Aggregation and sedimentation in gas-fluidized beds of cohesive powders

Aggregation and sedimentation in gas-fluidized beds of cohesive powders A. Castellanos, J. M. Valverde, and M. A. S. Quintanilla Departamento Electronica y Electromagnetismo, Universidad de Sevilla, Avenida Reina Mercedes s/n, 41012 Sevilla, Spain 共Received 12 March 2001; revised manuscript received 14 May 2001; published 24 September 2001兲 We present measurements on the settling velocity of gas-fluidized beds of fine cohesive powders. In the solidlike regime 共solid volume fraction ␾ ⬎ ␾ c) particles are static, sustained by enduring contacts. The settling is hindered by interparticle contacts and is a very slow process. In the fluidlike regime ( ␾ ⬍ ␾ c) permanent contacts no longer exist, and the bed displays a diffusive dynamics. The interparticle adhesive force leads to the formation of particle aggregates, and for this reason the sedimentation velocity exceeds the predicted value by empirical or theoretical laws on the settling of individual particles. We use an extension of the Richardson-Zaki empirical law for the settling of aggregates in the fluidlike regime to fit the experimental data. Aggregates are characterized by the number of aggregated particles Nand by an effective radius R. The trend followed by these parameters with particle size is confirmed by direct visualization of the aggregates, and shows that cohesive effects become important when the adhesion force between particles is above particle weight. Results show that aggregates form open structures with a fractal dimension close to the predicted one in the diffusionlimited-aggregation model (D⫽2.5). DOI: 10.1103/PhysRevE.64.041304 PACS number共s兲: 47.70.⫺n, 47.55.Kf, 47.55.Mh I. INTRODUCTION Gas-fluidized beds are widely used in industry, and their properties have been the subject of research for some decades. An important issue is to understand the settling rate of the particles when the fluidizing gas is suddenly stopped. The problem is analogous to that of the sedimentation of suspensions of solid particles in a liquid, which, in spite of having a long and well-deserved history for its ubiquitous applications 关1兴, is not yet well understood. For a multiparticle system at low solid volume fractions ( ␾ ), long ranged hydrodynamic interactions hinder sedimentation 关2兴.At higher values of ␾ the presence of a fluid backflow arising from the sedimenting particles, interparticle collisions, and crowding become increasingly important 关3兴. The relationship between the settling velocity in sedimentation vsand the solid volume fraction ␾ for a homogeneous gas-fluidized powder of noncohesive spheres was well described by the phenomenological Richardson-Zaki law 关4兴 vs vp0 ⫽共1⫺ ␾ 兲n.共1兲 Here vp0is the Stokes settling velocity of a single particle, vp0⫽(2/9)( ␳ p⫺ ␳ f)a2g/ ␮ in the laminar regime, where ␳ p is the particle density, ␳ fis the gas density, gis the gravity field, ais the particle radius, ␮ is the viscosity of the gas, and nis a parameter of order 5 in the small Reynolds number limit. Buscall et al. 关5兴reported on measurements of the mean fall velocity of sedimenting polystyrene spheres in water in a small Reynolds number range over the whole range of values of ␾ which fitted to the Richardson-Zaki law for n⫽5.5. Theoretical considerations by Batchelor 关6兴led to vs/vp0⯝1⫺5.6 ␾ 关7兴for very diluted suspensions of nonBrownian monodisperse hard spheres in the limit of small Reynolds numbers. Thus the sedimentation velocity predicted by Eq. 共1兲conformed with the dilute limit of Batchelor for n⫽5.6. The simplicity of the Richardson-Zaki equation, although satisfying from a practical point of view, lacked any theoretical justification at high values of ␾ . Mills and Snabre 关8兴theoretically derived the settling velocity for a system of concentrated noncohesive spheres from the energy balance during the sedimentation process. They used mean field and scaling arguments to arrive at the expression vs vp0 ⫽1⫺ ␾ 1⫹k ␾ /共1⫺ ␾ 兲3,共2兲 and reported on excellent agreement with experimental data by choosing the parameter k⫽4.6 to match Eq. 