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Brazilian Journal of Chemical Engineering
Print version ISSN 01046632Online version ISSN 16784383
Braz. J. Chem. Eng. vol.25 no.4 São Paulo Oct./Dec. 2008
http://dx.doi.org/10.1590/S010466322008000400007
FLUID DYNAMICS; HEAT AND MASS TRANSFER; AND OTHER TOPICS
Behavior and fluidization of the cohesive powders: agglomerates sizes approach
D. Turki^{I,} ^{*}; N. Fatah^{II}
^{I}Université IbnKhaldoun de Tiaret, Laboratoire de Chimie et Environnement, BP 78, 14000 Tiaret, Algeria
^{II}Unité de Catalyse et de Chimie du Solide, UMR CNRS 8181, ENSC Lille, bat. C7, 59655, Villeneuve d'Ascq Cedex, France. Email: turkidjamel@yahoo.fr
ABSTRACT
This work focuses on the fluidization of three types of TiO_{2 }powders: Anatase (99% TiO_{2}), Rutile 1 (95% TiO_{2} and 5% Al) and Rutile 2 (96.5% TiO_{2 }and 3.5% Al and Si); the average diameters of the powders are 204 nm, 159 nm and 167 nm, respectively. These powders belong to group C of the Geldart classification and are characterized as cohesive powders with a nonfree flow and a difficult fluidization. The fluidization of the powders was carried out in a glass column of 103 mm inner diameter and 1500 mm height. The experiments and analysis performed included measurements of the physical properties of the powders such as the particle size, density, specific surface area and the flow properties of the powders like the Hausner's index, the angle of repose, the angle of slide, consolidation and shearing (via shear cell testing). The results obtained with the nanometric TiO_{2} powders show a more complex behavior than the micronic powders; with a low strength value (Hausner index, angle of repose and angle of slide), the TiO_{2} powders have a free flow or intermediateflow and a nonfreeflow for higher strength intensities (consolidation and shearing). This behavior is related to the structure of the nanometric particles in the packed bed; the evolution of this structure is made up of individualized and spherical agglomerate shapes and is not perturbed by stresses of low intensities. Indeed, the latter seems to modify the structure of the powder (group C of Geldart classification) to acquire a behavior typical of group A, B or D in the Geldart classification. With high stress values, the individualized agglomerates are disintegrated and the powder is reduced to a more compact structure. The fluidization of TiO_{2} powders seems to evolve in a more homogeneous way than the micronic powders. This behavior is related to the initial structure being made up of stable agglomerates. Thus, this fluidization is made by agglomerates with a gas velocity of 3×10^{6} to 4.6×10^{6} times the gas velocity for fluidizing the primary particles.A numerical approach based on a force balance in agglomerating fluidized beds was developed in order to estimate the agglomerates sizes.
Keywords: Fluidization; Cohesive powder; Agglomerate; Nanometric powder; Interparticle Forces.
INTRODUCTION
In the last years, fluidization has progressed through systematic research, with various specialized functions such as: combustion, mixing, chemical reaction, heat and mass transfer, coating, granulation, encapsulation and CVD (Chemical Vapor Deposition) processes. This research development persuaded the researchers to become interested in the use of fine powders (cohesive powder), micronic and nanometric powders. Many industrial processes require the use of fine powders in the manufacture of catalysts for reactors, ceramics, pharmaceuticals, paints and cosmetics.
According to research performed for nearly fifty years, the fluidization of fine powders (diameter of the primary particles less than 50 µm) is complex. The difficulty of putting these powders in suspension in the fluidizing gas is related to the cohesive structure and, in particular, to the physical forces between the primary particles. In general, the experiments are complex during the handling or fluidizing of these powders. This is due to the unpredictable behaviors of the cohesive powders. This category of powders belongs to group C of Geldart's classification (Geldart, 1973). The complexity of this behavior is due to the small size of the fine particles, i.e., the micronic or nanometric particles. These tend to form agglomerates (selfagglomeration) of completely random size and shape by the action of the interparticle forces between the primary particles. This is related to the high ratio between the surface and the volume of the particles and to the short distance between the particles. This strong interaction between the powders affects the flow properties of the powders.
The smooth fluidization of gassolid particles is the result of equilibrium between the hydrodynamic, gravitational and interparticle forces. However, in the case of cohesive powders, the interparticle forces are considerable and they control the behavior of a bed composed of fine particles. Thus, during fluidization, the bed of powder cracks into large portions and the gas tends to flow into the gap between the fissures. Then, channeling occurs in the bed and, eventually, the gassolid contact is very low and any heat and mass transfer operation is weakened.
