Estimation of hindered settling velocity of suspensions

Four effective-medium models (EM-I, II, III, IV) are utilized and compared for determining hindered settling velocity of equi-sized particles in a viscous fluid. Among the models, EM-IV model is found to accurately predict the effective viscosity and the hindered settling velocity of monodisperse suspensions. In EM-IV model which was developed for determining the diffusivity of proteins in a biological membrane by Dodd et al. [T.L. Dodd, D. A. Hammer, A.S. Sangani, D.L. Koch, J. Fluid Mech. 293 (1995) 147], the effective-medium region begins at the distance R = a[(1 ? S(0))/?]^1/3 from the origin where the center of the test particle is located, where a is the radius of the particle, ? is the volume fraction of the particles in the suspension, and S(0) is the zero wavenumber limit of the structure factor. The estimations by EM-IV model agree very well with the exact calculations and the experimental observations. The hindered settling velocity U of the particles is given, in Richardson–Zaki form, by U/U₀ = (1 ? ?)^5.5, where U₀ is the settling velocity for an isolated particle.     

The minimum fluidisation velocity,bed expansion and pressure-drop profile of binary particle mixtures

The total bed pressure drop, the pressure-drop profile, bed expansion and bed voidage have been measured for a variety of binary particle mixtures over a wide range of gas velocities.Apparent minimum fluidisation velosities have been defined for segregating systems, and the addition of dense particles of lower minimum fluidisation velocity can cause a decrease in apparent minimum fluidisation velocity of the mixture in a very similar fashion to the addition of finer particles to larger ones of the same density.The measured u_mf s are compared with presently derived simplified theoretical equations and with equations from the literature. It is clearly shown that because of the sensitivity of u_mf determination to voidage, such relationships cannot be used with confidence. However, the empirical equation of Cheung on average follows the shape of the experimental curves well, includig those for binary systems of different density, provided the bed is in a well-mixed condition.Bed pressure-drop profiles are related to the mixing/segregation state and to the amount of fluidisation of the bed and may offer a simple indirect method of determining these conditions in practice.     

Sedimentation of supsensions: implications of theories of hindered settling

Expressions are derived for the parameters A in Steinour's theory and n Richardson and Zaki's expression dealing with the hindered settling of powdered materials.It is shown that n is simply related to ?₁, the initial liquid volume fraction of a uniformly mixed suspension for which [Q(1 — ?)] was maximum value, Q being the linear settling rate of the suspension/supernatant interface and ? the initial liquid volume fraction of the suspension. For systems obeying the Richardson-Zaki expression, the settling rate at ?₁ approaches the limit V_s exp^?1, where V_s is the estimated Stokes' law limiting velocity for the system, when ?₁ approaches unity (i.e. infinite dilution).Highly hindered systems have large values for n, and ?₁ values approaching unity; for such systems Q_?1 ? V_s exp^?1. It is suggested that such behaviour implies the existence of relatively long-range forces within suspensions, hindering settling, and that particle-liquid (including particle-liquid-particle) forces are of importance in addition to particle-particle interactions.Evidence is presented that hindrance to settling is directly proportional to the polarity of the solid/liquid system and that the density of charge on the superficial particle surface is the dominant factor in determining hindrance. Maximum reduction of surface polarity without causing flocculation is suggested as the most efficient condition for separation by settling under gravity and pumping off supernatant liquor.n and ?₁ can be considered as useful indices of hindrance to sedimentation. ?₁ has particular significance as the initial liquid volume fraction at which solids flux has maximum value (for any suspension which obeys the Richardson-Zaki equation).     

The Voidage Function for the Drag Force Ratio in a Liquid‐Fluidized Bed

Di Felice (1994) has shown that the ratio of the drag coefficient, C_D, on a sphere in a liquid‐fluidized bed of uniform spheres to the drag coefficient, C_DS, on the same sphere in isolation and subjected to the same superficial liquid velocity, u, is given by a function ?^?β, where β was expressed as an empirical function of the particle Reynolds number, Re = duρ/µ. Here it is shown that C_D/C_DS is well approximated by ?^?mm, where the Richardson‐Zaki index n is a function of the terminal free‐settling Reynolds number, Re_t = du_tρ/µ, and m is 2 plus the slope of the standard log C_DS vs. log Re plot at plot at Re = Re_t. The present model, using the best experimentally confirmed equation for n and a new simple equation for and a new simple equation for m, is compared with that of Di Felice in their respective abilities to predict liquid‐fluidized bed expansion.     