共2兲in the diluted limit 关vs/vp0⫽1⫺(k⫹1) ␾ 兴to Batchelor’s result. Equation 共2兲yields, in the relevant range ␾ ⬍⬃0.6, results almost identical to those of the empirical Richardson-Zaki equation. When the particle size is smaller than some critical value, the gravity is dominated by interparticle forces. This makes an understanding of the settling process more complicated 关22兴. In situations dealing with the flow of the bulk powder, it is generally observed that the tendency of particles to aggregate leads to a substantial departure from the behavior of free-flowing particulate systems 关9–11兴. From numerical simulations and experiments Nase et al. 关11兴found that the static angle of repose of a granular heap did not depend on the particle size if the particle weight Wexceeded the interparticle adhesion force Ft, but increased markedly as Ft/W⬅Bogwas increased above 1. As with the static heap angle, a sharp decrease was observed to occur in the discharge rate through a hopper when Bogwas increased above 1. In that work cohesion arose from capillary forces, and the nondimensonal number Bogwas reminiscent of the Bond number in fluid mechanics. For this reason it was referred to as the granular Bond number 关11兴. In the case of dry fine particles 共particle size ⬍⬃30 ␮ m) the van der Waals attractive force is dominant and may exceed the particle weight 关14,15兴by several orders of magnitude. Numerical simulations and experiments 关12兴revealed the important role of the ratio of this force to particle weight 共for which, hereafter, we PHYSICAL REVIEW E, VOLUME 64, 041304 1063-651X/2001/64共4兲/041304共7兲/$20.00 ©2001 The American Physical Society64 041304-1 keep the denomination of the granular Bond number兲on the packing of fine particles. The solid volume fraction of the packing was shown to decrease as Bogwas increased above 1. However, the decrease of ␾ became significant only when Bog⭓10. For Bog⭐10, the value ␾ closely approached 0.64, which is the limit of random close packing of noncohesive spheres 关16兴. The agglomeration of fine particles in fluidized beds due to attractive van der Waals force is a well known phenomenon 关17–19兴, but the effect of aggregation on sedimentation has scarcely been addressed in the literature. Lettieri et al. 关13兴reported on deviations from the Richardson-Zaki law in the settling of fcc catalysts powders 共average particle size 50 ␮ m) when contact forces between particles were increased by the presence of potassium acetate. Allain et al. 关20,21兴showed that the settling of a fractured gelled suspension of calcium carbonate particles 共particle diameter 0.07 ␮ m) is well described by a modified Richardson Zaki correlation law vs⫽0.33(1⫺ ␾ / ␾ *)5.1⫾0.4, where ␾ *is the particle volume fraction inside the sediment. Elsewhere we proposed 关22兴a similar equation to account for the effect of aggregation in gas-fluidized beds. Assuming that the aggregates behave like hard spheres with a hydrodynamic radius equal to their radius of gyration R, we arrived at vs vp0 ⫽N ␬ 冉 1⫺ ␾␬ 3 N 冊 n ,共3兲 where Nis the number of aggregated particles and ␬ ⬅R/a. Equation 共3兲is valid as long as aggregates can be considered as individuals. We shared elsewhere 关23兴that a stable gas-fluidized bed presents two distinct regimes of dynamical behavior. For solid volume fractions above a certain critical value ␾ cthe particles in the fluidized bed remain static, sustained in part by enduring contacts. The bed then behaves as a weak solid 关24兴. For example, when it is tilted the free surface remains stable because of the existence of a certain mechanical strength. In the second regime, for ␾ ⬍ ␾ c, permanent contacts no longer exist, the mechanical strength of the bed vanishes, and stresses are carried by interstitial gas and aggregate collisions due to fluctuations in their motion. The powder then takes on many of the properties of a fluid, its upper surface remaining horizontal when the container is tilted. The boundary between both regimes can be found by measuring the tensile strength of the fluidized powder, that becomes zero when the bed enters the fluidlike regime 关23兴. We will report measurements of the settling velocity on a set of cohesive powders differing in particle surface structure and particle size. The aggregation of these cohesive particles is investigated through the coupling between aggregation and sedimentation. Fitting the experimental results to Eq. 