (µm)

Log-normal law

Figure 16 shows the volume shape factor number distribution of these particles. The average value is 0.17. Observation of the particles with the naked eye showed a ratio length/width =10 to 30, which corresponds to a theoretical value of 0.29. The experimental values were in the range of 0.04 to 0.24.

The deviation comes from the utilization of the Equivalent Circular Diameter instead of a length to width ratio. Nevertheless, the experimental value of the volume shape factor measured exhibits a highly dissymmetric shape.

It should be noted that 0.17 is an expected value for needles.

Figure 16: Volume shape factor number distribution of precipitated needles

To enable comparison, the particle characteristics are summarized in Table 3: These compounds were used as standards or reference materials in the filterability test.

Implementation and assessment of the simulation

Validation of the absence of wall effects

A cell diameter of 1cm was used in the experiments. A comparison was made between cake specific resistance measured at 3bar from a 1cm-diameter cell and a 3-cm cell, and is presented in Table 4. A statistical Student test showed that the mean values of specific cake resistance at 1 and 3 cm are not significantly different. The wall effect can be ignored for this 1cm-diameter cell.

Particle properties used for the model validation

The cubic, spherical and needle-like precipitated calcium carbonate particles were used to validate the model. As the plate-like particles of calcium carbonate were too agglomerated, they were replaced by Uranium oxalate particles which also exhibited a plate-like shape. Their main features were determined as described in Part 4, and are summarized in Table 3.

Particle physical properties, such as bulk density, were also measured to be integrated in the model:

Calibration and determination of β and γ

The exponents β and γ needed to be obtained for each shape. Two particle populations with similar shapes were obtained by performing two successive trials. Their particle size distribution features were measured. Filtrations with each population were carried out in order to measure the cake porosity and compressibility. With this data, it was then possible to calculate the value of the exponents by solving the following system of two equations:

� ( 𝜀𝜀 1 1 -𝜀𝜀 1 ) 𝛽𝛽 . 𝑉𝑉𝐸𝐸 1 𝛾𝛾 = 𝑛𝑛 1 ( 𝜀𝜀 2 1 -𝜀𝜀 2 ) 𝛽𝛽 . 𝑉𝑉𝐸𝐸 2 𝛾𝛾 = 𝑛𝑛 2 (16)
The resolution of the system led to:

⎩ ⎪ ⎨ ⎪ ⎧ 𝛾𝛾 = [(ln(𝜀𝜀 1 ) -ln(1 -𝜀𝜀 1 )). ln(𝑛𝑛 2 )] -[(ln(𝜀𝜀 2 ) -ln(1 -𝜀𝜀 2 )). ln(𝑛𝑛 1 )] [(ln(𝜀𝜀 1 ) -ln(1 -𝜀𝜀 1 )). ln(𝑉𝑉𝐸𝐸 2 )] -[(ln(𝜀𝜀 2 ) -ln(1 -𝜀𝜀 2 )). ln(𝑉𝑉𝐸𝐸 1 )] 𝛽𝛽 = ln(𝑛𝑛 1 ) -𝛾𝛾ln (𝑉𝑉𝐸𝐸 1 ) ln(𝜀𝜀 1 ) -ln(1 -𝜀𝜀 1 ) (17 )
The logarithmic derivation of Equation 17gives the uncertainty. The calculation shows that the uncertainty on the β and γ coefficients is respectively 0.15 and 0.25. The table 6 summarizes the constants calculated for the four shapes studied.

Assessment of the resistance additivity hypothesis

To validate the additivity hypothesis, experiments were carried out using calibrated spheres of PMMA dispersed in ethanol. Two monodisperse distributions centered on 20µm and 50µm were used.

