Elodie Chabanon
email: elodie.chabanon@univ-lyon1.fr

Nida Sheibat-Othman

O Mdere

Jean-Pierre P Valour

Sébastien Urbaniak

François Puel

Drop size distribution monitoring of oil-in-water emulsions in

Keywords: Oil-in-water emulsions, SMX+ static mixers, viscous fluids, high volume fraction, Process Modelling

come

INTRODUCTION

The development of multiphase systems, such as mixing immiscible gas-liquid or liquid-liquid fluids, is one of the major challenges of engineering processes in different domains like cosmetic, pharmaceutical, chemical and food industries [START_REF] Leng | Immiscible Liquid-Liquid Systems[END_REF]. Amongst them, Oil-in-Water (O/W) emulsions represent an important sub-group where the dispersed phase is organic, i.e. the oil, and the continuous phase is aqueous. Controlling the drop size distribution (DSD), at lower power consumption, is important in these systems as it might affect mass transfer and potential reactions during the formation as well as the quality of many finished emulsion products [START_REF] Becker | Monitoring silicone oil droplets during emulsification in stirred vessel: Effect of dispersed phase concentration and viscosity[END_REF].

Emulsions can be realized using mechanical stirrers (such as turbines and rotor stators), high pressure homogenizers or static mixers. Stirred tanks are still assumed to be the reference for emulsification in most industrial applications because of their flexibility and the possibility of mixing viscous products which are difficult to pump. But, the emulsification efficiency is limited and spatially non-uniform (due to higher shear around the stirrer). Moreover, they suffer from space requirement, operating cost (OPEX) and capital cost (CAPEX), long residence-time distributions, non-isothermal operating condition and the decrease of safety conditions and process control due to the big volume [START_REF] Ghanem | Mixing performances of swirl flow and corrugated channel reactors[END_REF]. Consequently, the development of continuous processes have been promoted in order to reduce the size of the process unit, increase the productivity, homogenize the shear rates, enhance heat transfer and allow faster scale-up from the lab-scale to the industrial-scale [START_REF] Taweel | Intensifying Mass Transfer Between Immiscible Liquids: Using Screen-type Static Mixers[END_REF][START_REF] Ghanem | Mixing performances of swirl flow and corrugated channel reactors[END_REF][START_REF] Laporte | Flow process conditions to control the void fraction of food foams in static mixers[END_REF][START_REF] Thakur | Static Mixers in the Process Industries-A Review[END_REF][START_REF] Van Gerven | Structure, Energy, Synergy, Time-The Fundamentals of Process Intensification[END_REF].

Static mixers allow continuous production with lower energy consumption than mechanical agitators (mainly compared to rotor stators or high pressure homogenizers) while maintaining good mixing performances [START_REF] Anxionnaz | Heat exchanger/reactors (HEX reactors): Concepts, technologies: State-of-the-art[END_REF][START_REF] Bayat | A comparative study of two different configurations for exothermic-endothermic heat exchanger reactor[END_REF][START_REF] Couvert | Static mixers with a gas continuous phase[END_REF][START_REF] Thakur | Static Mixers in the Process Industries-A Review[END_REF]. They consist of small motionless elements, with complex porous structures.

They are inserted in a pipe, where a fluid is pumped, to generate shear by redistributing the fluid in directions transverse to the main flow. The main energy consumption consists therefore of pumping the fluid in the unit, which increases with the pressure drop. The pressure drop thus and fluid velocity constitute the driving force for dispersing the two immiscible liquids when flowing through the mixing elements. Static mixing units require therefore optimization of energy (pressure drop, time and mixing length) as well as drop size. The mixing length is modified through the number of static mixers or the number of passes (that are equivalent if pumping effects are negligible).

Different designs of static mixers are available commercially, and the most widely used are Kenics® helical mixer from Chemineer [START_REF] Berkman | Dispersion of viscous liquids by turbulent flow in a static mixer[END_REF][START_REF] Chen | Gas-liquid and liquid-liquid dispersions in a Kenics mixer[END_REF][START_REF] Haas | Turbulent dispersion of aqueous drops in organic liquids[END_REF][START_REF] Middleman | Drop Size Distributions Produced by Turbulent Pipe Flow of Immiscible Fluids through a Static Mixer[END_REF][START_REF] Yamamoto | Relationship between the Dispersed Droplet Diameter and the Mean Power Input for Emulsification in Three Different Types of Motionless Mixers[END_REF] and more recently SMV ® [START_REF] Paglianti | A mechanistic model for pressure drops in corrugated plates static mixers[END_REF] and SMX ® static mixers from Sulzer. SMX static mixers were used for mixing miscible fluids, as well as non-miscible fluids (gas-liquid and liquid-liquid), either in laminar [START_REF] Anxionnaz | Heat exchanger/reactors (HEX reactors): Concepts, technologies: State-of-the-art[END_REF][START_REF] Das | Laminar liquid-liquid dispersion in the SMX static mixer[END_REF][START_REF] Das | Drop breakage model in static mixers at low and intermediate Reynolds number[END_REF][START_REF] Fradette | Comparison of Mixing Technologies for the Production of Concentrated Emulsions[END_REF][START_REF] Kiss | Formation of O/W emulsions by static mixers for pharmaceutical applications[END_REF][START_REF] Laporte | Flow process conditions to control the void fraction of food foams in static mixers[END_REF][START_REF] Legrand | Liquid-Liquid Dispersion in an SMX-Sulzer Static Mixer[END_REF][START_REF] Liu | Drop Breakup in an SMX Static Mixer in Laminar Flow[END_REF]Rama Rao et al., 2007;[START_REF] Thakur | Static Mixers in the Process Industries-A Review[END_REF] or turbulent regimes [START_REF] Anxionnaz | Heat exchanger/reactors (HEX reactors): Concepts, technologies: State-of-the-art[END_REF][START_REF] Lobry | Turbulent liquid-liquid dispersion in SMV static mixer at high dispersed phase concentration[END_REF][START_REF] Streiff | New fundamentals for liquid-liquid dispersion using static mixers[END_REF][START_REF] Thakur | Static Mixers in the Process Industries-A Review[END_REF][START_REF] Theron | Turbulent Liquid-Liquid Dispersion in Sulzer SMX Mixer[END_REF][START_REF] Theron | Comparison between three static mixers for emulsification in turbulent flow[END_REF]. SMX+ and SMX static mixers have the same X geometry but in SMX+ the crossed bars are thinner and less numerous which leads to a reduction in the pressure drop to about 50 % compared to SMX [START_REF] Hirschberg | An improvement of the Sulzer SMX TM static mixer significantly reducing the pressure drop[END_REF]. SMX+ static mixers have already been used for liquid-liquid and gas-liquid systems [START_REF] Baumann | Flow regimes and drop break-up in SMX and packed bed static mixers[END_REF][START_REF] Hirschberg | An improvement of the Sulzer SMX TM static mixer significantly reducing the pressure drop[END_REF][START_REF] Laporte | Flow process conditions to control the void fraction of food foams in static mixers[END_REF][START_REF] Meijer | On the performance of static mixers: A quantitative comparison[END_REF][START_REF] Theron | Comparison between three static mixers for emulsification in turbulent flow[END_REF].

Understanding and modelling the mechanisms taking place in liquid-liquid emulsification using static mixers are still not completely achieved. This is due to the wide range of geometries and working conditions, such as the flow rates, flow regime (laminar to turbulent), surfactant dynamics, fluid type (Newtonian or non-Newtonian), volume fraction of the dispersed phase, or the ratios of densities and viscosities of the dispersed phase to the continuous phase. Due to the lack of complete experimental data, the available models are still not universal for the various static mixers and fluids. In particular, the available literature of liquid-liquid emulsification using static mixers is limited to dilute systems, with a dispersed phase volume fraction (Ф d ) rarely higher than 35 %vol. However, the increase of the dispersed phase volume fraction might have an effect on both the droplet size and the pressure drop, for instance due to an increase in the apparent viscosity (if the dispersed phase is more viscous than the continuous one).