The powders are generally presented in the form of heterogeneous structures in particle size, size distribution, morphology (shape) and physical properties (density, specific surface area and porosity). These characteristics are related to the local structure of the particulate system and the type of gas flowing around these particles. In the case of the cohesive powders, this particulate system appears to be very complex and uncontrollable. The interparticle forces are considerable and control the micronic and nanometric particle behavior of the bed. This physical attraction is the effect of the great intensity of internal forces between primary particles. In the case where the particulate system is handled in air and when the size and the distance between the particles are very small, the van der Waals forces are controlling rather than the other interactions forces (capillary and electrostatic forces). The particleparticle interactions are achieved primarily by contact. If the particles are not in contact, no force is exerted between them. The van der Waals forces between two macroscopic bodies are calculated in a different way from the microscopic one. Indeed, the force of interaction between two solids depends on the size, the shape and the roughness of the particles, as well as the distance between the particles. These forces are calculated starting from the Derjaguin approximation (Derjaguin, 1934) and the theory of Lifshitz (Lifshitz, 1956). The estimation of the van der Waals forces is defined as the sum of all interactions between molecules held on the surface of the particles face to face. The magnitude of these forces increases with the size reduction of the particles and becomes dominating compared to the weight of the particles.
In many industrial processes, the cohesive powders are fluidized in the form of agglomerates. For this, it is necessary to apply either a high gas velocity (Fatah and Cavrois, 1998) compared to the velocity necessary for fluidizing primary particles, or a lower velocity but with addition of external energy. (Chirone et al., 1993) drew up a list of techniques used as additional energy (vibration, magnetic pulsation and acoustic energy) to disintegrate the cohesive structure and to facilitate the fluidization of the agglomerates. The mechanism of agglomerate fluidization supposes the passage from group C to a group A powder or even a group B powder of the Geldart classification. (Leu and Huang, 1994) note a clear improvement of the quality of fluidization and expansion of the bed for powders with low particle size under the effect of the acoustic pressure. Indeed, this seems to modify (Russo and al., 1995) the structure of the fluidized bed (group C of Geldart) and to acquire a behavior typical of group A of Geldart. (Nezzal, 1996) describes that the use of mechanical agitation (intrusive method) modifies the hydrodynamic behavior of the bed powder. This provides a better aeration of the bed particles and homogenizes the fluidization in the form of agglomerates. Thus, the size of the agglomerates decreases when the agitating velocity increases.
The aim of this work is to study an important feature of powder structure and the use of different measurement techniques that can help to understand the behavior of fine particles. The experiments and analysis performed included measurements of the physical properties of the powders such as the particle size, density, specific surface area and the flow properties of the powders like the Hausner's index, the angle of repose, the angle of slide, the consolidation and the shearing (via annular shear cell).
The flowability and the physical properties of the powders seem to be an interesting way of describing the hydrodynamic behavior of the very cohesive structure.
The interparticle forces for the cohesive interaction are determined by a force balance (kinetic, van der Waals, collision, gravitational and buoyancy forces). A numerical approach based on a force balance in agglomerating fluidized beds was developed in order to estimate the agglomerate's sizes.
The experimental results indicate that there is a close correlation between the hydrodynamics and the local structure of the powder bed.
MATERIALS AND METHODS
In this work three types of TiO_{2} powders of the Kronos company were studied:

Anatase (99% TiO_{2}): labeled TiO_{2} (A);

Rutile1 (95% TiO_{2 }and ≈ 5% Al): labeled TiO_{2} (R1);

Rutile2 (96.5% TiO_{2} and ≈ 3.5% Al and Si): labeled TiO_{2} (R2).
The powder size distribution was measured with the lightscattering laser instrument (BeckmanCoulter, LS230). This experiment measures the particle size. It should be noted that, due to the complexity of this experiment, each batch of powder was analyzed according to a suitable protocol that gives the primary particle size and not the size of agglomerate. During these experiments, the optimal conditions obtained to reach the lowest size of TiO_{2} particles without degrading the powder structure, are as follows: the cohesive powders were dispersed in demineralised water with addition of 0.5% tetrasodium diphosphate as dispersant in a flask. Before the introduction of this suspension into the laser instrument and for a better dispersion of the primary particles, a manual mixing of TiO_{2} (R1) powders and the use of ultrasound for TiO_{2} (A) and TiO_{2} (R2) was carried out in a flask. The average diameter of the three powders of TiO_{2} was calculated according to the diameter definition "surfacevolume"or Sauter diameter:
where N_{i} is the number of particles in class i and d_{i} the average diameter of the particle in this class.
