Electromechanics of fluidized beds of nanoparticles

Electromechanics of fluidized beds of nanoparticles M. J. Espin,1J. M. Valverde,2M. A. S. Quintanilla,2and A. Castellanos2 1Department of Applied Physics II, University of Seville, Avenida Reina Mercedes, 41012 Seville, Spain 2Department of Electronics and Electromagnetism, University of Seville, Avenida Reina Mercedes, 41012 Seville, Spain 共Received 9 May 2008; revised manuscript received 16 December 2008; published 27 January 2009兲 The electromechanical behavior of a gas-fluidized bed of insulating silica nanoparticles is investigated. When fluidized by gas, these nanoparticles form highly porous agglomerates of size of the order of hundreds of microns, which gives rise to a nonbubbling fluidization regime. Bed expansion is enhanced by an imposed alternating electric field for oscillation frequencies in the range between tens and hundreds of hertzs and field strengths of about 1 kV/cm. Nanoparticle agglomerates are naturally charged and experience forced oscillations that cause an increase of the gas flow shear on their surface. As a consequence, the agglomerate size is expected to decrease, which can explain the observed behavior. A model based on the balance between attractive and flow shear forces is presented that accounts for agglomerate size reduction as the strength of the field is increased. DOI: 10.1103/PhysRevE.79.011304 PACS number共s兲: 45.70.⫺n, 47.55.Lm, 81.07.Wx, 84.30.Vn I. INTRODUCTION Fundamental problems in our modern world, such as reducing air pollution from diesel exhaust and filtration of smoke emissions from asphaltic pavement recycling process, motivated Melcher to develop electrofluidized beds for collection of submicron particles 关1兴. In a fluidized bed the granular material rests on a porous plate through which gas is distributed. At a certain gas flow the drag force balances the material weight per unit area and the grains become fluidized. Melcher stressed a fluidized bed by imposing an electric field that effectively polarized the grains, which thus acted as collection sites for the charged pollutants entrained in the fluidizing gas, serving the function of electrostatic precipitators. In Melcher’s investigations, semiinsulating wet grains were used to allow for charge exchange during collisions, thus preventing charge accumulation on the grains. Tardos et al. 关2兴investigated the efficiency of contaminant collection in a dry fluidized bed due to the charges generated naturally on insulating polystyrene spheres. The dry fluidized bed demonstrated very high cleaning efficiencies, even without an externally imposed electric field, due to the strong charge buildup on the granules 关2兴. Elsdon and Shearer 关3兴 reported that the application of an ac electric field to a dry bubbling bed of 0.24 mm polymer particles caused an appreciable increase of the net transport of heat for field oscillation frequencies of about 100 Hz. This behavior could be explained semiquantitatively in terms of the enhanced agitation of the particles, which were charged naturally by contact electrification and, thus, experienced forced oscillations around their mean position. However, a major concern that besets fluidized bed filters lies in the bypassing of the contaminants through large gas bubbles that usually develop in gas-fluidized beds of coarse grains 共grain size dpⲏ100 ␮ m兲 关4兴. In contrast, it has been shown that gas-fluidized beds of some micron and submicron particles exhibit nonbubbling fluidlike behavior 关5,6兴. For example, powders of moderate density nanoparticles 共such as silica兲can be uniformly fluidized by gas 关5,7兴. Fluidization of nanoparticles, providing extremely high specific surface area, has become thus one of the most promising techniques for emerging industrial applications. The use of fluidized beds of titania nanoparticles has been proposed to remove pollutant nitrogen oxides from air by photocatalytic reaction 关8兴. Fluidization of Co/Al2O3 nanoparticles has been employed to improve the catalytic activity of reforming methane with CO2关9兴. Fluidized beds of silica nanoparticles coated with catalysts have been thought to be an alternative to bubbling beds used in the oil industry 关10兴among other potential uses. In this paper we investigate the electromechanical behavior of a dry fluidized bed of silica nanoparticles stressed by a cross-flow ac electric field. II. EXPERIMENTAL SETUP