This work aims to characterize oil-in-water emulsions through SMX+ static mixers involving no mass transfer between the two phases. It investigates the influence of the fluids viscosities and densities, the dispersed phase volume fraction (from diluted systems to highly concentrated ones), the Reynolds number (from transitional to turbulent flows) and the number of SMX+ static mixers on the drop size distribution and the Sauter mean diameters. These impacts are then described using a new optimized correlation. The correlation accounts thus for the number of mixers and hydrodynamic and physicochemical parameters.

MATERIALS AND METHODS

Emulsion ingredients

The investigated oil-in-water emulsions are made up of silicon oil, which constitutes the dispersed phase (supplied by Bluestar Silicone/France), and distilled water as the continuous phase. Three viscosities of silicon oils are studied, 20, 100 and 350 mPa s, hereafter denoted V20, V100 and V350 respectively. The emulsions are stabilized by the surfactant polyoxyethylene (20) sorbitan monolaurate (Montanox 20 ® ), C58H114O26 (supplied by SEPPIC). The physicochemical properties of the different fluids are summarized in Table 1.

Static mixers

The recently developed SMX+ static mixers (supplied by Sulzer, cf. Figure 1) are used in this work for their ability to limit the pressure drop and improve the mixing performance of viscous fluids [START_REF] Hirschberg | An improvement of the Sulzer SMX TM static mixer significantly reducing the pressure drop[END_REF]. Each element is composed of an array of crossed bars (6 in each direction) arranged at an angle of 45° relative to the flow direction. Note that SMX mixers have 8 crossed bars in each direction which causes higher pressure drop [START_REF] Hirschberg | An improvement of the Sulzer SMX TM static mixer significantly reducing the pressure drop[END_REF]. The diameter and the height of each SMX+ is about 5 mm, which results in an aspect ratio of length to diameter per static mixer of L/D = 1. They have a 75% global porosity (ϕ), a hydraulic diameter of 𝐷𝐷 ℎ = 1.42×10 -3 m (taking into account the tube walls) and a specific area of 𝑎𝑎 𝑔𝑔 = 1527 m 2 m -3 (neglecting the tube surface). The used mixers are made of stainless steel.

Experimental Setup

The used experimental set-up is described in Figure 2. The SMX+ static mixers are placed into a transparent tube of equivalent diameter (5 mm) made of PVC. The tube slightly compresses the mixers which avoids radial rotations. Consecutive static mixers are rotated by an angle of 90° each with respect to the previous one. No space is allowed between the mixers. A piece of copper pipe is inserted in the pipe after the last static mixer in order to stick it and to avoid any axial displacement of the mixers in the pipe.

The continuous and the dispersed phases are stored in separate feed tanks placed on electronic scales to measure their flow rates precisely during the experiment. These flow rates are used to calculate the volume fraction of the dispersed phase:

Ф d = 100 Q d (Q d +Q c ) (1)
Where Qd and Qc represent the volume flow rates (m 3 s -1 ) of the dispersed and continuous phases respectively. The fluids are pumped using two MCP-Z Ismatec ® gear pumps equipped with Micropump ® magnetically driven heads.

The phases enter in contact in the pipe thanks to a T-junction with a ϒ's form (cf. Figure 2) placed before the static mixers, which introduces the oil phase into the flowing continuous phase. This represents the first step of the emulsification process. Similar online pre-mixing strategies were used by [START_REF] Hirschberg | An improvement of the Sulzer SMX TM static mixer significantly reducing the pressure drop[END_REF][START_REF] Theron | Turbulent Liquid-Liquid Dispersion in Sulzer SMX Mixer[END_REF]. Theron et al. [START_REF] Theron | Turbulent Liquid-Liquid Dispersion in Sulzer SMX Mixer[END_REF] chose to inject the dispersed phase perpendicularly to the flowing continuous phase just before the first static mixer while Hirschberg et al. [START_REF] Hirschberg | An improvement of the Sulzer SMX TM static mixer significantly reducing the pressure drop[END_REF] introduced the dispersed phase in parallel of the continuous phase directly on the first static mixer. Note that some authors first realise a pre-emulsion using another device [START_REF] Hoevekamp | Experimental and Numerical Investigation of Porous Media Flow with Regard to the Emulsion Process[END_REF][START_REF] Van Der Zwan | Premix membrane emulsification by using a packed layer of glass beads[END_REF], but in this case, one should be careful to the pumping effects on the pre-emulsion size due to the generated shear and possible breakup, that should be checked at the inlet for each experiment.

Baumann et al. [START_REF] Baumann | Flow regimes and drop break-up in SMX and packed bed static mixers[END_REF] indicated that online pre-mixing gives similar drop sizes as off-line pre-mixing but higher standard deviations.

A pressure gauge (0-6 bar, ± 3 mbar) placed at the inlet of static mixers (between the T and the mixers) measures the differential pressure. The static mixers outlet pressure corresponds to the atmospheric pressure. An online Turbiscan (from Formulaction ® /France) is placed at the outlet of the static mixers tube to monitor the backscattering signal. A collecting vessel is placed after the Turbiscan. A sample of each experiment is also withdrawn at the outlet to measure the drop size distribution at-line with a video back lighting probe EZ Probe-D25-L1300 [START_REF] Becker | Investigation of Discrete Population Balance Models and Breakage Kernels for Dilute Emulsification Systems[END_REF] after eventual dilution with the continuous phase depending on the dispersed phase fraction. Indeed, the image treatment is limited to about 10%vol. dispersed phase fraction for the considered drop size.

Emulsification procedure

Emulsifications were performed at room temperature. The dispersed phase was put on the scale without any additives. The continuous phase was also put on a scale, and consisted of distilled water in which the surfactant was dissolved.

The surfactant (Montanox 20) hydrophilic-lipophylic balance (HLB) is 16.7, which is large enough to form direct O/W emulsion when dissolved in the aqueous phase. Its critical micellar concentration (CMC), and therefore saturation concentration at 21°C in pure water, is 8.04×10 -5 mol L -1 . In order to ensure an excess of surfactant, two Montanox 20 concentrations that are higher than the CMC were prepared: 10 g L -1 (1%wt., or 8.15×10 -3 mol L -1 ) and 25 g L -1

(2.5 %wt., or 2.037×10 -2 mol L -1 ). According to Becker et al. [START_REF] Becker | Monitoring silicone oil droplets during emulsification in stirred vessel: Effect of dispersed phase concentration and viscosity[END_REF], the interfacial tension of silicon oil-water containing between 0.5 to 3.0%wt. of surfactant does not vary significantly. The surfactant has a polar head surface of about 120 Å [START_REF] Rosen | Pressure drop for laminar flow of non-Newtonian fluids in static mixers[END_REF]. Using this value, the surface coverage of droplets by the surfactant was calculated to be much higher than 100% for all experiments.

The volume fraction of the dispersed phase was varied from 5 to 60%vol., which is close to the theoretical value of phase inversion [START_REF] Brochette | Émulsification -Élaboration et étude des émulsions[END_REF]. Experiments with higher volume fraction of the dispersed phase (75%vol.) were investigated but phase inversion of the samples was observed with naked eyes. As mentioned above, three different silicon oils, with different viscosities and densities, were used (cf. Table 1). The number of SMX+ static mixers was varied from 1 to 15. The total flow rate was varied between 120 and 1627 mL min -1 .

For each experiment, care was taken to achieve steady state conditions, reflected by a constant backscattering signal using the Turbiscan. Typically, steady state in transitional or turbulent flow regimes was achieved very quickly, after 2 minutes.