The range of the particle size distribution was quantified using the index of uniformity C_{u} (Schlosser, 1998) obtained from the cumulative percentage undersize curve (in volume) and defined by the following equation:
C_{u} < 2: The particle size distribution is known as uniform.
C_{u} > 2: The particle size distribution is known as nonuniform or large.
d_{i60% }and d_{i10% }are defined as dimensions of the particles corresponding respectively to 60% and 10% of the particle size distribution of the cumulative curve.
The density (ρ_{p}) and the BET surface area (S_{s}) of powders were, respectively, measured by helium gas pycnometer (Micromeritics, AccuPyc 1330) and nitrogen gas adsorption (Micromeritics, Asap 2000).
MEASUREMENTS OF FLOW PROPERTIES
Measurements were carried out under the same working conditions, i.e., with a relative humidity ranging between 40% and 53% and at ambient temperature. The tests were carried out according to a specific protocol in order to preserve the reproducibility of the results. With regard to the following techniques:

Ensuring that the initial structure of particles is homogeneous during filling. In this test, the loading of powder is carried out with a vibrating sieve of 1 mm in aperture.

Using a smooth and clean cell wall to decrease friction forces and contamination of the powder.

Controlling humidity related to each test, to avoid capillary interactions.
For most measured flow properties, the average of 10 successive measurements was recorded except for the test of shearing, where measurements were repeated three times. Table 1 shows the average flow measurements and the standard deviation representing the average error of these measurements.
Hausner's Index
The Hausner index (HR) is defined as the ratio of bulk density ρ_{t} of fine powders after prolonged tapping to aerated density ρ_{ae}.
HR describes the powder flowability. The various flows can be classified as follows:
HR > 1.4 characterizes cohesive powders (nonfree flowing), HR < 1.2 indicates a free flowing (noncohesive powders) and 1.2 < HR < 1.4 shows an intermediate behavior of the powders. This test measures the reduction of the volume of packed bed of particles. This is carried out in a graduated cylinder of 100 ml. The initial volume of the particles is fixed at 40 ml. Finally, the cylinder is subjected to vertical tapping until no change in the volume related to time is observed. The packed volume of the powders is recorded when that volume remains constant with time. The tapping technique was achieved using two methods: manual (manual tapping of the powder in a testtube) and automatic with an ERWEKA SVM22 device.
Angle of Repose Measurement
The angle of repose (α) was measured by applying the flow procedure through a funnel. This measurement consists of passing a known mass of powder through a funnel with a sieve of 1mm in aperture and 35 mm in diameter, the funnel is submitted to a mechanical vibration to form a uniform powder heap on a glass base of 40 mm in diameter. The funnel is fixed at a distance of 45 mm from the base. The angle of repose is the constant angle formed by the base and the tangent to the heap formed by the powder. The flow is known as free flowing for α < 40° and nonfree for α > 40°.
Angle of Slide Measurement
The device for measurement of the angle of slide (β) is composed of a glass plate 40 mm in width, 145 mm in length and 5 mm thick revolving around a hinge of 0° to 180° from a fixed support (Figure 1). After depositing a uniform particle layer 5 mm thick, gradually we inclined the plate at a practically constant velocity until no particle is on the surface plate (or covered of a monolayer of particles that adhere to the wall). The slide angle is the angle formed by the plate and the fixed support. The flow is known as free for β < 30° and nonfree for β > 30°.
Consolidation Test
The consolidation test shows the aptitude that a powder has to consolidate, i.e., to decrease the volume of the packed bed of powder under the effect of an increasing normal stress (σ). A known mass of powder (M) is introduced into a graduated 5.3 cm diameter cylinder. The base of the cylinder is closed by a porous plate to allow the evacuation of the air stored in the powder during consolidation. A twister is introduced into the cylinder to homogenize the surface of the powder. A glass piston is applied on the powder and the variation of volume is measured according to the force (F) related to the cylinder (Figure 2). The solid fraction (1 ε ) related to the strain s is noted as:
M and V_{t }are the mass and the total volume of the packed bed of particles, respectively.
S is the crosssectional area of the glass piston.
Annular Shearing Cell
The annular shear cell (RSTXS) (Schulze, 1995) is an alternative version of Jenike's cell (Jenike, 1954). The illustration follows the same procedure and theory as Jenike's cell (Jenike, 1954), where the yield locus is obtained starting from at least four characteristic points of rupture for a given consolidation state. With the annular shearing cell, a complete yield locus is usually measured with one sample (in contrast to the Jenike shear tester where only one point can be measured with one sample). This cell is automatic and less time consuming.