Three other distributions were obtained by mixing the two previous ones, in different ratios 25%/75%, 50%/50% and 75/25% wt of the content respectively. The standard deviation for the mixture was calculated as follows:

𝜎𝜎 𝑚𝑚 = �𝜎𝜎 1 2 + 𝜎𝜎 2 2 (18)
with σ1 and σ2 being the standard deviations of the 20µm and 50µm monodispersed distributions The porosity and the filter cake bulk density were measured for each experiment. Table 7 presents the main features of the five particle populations considered, and of their cake and filtrate. Cake density ρSapp 0.658 g.cm -3 0.658 g.cm -3 0.748 g.cm -3 0.715 g.cm -3 0.794 g.cm -3

Filtering media area Ω

cm²

Filtering media resistance 10 10 m -1

Pressure drop 1, 3 and 5 bar Table 8 summarizes the experimental and calculated cake specific resistance and compressibility for the three particle distributions. A relative deviation between the experimental and calculated values is also reported.

Table 8: Experimental and calculated cake specific resistance for the five particle distributions 1.6 10 9 m.kg -1 6.2 % ∆P = 3 bar 2.9 10 9 m.kg -1 3.1 10 9 m.kg -1 6.4 % Distribution 4 ∆P = 5 bar 4.1 10 9 m.kg -1 4.2 10 9 m.kg -1 2.4 % Compressibility 0.62 0.61 1.6 % Distribution 5 ∆P = 1 bar 3.9 10 9 m.kg -1 3.9 10 9 m.kg -1 0.0 % ∆P = 3 bar 6.3 10 9 m.kg -1 6.0 10 9 m.kg -1 5.0 % ∆P = 5 bar 7.7 10 9 m.kg -1 7.5 10 9 m.kg -1 2.6 % Compressibility 0.42 0.43 2.3 %

Experiment

The experimental and calculated values were close, with a relative deviation of up to 4.2 6.4 %, whatever the cake feature considered (specific resistance or compressibility). The relative precision was also maintained with the distributions 3, 4 and 5, even though these were bimodal. However, we did not have the opportunity to test this additivity hypothesis with particles exhibiting shapes other than spheres.

Results and discussion

Compressibility and cake resistance measurements

Figure 17 (see values in Table 9) presents the evolution of the cake specific resistance with the filtration pressure for the four samples. It appears quite clearly that the smaller the particles were in size, the higher the cake specific resistance. It could be emphasized that the shape also influences the specific resistance, since spheres showed a lower resistance than platelets, for quite similar sizes. It was seen that the higher the volume shape factor, the lower the resistance. From these observations, the first order parameter impacting cake resistance is nevertheless the size, while the shape factor acts as a parameter of secondary importance. The effect of the shape factor on the cake resistance value has already been noted by [START_REF] Beck | The influence of crystallization conditions on the onset of dendritic growth of calcium carbonate[END_REF], but without considering the particle size more specifically. Figure 17 shows that the compressibility factor, i.e. the slope of each curve, was almost the same for the spheres, cubes and platelets. On the contrary, the needle particle cake exhibited a much higher compressibility factor, even though the average particle diameter was quite close to those of the other populations of spheres, cubes and platelets. It is also interesting to note that the cake porosity was not really affected by the pressure rise. Two physical considerations can be discussed. First, it is assumed that the needles may reorganise in each cake layer, thus leading to a decrease in the mean pore size without affecting the total porosity. Second, a population of particles with an extended size distribution probably leads to a cake made up of uneven layers containing different-sized particles. The smallest particles may move from one layer to another, especially as the filtration pressure increases.

Consequently the liquors may have to flow through more twisted paths. A relationship between the compressibility factor and the coefficient of variation could be supposed. To our knowledge, this is the first time that such a relationship between the compressibility factor and the size variation coefficient is reported in a quantitative way. Nevertheless, the impact of the volume shape factor on the compressibility factor remains lower than that induced by the particle size distribution variation coefficient.

The quantity of filtrate collected is highly dependent on the cake specific resistance and on the pressure applied. Figure 18 shows the profile with time of the filtrate volume collected as a function of the particle shape at a pressure of 3 bar. exhibited a very porous cake and a larger mean diameter compared to other shapes. This explains that the quickest filtration was obtained with this shape. The figure 19 shows the evolution of the filtrate volume collected as a function of time and pressure for the spheres.