共3兲,we will obtain the number of particles per aggregate as well as the aggregate size as a function of the granular Bond number. We will show that at a critical value of the solid volume fraction the settling rate slows down because the bed enters a solidlike regime where enduring interparticle contacts hinder sedimentation. Thus our experiments will also serve to determine the transition from fluidlike to solidlike behavior. II. MATERIALS The powders used in this study were made from the random copolymer styrene n-butylmethacrylate 共particle density 1135 kg/m3兲. The particles were produced by an attrition process followed by size classification using a cyclone separator apparatus. By this technique four sets of powders were produced whose particles are of volume average diameters 19.1, 15.4, 11.8, and 7.8 ␮ m. Particles are irregularly shaped, since they are formed by a grinding process and are rather monodisperse 共the typical standard deviation from the average is 1 ␮ m). These powders were blended with a surface additive such as AerosilR共nanoparticles of fumed silica兲, which is well known for its ability to control interparticle cohesive forces 关14兴. The present paper includes powders with additive concentrations of 8% and 32% surface area coverage 共SAC兲, making a matrix of eight powders covering a range of four particle sizes and two levels of cohesivity. Measurements on the tensile strength of these powders are reported elsewhere 关25,26兴. In Fig. 1 we present experimental results of the tensile strength versus the solid volume fraction for a set of powders used in the present study. A marked trend of the packing fraction with particle size is found. Powders that flow well 共higher particle size兲pack in closer structures, while as the particle size decreases the powder flowability decreases and particles pack in very open structures. The effect of the fumed silica on the packing is also clearly seen in Fig. 1. Increasing SAC causes a better flow of the powder, and results in a closer packing. The effect of the additive is more pronounced for finer powders 关26兴, since in this case the bare particles are very cohesive, i.e., the interparticle van der Waals force is much larger than the weight of the particles. From the measured macroscopic stresses we have estimated 关14,26兴the interparticle contact forces under the assumption of a uniform distribution of stresses inside the material. Results of the estimated adhesion force per contact Ft, as a function of the estimated load force per contact Fcin the range of low load forces, are drawn in Fig. 2. In the range of low stresses, Ftdoes not depend on particle size, implying that only stresses in the contact region are important for the contact forces. The addition of fumed silica nanoparticles results in an increase of FIG. 1. Tensile strength measured by a powder bed technique 关25兴for a set of toners used as a function of the solid volume fraction 共particle size is indicated in ␮ m). Open symbols correspond to 32% SAC and solid symbols to 8% SAC. The estimated vertical error bar is 2 Pa. CASTELLANOS, VALVERDE, AND QUINTANILLA PHYSICAL REVIEW E 64 041304 041304-2 the contact hardness, because the particles are made of a harder material than polymer. The increase in hardness cause a significant reduction of the slope Ft⫺Fc共see Fig. 2兲,as predicted by theory 关26兴. The additive could also play a role in the fluidized regime, where the load force on the contacts due to gravity is removed by the gas flow. The reason for this is the reduction of the size of the contacts by the increase of surface rugosity. However, as can be observed from Fig. 2, the reduction in the adhesion force extrapolated to Fc⫽0, due to the increase of surface silica coverage, is quite small 共the measured reduction is of the order of the error bar of the measurements兲. From this