Stochastic sedimentation and hydrodynamic diffusion

Molecular collisions with very small particles induce Brownian motion. Consequently, such particles exhibit classical diffusion during their sedimentation. However, identical particles too large to be affected by Brownian motion also change their relative positions. This phenomenon is called hydrodynamic diffusion. Long before this term was coined, the variability of individual particle trajectories had been recognized and a stochastic model had been formulated. In general, stochastic and diffusion approaches are formally equivalent. The convective and diffusive terms in a diffusion equation correspond formally to the drift and diffusion terms of a Fokker–Planck equation (FPE). This FPE can be cast in the form of a stochastic differential equation (SDE) that is much easier to solve numerically. The solution of the associated SDE, via a large number of stochastic paths, yields the solution of the original equation. The three-parameter Markov model, formulated a decade before hydrodynamic diffusion became fashionable, describes one-dimensional sedimentation as a simple SDE for the velocity process {V(t)}. It predicts correctly that the steady-state distribution of particle velocities is Gaussian and that the autocorrelation of velocities decays exponentially. The corresponding position process {X(t)} is not Markov, but the bivariate process {X(t), V(t)} is both Gaussian and Markov. The SDE pair yields continuous velocities and sample paths. The other approach does not use the diffusion process corresponding to the FPE for the three-parameter model; rather, it uses an analogy to Fickian diffusion of molecules. By focusing on velocity rather than position, the stochastic model has several advantages. It subsumes Kynch’s theory as a first approximation, but corresponds to the reality that particle velocities are, in fact, continuous. It also profits from powerful theorems about stochastic processes in general and Markov processes in particular. It allows transient phenomena to be modeled by using parameters determined from the steady-state. It is very simple and efficient to simulate, but the three parameters must be determined experimentally or computationally. Relevant data are still sparse, but recent experimental and computational work is beginning to determine values of the three parameters and even the additional two parameters needed to simulate three-dimensional motion. If the dependence of the parameters on solids concentration is known, this model can simulate the sedimentation of the entire slurry, including the packed bed and the slurry–supernate interface. Simulations using half a million particles are already feasible with a desktop computer.     

Settling Velocities of Polydisperse Concentrated Suspensions

A model for particle settling velocities in polydisperse suspensions is proposed alongside normalized particle self and interaction mobilities. The model predictions are compared with other models (Davis and Gecol, 1994; and Al‐Naafa and Selim, 1989), as well as published experimental data. A general agreement was found between the current model and the existing data for binary dispersed particle systems except for one case where Φ_L is fixed. For ternary dispersed particle systems, predictions from both the A‐S model and the current model agree well with the existing data. With the improved sedimentation coefficients, the D‐G model also produces satisfactory results.     

Sedimentation of Binary Particle Mixtures

An equation is given, based on proposals of previous workers, for the slip velocity of each particle species within a polydisperse particle mixture moving vertically through a liquid. It is demonstrated how the initial settling velocity of each particle species may be determined when such a mixture undergoes sedimentation. A simplified graphical construction is developed for a binary particle mixture similar to that of Kynch, which was for a single species. It is shown how the zone boundary velocities and the sediment rise velocities may be determined using the proposed construction. The method is confirmed by experiments in which a mixture of particles consisting of two distinct sizes was allowed to settle to completion.     

A new method for the control of dilute suspension sedimentation by horizontal movement

This research addresses some factors that control the stability of dilute suspensions in sedimentation processes under a dynamic environment. Experimental and numerical studies were conducted about the sedimentation velocity control of dilute suspensions by horizontal movement for particle concentrations up to 8 wt.%. Nearly monodispersed particles were used as test particles. The effects of horizontal movement speed and amplitude on particle sedimentation process were investigated. Under stationary conditions, particles settle in only one vertical direction because of gravitational force. However, complicated particle motions arise under moving conditions due to circulation flow in horizontal moving conditions. The results show that horizontal movement can reduce the particle settling velocity or maintain the stability of the suspension.     