The shearing test (annular cell) is composed of an annular channel, carefully filled with the powders. A lid is placed on the annular channel and the unit is then placed on a revolving plate. Then a piston is lowered on the lid. The lid has as the role of transmitting at the normal strain (σ) and the shear stress (τ) simultaneously to the powder. Only the lower plate is in rotation and engages the annular channel (Figure 3). The upper part, i.e., the lid, is maintained fixed using two tie rods transmitting the shearing force to the sensors. The first stage consists of shearing the powder until a stationary state corresponding to a plate marking the flow, then the cell returns to its initial state. In this case the critical point (stationary flow) is reached (σ_{sf},τ_{sf}) in the coordinate system (σ, τ) (Figure 4), i.e., the limiting point of flow. The second point in this same system is obtained by applying σ < σ_{sf} and the powder is sheared until rupture that corresponds to the start of flow, the value of t is noted and the second point (σ, τ) is obtained. The whole procedure is repeated for three or four other points. The curve, called yield locus (Figure 4), is obtained. This illustrates the limit between two zones corresponding to the pressures applied: the zone in the lower part of the yield locus corresponds to an impossible flow of the powders and the zone above the yield locus curve represents the collapse of the powder and flow.
From the yield locus two properties of flow can be obtained:

The cohesion C is the value of the ordinate at the origin (σ =0). This parameter gives the true cohesion between the particles apart from particleparticle friction. For C=0, the powder shows a free flow; for C > 0 the powder is said to have a nonfree flow.

The compressive resistance F_{c} is obtained from Mohr's circle passing through the origin and tangent to the yield locus.

The principal stress σ1 corresponds to the maximum stress that the powder has undergone for the first time. It is obtained from Mohr's circle passing through the point (σ_{sf},τ_{sf}) and tangent to the yield locus.
From the values of σ_{1} and F_{c}, the parameter FF_{c} =σ_{1}/F_{c} can be defined and this characterizes the flow property of the powders according to the following classification:

FF_{c} < 2: very cohesive powder, nonfree flow.

2 < FF_{c} < 4: cohesive powder, difficult flow.

4 < FF_{c} < 10: intermediate flow.

10 < FF_{c}: non cohesive powder and free flow.
EXPERIMENTAL APPARATUS: FLUIDIZATION COLUMN
The tests of fluidization were carried out in a glass column of 103 mm inner diameter and 1500 mm height (Figure 5). Air was used as the fluidizing gas; the flow was controlled by a set of rotameters. The diffusion of air through the bed of particles is ensured by using a porous stainless steel plate (gas distributor) placed at the base of the column. The filtering capacity of the porous plate is 98 % and 99.9 % to stop particles of 1.2 µm and 3.6 µm in size, respectively. The pressure drop (ΔP) through the bed is measured using a pressure probe made up of a stainless steel tube of 1mm inner diameter, immersed in the column and placed just above the distributor. To avoid clogging the pressure probe by fine powders, the end of this tube is protected by a double steel sheet of 50 µm in aperture. The tube is connected to a differential pressure sensor (Societé Sedeme) where the second pressure probe corresponds to atmospheric pressure. The fluctuation of the recorded pressure drop (ΔP) with time and velocity of the gas are achieved using a Sefram recorder. A cyclone is connected to the top of the column to collect the fine particles elutriated during fluidization.
RESULTS AND DISCUSSION
The studies of the cohesive powders (Fatah et al., 1998; Fatah and Sanchez, 2004) reveal the existence of two categories of powders: micronic particles (diameter of the particle varying between a value of ≈ 500 nm and ≈ 50 µm) and nanometric particles (diameter of the particle less than ≈ 500 nm). The nanometric particles are more complex and show a different behavior from the micronic particles under the action of the variation of the external forces. Thus, this present work is concentrated on the study of the behavior of nanometric powders.
The particle size distributions of the three TiO_{2} powders were obtained according to an optimal procedure of dispersion in order to measure the size of the primary particles and not that of agglomerates. The particle size distributions of the TiO_{2} powders are given in Figure 6. The average sizes of the particles are d_{p}=204 nm, d_{p}=159 nm and d_{p}=167 nm for TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2), respectively. The C_{u} index is obtained from the cumulative percentage curve and defined by equation (2). This parameter indicates a broad size distribution corresponding to Cu= 3.6, Cu=3 and Cu= 3 for the three TiO_{2} powders (TiO_{2} (A), TiO_{2 }(R1) and TiO_{2} (R2)), respectively. This large interval is due to the presence of agglomerates, as shown in Figure 6. For the three powders, a first mode towards 225 nm is observed, that seems rather close to the average diameter calculated.
The particle density (ρ_{p}) is 3900 kg/m^{3}, 4200 kg/m^{3} and 4200 kg/m^{3} for TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2), respectively. The BET surface area seems to be identical for the three powders and is about 9 m^{2}/g.