THEORETICAL ASPECTS

Different correlations have been proposed in the literature under equilibrium for both the pressure drop, which determines the energy requirements, and the droplet size determining the product quality, stability and eventual kinetics.

Flow characteristics

The pressure drop (∆P) caused by friction between the fluid and the surfaces is useful to predict the energy characteristics of the flow. Correlations for Newtonian single phase fluids in porous media are present for laminar [START_REF] Liu | Laminar mixing of shear thinning fluids in a SMX static mixer[END_REF] and turbulent regimes [START_REF] Cybulski | Static mixers-criteria for applications and selection[END_REF][START_REF] Pahl | Static Mixers and Their Applications[END_REF][START_REF] Streiff | The design and application of static mixer technology[END_REF], as well as for non-Newtonian fluids [START_REF] Li | Pressure Drop of Newtonian and Non-Newtonian Fluids Across a Sulzer SMX Static Mixer[END_REF][START_REF] Li | Hydrodynamics and heat transfer of rheologically complex fluids in a Sulzer SMX static mixer[END_REF]Shah and Kale, 1991). For turbulent flow in a pipe, the pressure drop is described by dimensionless ratio of resistance to inertial forces using Newton number (Ne) and the Reynolds number (Re) as the ratio of inertial to viscous forces:

Ne = 2f F = ∆P ρu s 2 D L (2) Re = ρu s D µ (3)
Where f F represents the friction factor (-), ∆P is the pressure drop (Pa), D is the tube diameter (m) and L its length (m), ρ is the apparent fluid density (kg m -3 ), µ is the apparent fluid viscosity (Pa s), and u s is the superficial velocity (m s -1 ) defined by:

u s = 4Q πD 2 (4)
Where Q is the volume flow rate of the fluid (m 3 s -1 ).

For pipelines containing mixer elements, pressure drop is usually assumed as uniformly distributed along the pipeline. Therefore, similar correlations were developed for single phase turbulent flow in static mixers, while accounting for the global porosity (ϕ) of the pipes containing static mixers [START_REF] Cavatorta | Fluid-dynamic and mass-transfer behavior of static mixers and regular packings[END_REF][START_REF] Couvert | Pressure Drop and Mass Transfer Study in Static Mixers with Gas Continuous Phase[END_REF][START_REF] Morancais | Friction Factor in Static Mixer and Determination of Geometric Parameters of Smx Sulzer Mixers[END_REF][START_REF] Pahl | Static Mixers and Their Applications[END_REF]Shah and Kale, 1991;[START_REF] Streiff | Statisches Mischen[END_REF][START_REF] Streiff | The design and application of static mixer technology[END_REF][START_REF] Streiff | New fundamentals for liquid-liquid dispersion using static mixers[END_REF][START_REF] Tallmadge | Packed bed pressure drop-an extension to higher Reynolds numbers[END_REF][START_REF] Thakur | Static Mixers in the Process Industries-A Review[END_REF][START_REF] Theron | Turbulent Liquid-Liquid Dispersion in Sulzer SMX Mixer[END_REF]. This allows comparison between differently structured mixers.

The superficial velocity (u s , expressed in m s -1 ), i.e. in an empty tube, is hence replaced by the interstitial velocity (u i , in m s -1 ) according to:

u i = u s φ (5)
And the pipe diameter (D, m) is replaced by the hydraulic diameter of the pipe (D h , m):

D h = 4φ a g (6)
Where a g is the mixer specific surface area (m 2 m -3 ) [START_REF] Streiff | The design and application of static mixer technology[END_REF]. In this case, the hydraulic Newton number (Neh, dimensionless) for a circular pipe containing mixers becomes [START_REF] Etchells | Mixing in Pipelines[END_REF]:

Ne h = ∆P φ 2 ρ u s 2 D h L (7)
And the hydraulic Reynold number (Reh, dimensionless) is given by:

Re h = ρ u s D h φµ (8)
In two phase fluids, the apparent viscosity of the emulsion is defined by [START_REF] Taylor | The Viscosity of a Fluid Containing Small Drops of Another Fluid[END_REF]:

µ = µ c �1 + 2.5 Ф d � µ d + 2 5 � µ c µ c +µ d �� (9)
And the apparent fluid density of the emulsion is defined according to [START_REF] Legrand | Liquid-Liquid Dispersion in an SMX-Sulzer Static Mixer[END_REF]:

ρ = Ф d ρ d + (1 -Ф d )ρ c ( 10
)
Where ρd and ρc (kg m -3 ) are the densities of the dispersed phase and the continuous phase respectively while µc and µd (Pa s) are the dynamic viscosities of the dispersed phase and the continuous phase respectively (cf. Table 1).

In this work, the interstitial velocity in a pipe containing the mixers varies from u i = 0.13 -1.82 m s -1 (i.e. u s = 0.10 -1.38 m s -1 in an empty pipe), which gives a hydraulic Reynolds number ranging between Re h = 160 -1000 (i.e. Re = 425 -2642). According to Theron and Sauze [START_REF] Theron | Comparison between three static mixers for emulsification in turbulent flow[END_REF], the turbulent regime in SMX+ mixers is achieved when

Re > 710 (i.e. Re h > 260, with the porosity considered in this work).

Correlations for the drop size in turbulent flow

Based on the theory of Kolmogorov (1949) and Hinze (1955), drop breakup is caused by large shear and pressure gradients associated with eddies generated in turbulent flow field. In this theory, breakup takes place either at the turbulent-inertial sub-range (i.e. drops are larger than the smallest eddies, where inertial stresses act on the drop surfaces) or turbulent-viscous dissipation sub-range (i.e. drops are smaller than the smallest eddies, where viscous stresses act on the drop surfaces). The size of smallest eddies is proportional to the Kolmogorov length scale (λ K ) (m), defined by:

λ K = � µ 3 ρ 3 ϵ � 1/4 ( 11
)
Where ϵ is the average rate of energy dissipation per unit mass of fluid (W kg -1 ), expressed by [START_REF] Ghanem | Mixing performances of swirl flow and corrugated channel reactors[END_REF]:

ϵ = Δp u s ρ c Lφ (12)
In this work, λ K varies from 2 to 10µm, and is therefore smaller than the drop size in all experiments which corresponds to the turbulent-inertial sub-range.

Static mixers redistribute the fluid in the radial and tangential directions (transverse to the main flow) using the pumping energy of the flowing fluid which causes distributive mixing by convection. Therefore, they can play the role of promoters of turbulence formation leading to an increase in the energy of eddies, shear and pressure gradients locally, therefore causing drop breakup.

Assuming isotropic flow field and uniform shear and pressure gradients, it is possible to use Kolmogorov-Hinze theory and relate the drop size to the power input and interfacial tension (σ)

between the dispersed phase and continuous phase (see [START_REF] Legrand | Liquid-Liquid Dispersion in an SMX-Sulzer Static Mixer[END_REF][START_REF] Streiff | New fundamentals for liquid-liquid dispersion using static mixers[END_REF][START_REF] Theron | Turbulent Liquid-Liquid Dispersion in Sulzer SMX Mixer[END_REF] for SMX and [START_REF] Hirschberg | An improvement of the Sulzer SMX TM static mixer significantly reducing the pressure drop[END_REF] for SMX+). Most correlations assume drop size equilibrium (equal breakup and coalescence rates), except those explicitly accounting for the number of static mixers (n e ).

According to the Kolmogorov-Hinze theory, the maximum diameter of stable drops (d max ) in the turbulent-inertial regime is given by [START_REF] Baumann | Flow regimes and drop break-up in SMX and packed bed static mixers[END_REF]:

d max = C 1 � σ ρ c � 0.6 ϵ -0.4 (13)
Where C 1 is a tuning parameter and σ is the interfacial tension (N m -1 ).