result it can be anticipated that the initial settling velocity of the fluidized bed will not be very sensitive to the surface properties of our dry fine powders. This will be confirmed by our experimental work. For comparison, spherical soda lime glass beads 共particle size ⬃40⫾10 ␮ m and density 2500 kg/m3) were also tested. In spite of their large density these particles could be fluidized. Larger glass beads, although available, did not fluidize homogeneously 关27兴, and their settling velocity could not be measured. III. EXPERIMENTAL SETUP The experimental setup is shown in Fig. 3. The powder sample is held in a 4.45-cm diameter, 17-cm height cylinder made of polycarbonate, the base of which is a sintered metal filter of 5- ␮ m pore size, and of 1.5-mm width. A dry nitrogen tank, furnished with a mass flow controller, supplies an adjustable gas flow. By using dry nitrogen, the effects of moisture on the interparticle adhesion are minimized. All the measurements start by driving the powder into the freely bubbling phase. Once the powder is in a bubbling phase the fluidizing gas is set down to a given value below the minimum bubbling point, and the powder settles to a reproducible and stable fluidized state. The solid volume fraction of the system is conveniently varied by controlling the gas flow rate. The bed height is read from an ultrasonic sensor, and from these data the solid volume fraction is calculated. Once the powder reaches a steady state for a given fluidization velocity, the gas flow is abruptly reduced to zero and the powder settles down. During the sedimentation process the bed height is continuously acquired by the ultrasonic sensor. Figure 4 displays a typical sedimentation curve. The head of the bed falls initially at a constant velocity, as long as aggregates settle individually. However, when the bed reaches a critical height 共indicated by the arrow in the plot兲, the settling process is hindered by stresses carried by enduring interparticle contacts. At this point the bed enters the solidlike regime. The initial settling velocity of the head of the bed vs is taken as the initial slope of the sedimentation curve. As we showed elsewhere 关22兴we find that vsis approximately equal to the gas velocity vg. This should be expected, since the particles in the fluidization phase are moving down from the gas frame at a velocity equal to the superficial gas velocity. Hence when the gas supply is suddenly stopped, the initial settling velocity of the particles must coincide with the gas velocity. IV. EXPERIMENTAL RESULTS A. Effect of surface additives In Fig. 5 results of the initial settling velocity are plotted against the initial solid volume fraction of the fluidized powFIG. 2. Estimated adhesion force per contact Ftas a function of the estimated load for per contact Fcfrom measured bulk stresses 关14,26兴共particle size is indicated in ␮ m and the surface silica coverage is indicated for each group of data兲. The estimated error bar is 1nN. FIG. 3. Experimental setup. Immediately after the gas flow is turned off the bed height is continuously acquired by the ultrasonic sensor placed at the top of the bed. FIG. 4. Powder bed height 共7.8- ␮ m particle size powder, 32% SAC兲as a function of time after the gas supply is suddenly stopped for an initial superficial gas velocity of 0.56 cm/s. The height is measured by means of an ultrasonic device, having a typical indeterminacy of 0.05 cm. AGGREGATIONANDSEDIMENTATIONINGAS-... PHYSICAL REVIEW E 64 041304 041304-3 der. Results for powders with the same particle size (11.8 ␮ m) but different values of surface additive coverage are compared. It is observed that the settling velocity increases slightly when the SAC value is reduced from 32% to 8%. The same is obtained for other powders of different particle size when the SAC value is reduced. This result is consistent with the formation of aggregates due to interparticle attractive forces. As the additive surface coverage is decreased, the interparticle adhesion force increases and particles tend to be more aggregated. Larger aggregates would give larger values of the settling velocity, as we obtain experimentally. However, the increase of interparticle adhesion in the absence of an external load 共fluidized state兲is very small 共see Fig. 2兲. Accordingly, the effect on the increase of the settling velocity is not pronounced. B. Effect of particle size In Fig. 6 the initial settling velocities are represented against the solid volume fraction of the fluidized powder for the set of powders with 32% SAC and different particle size. We observe that for a given value of ␾ the initial settling velocity increases markedly as the particle size is increased. The predicted curves from Eq. 