AND PROCEDURE The fluidized bed apparatus consists of a vertical 2.54 cm internal diameter polycarbonate vessel, 16.2 cm in height, with a sintered stainless steel plate at the bottom that acts as a gas distributor. The fluidization cell is placed between two parallel electrodes 8 cm width apart. One of the electrodes is grounded and alternating high voltage Vis applied to the other from an oscillator and amplifier network. According to three-dimensional 共3D兲numerical calculations the field strength within the bed is approximately uniform. A cross section with the electric field and equipotential lines obtained from the numerical calculations are shown in Fig. 1. As can be seen, the field inside the vessel is quite homogeneous with a slightly smaller strength than outside the fluidization vessel due to wall effects. The calculations performed in the horizontal plane at different levels show that the disturbance created by the metallic distributor is important only in the very close vicinity of the distributor, and that at the level of the lower edge of the electrodes this perturbation is not significant. The power supply system allowed us to provide peak field strengths up to E0⯝2kV/cm over a frequency range up to 10 kHz. Different types of wave form, sinusoidal, square and triangular shape, were applied. The gas flow 共dry air兲to the cell is controlled by a MKS Instruments mass flow controller with a full range from 0 to 2000 cm3/min. The nanopowder PHYSICAL REVIEW E 79, 011304 共2009兲 1539-3755/2009/79共1兲/011304共7兲©2009 The American Physical Society011304-1 tested is Aerosil R974, which consists of hydrophobic silica 共particle density ␳ p=2250 kg/m3and particle size dp =12 nm兲. The nanopowder is presieved 共sieve opening of 425 ␮ m兲, and initialized in the fluidization cell by applying a sufficiently high gas flow. Once the initialized bed reaches a uniform state, the gas flow is set to a given value and the bed height in the stationary state is measured. Then the electric field is turned on and the stationary bed height after application of the field is measured. Additionally, scanning electron microscopy of the nanopowder has been performed. III. EXPERIMENTAL RESULTS AND DISCUSSION A. Bed expansion in the absence of applied electric field In Fig. 2we have plotted experimental data of the particle volume fraction ␾ 0as a function of the superficial gas velocity vgwithout electric field applied. Note that small values of vgare required for uniform fluidization 共vg⬃1cm/s兲as compared to values typically required for fluidization of bubbling beds of coarse grains 共vg⬃1m/s兲关2兴. Figure 2shows the different fluidization regimes displayed by the nanofluidized bed. For vgⲏ1cm/s the nanofluidized bed shows a fluidlike behavior and nanoparticle agglomerates are seen to display a pseudoturbulent random motion. In this uniform fluidlike regime, macroscopic bubbles are not visible, and the gas pressure adjusts to the powder weight per unit area. The height of the free surface and the mass of elutriated powder increase monotonically as the gas velocity is increased and we observe a full suppression of the bubbling regime, i.e., the fluidized bed transits gradually to elutriation. Since bed expansion is not curtailed by the development of macroscopic gas bubbles, the bed is seen to expand uniformly as vgis increased. Nanoparticle agglomerates elutriate off the bed appreciably when the gas velocity is further increased above vg⯝2.7 cm/s. For vgⱗ1cm/s, free surface instabilities are observed due to the formation of transient solid networks that are increasingly predominant as the gas velocity is further decreased. This is the transitional regime. For vgⱗ0.6 cm/s, the solidlike structure is just perturbed by little volcanoes that are visible at the free surface and for vgⱗ0.5 cm/s, the nanofluidized bed is fully jammed. This is the solidlike regime. In the solidlike regime the bed retains a small, but nonvanishing, yield strength due to the enduring adhesive contacts established between the jammed agglomerates. Note that the particle volume fraction in the jammed state is rather low, which is attributable to the extremely high porosity of the nanoparticle agglomerates 共about 99% of the agglomerate volume is free of particles 关7兴兲. The existence of these fluidization regimes has been previously reported 关11兴 and can be inferred either from bed expansion observations or measurements of the mechanical strength of the fluidized bed and local reflectance 共see Ref. 