Thereafter, different correlations were proposed using several process parameters as driving force for drop breakup (e.g. We h , Ne h , Re h , ϵ, u s , number of static mixers) and accounting for different fluid properties (dispersed phase fraction, viscosity ratio, density ratio).

Streiff [START_REF] Streiff | New fundamentals for liquid-liquid dispersion using static mixers[END_REF] introduced to Eq. 13 the effect of density ratios between the continuous and dispersed phase � ρ c ρ d � based on the energy dissipation as driving force for breakup in SMX mixers:

d = C 2 � σ ρ c � 0.6 � ρ c ρ d � 0.1 ϵ -0.4 (14)
Middleman [START_REF] Middleman | Drop Size Distributions Produced by Turbulent Pipe Flow of Immiscible Fluids through a Static Mixer[END_REF] adapted Eq. 13 of Kolmogorov-Hinze for Kenics static mixers using as driving forces for drop breakup the hydraulic Newton number (Eq. 7) and the hydraulic Weber number (Weh, ratio of inertial to interfacial forces, Eq.16), but both correlations may be considered equivalent if the energy dissipation is assumed to be given by Eq.12:

d max D h = C 1 We h -0.6 Ne h -0.4 (15)
We

h = ρ c u s 2 D h φ 2 σ (16)
Assuming Ne h ∝ Re h -0.25 (Blasius law), Eq. 15 is reduced to:

d max D h = C 3 We h -0.6 Re h 0.1 (17)
The volume fraction of the dispersed phased (Ф d ) was accounted for using hydraulic Weber number alone as a driving force for drop breakup, [START_REF] Poux | Dispersion of high-viscosity liquid-liquid systems by flow through SMX static mixer elements[END_REF], which added a second tuning parameter:

d 32 D h = C 4 s(1 + C 5 Ф d )We -3/5 (18)
Calabrese et al. [START_REF] Calabrese | Drop breakup in turbulent stirred-tank contactors. Part I: Effect of dispersed-phase viscosity[END_REF] and Davies [START_REF] Davies | Drop sizes of emulsions related to turbulent energy dissipation rates[END_REF] introduced the effect of the ratio of the viscosity of the dispersed phase to the continuous phase to the Middleman equation (Eq. 15):

d max D h = C 6 �1 + C 7 � µ d µ c � � d D h � 1/3 We h Ne h 1 3 ⁄ Re h � 0.6 We h -0.6 Ne h -0.4 (19)
The combined effect of dispersed phased fraction and density ratio was accounted for by Streiff et al. [START_REF] Streiff | New fundamentals for liquid-liquid dispersion using static mixers[END_REF] based on the drop breakup driving forces of energy dissipation and interstitial velocity:

d = C 8 (1 + C 9 Ф d ) � (1+ C 10 u i )We c 2 � 0.6 � σ ρ c � 0.6 � ρ c ρ d � 0.1 ϵ -0.4 (20)
Where Wec is the critical Weber number (-). Note that this equation has three tuning parameters.

The superficial velocity was also directly considered as a driving force for breakup [START_REF] Barega | Influence of Physical Properties and Process Conditions on Entrainment Behavior in a Static-Mixer Settler Setup[END_REF]:

d 32 D = (1 + C 11 Ф d ) � ρ u s 2 D σ � α(Ф d ) � ρ u s D µ c � β(Ф d ) � µ d µ c � γ (21)
Where α, β, γ and C11 are tuning parameters.

Theron et al. [START_REF] Theron | Turbulent Liquid-Liquid Dispersion in Sulzer SMX Mixer[END_REF], as Maa and Hsu before [START_REF] Maa | Liquid-liquid emulsification by static mixers for use in microencapsulation[END_REF], introduced the number of SMX static mixer elements (n e ) in Middleman equation as follows:

d 32 D = C 12 We -0.6 Re 0.1 n e -0.2 (22)
The mean diameters, d32 (i.e. the Sauter mean diameter) and d43, can be calculated using the drop size distribution in number, n(d) [START_REF] Middleman | Drop Size Distributions Produced by Turbulent Pipe Flow of Immiscible Fluids through a Static Mixer[END_REF]:

d ab = � ∑ n i d i a n i=1 ∑ n i d i b n i=1 � � 1 a-b � (23)
As the objective of this work is to develop a correlation that describes the drop size for flow regimes going from transitional to turbulent, for low and high Reynolds numbers, then the approximation of Blasius will not be used, as the Reynolds and Newton numbers may not be correlated. The proposed correlation should thus be based on either the energy dissipation or Newton number. Moreover, it is aimed to take into account the number of static mixers in the correlation.

RESULTS AND DISCUSSION

The evolution of the droplets diameter and the droplets size distribution in static mixer processes is governed by physicochemical parameters (such as the viscosity and density ratios, the surfactant dynamics and efficiency), concentrations (fraction of the dispersed phase, surfactant), and hydrodynamic parameters. These latter are affected by the flow rate and the geometrical parameters (mixer diameter, length, porosity and structure).

The influence of key parameters was investigated as follows: a process configuration was selected, i.e. a number of SMX+ static mixers and a series of oil-in-water emulsion experiments were performed for each oil viscosity according to the protocol detailed in the materials and method section. For each series of experiments, the Reynolds number and the volume fraction of the dispersed phase were varied randomly to avoid any memory effect of one experiment on the following one. At the end of the series, the first experiment was repeated to validate the whole series. And between each series, the experimental setup was abundantly rinsed with distilled water.

Droplet size distribution measurement

The droplet size distribution was measured at-line by the video probe for all experiments. This measurement technique was widely used for emulsion size characterisation [START_REF] Becker | Monitoring silicone oil droplets during emulsification in stirred vessel: Effect of dispersed phase concentration and viscosity[END_REF].

It has the advantage of working online for diluted emulsions, up to 10%vol. As the volume fraction in this work is increased up to 60%vol., a sample is withdrawn, diluted 10 times, e.g. 10mL of sample is diluted in a 100mL volumetric flask with the continuous phase, and measured using the video probe under gentle mixing (at-line). The attainment of reproducible results is indeed an essential pre-requisite for any emulsification experiment with a volume fraction of the dispersed phase higher than 10%vol.

Turbiscan was used for backscattering measurement online in order to detect equilibrium in drop size (equal breakup and coalescence). Note that backscattering measurements may be used to predict the mean drop size after three-dimensional calibration (as a function of the volume fraction of the dispersed phase, drop size and index), which quickly becomes timeconsuming. Moreover, this would only give the mean diameter and not the drop size distribution.

Figure 3 shows the cumulative size distributions measured using the video probe with varying operating conditions. It can be seen that the reproducibility of the experiments is good regardless the volume fraction, viscosity or Reynolds number. The dilution of the sample for video probe measurements can also be assumed not to influence the measurement of the drop size distribution.

Influence of the number of SMX+ mixers elements

The influence of the geometrical parameter is studied by varying the number of SMX+ in the pipe from 1 to 15. Both figures show that increasing the number of SMX+ static mixers in the pipe allows decreasing the drop size. This is due to the increase in the residence time and therefore dispersion kinetics. Figure 4 a) shows that regardless the volume fraction of the dispersed phase; the drop size reaches an equilibrium value after 5 SMX+ which means that the breakup is importantly reduced after the 5 th mixer (equilibrium between breakup and coalescence). This indicates that the first five SMX+ are mainly efficient for breakup, and the following mixers will be responsible for an increase in the pressure drop, which has to be avoided. By the same way, much more breakup takes place between the 1 st and 3 rd mixer than between the 3 rd and 5 th one. In this work, we will be interested in modelling the part of this curve where drop size evolves. It can be seen on Figure 4 a) that the oil volume fraction has a negligible effect on the drop size.