The study of TiO_{2 }powder flows shows a free to intermediary flow under the action of low strength, i.e., in the case of using the automatic tapping test measurement (Hausner's index, angle of repose and angle of slide) (Figures 7, 8 and 9) and Table 1. The values of HR seem to be identical and slightly higher than 1.2. It is the same for the angle of repose and the angle of slide, which are close to 41°51° and 37°43°, respectively. This behavior is related to the initial structure of the nanometric powder in the packed bed that appears to be (visually) made up of stable agglomerates with different diameters and close to a spherical shape. This structure seems to depend on the history of the packed bed (selfagglomeration) and the process of microconsolidation. This phenomenon is due to the great intensity of the interparticle forces and to the very short distance between primary particles. Such a structure generates a number of agglomerates with a large size distribution that can be classified in group A, B or D of Geldart's classification. Finally, this powder system does not seem to evolve under the action of forces with low intensities.
However the application of higher forces by manual tapping test, consolidation test and annular shear cell, indicates a nonfree flow (Figures 7, 10 and 11). The values of HR are much higher than 1.4 (results more coherent for this type of powders) and do not seem to evolve identically for powders when using automatic tapping. It is more advisable to link this test with consolidation and shearing tests to categorize the powders Thus, it is noted that the powders do not present the same degree of cohesion.
Figure 10 shows the variation of the solid fraction (1ε) according to the normal strain (σ) (test of consolidation). In the initial state, a high rate of aeration is noted, indicating a solid fraction of about 15%, 18% and 19% for TiO_{2 }(A), TiO_{2 }(R1) and TiO_{2 }(R2), respectively, and formation of porous agglomerates with an internal porosity ξ. Experimental error for the calculation of the solid fractions came from the estimation of the standard deviation. Thus, values of 0.029, 0.0419 and 0.050 are obtained for TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2) powders, respectively. Then, for a low consolidation, (1ε) increases with σ , according to a slightly linear evolution. This corresponds to the diffusion of air in the structure of the packed bed of particles. For higher values of σ , the transition zone is reached, corresponding to the load held by the solid structure; then packing and rearrangement of the particles occurs. Beyond this zone, the load is spread out on the solid; thus, there is consolidation of the packed bed of powder. Low solid fraction values (in the case of TiO_{2 }(A)) show a very cohesive behavior of the powders. However, TiO_{2 }(R1) powder seems to be less cohesive than TiO_{2 }(A) and TiO_{2 }(R2) powders. In the same way, Figure 11 represents experimental results obtained from the annular shear cell, where the FF_{c} value characterizes the type of flow (deformation of the powder's packed bed) and C indicates cohesion between particles. For all powders, the results indicate a nonfree flow (FF_{c} < 2 and C > 0). The TiO_{2 }(A) and TiO_{2 }(R2) powders show higher degrees of cohesion than TiO_{2 }(R1). These results are in perfect agreement with those obtained with the test of consolidation. However, they seem to be in contradiction with the results obtained when using the manual tapping test, due to the fact that the stress applied with this test still remains low compared to the test of consolidation and shearing. This phenomenon shows that the study of nanometric powders should be carried out with care. Thus, it is sensible to link several tests.
In this work, the TiO_{2 }powder structure loses its individualized character (formation of dispersed agglomerates) under the action of high values of external force. Thus, there is a rupture of agglomerates and the structure changes to a more compact powder system, identical to a dense bulk that makes the handling of these powders very complex. The cohesive character of these powders is detectable only under high external forces that overcome the effect of the internal forces between particles that form agglomerates.
Thus, to fluidize the TiO_{2} powders, an important energy must be applied not only to disintegrate the cohesive structure but also to set in suspension the agglomerates. Referring to these results the TiO_{2} (R1) powder should be fluidized with a gas velocity lower than the two other powders.
The suspension of the TiO_{2} powders was carried out with air and at ambient temperature. The disintegration of the cohesive structure in the fluidized bed was carried out under high gas velocity (≈ 0.5m/s).
During the fluidization of Geldart's group C powders, the following phases often observed can be listed below:
1) For low gas velocity, the fluid crosses the cohesive structure by creating a preferential passage (channeling). The totality of the gas passes through a channel, observed especially for beds that are not deep enough (the static height of the bed of particles lower or equal to the diameter of the column). For deep beds, the channeling is preceded by the appearance of the phenomenon of slugging. This same behavior was observed by (Pacek and Nienow, 1990).
2) For higher gas velocity, the formation of cracks and channels at various orientations is observed and does not seem to be reproducible.