共1兲for the settling of individual particles are also plotted in Fig. 6. Equation 共1兲underestimates experimental results for ␾ ⬍ ␾ c, and overestimates them for ␾ ⬎ ␾ c. Table I shows the values of ␾ cas a function of particle size. In a single settling experiment we observe that the sedimentation velocity decreases at ␾ ⫽ ␾ c. Thus at ␾ ⫽ ␾ cthe transition form fluidlike to solidlike behavior is expected. For ␾ ⬎ ␾ cthe bed would have a mechanical strength caused by permanent particle contacts that hinder the sedimentation process. Conversely, for ␾ ⬍ ␾ c, particles aggregate and display a collisional dynamics. The effect of aggregation in this regime is an increase of the settling velocity over the predicted value by Eq. 共1兲, where particles are considered free floating individuals. Although the measurements of the tensile strength ␴ tnear the solidfluid transition are quite difficult, the extrapolation of our experimental results 共Fig. 1兲to ␴ t⫽0 gives a similar trend of the critical solid volume fraction ␾ cwith particle size 共see Table I兲. Elsewhere we reported 关22兴results on the settling velocity obtained for the commercial toner Canon CLC700 共particle size 8.53⫾2.32 ␮ m). According to that work the value of ␾ cfor the powder would be ␾ c⯝0.28, which matches the value obtained from independent measurements on the diffusion coefficient, tensile strength, and maximum stable angle of inclination 关23兴. The best fit curves from Eq. 共3兲, where the aggregation of particles is taken into account, to the experimental results in the fluidlike regime are also plotted in Fig. 6. Table II shows the best fitting parameters to the experimental data. As should be expected, the number of aggregated particles N and the ratio of the aggregate radius to particle radius ␬ ⫽R/aincrease as the particle size is decreased. Interestingly, Rshows a weak dependence on the particle size, being around 40 ␮ m for all powders investigated. The scaling of ␬ with the number of aggregated particles can be used to determine the fractal dimension Dof the aggregates, FIG. 5. Initial settling velocity of the head of the powder bed as a function of the solid volume fraction of the powder before the gas supply was stopped for powders of the same particle size (11.8 ␮ m) but different surface additive coverage. The estimated error bar is of the order of symbol size. FIG. 6. Initial settling velocity of the head of the powder bed as a function of the solid volume fraction of the powder before the gas supply was stopped for the set of powders used 共particle size is indicated in ␮ m). Predicted curves from Eq. 共1兲共---兲and best fit curves from Eq. 共3兲共—兲in the fluidlike regime are also shown. The estimated error bar is of the order of symbol size. TABLE I. Critical solid volume fraction predicted from the settling experiments for the transition from the fluidlike to solidlike fluidization regime as a function of the particle diameter (2a) for 32% SAC. 2a( ␮ m) ␾ c 7.8 0.22 11.8 0.26 15.4 0.28 19.1 0.29 TABLE II. Average number of aggregated particles N, ratio of aggregate radius Rto particle radius aand fractal dimension D ⫽ln N/ln(R/a) for particle aggregates 共32% SAC兲. 2a( ␮ m) NR/aD 7.8 63 5.218 2.523 11.8 23.7 3.549 2.512 15.4 12.4 2.724 2.499 19.1 9.6 2.448 2.508 CASTELLANOS, VALVERDE, AND QUINTANILLA PHYSICAL REVIEW E 64 041304 041304-4 D⫽lnN/ln ␬ . We obtain D⯝2.5 independently of the particle size 共see Table II兲. This is approximately the value of D given by the diffusion-limited aggregation model