Figure 4 b) also shows that the increase of the number of SMX+ induces a decrease in the drop size with the V20 silicon oil. Here again, the drop size reaches an equilibrium value after 5 SMX+ confirming the result presented above. However, it has to be noted that surprisingly the increase of the oil viscosity is responsible of a decrease of the drop size. This effect is more pronounced when the oil viscosity is increased from 20 to 100 mPa s; while the effect is nonexistent when the increase of the oil viscosity is from 100 to 350 mPa s. Note that these experiments were realized with 20%vol dispersed phased fraction. These experiments were repeated for the different oil concentrations. At low oil fractions, the tendency was previously found to be almost opposite [START_REF] Becker | Investigation of Discrete Population Balance Models and Breakage Kernels for Dilute Emulsification Systems[END_REF], and so in accordance to the literature [START_REF] Baumann | Flow regimes and drop break-up in SMX and packed bed static mixers[END_REF]. This reveals a combined effect of concentration and viscosity.

Influence of the hydrodynamic parameter

The influence of the hydrodynamic parameter on the drop size is investigated by changing the total input flow rate, leading to different energy dissipation levels. The volume fractions of 20%vol., 35%vol., 50%vol. and 60%vol. of the dispersed phase with a V20 viscosity are considered in these series of experiments. The number of SMX+ static mixers is fixed at 10-11 in order to ensure a constant drop size. In accordance with the literature [START_REF] Barega | Influence of Physical Properties and Process Conditions on Entrainment Behavior in a Static-Mixer Settler Setup[END_REF][START_REF] Das | Laminar liquid-liquid dispersion in the SMX static mixer[END_REF][START_REF] Theron | Turbulent Liquid-Liquid Dispersion in Sulzer SMX Mixer[END_REF], both figures show that increasing ϵ (or the velocity or the Reynolds number) induces a slight decrease in d32 due to the increase in the level of turbulence involving higher inertial forces and pressure gradients. However, d32 is constant above hydraulic ϵ= 1000 W kg -1 , which could be explained by a balance between breakup and coalescence mechanisms. The effects of the dispersed phase fraction or viscosity are found to be negligible in front of that of ϵ. Thus, the influence of the energy dissipation is to be of first order compared to the volume fraction or viscosity of the dispersed phase, as will be confirmed in the next section.

Note that the apparent density and viscosity of the fluid phase, used in the calculation of the energy dissipation rate and the Reynolds number, are quite the same for all the experiments and ranged respectively between 958 to 998 kg m -3 and 1.015 to 3.158 mPa s respectively, while the flow rate varied from 120 to 1627 mL min -1 .

Correlation to predict the Sauter mean diameter (d32)

This part of the study intends to propose a new form of the Middleman correlation which considers the flow properties through the hydraulic Newton number and properties of the droplets surface through the hydraulic Weber number. The correlation is aimed to be able to forecast the drop size evolution until a constant size is reached. Thus, the influence of the number of SMX+ static mixer (ne) (equivalent to a residence time) is added to these two dimensionless numbers. As reported above, the experimental results highlighted that the influence of the viscosity is of the second order and the influence of the volume fraction of the dispersed phase is negligible on the range studied; while the influence of the number of SMX+ and the Reynolds number are of first order. Hence, the influence of the viscosity ratio and the volume fraction of the dispersed phase are lumped into the tuning parameter C1. Note also that it was preferred to use the original correlation of Middleman that uses the hydraulic Newton number, rather than the latter correlations based on the Reynolds number, in order to avoid adding a supplementary assumption that Ne h ∝ Re h -0.25 (Blasius law). Indeed, some of the results are reached in a transitional flow while the other part is achieved in turbulent flow; thus, the linearity relationship between the Reynolds number and the Newton number is not confirmed. Note also that the Middleman correlation constitutes the equivalence of the original Kolmogorov-Hinze correlation proposed for turbulent systems, for static mixers. The results are presented on Figure 6 and the obtained correlation is:

d 32 D h = C 1 We h -0.6 Ne h -0.4 n e -C 2 = 1 D h C 1 � σ ρ c � 0.6 ϵ -0.4 n e -C 2 (24)
The correlation was established by using experimental results obtained with 1 to 5 SMX+ , according to the results presented above which highlighted that only the first five SMX+ are efficient to breakup. A volume fraction of the dispersed phase ranging from 5 to 60%vol. was considered, and the three silicon oil viscosities (20, 100 and 350 mPa s) were considered. Finally, the hydraulic Newton number was varied between 5 and 130, corresponding to a hydraulic

Reynolds number ranging from 220 to 950. Thus, the correlation is usable both in the transitional and turbulent flow regimes.

Figure 6 shows the modelling results. The values of the parameters, obtained by linear regression, are C 1 = 1.7 and C 2 = 0.27. Only the experiments done at low energy dissipation are not caught by the model. This is due to the fact that at low energy dissipation, the drop size distribution is very wide. Therefore, the calculation of d 32 might be disturbed by the presence of small number of big drops detected by image treatment. Moreover, at very low flow rates, controlling precisely the flow rates and therefore the volume fraction of oil is more difficult with the used pumps. It has to be noted that the exponent values of the hydraulic Weber number (i.e. -0.6) and the hydraulic Newton number (i.e. -0.4) are the same as those reported in the Middleman correlation [START_REF] Middleman | Drop Size Distributions Produced by Turbulent Pipe Flow of Immiscible Fluids through a Static Mixer[END_REF]. The obtained exponent value of ne is close to those reported in the literature for other static mixers [START_REF] Baumann | Flow regimes and drop break-up in SMX and packed bed static mixers[END_REF][START_REF] Theron | Turbulent Liquid-Liquid Dispersion in Sulzer SMX Mixer[END_REF]. Hence, the influence of the static mixers type (SMX, SMX+, Kenics) on the process performances seems to be negligible.

CONCLUSIONS

The objective of this study was to push the limit of modelling the emulsification process by new SMX+ static mixers. This kind of static mixers is particularly recommended for viscous liquid-liquid dispersion causing limited pressure drop. The reported experimental results allow investigating the influence of some physicochemical parameters (the volume fraction of the dispersed phase and its viscosity), the geometric parameter through the number of SMX+ in the pipe and the hydrodynamic parameter (the Reynolds diameter) on the drop size. The comparison was based on the Sauter mean diameter (d32).

The obtained results show that the increase of the energy dissipation rate or the number of SMX+ static mixers induce a decrease in the d32 diameter while the increase of the dispersed fraction has a negligible effect and its viscosity seems to have no significant effect when it is higher than 100 mPa s. More importantly, the decrease in the d32 with the increase of the number of SMX+ is mainly limited to the first five SMX+, whatever the value of the energy dissipation rate (or the Newton number or the Reynolds number), in the studied range. In fact, the following SMX+ static mixer added after the five first ones will only increase the pressure drop but will not have any influence on the d32. Hence, it clearly appears that the influence of the energy dissipation rate and the number of the SMX+ static mixer are of the same order of magnitude whereas the influence of the volume fraction of the dispersed phase and its viscosity are lower.

Figure 4 a

 4 Figure 4 a) shows the effect of the number of SMX+ static mixers on d32 for different oil

Figure 5 a

 5 Figure 5 a) shows the effect of the energy dissipation on d32 for different dispersed phase

Finally, a correlation

  of type Middleman was established. This one takes into consideration the number of SMX+ mixer elements. The correlation is adapted to situations where the drop size distribution is not yet fully established. The exponents of each parameter of the correlation confirmed that the number of SMX+ has almost the same influence on the d32 as the Newton number.