3) A defluidization is noted in the bottom zone of the bed but, beyond this, a pseudofluidization is observed as if the various channels formed acted as gas jets. On top of the jet there is formation of bubbles (behavior similar to a spouting bed) that maintains this zone in fluidization. The bottom zone of the bed seems to be reduced with increasing gas velocity to form a structure made up of agglomerates.
During fluidization of the three types of TiO_{2 }powders, only phase 2 and phase 3 were observed for gas velocities much higher than the necessary fluidization velocity of the primary particles. This high velocity disintegrates the cohesive structure and eases the fluidization of agglomerates.
For the TiO_{2 }powders, the preferential passage of the gas through the bed of particles seems to be nonexistent; this is due to the initial structure of the TiO_{2 }powder bed formed by individualized agglomerates (selfagglomeration). Figure 12 illustrates all the curves related to the variation of the pressure drop (ΔP) according to the gas velocity (U) for the three powders (TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2)).
Fluidization of cohesive powder highlights the suspension of the agglomerates at a very high gas velocity (above the minimum velocity). Thus, the cohesive powders are arranged to form a new structure of agglomerates (made up of primary particles). The apparent minimum fluidization velocity (U_{mfa}) has been determined approximately from the third zone by the standardized method of Richardson (Richardson, 1971), that corresponds to the intersection with the horizontal line (fluidization of agglomerates) and the relative curve of the bed that exhibits the formation of cracks at different orientations. The horizontal line and the relative curve are estimated with a regression analysis to fit approximately the corresponding values. As an example, Figure 13 illustrates the determination of the apparent minimum fluidization velocity (U_{mfa}) of T_{i}O_{2 }(A). Values of 0.176 m/s, 0.11 m/s and 0.175 m/s for the TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2) powders are noted, respectively. The results are compared with those of references to calculate the fluidization index defined as the ratio between U_{mfa }and the fluidization velocity calculated (U_{mfc}) from the (Wen and Yu, 1966) correlation.
where µ_{g} and ρ_{g} indicate the viscosity and the density of the air, respectively.
However, it is important to note that the minimum fluidization velocity (U_{mfc}) is calculated from the physical properties of the primary particles: TiO_{2} (A) (d_{p}=204 nm and ρ_{p}=3900 kg/m^{3}_{)}, TiO_{2} (R1) (d_{p}=159 nm and ρ_{p}=4200 kg/m^{3}) and TiO_{2} (R2) (d_{p}=167 nm and ρ_{p}=4200 kg/m^{3}). That is around 5.2 ´ 10^{8} m/s, 3.4´ 10^{8} m/s and 3.76´ 10^{8} m/s for TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2), respectively. Finally, the index of fluidization (U_{mfa}/U_{mfc}) is about 3.4´ 10^{6}, 3.2´ 10^{6} and 4.6´ 10^{6} for TiO_{2 }(A), TiO_{2} (R1) and TiO_{2} (R2), respectively. These results indicate that it is necessary to inject a gas velocity about 4´ 10^{6} times higher than the velocity (U_{mfc}) necessary to fluidize the primary particles for the three powders. In addition, the dynamic diameter of the agglomerates, d_{pa}, was estimated using equation (6) according to the minimum gas velocity of the experimental fluidization or U_{mfa}. Values of d_{pa}=375 µm, d_{pa}=285 µm and d_{pa}=360 µm were obtained, i.e., 1838, 1792 and 2155 times larger than the diameter of the primary particles (d_{p}) for the three powders TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2), respectively.
Supposing that the agglomerates thus formed are porous, this implies that the density of the agglomerate (ρ_{a}) is less than that (ρ_{p}) of the primary particles. A simple calculation of ρ_{a} (ρ_{a} = ρ_{p} (1ξ)) can be done by estimating the internal porosity (ξ) of the agglomerates (the mean value bed powder porosity between tapped and aerated treatment of the bulk powder bed). Thus, values of 0.8, 0.75 and 0.76 were obtained for TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2)), respectively. The corresponding agglomerate densities (ρ_{a}) for the three powder of TiO_{2} are 780 kg/m^{3}, 1050 kg/m^{3} and 1029 kg/m^{3 }for TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2), respectively. Taking into account the dynamic diameter of the agglomerates (d_{pa}) and the values of ρ_{a,} this hydrodynamic behavior for the three powders can be classified in group B of Geldart's classification.
A model is developed to estimate the agglomerate size in a fluidized bed, taking into account the various forces that control the formation of agglomerates in this bed of Geldart's group C powders, and assuming that cohesiveness is principally due to van der Waals forces rather than the other interaction forces (capillary and electrostatic forces).