关11兴for further details兲. According to the scanning electron microscopy analysis by Yao et al. 关12兴on 7–16 nm silica particles, the primary nanoparticles form multistage agglomerates by three steps. First, primary nanoparticles are seen to be agglomerated into 3D netlike structures 共let us call them subagglomerates兲. Second, the subagglomerates are agglomerated into simple agglomerates of size dSof the order of tens of microns and existing before fluidization. Scanning electron microscope photographs performed by us of simple agglomerates of Aerosil R974 are shown in Fig. 3, where it is seen that their size fits in this typical size. When the nanopowder is fluidized, the pre-existing simple agglomerates further agglomerate into so-called complex agglomerates of size dCof the order of hundreds of microns. The suppression of gas bubbles in nanofluidized beds is causally related to the formation of these highly porous multistage agglomerates 关12兴. The hydrodynamic and gyration radius of fine particle agglomerates are similar since they screen the gas flow quite efficiently 关13兴. Thus, a phenomenological interpretation of the expanded uniform state of nanofluidized beds carried out in previous works has been to consider complex agglomerates as effective low-density spheres, which may exhibit nonbubbling fluidization 关11兴. FIG. 2. Particle volume fraction ␾ 0vs superficial gas velocity vg without electric field applied. Fluidization regimes have been delineated. The solid line represents the modified RZ equation fit to the data in the fluidlike regime. Photographs in the inset illustrate bed expansion. FIG. 1. 共Color online兲Cross section of the experimental system showing the electric field and equipotential lines obtained from numerical calculations. The electrodes and the cell have been drawn to scale, with the distances in cm. One electrode was set to voltage 1 and the other one was grounded. ESPIN et al. PHYSICAL REVIEW E 79, 011304 共2009兲 011304-2 The Richardson-Zaki 共hereafter termed as RZ兲empirical equation 关14兴 vg vp =共1− ␾ 兲n共1兲 relates the particle volume fraction ␾ of the fluidized bed to the superficial gas velocity vg共or the settling velocity in sedimentation兲. Here vpis the sedimentation velocity of a single particle, vp=共1/18兲共 ␳ p− ␳ g兲gdp 2/ ␮ in the laminar regime, where ␳ pis the particle density, ␳ gis the gas density, g is the gravity field, dpis the particle size, ␮ is the viscosity of the gas and nis a parameter of order 5 in the small Reynolds number limit. Theoretical considerations by Batchelor 关15兴 leaded to vg/vp⯝1–5.6 ␾ for very diluted suspensions of non-Brownian hard spheres in the limit of small Reynolds number. Thus, the sedimentation velocity predicted by Eq. 共1兲conforms with the theoretical dilute limit by Batchelor for n=5.6. The RZ equation was originally intended to describe the expansion of liquid-fluidized beds 关14兴, where the fluidized particles were not agglomerated, and fluidization was usually uniform. Thus, Eq. 共1兲is not suitable for fluidized beds of fine particles, which are usually agglomerated. The simplest approximation to account for agglomeration in fluidized beds of nanoparticles is to assume an ideal system of monodisperse complex agglomerates with a hydrodynamic radius equal to their radius of gyration due to the effective screening of the gas flow by the agglomerates 关13兴. Then the Richardson-Zaki law would be vg vC =共1− ␾ C兲n,共2兲 where vCis the settling velocity of an individual complex agglomerate and ␾ Cis the volume fraction occupied by the complex agglomerates. This is the so-called modified Richardson-Zaki 共hereafter termed as mRZ兲equation. The extension of the RZ equation for fluidized systems of agglomerated fine particles was first reported in Ref. 关16兴. The settling velocity of an individual complex agglomerate can be calculated from the balance of the weight of the complex agglomerate, WC=N共1/6兲 ␲ dp 3 ␳ pg, and the Stokes force, FS=3 ␲ ␮ dCvC, where Nis the number of nanoparticles in the complex agglomerate. Note that the buoyancy force is neglected since the density of the gas is much smaller than the density of the particle. From this we have vC=vpN/k, where k=dC/dpis the ratio of complex-agglomerate size dC to primary particle size. The complex-agglomerate volume fraction ␾ Cwould be given by ␾ C=共N0/N兲共1/6兲 ␲ dC 3, where N0is the number density of particles 关 ␾ =N0共1/6兲 ␲ dp 3兴. Thus ␾ C= ␾ k3/Nand Eq. 