Figure 1 :

 1 Figure captions:

Figure 2 :

 2 Figure 2: Schematic diagram of the experimental set-up.

Figure 3 :

 3 Figure 3: Examples of cumulative drop size protocol validation on emulsification

Figure 4 :

 4 Figure 4: Influence of the number of SMX+ mixers: a) Effect of numbers of

Figure 5 :

 5 Figure 5: Influence of the hydrodynamic parameters: a) Effect of the volume

Figure 6 :

 6 Figure 6: Middleman type correlation for the experimental data in terms of Weber

  Figure 3

Table 1

 1 Physico-Chemical properties of silicon oil in distilled water emulsion stabilized using Montanox 20, at 20°C

						Silicon Oil
	Properties
					V20	V100	V350
	Dispersed phase density ρd (kg m -3 )	948.8	964.4	967.6
	Continuous phase density ρc (kg m -3 )		998.4
	Density ratio	𝜌𝜌 𝑑𝑑 𝜌𝜌 𝑐𝑐	(-)	0.95	0.97	0.97
	Dispersed phase dynamic viscosity µd (Pa s)	19 10 -3	96.5 10 -3	339.5 10 -3
	Continuous phase dynamic viscosity µc (Pa s)		1.01 10 -3
	Viscosity ratio	µ 𝑑𝑑 µ 𝑐𝑐	(-)	18.8	95.4	335.6
	Refractive index (-)	1.400	1.402	1.402
	Interfacial tension for 1%wt. surfactant
					9.65	10.14	10.60
	(mN m)

ACKNOWLEDGEMENTS

The authors would gratefully thank Alain RIVOIRE, Jean-Marc GALVAN and Julian BECKER, for their help in the field of in situ video monitoring and image treatment, Mathilde ESCHMANN and François WARRICK for their experimental work.


Damien Bourcier
email: damien.bourcier@cpe.fr

Yan Kong

Siong

Filtration process optimization: rheological behavior of a filter cake at different moisture contents

Keywords: rheology, moisture content, filtration, vane geometry

Transferring a wet cake from a filter to a dryer or furnace could severely lower the productivity of a whole process when the wet cake exhibits high viscosity and a sticky paste behavior. Depending on their moisture content, three distinct regions of rheological behavior were identified. Among them, only the granular solid type had sufficient flowability during the transfer operation. It was possible to discriminate the flow behavior more quantitatively by determining two transition values of moisture content. Rheological measurements were carried out at a constant shear rate representative of that encountered in the mechanical transfer system used at plant scale. The use of a vane geometry for the rheometer was necessary to ensure an accurate repeatability of the viscosity measurements.

1-Introduction

Solid-liquid separation technologies are needed in many industries: bio-technology, chemistry, oil and pharmaceutics for instance. Solid-liquid separation involves the separation of a solid and a liquid phase from a slurry. The filtrate viscosity is generally very low and is easily transported. On the contrary, the transferability of the residual cake from the filter or centrifuge to the dryer may present some difficulties, severely reducing plant productivity.

The cake obtained contains a residual moisture content induced by its porosity. If this moisture content is high, a suspension is obtained which behaves like a liquid, leading to a long drying time. On the other hand, if the moisture content is sufficiently low, a cake with a solid-like behavior is formed, resulting in an easier transfer process. When the moisture content has an intermediate value, however, a sticky paste-like substance is formed, causing problems during the transfer step. Fig. 1 shows the different behaviors of a suspension depending on its moisture content. X1 and X2 are the two moisture content values for which the cake flowability changes drastically. This paper aims at developing an experimental protocol for the determination of the moisture content transition values thanks to a rheological approach. Granular paste is by far the most complex system to study with rheometer techniques, due to various effects such as wall slip, migration, sedimentation, fracture and evaporation which may compromise the results [START_REF] Coussot | Rheophysical classification of concentrated suspensions and granular pastes[END_REF]. Nevertheless, later in this paper we describe which rheometer geometry can give satisfactory quality, enabling the identification of the two transition moisture content values (X1 and X2) presented above.

The first part of this paper outlines the theoretical background of rheology. In the second part, the most commonly-used rheometer geometries are compared. Various tests were carried out to determine the most suitable geometry and experimental procedure in terms of repeatability. Finally, the two transition moisture content values are given to conclude our findings on the model system.

Theoretical considerations regarding granular paste viscosity

The term 'granular paste' or 'granular suspension' has been widely used in the literature to describe a highly concentrated mixture of solid particles in a Newtonian fluid [START_REF] Ancey | A theoretical framework for granular suspensions in a steady simple shear flow[END_REF]. If the particles are large enough, Brownian and colloidal interactions are negligible and the whole system is governed by interactions at particle scale. Three mechanisms can generally be considered: friction, lubricated interaction and collision. A few dimensionless numbers have been proposed in recent decades to determine the governing mechanism within a particular system, including the Leighton number, Le [START_REF] Leighton | The shear-induced migration of particles in concentrated suspensions[END_REF] and the Bagnold number, Ba [START_REF] Bagnold | Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear[END_REF]. The Fig. 2 [START_REF] Coussot | Rheophysical classification of concentrated suspensions and granular pastes[END_REF] summarizes the different situations depending on the solid fraction and shear rate applied. When the volume fraction ϕ is higher than a critical concentration ϕc dependant of the shape [START_REF] Coussot | Rheophysical classification of concentrated suspensions and granular pastes[END_REF], interactions in the cake could depend on three mechanisms: friction, lubrication and collision. To identify the dominant interactions on our system, the dimensionless number of Leighton is proposed:

𝐿𝐿𝐿𝐿 = 𝜂𝜂 𝑠𝑠 𝛾𝛾ḃ𝑏 𝜎𝜎 𝑁𝑁 𝜀𝜀 𝑟𝑟 (1)
Le represents the ratio of a characteristic time for the particle to enter into direct contact with another particle when moving through the fluid under the action of the external normal stress to the characteristic time of flow [START_REF] Coussot | Rheophysical classification of concentrated suspensions and granular pastes[END_REF].

In our system, the interstitial fluid viscosity 𝜂𝜂s was 1 mPa.s, the shear rate 𝛾𝛾̇ 100 s -1 , the normal constraint σN was 20 Pa, the mean distance between the two centers of neighboring particles b was 9µm, and the roughness εr about 1 µm. After calculation, the Leighton number Le is 0,045. Considering the solid fractions experimented, the low shear rate and the Le number value, it is likely that frictional interactions were preponderant (see Fig 2 .) and the particles were in direct contact with each other. Frictional flow was established.

In the literature [1; 2; 5; 6], these numbers have proven to be useful in providing an insight into the governing mechanism within a system. However, it should be noted that these numbers were formulated by considering particle dimensions, the average mean distance between the particles and their roughness [START_REF] Ancey | A theoretical framework for granular suspensions in a steady simple shear flow[END_REF].

Hence, some approximations are necessary when the particles have a complex size distribution and an asymmetric shape, which could possibly lead to a misinterpretation of the result obtained.

There are a few classical formulas for granular paste viscosity estimation. The simplest one states the linear dependence of the relative viscosity on the solid volume fraction of the paste [START_REF] Einstein | Investigation on the Theory of the Brownian Movement[END_REF]:

𝜂𝜂 𝜂𝜂 𝑠𝑠 = 1 + 2.5𝜙𝜙 (2)
where η (Pa.s) is the dynamic viscosity of the paste, ηs (Pa.s) is the dynamic viscosity of the Newtonian fluid and ϕ is the solid volume fraction of the paste. Later, the coefficient 2.5 was named as the intrinsic viscosity of a suspension composed of spherical and hard particles.