The forces acting on an agglomerate in a fluidized bed are determined by the force balance (Zhou and Li, 1999):
Drag force(F_{y})+Collision force(F_{co}) = Gravitational  Buoyancy force(F_{g})+Cohesive force (F_{va}).
The drag force exerted by a fluid with density ρ_{g} and a superficial gas velocity (U) flowing past an isolated agglomerate of diameter (d_{a}) is given by:
where is the projected area of the sphere and C_{d} is the drag coefficient; this coefficient
depends on the Reynolds number (Re). Significant research has been done on flow past fixed and fluidized beds and it is possible to calculate drag forces over agglomerates in the bulk of a uniform suspension of agglomerates with reasonable accuracy. The corresponding drag force according to (Khan and Richardson, 1990) is given by:
where (d_{a}) is the agglomerate diameter and ε_{f} is the fluidized bed voidage.
The collision force between two agglomerates '1' and '2' vertically aligned in a fluidized bed that collide inelastically with a relative velocity (q) is according to (Timoshenko and Goodier, 1970), given by:
where α_{d} is the displacement of the maximum compression and is given by:
The parameters and are expressed by:
and
where m_{1} and m_{2 }are the masses and d_{a1} and d_{a2} are the diameters of agglomerates '1' and '2' respectively, k is a function of Poisson's ratio and Young's modulus. Since the two agglomerates are assumed to have the same properties, then m_{1}=m_{2}=m and k_{1}=k_{2}=k and therefore:
where ρ_{a} is the density of agglomerate estimated by measuring the internal porosity of the agglomerate (mean value powder bed porosity between tapped and aerated powder). The factor k is given by (Timoshenko and Goodier, 1970).
where υ is the Poisson's ratio and E is the Young's modulus.
Substituting equations (12) and (11) into equation (9) we obtain:
If d_{a1} = d_{a2} = d_{a} then
The difference in gravitational and buoyancy forces acting on one agglomerate can be expressed by:
The cohesive force is controlled mainly by the van der Waals force, given by (Derjaguin, 1934) as:
where δ is the distance between the agglomerates' surfaces and A is the Hamaker constant. According to (Israelachvili, 1985), this constant can be evaluated when the medium is a vacuum from:
where h is Plank's constant, B is Boltzmann's constant, T is the absolute temperature, N_{1} is the index of refraction of the particles and ν_{c} is the UV adsorption frequency.
Taking d_{a1}=d_{a2} =d_{a} and substituting these equations into the force balance equation we have:
It has been observed that the significant parameters that disturb the solution of equation (19) are: the density of the agglomerates and the fluid, the relative collision velocity and the distance between the agglomerate surfaces. The solution of equation (19) depends on the exact estimation of the parameters, so some parameters have been experimentally obtained from different powders of TiO_{2}:ε_{f} (fluidized bed porosities), ρ_{a} (densities of agglomerates), U (superficial gas velocities, as explained above). For TiO_{2} powders, the parameters: N_{1}, ε_{1}, ν_{c}, B and h are obtained from (Perry and Green, 1984; Israelachvili, 1985), where h is Plank's constant h=6.626×10^{34 }J.s and B Boltzmann's constant B=1.381×10^{23}J/K. The index of refraction (N_{1}) and the dielectric constant (ε_{1}) of TiO_{2} particles are 2.493 and 4060, respectively. The UV adsorption frequency is ν_{c}=3×10^{15} s^{1}. With an absolute ambient temperature, the Hamaker (A) constant calculated from equation (18) is A=3.97×10^{19 }J. The distance between agglomerate surfaces (δ) is taken as 4×10^{10} m (Krupp, 1967). The parameter k calculated with Poisson's ratio and Young's modulus is k=3.0×10^{06 }Pa^{1 } (Krupp, 1967).
For an estimated porosity ε _{=} _{ε} _{aerated }(ε _{=} _{ε} _{aerated }= 1 ρ_{ae }/ ρ_{p}) of 0.85, 0.82 and 0.81 and an experimental superficial gas velocity (U) of 0.5, 0.21 and 0.39 m/s for TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2), respectively, the corresponding velocities (q) (Horio and Iwadate, 1996) are 0.24 m/s, 0.188 m/s, and 0.218 m/s, respectively.
A successive approximation method was applied to solve equation (19). Diameters of d_{a}=310 µm, d_{a}=352 µm and d_{a}=298 m m were found for the three different powders (TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2)), respectively. The numerical prediction of the agglomerate size (d_{a}) is a factor of 1520, 2213 and 1784 higher than the diameter of the primary particles (d_{p}) for the three powders TiO_{2} (A), TiO_{2} (R1) and TiO_{2} (R2), respectively. These results are in accord with the agglomerates diameters (d_{pa}) estimated from equation (6), with a variation of 17 to 20%.