introduced by Witten and Sander 关28兴in three-dimensional systems, where self-similar fractal ramified patterns grow via irreversible sticking of particles in a random motion. Indeed, fine particles display a diffusive dynamics in the fluidized bed 关23兴, and stick to aggregates due to the domination of interparticle adhesion force over particle weight. Thus aggregates should appear as branched structures whose growth is controlled by a diffusive process. C. Visualization of aggregates To further investigate the aggregation properties of fine particles, we have visualized the structure of the fluidized bed. We made use of the great tendency of powder particles to adhere to a paper surface. A paper card was carefully lowered edgewise into the fluidized bed, so as to cleave it perpendicular to the gas flow path. Then, on carefully and slowly withdrawing the card, it was found to carry a replica of the fluidized powder. Although this technique introduces an unavoidable perturbation, we expect it to be small due to the low values of the extracting velocity and solid volume fraction. Furthermore, it has been proven elsewhere 关10,22兴 to be very effective in yielding a good replica of the powder. Figures 7共a兲and 7共b兲show sample pictures from the optical microscope. As anticipated, we observe that particles are aggregated, forming branched structures. Furthermore, in the case of the larger particles, we can distinguish that aggregates are formed by a number of the order of ten particles, which agrees with our estimation from the settling experiment 共see Table II兲. It is also apparent from Fig. 7共b兲that when the particle size is reduced the number of aggregated particles increases. V. DISCUSSION Cohesion between particles may arise from a variety of sources. Dispersion or van der Walls forces, electrostatic forces, and capillary forces are regarded as the main forces in particle adhesion. We have minimized capillary and electrostatic forces, working with uncharged particles in an ambient of dry nitrogen gas. Consequently, van der Waals forces are dominant between our dry fine particles. These forces have their origin in the interaction of molecules through fluctuating dipole-dipole fields. Summing up all the interactions between two spherical and rigid particles in contact with diameters d1and d2leads to an attractive force 关29兴 FvdW⫽A 12z0 2 冉 d1d2 d1⫹d2 冊 ,共4兲 where Ais the Hamaker constant and z0is the distance of closest approach between two molecules which is about 4Å关30兴. At very small loads, the deformation of the contact is negligible, and Eq. 共4兲can be taken approximately as the interparticle adhesion force. Most powder particles have a rather rough surface with many asperities 关31兴of typical sizes often not larger than 0.2 ␮ m关32兴. Thus the effective contact area is given by the asperities at contact, and the interparticle attractive force can be approximately obtained 关33兴inserting the typical diameter of the asperities dain Eq. 共4兲. One of the reasons for which silica additive is used in xerographic toners is to increase surface rugosity, thus decreasing the typical size of the asperities at contact. From typical scanning electron microscopy micrographs of xerographic toner particles, it is observed that the additive particles are distributed in agglomerates with estimated diameters of 40⫾5nm关34兴. If the surface coverage of additives is 32%, we would expect that a large part of contacts would be silica to silica. Thus we use the average diameter of silica aggregates as the diameter of the surface asperities, and A ⬇1.5⫻10⫺19 J for silica 关35兴is inserted into Eq. 共4兲. On the other hand, for only a 8% of surface coverage of silica the contacts are most likely to occur between the polymer surfaces of the particles. In such a case A⬇0.65⫻10⫺19 J关35兴, and we will take da⯝0.2 ␮ m. According to these numbers, Eq. 共4兲yields FvdW⯝1.6 nN for 32% SAC, and FvdW ⯝3.4 nN for 8% SAC. Note that these results agree, within experimental scatter, with extrapolation to zero load of the estimated interparticle adhesion force from the measured bulk stresses 共see Fig. 2兲. Thus the change in the interparticle adhesion force at zero load 共fluidized state兲, when the surface additive coverage is reduced, is quite small. As long as the FIG. 7. 