In recent decades, many researchers have attempted to evaluate the viscosity of such pastes through various much more complex formulas, such as the formula of Van Der Werff and De Kruif (3) presented below [START_REF] Van Der Werff | Hardsphere Colloidal Dispersions : The Scaling of Rheological Properties with Particle Size, Volume Fraction and Shear Rate[END_REF] and the formula of Krieger and Dougherty [START_REF] Krieger | A mechanism for non-Newtonian flow in suspensions of rigid spheres[END_REF]:

𝜂𝜂 𝑟𝑟 = 𝜂𝜂 𝜂𝜂 𝑠𝑠 = 1 + 2,5 𝜙𝜙 + 6,2 𝜙𝜙 2 + 𝛿𝛿(𝜙𝜙 2 ) ( 3
)
where ηr is the relative viscosity of the suspension and 𝛿𝛿(𝜙𝜙 2 ) an inosotropic term usually neglected. These formulas were established through various approaches, ranging from models assuming only hydrodynamic interactions (2) to models attempting to include colloidal [START_REF] Ogawa | Viscosity equation for concentrated suspensions of charged colloidal particles[END_REF] and Brownian interaction by varying the effective solid volume fraction [11; 12]. However, it is worth mentioning that the existing formulas are far from being capable of predicting and evaluating all the rheological properties of granular paste. All of them were tested to estimate the viscosity of our suspension of known solid volume fraction. The results obtained are shown on Tab.

1 for a solid volume fraction of 0.48. The results obtained, however, differed from the measured results. As the solid volume fraction increased, the difference observed enlarged, showing the inaccuracy of the formulas outside their domain of validity as the particle interactions intensify. The prediction of the viscosity of high concentrated suspension must take into account a lot of parameters and interactions between particles and the fluid.

Materials, set-up and methods

Model substance

Cerium (III) oxalate is a solid currently used in the nuclear industry to model the behavior of reprocessed nuclear fuel. It was purchased from Sigma-Aldrich. It is hydrated, having a chemical formula of C6Ce2O12.xH2O.

The degree of hydration is around 9. This oxalate exhibits a platelets-like shape and the particle size distribution is centered on 38µm and the standard deviation is 28.9µm. The molecular weight of cerium (III) oxalate is 544.29 g/mol on an anhydrous basis and its CAS number is 15750-47-7.

The cake is replaced in the rheometer by a mix between distillated water and dry cerium oxalate. As the cake flows through a worm drive after filtration in our application, the integrity is lost and this is the mix rheology that is important.

Rheometer set-up

All the viscosity measurements were carried out with an Anton Paar MCR 301 rotational rheometer equipped with the Rheoplus32 software. All suspensions were sheared at the given shear rates and the shear stresses were measured.

A rheometer can run with various geometries. Hence, it is important to select the most suitable geometry for cerium (III) oxalate suspension for the most accurate result interpretation. Three geometries were tested and will be discussed hereafter regarding their repeatability and reproducibility: the parallel disk, the concentric cylinder and the vane.

The parallel disk geometry used had a diameter of 25 mm. Both surfaces of the disk were covered by a Metalite K225 P80 sandpaper with a double tape to prevent wall slip. The gap between the two disks was 1 mm.

All measurements were carried out with a logarithmic increase of the shear rate from 1 s -1 to 200 s -1 for 150 s.

The induced shear stresses were measured.

The two concentric cylinders used had inner and outer diameters of 16 mm and 18 mm respectively.

The smallest had a conical bottom shape to limit the edge effect.

A new geometry called vane has been developed and used by many researchers. The rotation of the thin blades around the axis creates a fictive cylinder (see Fig. 3). This geometry may be considered as equivalent to the classic concentric cylinder geometry. It has become popular in recent decades as a simple and effective means to study the rheological properties of any complex fluid [START_REF] Barnes | Rotating Vane Rheometry-A Review[END_REF]. Da Cruz et al., for instance, have used this geometry to study foams and emulsions [START_REF] Cruz | Viscosity bifurcation in granular materials, foams and emulsions[END_REF]. Coussot and al., on the other hand, have used it to measure the viscosity of a bentonite suspension with a solid volume fraction of 0.045 [START_REF] Hu | Rhéologie des bétons fluides[END_REF]. In this study, the geometry had 4 blades and the blades had a diameter RA of 10 mm whereas the diameter of the outer cylinder RB was 13.5 mm.

All measurements were carried out at 25°C by varying the shear rate from 1 s -1 to 200 s -1 logarithmically during 150 s.

Figure 3: Vane geometry scheme

The usual recommendation for a vane-in-cup ratio to neglect wall effect is 3:1 [START_REF] Stickland | The effect of premature wall yield on creep testing of strongly flocculated suspensions[END_REF]. The vane-in-cup used in the experiments had a ratio of 1.3:1. Referring to the literature, wall effect can occur and lead to a more complex measure. In our case, with the ratio and the material used, wall effects such as slip didn't appear and the choice to use a lower ratio than recommended was made.

All the measurements on various geometries were repeated three times to assess their repeatability.

Before each measurement, the samples were stirred in order to limit sedimentation phenomena (see next section). The associated mean viscosity and the standard deviation were calculated.

Operating conditions

The preparation procedure for such solid/liquid media influences the rheological properties. When the sample is pre-sheared during a short period before the measurement, the microstructure of the sample is only partially destroyed, resulting in high viscosity during the measurement and poor measurement reproducibility.

However the sample exhibits low viscosity if it is pre-sheared during a significantly long period before the measurement, since the microstructure of the sample requires a long time to be restored. Several shearing protocols were tested in order to evaluate the influence of a pre-shear followed by a rest time prior to the rheological measurements.

The operating conditions can be summarized in three steps (i) preparation of the slurry (ii) filling the rheometer and applying the shearing protocol (iii) measurement. The slurry preparation consisted in stirring the suspension of cerium (III) oxalate with a known moisture content in demineralized water in a beaker with a magnetic stirrer for 20 minutes. Three shearing protocols were tested: no pre-shear, application of a pre-shear at 50 s -1 for 100 s, and application of a pre-shear at 50 s -1 for 30 s and the sample then left to rest for 30 s.

Rheological measurements were then made at 25°C in triplicate to evaluate the repeatability of the protocol, varying the shear rate from 1 s -1 to 200 s -1 logarithmically for 150 s.

Results and discussion

Choice of the rheometer geometry

The mean values and the standard deviations of the viscosity measurement performed with the three possible geometries are plotted in Fig. 4 for a cake with 30% wt moisture content. All measurements were made with the pre-sheared protocol with rest time described in section 4.2.

Figure 4:

Variation of the viscosity at 30% humidity on parallel disk, concentric cylinder and vane geometries.

The layer of sandpaper used with the parallel disk set-up greatly intensified the frictional interactions between the particles and the sandpaper. However this layer was necessary, given that the wall slip was too significant at 30% moisture content on the parallel disk. Measurement was not possible without it as the suspensions of cerium (III) oxalate began to age and solidify immediately after the sample preparation process, making the disk slide on the hardened solid. In addition, the cake broke easily on the parallel disk geometry, during which the shear rate was not transmitted to the lower layer of the suspension. With the concentric cylinders geometry, the small 1 mm gap between the two cylinders was not completely filled by the paste. This effect was enhanced when the moisture content was below 30% wt. On the contrary, these operating problems were not encountered with the vane geometry, whatever the paste moisture content.

Concentric cylinder

Parallel disk

Vane

The quasi linear evolution of the viscosity with the shear rate in a log-log plot shows that the granular paste had a sheer-thinning behavior. The viscosity measurements performed with the parallel disk geometry were significantly higher, by at least one order of magnitude, than those obtained with the two other geometries.

On the other hand, the values obtained with the two other geometries were in the same range of magnitude, the concentric cylinder values being about 4 times higher than the vane values.

For the viscosity measurement, a variation coefficient (VC) was defined as the ratio of the standard deviation to the mean value for each shear rate. This VC was calculated for measurements with the three geometries. Whatever the geometry, the variation coefficient was higher at low shear rate. The parallel disk gave the largest variation coefficients, from 53% to 121%. The VC values were from 13% up to 60% with the concentric cylinders, and 0.6% up to 15% with the vane geometry. In the field of granular paste rheology, a VC ranging from 10% to 20% is very common due to various interactions between the particles and to geometry defects. [START_REF] Hu | Rhéologie des bétons fluides[END_REF] According to these observations and measurements, accurate reproducibility was obtained when the shear rate applied was above 10 s -1 . This was particularly the case for the vane geometry, which appears to be the best approach for studying granular paste viscosity.