CFD approach for simulation of bitumen froth settling process - Part I: Hindered settling of aggregates

Bitumen or heavy oil aggregates are formed when bitumen emulsions, consisting of emulsified water droplets dispersed solids and precipitated asphaltenes, are treated with aliphatic solvents. While settling, the aggregates exhibit zone settling mode with the development of a sharp oil and settling zone interface. Previous research [Long Y, Dabros T, Hamza H. Structure of water/solids/asphaltenes aggregates and effect of mixing temperature on settling rate in solvent-diluted bitumen. Fuel 2004;83:823-32] provides settling experimental data for bitumen aggregate settling and a Richardson-Zaki approximation was proposed by the authors with modified exponents for simulating the settling behavior of aggregates. However, the need for modified exponents and their dependence on aggregates and solvents used was not explained. Since the aggregates exhibit hindered settling, where the settling rate is different from that of individual particles or aggregates, numerous settling models have been proposed to correlate particle swarms to single particle via drag correlations [Richardson JF, Zaki WN. Sedimentation and fluidization: Part I. Trans Inst Chem Eng 1954;32:35-53; John G, Maan RA. Velocity-voidage relationships for fluidization and sedimentation in solid-liquid systems. Ind Eng Chem Proc Des Dev 1977;16:206-14]. The compaction zone, due to high concentrations of solids and the effect of their resultant weight during settling is also often ignored. In this work, the hindered settling behavior of bitumen aggregates is first studied in a CFD framework using two models: (a) modified Richardson-Zaki approximation by Long et al. [Richardson JF, Zaki WN. Sedimentation and fluidization: Part I. Trans Inst Chem Eng 1954;32:35-53] and (b) the Syamlal-O’Brien model [John G, Maan RA. Velocity-voidage relationships for fluidization and sedimentation in solid-liquid systems. Ind Eng Chem Proc Des Dev 1977;16:206-14]. To address the limitations of the two models, a new model is proposed that incorporates the irregular (fractal) structure of aggregates by considering the aggregates as porous liquid-filled solids that are fractal in nature. Results, from the new fractal model, are found to be in good agreement with empirical data.     

CFD modeling of solid-liquid fluidized beds of mono and binary particle mixtures

CFD simulation of bed expansion of mono size solid-liquid fluidized beds has been performed in creeping, transition and turbulent flow regimes, where Reynolds number (Re_∞=d_pV_S∞ρ_L/μ_L) has been varied from 0.138 to 1718. It has been observed that the predicted values of bed voidage using the drag law of Joshi [1983. Solid-liquid fluidized beds: some design aspects. Chemical Engineering Research and Design 61, 143-161] and Pandit and Joshi [1998. Pressure drop in packed, expanded and fluidized beds, packed columns and static mixers—a unified approach. Reviews in Chemical Engineering 14, 321-371] (which has been derived from the first principals), exhibited an excellent agreement with the Richardson and Zaki equation. CFD simulations have also been performed for the prediction of segregation and/or intermixing of binary particle systems having the ratio of terminal settling velocity over a range from 3.2 to 1.06. The Reynolds number has also been varied over the range of 0.33 to 2080. It has been observed that the present CFD model explains all the qualitative and quantitative observations reported in the published literature (complete segregation, partial segregation, complete intermixing, etc) and these predictions are in good agreement with the experimental results. The present CFD model also predicts successfully the layer inversion phenomena which occur in the binary particle mixtures of different size as well as density. Further, the critical velocity at which the complete mixing of the two particle species occurs has also been predicted.     