Figure 19: Evolution of the filtrate volume profile with pressure drop for the spherical particles

The filtration time t is the total time necessary to obtain the whole filtrate volume (in this study, about 60g). A filtration time factor was defined as the ratio of t1 on t2, t1 and t2 being the times measured when the filtrations were carried out at filtration pressures P1 and P2 respectively. The filtration time decreased with the filtration pressure applied. The filtration time factor was reduced by 2.5 between 1 and 3 bars, and by 1.2 between 3 and 5 bars. The time reduction is lower at higher pressure, meaning that the effect of the pressure drop was attenuated by the increase in the cake specific resistance.

Nevertheless between 1 and 5 bars, a quicker filtration was obtained for the spheres at the highest pressure drop. The previous statement can be repeated: the filtration time is linked to the cake specific resistance.

The value of the cake specific resistance is governed by the particle size, the shape factor and the compressibility. If the compressibility is very high, an increase in the pressure can lead to a higher filtration time, due to a considerable increase in the resistance.

Comparison between experiments and the model

The suspensions tested are presented in Part 4. Table 9 reports the experimental and calculated cake specific resistance and compressibility, and their relative deviation. ∆P = 1 bar 1.4 10 9 m.kg -1 1.4 10 9 m.kg -1 0.0 % ∆P = 3 bar 3.8 10 9 m.kg -1 3.9 10 9 m.kg -1 3.3 % ∆P = 5 bar 6.4 10 9 m.kg -1 6.3 10 9 m.kg

Conclusion

The controlled synthesis of different shapes for a same chemical product was used in order to establish relationships between particle characteristics (size and shape) and cake filtration behaviours (specific resistance and compressibility). Indeed, it allows to overcome the chemical composition of particles and to focus only on physical particle considerations. Calcium carbonate is a perfect model system, since the experiments can be carried out in a non-hazardous and repeatable way. The minitest of filterability needing very small amounts of particles gave repeatable measurements whatever the particle shapes and size distributions.

Both PSD and shape have an impact on the cake resistance and compressibility. Their respective effects were clearly highlighted in a quantitative way. This study also showed that the parameter of primary importance impacting specific resistance and the compressibility factor is the particle size distribution (mean diameter and variation coefficient). The effect of the shape on the compressibility factor is significan when the shape factor is far from 1 (needles). This demonstrates that considering the mean particle diameter when a filtration operation has to be performed at plant scale is not enough. Both the entire PSD and the shape have to be considered in order to choose the best pressure drop conditions if seeking to reduce the filtration time. Mastering the PSD by controlling the operation of solid generations is often possible, but shape control is much trickier. This is why considering also the shape is a key point when a process with a filtration step needs to be developed.

Here the modelling is based on the classical filterability law, but also incorporates a new correlation for compressibility factor estimation. The cake construction and its behaviour under pressure were correctly calculated. The new correlation is based on porosity and PSD for a given shape, which are two relatively easily-measured features of the particle population and of the filtrated cake. Usually, the particle shape for a given separation is already experimentally known. Moreover, only two trials are necessary for the identification of the two parameters β and γ. This is particularly interesting when there is only a small amount of slurry, or when the materials to be handled are hazardous and highly toxic. Such a model is also useful when needing to estimate the impact of modified solid features which may occur after process changes in the upstream operations prior to the filtration step.

  Figure 1: Experimental batch precipitation vessel

Figure 2 :

 2 Figure 2: Definition of Equivalent Circular Diameter (ECD) for different particles.

Figure 3 :

 3 Figure 3: Simulation flowchart

Figure 5 :Figure 6 :

 56 Figure 5: Size volume distribution of precipitated spheres

Figure 7 :Figure 8 :Figure 9 :

 789 Figure 7: SEM image of dried cubic particles

Figure 11 :

 11 Figure 11: SEM image of dried platelets of uranium oxalate

Figure 12 :

 12 Figure 12: Size volume distribution of uranium oxalate

Figure 13 :

 13 Figure 13: Volume shape factor distribution of uranium oxalate

Figure 15 :

 15 Figure 15: Size Volume distribution of precipitated needles

  9 m.kg -1

Figure 17 :

 17 Figure 17: Cake specific resistance and compressibility factor values depending on the pressure and

Figure 18 :

 18 Figure 18: Evolution of the filtrate volume collected as a function of time and shape at a pressure

Figure 20 :

 20 Figure 20: Time profile of the filtration volume collected as a function of shape and pressure drop

  , the curves show a comparison between physical parameters. They compare the evolution of the experimental and modeled filtrate volume collected as a function of time for a cube suspension at 5 bar (ε measured = 0.7, w=30g.L -1 ).