共a兲Picture taken from the optical microscope of particle aggregates of a gas-fluidized bed of particles of size 19.1 ␮ m, 32% SAC. 共b兲Same as 共a兲, but for particles of size 7.8 ␮ m. AGGREGATIONANDSEDIMENTATIONINGAS-... PHYSICAL REVIEW E 64 041304 041304-5 particle size is held constant, the decrease in the Bognumber is also small. Thus the average number of aggregated particles is expected to increase only slightly. This would explain why the effect of reducing the SAC value on the initial settling velocity of the powder is not marked 共Fig. 5兲. Indeed the main effect on the settling velocity is caused by a change in particle size 共Fig. 6兲. In Fig. 8, the calculated Bog, assuming contact between silica aggregates, is shown as a function of particle diameter. As can be seen, Bogis very sensitive to the particle size, increasing in more than two orders of magnitude when particle size is decreased from 19.1 ␮ m(Bo g⫽39) to 7.8 ␮ m(Bo g⫽5700). Thus the average number of aggregated particles should increase markedly as the particle size is reduced. In Fig. 9 we represent the fitting parameters Nand ␬ ⫽R/aas a function of the granular Bond number. Results for Nand ␬ for commercial toner Canon CLC700, reported elsewhere 关22兴, are also plotted in Fig. 9. We see from this figure that Nand ␬ increase following a potential law as the granular Bond number increases. Extrapolating these results we would predict that N⯝1 and R/a⯝1 for Bog⯝1. To make clear the fundamental role of particle size in the aggregation of gas-fluidized particles we have carried out the settling experiment with dry glass beads 共particle size ⬃40 ⫾10 ␮ m and density 2500 kg/m3). Taking A⯝10⫺19 J and da⯝0.2 ␮ m, the granular Bond number is ⬃6, and therefore we should expect almost no aggregation. Accordingly their settling velocity should be well described by the Richardson-Zaki law for individual particles. The measured solid volume fraction of the settled beads is ␾ ⯝0.51. The bed was fluidized and expanded up to ␾ ⯝0.44⫾0.01. Then the gas flow was suddenly stopped. The average value of the ratio of the measured sedimentation velocity to the settling velocity of an individual particle was vs/vp0⯝0.037 ⫾0.005. On the other hand, the predicted value by the Richardson-Zaki law for noncohesive spherical particles 关Eq. 共1兲兴 is vs/vp0⯝(1⫺0.44)5.6⯝0.039, which is in agreement with the experimental result. In sum these results suggest a change of behavior, from free-flowing to aggregated particulate systems, for Bog⬎ ⬃1, much in the same way as already found when cohesion was originated from capillary forces 关11兴. Our results are also in agreement with computer simulation results that predict a decrease in the packing fraction for Bog⬎⬃1关12兴. VI. CONCLUSION The strong interparticle van der Waals force existing between fine particles induce aggregation, and plays a fundamental role in the behavior of gas-fluidized beds of fine powders. This force depends on the surface structure of the particles, i.e., surface density and size distribution of asperities. We have used a set of fine powders differing in particle size and surface additive coverage. Settling experiments have served to identify the transition from a fluidlike to a solidlike fluidized regime at a given value of the solid volume fraction. This transition has been confirmed by measurements of the mechanical strength and diffusivity of the bed in the fluidized state. Our experimental results on the collapse of gas-fluidized beds in the fluid-like regime can be explained by a modified Richardson-Zaki law that takes into account the agglomeration of fine particles. In this way we have determined the average number of particles per aggregate Nand the ratio of the aggregate radius to particle radius ␬ , which result to be directly linked to the granular Bondnumber 共the ratio of the interparticle adhesive force to the particle weight兲. The trend followed by the fitting parameters Nand R/ahas been confirmed by direct visualization of the aggregates and suggests that particles aggregate when Bog ⬎⬃1. Aggregates are found to have a fractal structure that can be approximately described by the DLA model. ACKNOWLEDGMENTS This research was supported by the Xerox Foundation, Spanish Government Agency Ministerio de Ciencia y Tecnologia 共DGES兲under Contract No. BMF2000-1056, and NATO Grant LINKAGE PST.CLG.976575. FIG. 8. Theoretical granular Bond number calculated as the ratio of interparticle attractive van der Waals force 关Eq. 