A kinematic model of continuous separation and classification of polydisperse suspensions

Kinematic models for polydisperse suspensions are based on specifying the solid–fluid relative velocity for each solids species as a function of the local solids concentrations. One such model, the Masliyah–Lockett–Bassoon (MLB) model, is employed herein to simulate continuous separation and classification of polydisperse suspensions. To this end, the clarifier-thickener (CT) setup for the continuous separation of suspensions is extended to a generalized clarifier-thickener (GCT). Discharge streams (or products) are described by new singular sink terms. Combining the GCT setup with the MLB model yields a system of nonlinear conservation laws with a discontinuous flux and a new non-conservative transport term describing the sinks. A numerical algorithm for the solution of this equation is presented along with numerical examples. The model describes the GCT unit with all critical design parameters, and predicts the composition of the overflow, underflow and discharge streams and the spatio-temporal evolution of the solids species concentrations inside the unit.     

ENHANCEMENT OF THE SEDIMENTATION RATES OF FIBROUS SUSPENSIONS

The settling of suspensions of slender rod-like particles with large aspect ratios was studied. The novel technique of enhancement of sedimentation rates by the addition of buoyant particle promoters was applied to such suspensions. The settling rates could be increased by many times by this simple technique. Sedimentation velocity increases of between 5 to 7 times have been observed for fibrous suspensions of concentrations of about 0.5%. A statistical analysis of the experimental data was performed to establish a correlation for the enhancement in settling velocities. The sediment volume was unaffected by the presence of buoyant particles. The buoyant parjticles were found to adsorb some surfactants onto their surfaces.     

ENHANCEMENT OF THE SEDIMENTATION RATES OF FIBROUS SUSPENSIONS

The settling of suspensions of slender rod-like particles with large aspect ratios was studied. The novel technique of enhancement of sedimentation rates by the addition of buoyant particle promoters was applied to such suspensions. The settling rates could be increased by many times by this simple technique. Sedimentation velocity increases of between 5 to 7 times have been observed for fibrous suspensions of concentrations of about 0.5%. A statistical analysis of the experimental data was performed to establish a correlation for the enhancement in settling velocities. The sediment volume was unaffected by the presence of buoyant particles. The buoyant parjticles were found to adsorb some surfactants onto their surfaces.     

Dynamic and steady-state sedimentation of polydisperse suspension and prediction of outlets particle-size distribution

The Kynch theorem was extended to non-linear system of conservation laws of polydisperse suspensions of spherical particles. The simulation predicts overflow of light particles and heavy particles at steady state and dynamic mode of batch and continuous sedimentation. The model eliminated the need for imposed non-theoretical parameters or functions to predict hindered settling and effluent concentrations. Particle-size distribution is also predicted at the effluent and the underflow. We examined several cases and predicted dynamic behaviour of rarefaction waves and overloaded continuous settler. A new concept of dynamic flux curves is also used and introduced.     

Clusters Formation during Sedimentation of Dilute Suspensions

Abstract

Particle interactions in dilute monodispersed sedimenting suspensions of spherical particles are studied as a function of solid concentration. It is shown that in suspensions with solid concentrations below 0.83%, the interactions are too insignificant to effect the use of Stokes' law in sedimentation results. Beyond this concentration, however, a definite change in suspension behavior occurs, as particles come close enough to form clusters of varying sizes causing faster settling rates. Optimum clustering takes place around 4.5%-solid concentration, corresponding to mean interspacing of 2.2 particle diameter within suspension and giving settling rates 1.58 times faster than the Stokes' velocity for a mean particle. Clusters start breaking beyond this concentration as the sedimentation becomes more hindered and the return upward flow of liquid becomes increasingly tortuous. The probability of clusters formation and their stability as a function of particle size, concentration, and the Reynolds number of suspensions are also investigated. The studies are further extended to demonstrate the effect of “immobile” liquid within the clusters in interpreting the sedimentation results.     

Large-scale calculation of hydrodynamic transport properties for random suspensions of hard-sphere particles

A numerical method based on fast multipole summation scheme is used to calculate hydrodynamic interactions in random suspensions of non-colloidal hard-sphere particles. The calculation is carried out for suspensions of 1,024 particles randomly placed in periodic unit cell to determine hydrodynamic transport properties such as permeability of a viscous flow through porous medium, effective viscosity of suspension, and sedimentation velocity of the suspended particles. The particle volume fraction ø ranges from 0.01 to 0.25. Effect of particle number N on the transport properties was examined through the numerical calculations with N=64-1,024. It is shown that sedimentation velocity increases with N approaching an estimate for infinite N, and the finite N effect is negligible in effective viscosity and permeability problems. The present scheme is quite useful for obtaining a statistically-averaged quantity for random suspensions. As an example, ensemble-averaged velocity when position of one particle is fixed is numerically obtained in sedimentation problem. The numerical results are shown to be in excellent agreement with theoretical prediction.     