Particle-to-particle heat transfer in fluidized bed drying

Particle-to-emulsion and interparticle heat transfer rates were estimated in the range 1.5 ? u/u_mf ?3.5, 0.69 ? d_p ? 2.15 mm by drying wet refractory particles in fluidized beds of similar dry particles of the same sizes. Overall particle-to-emulsion heat transfer coefficients decrease roughly as the inverse of the particle diameter. Particle-to-particle heat transfer coefficients vary with the power-2 of the particle diameter and decrease as the fluidization velocity increases.     

Sedimentation Velocity of Solids in Finite Size Vessels

The Richardson‐Zaki equation is by far the most popular empirical equation used to describe the velocity‐voidage relationship for sedimenting solid‐liquid homogeneous suspensions, using only two empirical parameters. In this work some of Richardson and Zaki suggestions for the two parameters are challenged on the basis of new and old experimental evidence.     

Parametric study of fine particle fluidization under mechanical vibration

Investigations into the effects of vibration on fluidization of fine particles (4.8-216 μm average in size) show that the fluidization quality of fine particles can be enhanced under mechanical vibration, leading to larger bed pressure drops at low superficial gas velocities and lower values of u_mf. The effectiveness of vibration on improving fluidization is strongly dependent on the properties (Geldart particle type, size-distribution and shape) of the primary particles used and the vibration parameters (frequency, amplitude and angle) applied. The possible roles of mechanical vibration in fine particle fluidization have been studied with respect to bed voidage, pressure drop, agglomeration, and tensile strength of particle bed. Vibration is found to significantly reduce both the average size and the segregation of agglomerates in the bed, thus improving the fluidization quality of cohesive particles. Also, vibration can dramatically reduce the tensile strength of the particle bed. Obviously, vibration is an effective means to overcome the interparticle forces of fine powders in fluidization and enhance their fluidization quality.     

A simple correlation for solids holdup in a gas-liquid-solid fluidized bed

A simple empirical model was established which allows solids holdup in a gas-liquid-solid fluidized bed containing large and dense particles to be readily predicted based on the equation of Richardson and Zaki (1954, Trans. Inst. Chem. Engrs32, 35) for liquid-solid fluidized bed systems. The approach is applicable both to monocompnent particle systems and to binary mixtures of particles. For a monocomponent system, a correlation for model parameters was proposed which is expressed as a function of particle diameter, particle density, bed diameter and liquid density. For a binary mixture of particles, the averaging and serial approaches were shown to predict the solids holdup equally well within the range of the gas and liquid velocities considered. Experiments were also performed using eight solid particles for the monocomponent system and five binary mixtures of particles differing in diameter and/or density for the mixture system to substantiate the model.     

Characteristics of fluidisation behaviour in a pressurised bubbling fluidised bed

The experiments were carried out in a bench‐scale fluidised bed of 90 mm in diameter to determine the influence of pressure on fluidisation characteristics of Geldart A and B particles over the range of pressure 0.1–4.5 MPa. For Geldart B particles, the results indicate that minimum fluidisation velocity (u_mf) was found to decrease with pressure whilst bed voidage at u_mf was unaffected, and the bed expansion height increase with pressure at fixed value of gas velocity was observed for both Geldart B and A particles. For Geldart A particles, minimum bubbling velocity (u_mb) bed voidage at u_mb and dense phase voidage were found to increase obviously with pressure, but a slight influence of pressure on u_mf was observed. The prediction values of high‐pressure fluidisation characteristics from the references' correlations developed at pressure were in agreement with the experimental data. © 2012 Canadian Society for Chemical Engineering     

Voidage characteristics and prediction of bed expansion in liquid-solid inverse fluidized bed

Studies on voidage fluctuations, axial voidage profile and bed expansion are carried out by measuring the local void fraction using particles of wide ranging characteristics in liquid-solid inverse fluidized bed. The quality of fluidization is elucidated by the local voidage fluctuations. The RMS voidage fluctuation depicts a maximum with respect to average bed void fraction and increases with increase in Archimedes number. The fluidization quality has been quantified using average normalized RMS voidage fluctuation in terms of Transition number. The axial void fraction is almost uniform throughout the bed except for particles with size distribution. All the literature and present experimental data on bed expansion are unified in terms of Richardson and Zaki equation using experimental terminal velocities. A new correlation is proposed for predicting the wall effect corrected experimental terminal velocities, as a substitute for standard drag equation. The bed expansion data are also predicted using the drift flux model.     