共4兲with A ⫽1.5⫻10⫺19 J, d1⫽d2⫽0.05 ␮ m, and z0⫽4Å兴to particle weight 共assuming perfect spherical particles兲, as a function of particle diameter. FIG. 9. Number of aggregated particles (N:〫) and ratio of aggregate size to particle size (R/a:䊐) calculated from the fit of Eq. 共3兲to experimental data 共Fig. 6兲as a function of the granular Bond number of toner particles. Results from experiments on toner Canon CLC700 (N:䉭and R/a:䊊) reported elsewhere 关22兴are also shown. CASTELLANOS, VALVERDE, AND QUINTANILLA PHYSICAL REVIEW E 64 041304 041304-6 关1兴W. Russel, D. Saville, and W. Schowalter, Colloidal Dispersions 共Cambridge University Press, Cambridge, 1995兲. 关2兴A. J. C. Ladd, Phys. Fluids 9, 491 共1997兲. 关3兴P. N. Segre, F. Liu, P. Umbanhowar, and D. A. Weitz, Nature 共London兲409, 594 共2001兲; P. N. Segre, E. Herbolzheimer, and P. M. Chaikin, Phys. Rev. Lett. 79, 2574 共1997兲. 关4兴J. F. Richardson and W. N. Zaki, Trans. Inst. Chem. Eng. 32, 35 共1954兲. 关5兴R. Buscall, J. W. Godwin, R. H. Otewill, and T. F. Tadros, J. Colloid Interface Sci. 85,78共1982兲. 关6兴G. K. Batchelor, J. Fluid Mech. 119, 379 共1982兲. 关7兴G. K. Batchelor and C.-S. Wen, J. Fluid Mech. 124, 495 共1982兲. 关8兴P. Mills and P. Snabre, Europhys. Lett. 25, 651 共1994兲. 关9兴A. Castellanos, J. M. Valverde, A. T. Perez, A. Ramos, and P. K. Watson, Phys. Rev. Lett. 82, 1156 共1999兲. 关10兴J. M. Valverde, A. Castellanos, A. Ramos, and P. K. Watson, Phys. Rev. E 62, 6851 共2000兲. 关11兴S. T. Nase, W. L. Vargas, A. A. Abatan, and J. J. McCarthy, Powder Technol. 116, 214 共2001兲. 关12兴R. Y. Yang, R. P. Zou, and A. B. Yu, Phys. Rev. E 62, 3900 共2000兲. 关13兴P. Lettieri, J. G. Yates, and D. Newton, Powder Technol. 110, 117 共2000兲. 关14兴J. M. Valverde, A. Ramos, A. Castellanos, and P. K. Watson, Powder Technol. 97, 237 共1998兲. 关15兴J. P. K. Seville, C. D. Willett, and P. C. Knight, Powder Technol. 113, 261 共2000兲. 关16兴J. Berryman, Phys. Rev. A 27, 1053 共1983兲. 关17兴N. Hawley, J. Geophys. Res. 87, 9489 共1982兲. 关18兴A. F. Fortes, P. Caldas, and J. V. Gallo, Powder Technol. 98, 201 共1998兲. 关19兴T. Zhou and L. Hongzhong, Powder Technol. 101,57共1999兲. 关20兴C. Allain, M. Cloitre, and M. Wafra, Phys. Rev. Lett. 74, 1478 共1995兲. 关21兴C. Allain, M. Cloitre, and M. Wafra, in Physical Chemistry of Colloids and Interfaces in Oil Production, edited by H. Toulhoat and J. Lecourtier 共Technip, Paris, 1992兲, p. 259. 关22兴J. M. Valverde, M. A. S. Quintanilla, A. Castellanos, and P. Mills, Europhys. Lett. 54, 329 共2001兲. 关23兴J. M. Valverde, A. Castellanos, and M. A. S. Quintanilla, Phys. Rev. Lett. 86, 3020 共2001兲. 关24兴N. Menon and D. J. Durian, Phys. Rev. Lett. 79, 3407 共1997兲. 关25兴J. M. Valverde, A. Castellanos, A. Ramos, A. T. Perez, M. A. Morgan, and P. K. Watson, Rev. Sci. Instrum. 71, 2791 共2000兲. 关26兴P. K. Watson, J. M. Valverde, and A. Castellanos, Powder Technol. 115,44共2001兲; J. M. Valverde, A. Castellanos, and P. K. Watson, Powder Technol. 118, 240 共2001兲. 关27兴K. Rietema, The Dynamics of Fine Powders 共Elsevier, London 1991兲. 关28兴T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47, 1400 共1981兲. 关29兴H. C. Hamaker, Physica 共Amsterdam兲4, 1058 共1937兲. 关30兴H. Krupp, Adv. Colloid Interface Sci. 1, 111 共1967兲. 关31兴T. G. Mason, A. J. Levine, D. Ertas, and T. C. Halsey, Phys. Rev. E 60, 5944 共2000兲. 关32兴L. Massimilla and G. Donsi, Powder Technol. 15, 253 共1976兲. 关33兴A. J. Forsyth and M. J. Rhodes, J. Colloid Interface Sci. 223, 133 共2000兲. 关34兴M. L. Ott and H. A. Mizes, Colloids Surf., A 87, 245 共1994兲. 关35兴S. Ross and I. D. Morrison, Colloidal Systems and Interfaces 共Wiley-Interscience, New York, 1988兲. AGGREGATIONANDSEDIMENTATIONINGAS-... PHYSICAL REVIEW E 64 041304 041304-7