Solids distribution and rising velocity of buoyant solid particles in a vessel stirred with multiple impellers

The distribution of buoyant solid particles in agitated suspensions has been studied. The investigation was carried out in a baffled vessel characterised by an aspect ratio equal to four and stirred with four radial impellers. Dilute suspensions of single-sized spherical particles of expanded polystyrene (density equal to 90.7 kg/m³) in water were used. Solid concentration was measured with a non-intrusive optical technique. Measurements were performed along the axis of the reactor to obtain steady-state vertical profiles (that increase from the vessel base to the top) as well as at fixed elevations to determine their transient after a pulse of solids injected at the bottom.Both the steady-state profiles and the transient concentration curves were interpreted in terms of the axial dispersion model with sedimentation. By data treatment the rising velocity in the agitated system could be determined, which proved to be significantly smaller than the rising velocity in a still liquid. The ratio of these two velocities is in reasonable agreement with a correlation of the ratio of the settling velocities for heavy particles with the ratio of the Kolmogorov microscale to particle diameter established in the past.     

SIMULATION OF PARTICLE TRANSPORT AND DEPOSITION IN A COMBUSTOR

A computational model for Lagrangian particle tracking for studying dispersion and deposition of particles in a combustor with swirling flow and chemical reaction is developed. The model accounts for the effect of thermophoretic force, as well as the drag and lift forces acting on particles, in addition to the Brownian motion and gravitational sedimentation effects. The mean turbulent gas flow, temperature fields and chemical species concentration in the combustor are evaluated using the stress transport turbulent model of the FLUENT code. The instantaneous fluctuation velocity field is generated by a Gaussian filtered white noise model.

The simulated axial, radial and tangential mean gas velocities are compared with the existing experimental data. Ensembles of particle trajectories are generated and statistically analyzed. The effects of size and initial distribution on particle dispersion and deposition are studied. The particle concentration at different sections are also evaluated and discussed. The results shows that the turbulence dispersion effect is quite important, while the thermophoresis effect is small.     

In Situ X-Ray Radiography and Tomography Observations of the Solidification of Aqueous Alumina Particles Suspensions. Part II: Steady State

This paper investigates the behavior of colloidal suspensions of alumina particles during directional solidification, by in situ high-resolution observations using X-ray radiography and tomography. This second part is focussed on the evolution of ice crystals during steady-state growth (in terms of interface velocity) and on the particle redistribution taking place in this regime. In particular, it is shown that particle diffusion cannot determine the particle concentration profile in this regime of interface velocities (20–40 μm/s). Particles are redistributed by a direct interaction with the moving solidification interface. Several parameters controlling the particle redistribution were identified, namely the interface velocity, the particle size, the shape of the ice crystals, and the orientation relationships between the crystals and the temperature gradient.     

Particle dynamics in sheared particulate suspensions

Measurements of particle dynamics of neutrally buoyant suspensions of non-Brownian glass beads in a Couette cell using dynamic sound scattering are reported. The dynamics were studied under steady shear flow across the entire gap between the stator and rotor for shear rates from 0.26 to 6.59 s ⁻¹ and particle concentrations from 20% to 50%, thereby enabling a comprehensive investigation of the dynamics to be carried out. The average particle velocity profile varies linearly with depth inside the cell for all shear rates and concentrations. The fluctuations in the particle velocities are large, indicating that the particles are not confined to streamlines but continue to fluctuate substantially during steady flow. Our data indicate that the fluctuations are anisotropic. The components of the velocity fluctuations (granular temperature) parallel to the flow and in the vertical direction are much larger than in the radial direction. The fluctuation anisotropy decreases as the concentration increases. © 2018 American Institute of Chemical Engineers AIChE J, 65: 840–849, 2019     

Particle tracking velocimetry investigations on density dependent velocity vector profiles of a swirling fluidized bed