Fluidization Behavior of Cohesive Ca(OH)2 Powders Mixed with Hydrophobic Silica Nanoparticles

Dry mixing with hydrophobic silica nanopowders was used to improve the fluidization quality of Ca(OH)₂ particles which belong to the Geldart C group and, thus, normally cannot be fluidized. Three parameters, i.e., sieved size of Ca(OH)₂ particles and sieved size and weight percentage of SiO₂ nanoparticles were selected for experiments. A direct proportionality was found between the coverage quality of materials over each other and the fluidization behavior of their corresponding adsorbents. Optimum SiO₂ size and concentration values were determined for the improvement of Ca(OH)₂ fluidizability. The sieved size of Ca(OH)₂ powder had no consequential effect on the coverage quality. The Richarson‐Zaki equation and fractal analysis combined with a modified Richardson‐Zaki approach were proposed for prediction of the fluidization quality and agglomerate size.     

Discrete element method modeling of non-spherical granular flow in rectangular hopper

Discrete element method (DEM) was developed to simulate the corn-shaped particles flow in the hopper. The corn-shaped particle was described by four overlapping spheres. Contact force and gravity force were considered when establishing the model. In addition, the velocity distribution and voidage variance of corn-shaped and spherical particles were investigated. The results show that the vertical velocity difference between centre and side wall and the horizontal velocity of corn-shaped particles are relatively larger than that of spherical particles. The mean voidage for corn-shaped particles is smaller than for spherical particles in any hopper. And the mean voidage values decrease with the increase of the ratio of width and length (D/L) and the ratio of height and width (H/D) for both corn-shaped and spherical particles. The local voidage profiles in hoppers with different D/L were also studied. It demonstrates that the wall effect on the voidage of spherical particles is more remarkable than that of the corn-shaped particles. The voidage fluctuations of corn-shaped and spherical particles decrease obviously with increasing D/L when the particles are far away from the wall. And when the particles are discharging, the wall effect on the spherical particles is more remarkable than the condition of packing naturally.     

Packed bed structure: Evaluation of radial particle distribution

A model describing the radial distribution of monosized spheres in randomly packed beds up to distances of about two particle diameters from the vessel wall is presented here. The model is based on the existence of a highly ordered layer of particles adjacent to the wall followed by a more diffuse, but still identifiable, second layer. Expressions generated from simple geometrical concepts (intersection between a cylindrical surface and a sphere) straightforwardly allow calculating the radial voidage profile given the radial distribution of particle centers and vice versa. These expressions are employed to fit the model to measures of voidage profiles within a wide range of aspect ratios, a = (R_T/R_P). The model can be used to accurately predict radial voidage profiles, but it is stressed that the identification of particle distribution constitutes more valuable information than an empirical expression for describing voidage variations.     

Characteristics of anthraquinone hydrogenation catalysts in a liquid‐solid fluidized bed

The fluidization characteristics of anthraquinone hydrogenation catalysts were investigated in a liquid–solid fluidized bed. The effects of the initial bed conditions such as particle size, bed depth‐to‐column diameter ratio and liquid density and viscosity on the fluidization behaviour, bed expansion and applicability of the Richardson–Zaki equation were studied. The results reveal a strong particle size effect on the Richardson–Zaki (R‐Z) expansion index which in general decreased as the particle diameter increased. One type of particles exhibited two distinct bed expansion behaviours, depending mainly on the bed depth‐to‐column diameter ratio, with an experimentally established boundary at . This behaviour could be attributed to increasing wall friction and a tendency to exhibit slugging. The dependence of the Richardson–Zaki exponent on the liquid dynamic viscosity confirms the classic result .     