Determination of the sample preparation protocol

The viscosities measured at 18% moisture content on the vane geometry for three different preparation protocols are shown in Fig. 5. A paste moisture content value of 18% was chosen as it gave the highest viscosity. The viscosity measurements were similar and repeatable when the sample preparation protocols included a pre-shear. The application of a rest time did not have an impact. The pre-shear enabled the particle to point in the rotational direction, resulting in almost identical viscosity. The viscosity measured was higher when the sample was not pre-sheared, especially when the shear rate applied was low. When the shear rate was higher than 140 s -1 , identical paste viscosities were obtained for all protocols.

The variation coefficient VC and standard deviation were evaluated for each preparation protocol. At 18% moisture content, the VCs ranged from 9% to 31% without pre-shear, 4% to 26% with pre-shear and 3% to 10% for the pre-shear with rest time protocol, showing that the latter enabled the most repeatable viscosity measurement. It should be pointed out that this is also the most frequently reported by other researchers working with granular pastes [14; 15].

Since the better protocol was determined, a first observation on the slurry can be made. The viscosity profile with the moisture content was similar whatever the shear rate applied. The viscosity decreased with an increase in the shear rate, showing that the cerium (III) oxalate suspension exhibited a shear-thinning behavior.

It could be assumed that this effect was due to a better particle alignment with the shearing rate. In an industrial process, the wet cake is generally transferred from the filter to the dryer typically by a worm drive of 10 up to 20 cm diameter rotating at a speed of approximately 0.5 to 1 rev/s, which is equivalent to a range of shear rate from 50 to 200 s -1 on the vane geometry. The viscosity measured at a shear rate of 96.3 s -1 with the vane geometry was therefore taken to identify the transition moisture content values of the cake, even if the shear rate seemed to have no influence on the mixture behavior.

Fig. 6 shows that a cerium(III) oxalate cake with approximately 35% wt moisture content reached the end of a the viscosity plateau at a shear rate of 96.3 s -1 . Above 35%, the suspension behaved like a liquid with a low viscosity level. Below 35% wt moisture content, the interactions between the particles became more intense due to a higher solid concentration, steadily increasing the viscosity as the moisture content decreased. A sticky paste started to form. At 18% mean moisture content in particular, the value of X1, cerium (III) oxalate has a dual behavior: solid and paste (see Fig. 7). In the hydrated area, a paste can be observed, whereas in the non-hydrated area, solid aggregates can be identified. Consequently this method allows, the determination of the transition value X1 with a few percent accuracy. These values are only relevant for the model substance considered.

Conclusion

Non-hydrated area: Solid aggregates

Hydrated area: Paste

In industry, operations dealing with solids quite often limit plant productivity and necessitate a large labor force in order to ensure sufficient reliability. This study provides a new method to predict the transferability of a wet cake from a filter to a dryer or furnace that can be applied to any solid-liquid system. It was demonstrated on a model system that the moisture content of the wet cake obtained at the end of the filtration must be controlled in order to ensure an easier transfer operation. At the same time this minimizes the loss of valuable product throughout the different parts of the transfer system. With a sticky paste, a difficult transfer operation and significant loss with crusting of solids within the transfer mechanical system may occur.

The method is based on the analysis of wet cake rheological behavior. The first recommendation is to use a rheometer equipped with a vane geometry, since the measurements are sufficiently reproducible with the variation coefficient of 15%. A second recommendation is to define a sample preparation procedure prior to rheological measurement. On the model system, a pre-shear period followed by a rest time was selected. The third recommendation is to select a shear rate representative of that induced by the transfer technology used in the plant (often a worm drive). It is then possible to select two moisture content transition values for the wet cake. Below the first transition, a granular solid behavior was observed here due to the formation of a greater proportion of solid aggregates, resulting in low viscosity and good transferability. Between the two transitions, the cake behaved like a moderately hydrated sticky paste. It showed high viscosity levels due to the numerous interactions between the solid particles. A difficult transfer operation and significant loss with crusting of solid within the transfer mechanical system may occur. Beyond the second transition, the wet cake was sufficiently humid to behave like low viscosity slurry, since the particles were not concentrated enough to be in contact with each other and thus remained separate. For the model system investigated in this study, the two threshold values for moisture content were 18% wt and 35% wt relatively. The first value X1 of 18% wt moisture content is the maximum acceptable after the filtration step to insure the transferability in a solid form through the process. If a liquid form is wanted for the transfer, the value X2 of 35% wt moisture content is a minimum to reach.

Further studies should be carried out with other model systems in order to relate particle features (size distribution, morphology, etc.) to the transition values. A better understanding of relationships between particle characteristics and cake behavior may help in preventing transfer difficulties at plant scale.

Figure 1 :

 1 Figure 1: Different behaviors of cerium (III) oxalate depending on its humidity: (a) solid-like cake, (b) sticky paste, (c) fluid-like suspension

Figure 2 :

 2 Figure 2: Conceptual classification of the rheophysical regimes of a suspension as a function of shear rate and solid fraction on a logarithmic scale. The other characteristics of the suspension are fixed [1].

Figure 5 :

 5 Figure 5: Evolution of the viscosity with the shear rate for an 18% wt humidity paste, measured with vane geometry for three different sample preparation protocols.

Figure 6 :

 6 Figure 6: Evolution of the viscosity at four constant shear rates with different paste humidity levels.

Figure 7 :

 7 Figure 7: Dual behavior of cerium (III) oxalate at 16% humidity: non-hydrated solid aggregates and hydrated paste

Figure 8 :

 8 Figure 8: Formation of solid aggregates at 15% humidity, showing low viscosity

Table 1 .

 1 Comparison between the viscosities calculated by Eqs. (2) and (3) and the measured one.

	Method	Eq. (2)	Eq. (3)	Measurement at 46.4 s -1
	Viscosity [Pa s]	2.2 •10 -3	3.7 •10 -3	1.5

Moisture content transition values of a cerium(III) oxalate cake

All the measurements were made with the vane geometry by using the pre-shear with rest time sample preparation protocol. Fig. 1 presents three water-Cerium Oxalate mix at different moisture content from 15 to 70%. Three different behaviors can be highlighted. The driest cake behaved like a granular solid able to flow. A cake sufficiently humid exhibited a sticky paste behavior, unable to flow. The highest moisture content cake was a suspension with liquid-like appearance that flowed easily.

The evolution of the viscosity of cerium (III) oxalate cake according to its moisture content for four different shear rates is given in Fig. 6. The measurements were done on 12 points from 10 to 70% wt moisture content (10%, 15%, 18%, 20%, 22%, 26%, 30%, 35%, 40%, 50%, 60%, 70%). The different parts of the curves can be linked to the observations made in Fig. 1. Indeed, for moisture content lower than X1, the granular solid behavior implies difficulties to measure viscosity. The curve of the viscosity as a function of moisture content tends to 0 as the moisture content tends to 0, as the viscosity can't be measured precisely. For a wetter cake, the increase in moisture content resulted in a change in appearance and rheology to become a sticky paste. For moisture content X1, which represents the transition between a solid granular and a sticky paste behavior, the viscosity was at its maximum. When moisture content grew until X2, the viscosity decreased and reached a plateau value. For higher moisture content than X2, the cake behaved like a liquid and the more water was added, the lower the viscosity was. Two transition values can be highlighted: X1, value of the solid granular/sticky paste transition which corresponds to the highest value of the curve representing the viscosity as a function of moisture content, and X2, at the end of the plateau and that represents the transition between a sticky paste and a liquid behavior. In the range of investigated operating conditions, the viscosity profile with the moisture content was similar for the different shear rate applied.

List of notations