Swirling fluidized bed (SFB) is a newer version of the well-known bubbling bed and very little know. An insight study is therefore required for complete understanding of the hydrodynamics of a SFB operation. The current study was conducted on stable regime of a SFB operated at different blade fin angles, blade inclination angles, particle densities and superficial air velocities. Roughly one quarter of the fluidized bed was photographed and its velocity vector field plots were generated using a MATLAB supported particle tracking velocimetry (PTV) technique. At lower superficial velocities, Gaussian distribution of the velocity vectors was predicted along the radius of the bed. Particles in the vicinity of the bed walls moved relatively slower than those marching in the middle of the bed. However, at higher superficial velocities, the particles closer to the cone boundary were moving with velocities comparable to the particles in the middle of the bed. Unlikely, the particles closer to the outer bed wall kept on moving with lower velocities regardless of increasing superficial air velocity. A further look into individual velocity vector profiles revealed negligible influence of smaller blade angles (9° and 12°) on particles’ motion. The overall velocity magnitude decreased by 6% with 3° increase in blade fin angle and by 9% with 5° increase in inclination angle. Contrarily, the particle velocity underwent a monotonic decrease with particle density.     

Settling velocities of particles in bubbly gas-liquid mixtures

The axial dispersion-sedimentation model is commonly used to describe the axial concentrations of solids in three phase bubble columns at low liquid velocities. When the two parameters of this model, the particle settling velocity and the solids axial dispersion coefficient, are uncoupled by the use of various assumptions, physically unrealistic values of these parameters often result. Direct experimental measurements of solids settling rates in bubbly gas-liquid mixtures were carried out. The measured mean settling velocities decreased slightly with gas flow rate and were equal to or slightly less than the single particle free settling velocity in the liquid alone. Solids axial dispersion coefficients were also obtained from the solids settling rate distribution data, and gave values considerably less than the experimental liquid axial dispersion coefficient.     

The influence of inclined plates on expansion behaviour of solid suspensions in a liquid fluidised bed — A computational fluid dynamics study

A significant increase in the particle sedimentation rate can be achieved by introducing inclined plates into conventional fluidised beds. In turn, high suspension densities are possible at fluidisation velocities in excess of the particle terminal velocity. The installation of the inclined plates, however, alters the dynamic characteristics of the fluidised bed, in particular, impacting upon the expansion behaviour of the suspension. In the present work a Computational Fluid Dynamics (CFD) approach was employed to investigate the influence of inclined plates on the expansion behaviour of solids suspensions in liquid fluidised beds. The model is based on the solution of the Eulerian multiphase equations for up to two different particle sizes with a continuous phase of water. The momentum equations treat hindered settling behaviour via the inclusion of a volume fraction dependent drag law. The computational model was validated against our experimental data and compared with the predictions of a kinematic model developed in one of our earlier works. In general the predictions made by both the CFD and the kinematic models were found to be in good agreement with the experimental results.     

The use of statistical properties of transmission signals for particle characterization

The measurement of transmission is an often used measuring principle in technology. The evaluation algorithms of fluctuating transmission signals are well known in the case of single particle measurement. By evaluating fluctuating transmission signals the particle characterization is also possible at high particle concentrations (0.01 to 30 vol.-%). In comparison with conventional photometers a significant gain of information can be received. Fluctuating signals are caused by the statistical probability of a limited number of particles being present in a defined measuring volume. The evaluation of such signals allows the determination of particle concentration, particle size of monodisperse particles (extinction diameter) and information about structure of agglomerates independent from each other. Mathematical fundamentals and practical ways to measure these parameters are shown. Experimental results are examplarily presented for monodisperse, polydisperse and agglomerated suspensions.     

DEM simulation of particle mixing in flat-bottom spout-fluid bed

The particle mixing mechanism affects the rate of the process and the achievable homogeneity. This paper presents a numerical study of the particle motion and mixing in flat-bottom spout-fluid bed. In the numerical model, the particle motion is modeled by discrete element method (DEM) and the gas motion is modeled by κ–? two-equation turbulent model. Validation with experiments is first carried out by comparing solid flow pattern and bed pressure drop at various gas velocities. Then, particle velocities, obtained from DEM simulations, are presented to reveal the mixing mechanisms. On the basic, the dependence of mixing index on the time and the effect of gas velocity on mixing and dead zone (stagnant solid) are discussed, respectively. The results indicate that the spouting gas is the driving force for the formation of particle circulation roll, resulting in the mixing. The convective mixing caused by the motion of circulation roll, shear mixing induced by the relative move of circulation rolls and diffusive mixing generated by random walk of particle among circulation rolls are three different mixing mechanisms in spout-fluid bed. The increase of spouting gas velocity promotes the convective and shear mixing. While increasing the fluidizing gas velocity improves significantly the convective mixing and but weakens the shear mixing. Both of them yield a reduction in the dead zone.