共2兲can be rewritten as vg=vpkD−1共1−k3−D ␾ 兲n,共3兲 where the complex-agglomerate fractal dimension Dhas been defined as D=lnN/ln k. The extension of the RZ equation for fluidized beds of nanoparticles was first proposed in Ref. 关18兴and has been used to estimate the complexagglomerate size in these systems 共see Ref. 关7兴for a review兲. The best fit of Eq. 共3兲to our data in the uniform fluidlike regime is plotted in Fig. 2and gives dC=226 ␮ m共⫾3 ␮ m兲 and D=2.59 共⫾0.001兲, which is in good agreement with previous measurements on the same nanopowder 关7兴. Experimental works reported in the literature for fluidization of different fluidized systems of fine and ultrafine particles show that Dis close to 2.5, independently of particle size 关5,17,19兴. This finding is consistent with diffusion-limited aggregation 共DLA兲mechanism 关20兴, in which self-similar FIG. 3. SEM photographs of simple agglomerates of the nanopowder used in the experiments 共Aerosil R974兲. ELECTROMECHANICS OF FLUIDIZED BEDS OF …PHYSICAL REVIEW E 79, 011304 共2009兲 011304-3 fractal ramified patterns are grown via irreversible sticking of the aggregating units in a random motion. DLA seems to be the dominant mechanism of aggregation in fluidized beds of fine particles. In the case of nanoparticles, pre-existing simple agglomerates would agglomerate in the fluidized bed into complex agglomerates according to DLA mechanism 关18,19兴. Agglomerating units 共simple agglomerates in the case of nanoparticles兲are expected to display a diffusive dynamics in the fluidized bed, as inferred from diffusion experiments 关21兴, and stick to agglomerates due to the prevailing of interparticle adhesion force on weight force. Thus complex agglomerates should appear as branched structures whose growth is controlled by a diffusive process and size is limited by flow shear. B. Effect of an alternating electric field In the absence of humidity the dominant interaction between nanoparticle agglomerates at contact arises from the van der Waals attractive force FvdW⯝Ada/24z0 2共it was shown in Ref. 关22兴that the van der Waals force prevailed over the electrostatic force arising from the natural charge of the agglomerates兲. Here Ais the Hamaker constant 共A ⯝10−19 J兲and z0⯝3–4 Å is the minimum intermolecular distance. In order to estimate the van der Waals force between the type of agglomerates shown in Fig. 3,dashould be the typical size of their surface asperities 关24兴. Chaouki et al. 关23兴used das=0.2 ␮ m as typically used for the surface asperity size of micron-scale particles 关24兴. Taking das ⯝0.2 ␮ m we estimate FvdW⬃10 nN, which is susceptible to increase due to compaction 关25兴. When an alternating field E=E0cos共 ␻ t兲is applied, the naturally charged complex agglomerates are forced to oscillate in the stationary state at the same frequency of the field. The order of magnitude of complex-agglomerate charge, measured in a previous work by tracking complex-agglomerate trajectories under a dc electric field 关22兴,isQ=QC⬃10−14 C. Since the electric force QCE⬃1 nN is thus smaller than FvdW it is likely that, at small oscillation frequencies, a solid powder layer remain deposited on the wall upon field reversal. This type of behavior may occur when the complex-agglomerate oscillation length xC, defined as the horizontal distance traveled by the complex agglomerates during one oscillatory period due to the electric force, is comparable to the bed width 共⬃1cm兲. To have an estimation of xC, long-range interactions will be disregarded and complex agglomerates will be considered as effective spheres. The drag force on a isolated sphere undergoing oscillations in a fluid with a velocity v=v0cos共 ␻ t + ␸ 兲is given by FD=av+bdv/dt, where a=6 ␲ ␮ R共1+R/ ␦ 兲, b=3 ␲ R2冑2 ␮ ␳ / ␻ 共1+2R/9 ␦ 兲,Ris the sphere radius, ␮ is the fluid viscosity, ␳ is the fluid density, and ␦ =冑 ␮ / ␳␻ is the penetration depth of the rotational fluid flow field around the sphere 关26兴. In the limit ␦ /R→⬁, as, for example, for a static electric field, FDconverges to the Stokes drag force FS. For nanoparticle complex agglomerates 共R=RC⬃100 ␮ m兲fluidized by dry air, it is estimated ␦ ⬃Rfor f⬃100 Hz, which fits in the range of frequencies tested in our experiment, thus the Stokes equation is not applicable. Solving the equation of motion of a single sphere of mass m, the sphere