CONCLUSION
The disintegration of the cohesive structure in a fluidized bed was performed under a high gas velocity up to 4×10^{6} times the velocity necessary for fluidizing the primary particles.
In the fluidized state, the experiments show that the exchange is carried out particularly between gas and the agglomerates formed from primary particles of different size and shape. There is a close correlation between the hydrodynamics and the local structure, controlled primarily by the van der Waals forces. The fluidized bed of agglomerates constitutes a new structure characterized by new physical properties of the agglomerates (ρ_{a}, ξ, d_{a}). Indeed, this new hydrodynamic behavior can be classified in group B of Geldart's classification.
The integration of the flowability and the physical properties of the powders is an interesting way of describing the hydrodynamic behavior of these very cohesive powders.
This work constitutes a fundamental step in comprehending the behavior of cohesive powders in a fixed or fluidized bed. Indeed, these effects present a challenge due to the complexity of the microscopic processes such as the formation of agglomerates and the trajectory of gas within the solid structure.
The flow properties of the powders could not be predicted by only one indicator or inherent parameter describing the powders properties.
The connections of two or several flow tests are enough to predict the flow of powders: free flow, nonfree flow or intermediate flow. However, in some categories of powders such as TiO_{2} (nanometric powder) it is suitable to link all tests for different requests. Indeed, the cohesive character is established only under high external pressures. The distinction between the behavior of micronic and nanometric powders is an interesting area that needs to be further investigated.
NOMENCLATURE
A  Hamaker constant  J 
C  Cohesion of powder  N/m^{2} 
C_{d}  Drag coefficient  () 
C_{u}  Index of uniformity  () 
B  Boltzmann's constant  J/K 
d_{i}  Average diameter in class i  m 
d_{p}  Mean particule diameter  m 
d_{pa}  Dynamic diameter of agglomerate ( calculated by equation 6)  m 
d_{a}  Calculated particle diameter of agglomerate (numerical approach)  m 
E  Young's modulus  Pa 
F  Force  N 
F_{c}  Compressive resistance  N/m^{2} 
F_{co}  Collision forces  N 
FF_{c}  Parameter to characterize the flow properties  () 
F_{g}  Gravitational force  N 
F_{va}  Cohesive force  N 
F_{y}  Drag force  N 
g  Gravitational acceleration  m.s^{2} 
HR  Hausner's index  () 
h  Planck's constant  J.s 
K  Function of Poisson's ratio and Young's modulus  Pa^{1} 
M  Mass of the packed bed of particles  kg 
m_{i}  Mass of agglomerate i (i = 1, 2)  kg 
n  Function defined by equation 11  () 
n_{1}  Function defined by equation 11a  () 
Ni  Number of particles in class i  () 
N_{1}  Index of refraction of particles  () 
ΔP  Total pressure drop in fluidized bed  Pa 
q  Relative velocity of agglomerate  m/s 
S  Crosssectional area of the glass piston  m^{2} 
S_{s}  Specific surface area  m^{2}/g 
T  Temperature  K 
U  Superficial gas velocity  m/s 
U_{mfa}  Minimum apparent fluidization velocity  m/s 
U_{mfc}  Minimum fluidization velocity calculated by equation (6)  m/s 
V_{t}  Total volume of the packed bed of particles  m^{3} 
z  Projected area of a sphere  m^{2} 
Greek Letters
α  Angle of reposev  (°) 
α_{d}  Maximum compression displacement  m 
β  Angle of slide  (°) 
δ  Distance between agglomerates or particles  m 
ε  Bed voidage  () 
ε_{1}  Dielectric constant of particles  
ε_{aerated}  Aerated porosity of the bed of particles  () 
ε_{f}  Fluidized bed voidage  () 
µ_{g}  Gas viscosity  kg.m^{1}.s^{1} 
ρ_{a}  Agglomerate density  kg.m^{3} 
ρ_{ae}  Aerated powder bed density  kg.m^{3} 
ρ_{g}  Gas density  kg.m^{3} 
ρ_{p}  Particle density  kg.m^{3} 
ρ_{t}  Tapped density  kg.m^{3} 
σ  Normal stress  Pa 
σ_{sf}  Normal stress (stationary flow)  Pa 
σ_{1}  Principal stress  Pa 
τ  Shear stress  Pa 
τ_{sf}  Shear stress (stationary flow)  Pa 
ν_{c}  UV absorption frequency  s^{1} 
ξ  Internal porosity  () 
υ  Poisson's ratio  () 
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(Received: August 01, 2007 ; Accepted: November 01, 2007)
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