Figure 21 :

 21 Figure 21: Comparison between modelling and experiments for cubes at 5 bar

Table 1 : Precipitation conditions (from Beck and Andreassen (Beck and Andreassen 2012))

 1

Table 2 : Theoretical values of volume shape factors for known shapes

 2

	Shape	Volume shape factor
	Spherical	1
	Cubic	0.81
	Plate like (1)	0.51
	Needle	0.29 to 0.2

Table 3 : Volume shape factor and PSD measurements of the precipitated particles (CC: Calcium Carbonate, UO: Uranium Oxalate)

 3

	Particle size distribution

* defined as the standard deviation to the average diameter ratio

Table 4 : Comparison of cake specific resistance measurements for two cell diameters, 1 and 3cm

 4

	α (10 9 m.kg-1)

Table 5 : Physical properties of particles under atmospheric pressure used for validation (CC: Calcium Carbonate, UO: Uranium Oxalate)

 5

	Particles	Dry cake bulk density (kg/m 3 )
	Spheres (CC)	0.696
	Cubes (CC)	0.488
	Platelets (UO)	0.624
	Needles (CC)	0.454

Table 6 : Calculated values of the exponents β and γ of the compressibility law for each shape (the subscripts 1 and 2 are the two experiments used for the determination).

 6

	Particles	n1	ε1	σ1	dp1	VC1	n2	ε2	σ2	dp2	VC2	β	γ
	Spheres (CC) 0.4	0.67 1.36 5.97 0.23 0.38 0.7	1.17 5.8	0.2	0.29 0.76
	Cubes (CC)	0.47 0.72 1.57 9.57 0.16 0.48 0.67 2.05 10.72 0.19 0.41 0.62
	Platelet (CC) 0.41 0.71 0.47 3.15 0.15 0.36 0.66 0.40 3.05 0.13 0.21 0.57
	Needle (CC) 0.92 0.78 5.76 17.6 0.33 0.90 0.77 5.91 18.1 0.33 0.48 0.62

Table 7 : Experimental filtration conditions for calibrated spheres

 7

			Distribution 1 Distribution 2	Distribution 3	Distribution 4	Distribution 5
	Mean		20 µm	50 µm	A 50/50 w/w	A 25/75 w/w	A 75/25 w/w
	diameter				mixture of the	mixture of the	mixture of the
					distributions 1	distributions 1	distributions 1
					and 2	and 2	and 2
	Standard		0.2774 µm	0.786 µm	0.833 µm	0.833 µm	0.833 µm
	deviation	of
	the
	distribution
	Cake Porosity	0.43	0.43	0.35	0.38	0.31
	Filtrate
					1.2 mPa.s
	Viscosity
	Filtrate density			0.789 g.cm -3

Table 9 : Cake resistance comparison between experiments and model (particles of calcium carbonate (CC) and uranium oxalate (UO))

 9

	Particles		Experiment	Modelling	Relative
					deviation
		∆P = 1 bar	11.5 10 9 m.kg -1	11.2 10 9 m.kg -1	2.6 %
	Spheres (CC)	∆P = 3 bar	17.4 10 9 m.kg -1	18.4 10 9 m.kg -1	5.4 %
		∆P = 5 bar	22.1 10 9 m.kg -1	21.7 10 9 m.kg -1	1.8 %
		Compressibility	0.40	0.42	5.0 %
		∆P = 1 bar	2.7 10 9 m.kg -1	2.8 10 9 m.kg -1	5.6 %
	Cubes (CC)	∆P = 3 bar	4.6 10 9 m.kg -1	4.7 10 9 m.kg -1	2.5 %
		∆P = 5 bar	5.8 10 9 m.kg -1	5.9 10 9 m.kg -1	1.4 %
		Compressibility	0.47	0.45	4.3 %
	Needles (CC)