oscillation length is x=2QE0/ ␻ 冑a2+ ␻ 2共m+b兲2. Using typical values in our experiment, it can be estimated that xC⬃1 cm for fof the order of hertzs. On the other hand, in the limit of high frequency, the complex agglomerates will display very short oscillations. Since complex agglomerates are formed in an agglomerating process of pre-existing simple agglomerates, it is unlikely that this mechanism is affected by the electric field when xC is shorter that the size of smallest simple-agglomerate size 共dS⬃1 ␮ m关12兴兲. Accordingly, and using typical values in our experiment, the field is expected to have no effect for f above kilohertzs. Figure 4shows the relative variation of the particle volume fraction 共⌬ ␾ / ␾ 0兲as a function of the oscillation frequency ffor fixed values of the root-mean-squared field 共Erms=1.25 kV/cm兲and gas velocity. A square wave form has been used for practical convenience 共the effect of waveform type is discussed below兲. It is seen that, in the limits of small and high frequency, the fluidization behavior depends on fas it has been anticipated. For frequencies of the order of hertzs the main mechanism is electrophoretic deposition of the complex agglomerates onto the walls as seen in the case of a dc field, which gives rise to bed channeling and collapse 共inset of Fig. 4兲. For frequencies of the order of kilohertzs and larger, the field does not play a significant role on bed expansion. In the range of intermediate frequencies bed expansion is enhanced as the inset of Fig. 4illustrates. The rest of this paper is devoted to analyze this latter effect. The effect of the field strength and wave-form type in the range of intermediate frequencies is shown in Fig. 5共f =20 Hz兲. The data, when plotted as a function of the rootmean-squared field Erms, adjust to a single trend independent of the wave-form type, indicating that enhanced bed expansion is the result of a time averaged process. Bed expansion is further enhanced as the strength of the field is increased at constant gas velocity, and for Erms⬎1.5 kV/cm the mass of elutriated powder off the bed becomes appreciable. Since in FIG. 4. Relative variation of the particle volume fraction vs electric field frequency. Root-mean-squared field is 1.25 kV/cm. Photographs in the inset illustrate electrophoretic deposition 共lefthand side, vg=2.7 cm/s, f=1 Hz兲and enhanced expansion 共righthand side, vg=2.7 cm/s, f=20 Hz兲. ESPIN et al. PHYSICAL REVIEW E 79, 011304 共2009兲 011304-4 a uniformly fluidized bed, the fluctuation velocity of the fluidized units 共in our case complex agglomerates兲is expected to be similar to gas velocity 关27兴, it would be expected that the field effect becomes relevant for field strengths giving rise to horizontal velocities of the complex agglomerates comparable to gas velocity. From the above calculations, and using typical values in our experiment, the complexagglomerate root-mean-squared velocity vrms is estimated to be similar to the gas velocity 共vrms⬃vg⬃1cm/s兲for Erms =1 kV/cm. On the other hand, for E=0.2 kV/cm, it is vrms ⬃1mm/s, indicating that complex-agglomerate fluctuations overwhelm the complex-agglomerate forced oscillations for this field strength. Thus, no field effect should be observed for fields of this strength and smaller as seen experimentally. IV. ANALYTICAL MODELING According to the mRZ equation 关Eq. 共3兲兴, enhanced bed expansion by the electric field could be attributed to a variation of the complex-agglomerate size. The strong dependence of ⌬ ␾ / ␾ 0on the root-mean-squared field strength suggests that there must be a relationship between the complexagglomerate size and Erms. In Ref. 关25兴a model was derived to estimate the size of agglomerates of micrometric particles in a fluidized bed subjected to the gravity field. We will carry out in this work a parallel derivation to estimate complex agglomerate size in the electrofluidized bed of nanoparticles, as effective particles undergoing agglomeration due to attractive forces. The novelty of the present analysis consists of extending it to nanoparticle agglomeration, including the effect of the electric field. In order to estimate theoretically the complex-agglomerate size in the electrofluidized bed of nanoparticles, simple agglomerates, which exist before fluidization, will be considered as effective particles undergoing agglomeration due to attractive forces between each other. This attractive force