Effect of particle shape on liquid-fluidized beds of binary (and ternary) solids mixtures: segregation vs. mixing

Experiments were carried out in water-fluidized binary (and ternary) mixtures of teflon spheres, discs and rods. All particles had the same volume, while the discs and rods had nearly the same sphericity. It is shown that segregation can occur by shape, with similar segregated and mixed zones as when binary mixtures of different size or density are fluidized. The model of Pruden and Epstein (1964; Stratification by size in particulate fluidisation and in hindered settling. Chemical Engineering Science 19, 696), in which the degree of segregation depends on the bulk density difference(s) of the corresponding monocomponent beds at the same liquid velocity, is vindicated qualitatively for each system, but sphericity is not sufficient as a single shape factor to yield a single quantitative correlation of the transitions between segregation patterns for the different systems. Segregation by shape of non-isometric particles appears to require higher reduced density differences than sizing of spheres, probably because of the greater bed instabilities generated by the non-isometric particles. Overall bed voidage is predicted well by the serial model of Epstein et al. (1981; Liquid fluidisation of binary particle mixture- I.Overall bed expansion. Chemical Engineering Science 36, 1803).     

Model for Calculation of Agglomerate Sizes of Nanoparticles in a Vibro‐fluidized Bed

The behavior of SiO₂ nanoparticles and the effects of operating conditions on nanoparticle agglomerate sizes have been investigated under conditions created in a vibro‐fluidized bed (VFB). The experimental results reveal that the vibrations imposed in the bed can suppress slugging and/or channeling, in contrast to conventional fluidization with upflow only. The vibrations imposed in the particle bed affect both the minimum fluidization velocity and the agglomerate size, both of which decrease with increases in the energy introduced to the bed by the vibrations. The effect of vibrations on the agglomeration in vibro‐fluidized beds of nanoparticles depends on the critical vibration frequency corresponding to a minimum agglomerate size. Both the amplitude and the frequency of the applied vibrations have significant effects on the agglomerate size. The experimental results and the consequent analysis reveal that increasing levels of vibrations in the bed yields finer agglomerates. The Richardson‐Zaki scaling law combined with Stokes law permits the prediction of agglomerate sizes and the extent of initial bed voidage. The average agglomerate sizes predicted are in good agreement with those determined experimentally.     

Gravity separation of bidisperse suspensions: Light and heavy particle species

Gravity separation of bidisperse suspensions containing particle species lighter and heavier than the suspending fluid in a vertical tube has been studied both theoretically and experimentally. The suspension was relatively dilute having a total solids volume concentration of no more than 16%. In this dilute range, lateral segregation of light and heavy particles into clusters and formation of fingering flow structure do not occur. The settling velocities of both particle species are retarded.Five models published by Lockett and Al-Habbooby, Mirza and Richardson, Masliyah, Selim et al. and Patwardhan and Tien, were used to predict the settling velocities of the heavy and the light particle species in bidisperse suspensions. The last three of these models give good predictions for the experimental data obtained in this study.     

Prediction of single particle settling velocities through liquid fluidized beds

Single particle settling velocities through water fluidized beds of mono-sized glass spheres (d_p = 0.645, 1.20, 1.94, 2.98 and 5 mm in diameter) were studied experimentally using a column, 40 mm in diameter. The settling spherical particles (D_p = 10 and 19.5 mm) had different densities (1237 to 8320 kg/m³), while the settling particles (D_p = 5 and 2.98 mm) were glass spheres. The pseudo-fluid model, which considers a liquid fluidized bed as a homogenous pseudo-fluid, predicts single particle settling velocities quite well if the ratio D_p/d_p is larger than about 10. With decreasing ratio D_p/d_p, the overall friction between the settling particle and the fluidized media increases. A method for predicting single particle settling velocities through a liquid fluidized bed is proposed and discussed. Following the approach of Van der Wielen et al. [L.A.M. Van der Wielen, M.H.H Van Dam, K.C.A.M. Van Luyben, On the relative motion of a particle in a swarm of different particles, Chem. Eng. Sci. 51 (2006) 995-1008], the overall friction is decomposed into a particle-fluid and a particle-particle component. The effective buoyancy force is calculated using the transition function proposed by Ruzicka [M.C. Ruzicka, On buoyancy in dispersion, Chem. Eng. Sci. 61 (2006) 2437-2446]. A simple model for predicting the collision force is proposed, as well as a correlation for the collision coefficient. The mean absolute deviation between the experimental and calculated slip velocities was 5.08%.