is counterbalanced by the gas flow shear to support the weight of the complex agglomerate in the gravity field plus the shear due to their forced oscillations in the electric field. The balance between the overall timeaveraged shear and the attractive force will determine the size of the complex agglomerates. The vertical complex-agglomerate weight force WC, which acts uniformly through the agglomerate body, is balanced by the hydrodynamic friction from the surrounding gas, which acts mainly at its surface due to the flow screening effect. From this balance, and using a spring model 关25兴, the typical strain on the surface of the complex agglomerate can be estimated as ␥ g⬃WC/KCRC. Here WC=NSWS, where NSis the number of simple agglomerates in the complex agglomerate and WSis the simple-agglomerate weight force. KCis the complex-agglomerate spring constant, which, using a theory on the elastic properties of random percolating systems 关28兴, can be obtained from the simple-agglomerate spring constant KSand the elasticity exponent ␤ 关KC ⬃KS/共kC兲 ␤ , where kCis the ratio of complex-agglomerate radius RCto simple-agglomerate radius RSand ␤ =3 in 3D兴 关25兴. The complex-agglomerate oscillations forced by the horizontal electric field give rise to an additional shear force to balance the electric force QCE. The root-mean-squared shear strain is thus increased up to ␥ rms=冑共 ␥ g兲2+共 ␥ E兲2, where ␥ E⬃QCErms/KCRC. Thus, the shear force, which acts on the simple agglomerate attached to the outer layer of the complex agglomerate, is Fs⬃KS ␥ rmsRS ⬃共kC兲D+2冑WS 2+共QSErms兲2. Here it is assumed NS=共kC兲Dand that the complex-agglomerate charge is equally distributed among the simple agglomerates 共QC=NSQS, where QSis the simple-agglomerate charge兲. Simple agglomerates are attached to the complex agglomerate as long as the shear force is smaller than the attractive force. Thus, the limit condition Fs=FvdW leads to the equation kC⬃Bo1/共D+2兲共4兲 for estimating the complex-agglomerate size dC, where the agglomerate Bond number Bo is defined as the ratio of FvdW to 冑WS 2+共QSErms兲2. The simple-agglomerate weight is WS =NpWp, where Wpis the nanoparticle weight and Npis the number of primary nanoparticles in the simple agglomerate, which is assumed to be Np=共dS/dp兲D关7兴. Using a typical FIG. 5. Top: Relative variation of the particle volume fraction vs peak field strength E0for three different wave-form types. Bottom: The same data is shown as a function of the root-mean-squared field strength Erms. Field frequency and superficial gas velocity are fixed to 20 Hz and vg=2.7 cm/s, respectively. ELECTROMECHANICS OF FLUIDIZED BEDS OF …PHYSICAL REVIEW E 79, 011304 共2009兲 011304-5 value of dS=30 ␮ m, as inferred from the work of Nam et al. 关5兴, it is predicted that dC 0=kC 0dS⬃150 ␮ m共E=0兲for our silica nanoparticles, which is in accordance with experimental values reported in the literature from noninvasive imaging techniques 关7兴and with our previous estimation in this paper. When the electric field is turned on, the simple-agglomerate charge QSis needed to estimate the complex-agglomerate size. Provided that simple agglomerates are unaltered by the electric field, it is QS=QC 0/NS 0, where QC 0and NS 0are the complex-agglomerate charge and number of simple agglomerates in the complex agglomerate for E=0, respectively. Thus kC=kC 0 冋 1+ 冉 QC 0Erms 共kC 0兲D 冊 2 册 −1/关2共D+2兲兴 .共5兲 Typically 共Erms⬃1kV/cm, QC 0⬃10−14 C兲, the relative decrease of dCand QCare about 15% and 30%, respectively. The effect of the field on bed expansion can be taken into account by using the complex-agglomerate predicted size 关Eq. 共5兲兴 in the mRZ equation 关Eq. 共3兲兴. Figure 6shows ⌬ ␾ / ␾ 0as a function of the gas velocity for Erms =1.25 kV/cm. The model prediction that fits better to the data in the fluidlike regime is plotted. Complex-agglomerate size, in the absence of electric field, and Dare taken from our previous bed expansion measurements 共dC=226 ␮ m, D =2.59兲. Remarkably, the prediction fits well to the data for a value of the complex-agglomerate charge 共QC 0=1.9 ⫻10−14 C兲similar to the value directly measured elsewhere 关22兴. Moreover, this value gives a charge to mass ratio in the range 10−5–10−4 C/kg, which is comparable to the values indirectly measured by Tardos et al. 关2兴from the electrical current in a Faraday cage induced by coarse grains in dry fluidized beds. Interestingly, for this charge value, the electric force on the complex agglomerate is approximately equal to the complex-agglomerate weight 共QC 0E⯝WC 0 ⯝2nN兲. It is remarkable that the theoretical results fit reasonably well to experimental results for values of the agglomerate charge close to the experimentally measured values, which upholds the applicability of the analysis presented in Ref. 关25兴, and extended in this paper, to predict agglomerate size from fundamental properties such as interparticle attractive force, particle size and density, and external fields. Due to the emerging applications of fluidization of nanoparticles, it is very important for the engineering community to have a simple tool to estimate beforehand agglomerate size, type of fluidization expected, fluidized bed expansion curve and packing in the solid state from fundamental physical parameters 关7兴. Since silica nanoparticles fluidize well without external aids, the effect of the electric field on bed expansion is not pronounced in the range of field strengths applied 共less than 20%兲. Note, however, that higher fields give rise to elutriation of the powder, which can be of interest for powder applications involving powder transport. It is also expected that an alternating field will have a relevant effect on the bubbling behavior exhibited by other nanoparticle systems such as titania 关11兴. In that case it is expected that the alternating field contributes to destabilize bubbles and homogenize fluidization. Experiments to corroborate this expectation are in progress. In the transitional regime, from the fluidlike to solidlike behavior 共vgⱗ1cm/s兲, it is expected that the main physical mechanism by which the field affects fluidization is destabilization of channels that allowed gas to bypass the bed, thus favoring bed expansion. At the jamming transition 共vg ⯝0.5 cm/s兲, the weight of the agglomerates is just sustained by the gas flow. For slightly smaller gas velocities, the attractive van der Waals force between agglomerates holds them in a weak solid network that spans through the bed. The attractive van der Waals force between agglomerates 共FvdW ⬃10 nN兲is just slightly larger than the electric force, thus gas flow perturbations, helped by the electric field, may still drive the bed into the fluidlike regime locally in the close vicinity of the jamming transition, which leads to enhanced bed expansion effect. The effect of the field is definitely neutralized for gas velocities below vg⯝0.3 cm/s. V. CONCLUSION The behavior of a cross-flow-type electrofluidized bed of nanoparticles has been studied. It has been found that bed expansion is enhanced for electric field strengths of about 1kV/cm and oscillation frequencies of tens to hundreds of hertzs. For oscillation frequencies of the order of hertzs, or smaller, bed expansion is hindered due to electrophoretic deposition of the agglomerates onto the vessel walls, whereas for oscillation frequencies of the order of kilohertzs, or larger, electrophoresis is nullified and bed expansion is not affected. According to a proposed model, the size of nanoparticle agglomerates stems from the balance between shear, which depends on field strength, and van der Waals forces. The optimum field strength for enhancing bed expansion produces an electric force on the agglomerates similar to FIG. 6. Relative variation of the particle volume fraction vs the superficial gas velocity. Oscillation frequency and electric field strength are fixed to 500 Hz and Erms=1.25 kV/cm, respectively. The line is the model prediction. ESPIN et al. PHYSICAL REVIEW E 79, 011304 共2009兲 011304-6 their weight, while the oscillation velocity of the agglomerates is similar to the gas velocity. Higher field strengths give rise to excessive elutriation, while smaller ones do not affect appreciably bed expansion. The electric field is screened by the van der Waals attractive interaction when the superficial gas velocity is decreased below vg⯝0.3 cm/s and the fluidized bed of nanoparticles is jammed. It is foreseen that because of their huge specific surface area and enhanced uniform fluidization, electrofluidized bed of nanoparticles could improve the efficiency of fluidized beds based devices such as gas cleaning filters. ACKNOWLEDGMENTS We acknowledge Spanish Government 共Contract No. FIS2006-03645兲and Junta de Andalucia 共Contract No. FQM 421兲. 关1兴J R. Melcher, Proc. 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