Central difference solutions of the kinematic model of settling of polydisperse suspensions and three-dimensional particle-scale simulations

The extension of Kynch's kinematic theory of sedimentation of monodisperse suspensions to polydisperse mixtures leads to a nonlinear system of conservation laws for the volume fractions of each species. In this paper, we show that a second-order central (Riemann-solver-free) scheme for the solution of systems of conservation laws can be employed as an efficient tool for the simulation of the settling and the separation of polydisperse suspensions. This is demonstrated by comparison with a published experimental study of the settling of a bidisperse suspension. In addition, we compare the prediction of the one-dimensional kinematic sedimentation model with a three-dimensional particle-scale simulation.     

The stability mechanisms of an injectable calcium phosphate ceramic suspension

Calcium phosphate ceramics are widely used as bone substitutes in dentistry and orthopedic applications. For minimally invasive surgery an injectable calcium phosphate ceramic suspension (ICPCS) was developed. It consists in a biopolymer (hydroxypropylmethylcellulose: HPMC) as matrix and bioactive calcium phosphate ceramics (biphasic calcium phosphate: BCP) as fillers. The stability of the suspension is essential to this generation of “ready to use” injectable biomaterial. But, during storage, the particles settle down. The engineering sciences have long been interested in models describing the settling (or sedimentation) of particles in viscous fluids. Our work is dedicated to the comprehension of the effect of the formulation on the stability of calcium phosphate suspension before and after steam sterilization. The rheological characterization revealed the macromolecular behavior of the suspending medium. The investigations of settling kinetics showed the influence of the BCP particle size and the HPMC concentration on the settling velocity and sediment compactness before and after sterilization. To decrease the sedimentation process, the granule size has to be smaller and the polymer concentration has to increase. A much lower sedimentation velocity, as compared to Stokes law, is observed and interpreted in terms of interactions between the polymer network in solution and the particles. This experimentation highlights the granules spacer property of hydrophilic macromolecules that is a key issue for interconnection control, one of the better ways to improve osteoconduction and bioactivity.     

Numerical simulation of sedimentation in the presence of 2D compressible convection and reconstruction of the particle-radius distribution function

Numerical simulation of the sedimentation of a polydisperse suspension in a convectively unstable medium is presented. For the simulation of 2D compressible convection, the full system of hydrodynamic equations is solved by the explicit MacCormack scheme. Velocities and positions of suspension particles are calculated simultaneously with the solution of the equations. Initially, the particles are randomly distributed in the computational region. The total weight of sedimented matter is recorded during the numerical experiment. The results are compared with the sedimentation of the same suspension without convection. To reconstruct the particle-radius distribution function from the sedimentation curve, a new method is used. This method is based on the solution of the sedimentation integral equation by the Tikhonov regularization method and was recently developed by the author. To illustrate this technique, sedimentation of cement powder in air is simulated. The suspension contains 50000 particles. The particle radii are assumed to be log-normally distributed. Heat-driven convection is completely determined by the top and bottom boundary temperatures of the computational region and lateral boundary conditions. It is shown that convective motions of a medium with sedimented particles lead to the following effect: the fine disperse fraction of the suspension remains suspended much longer than without convection. Some particles will not sediment at all. The maximum radius of the particles of this fraction depends on the convection parameters (e.g. on convection cell size and convection velocities). These parameters, in their turn, depend only on the temperature difference of the top and bottom boundaries. The results of these calculations can be applied in geology and meteorology for studying dust sedimentation in air as well as in technology. Heat-driven convection can be used for separation of suspensions with the cut-off particle radius depending on temperature difference only.     

Analysis of electrokinetic sedimentation of dredged Welland River sediment

The Welland River is a tributary of the Niagara River. In the late 1980s it was discovered that a section of the Welland River was contaminated with heavy metals as a results of two sewer outfalls that has been used by a steel plant and local industrial and municipal operations for the last 50-60 years. One of the major problems encountered in the treatment of the dredged Welland River sediment is a slow rate of sedimentation due to the large proportion of fine solids in the sediment. In this study, the results of electrokinetic sedimentation of the Welland River sediment are analyzed based on the principles of gravitational and electrokinetic sedimentation. It was found that the effects of electric field intensity and the initial solid concentration of the suspension are the dominating factors governing the average particle settling velocity, the coefficient of free settling in the free settling stage and the coefficient of sedimentation in the hindered settling stage. The electrokinetic treatment is proven to be effective in terms of increasing the free and hindered settling velocities, reducing the overall sedimentation time and increasing the final solid concentration of the sediment. Thus, electrokinetics can be used to accelerate sedimentation of dilute solid suspensions, such as dredged sediment, wastewater and mine tailings.     

Study of sedimentation of non-cohesive particles via CFD–DEM simulations

The sedimentation process of granular materials exists ubiquitously in nature and many fields which involve the solid–liquid separation. This paper employs the coupled computational fluid dynamics and discrete element method (CFD–DEM) to investigate the sedimentation process of non-cohesive particles, including the hindered settling stage and the deposition stage. Firstly, the coupled CFD–DEM model for sedimentation is validated by the hindered settling velocity at different solid volume concentrations of suspension \(\phi _{0} \), i.e., \(\phi _0 =\) 0.05–0.6. Two typical modes of sedimentation are also presented by the concentration profiles and the equal-concentration lines. Then, the comparisons between mono- and poly-dispersed particle system are detailed. In the sedimentation of the poly-dispersed particle system, the segregation phenomenon is simulated. Furthermore, this segregation effect reduces with the increase of the initial solid concentration of suspension. From the simulations, the contact force between every pair of particles can be obtained, hence we demonstrate the “effective stress principle” from the view of the particle contact force by giving the correspondence between the particle contact force and the “effective stress”, which is a critical concept of soil mechanics. Moreover, the deposition stage can be simulated by CFD–DEM method, therefore the solid concentrations of sediment bed \(\phi _{\mathrm{max}} \) on different conditions are studied. Based on the simulation results of \(\phi _{\mathrm{max}} \) and the theory of sedimentation, this paper also discusses a method to calculate the critical time when sedimentation ends of two typical modes of sedimentation.     

The movement law and orientation control of rectangular particles in the viscous fluid domain based on IS-FEM

Understanding the movement law and orientation control mechanism of non-spherical particles are significant for industrial applications. In this work, the flow characteristics of rectangular particles, in the uniform and wedge viscous fluid domain, are simulated by the immersed smoothed finite element method (IS-FEM). The influences of mesh resolution and time-step on particle velocity are analyzed, and the numerical procedure is validated by the published model and sedimentation experiments. The operating parameters that affect the particle flow are systematically studied, including Reynolds number, initial angle, channel offset distance, and aspect ratio. Moreover, the particle angles are adjusted by the velocity gradient of fluid domains. The result indicates that the velocities, angle, and drag of rectangular particles are closely related to the working conditions. The long axis of rectangular particles is consistent with the flow direction in shrinking fluid domains and is perpendicular to the flow direction in expanding fluid domains. The angle distribution law of rectangular particles in moving wedge fluid domains is determined. These findings provide a theoretical foundation for particle sedimentation and suspension flow, which is helpful for the further separation and orientation control of mixed particles.     

Dispersing Characteristics of Graphite Suspension by Surface Modification with ABDM

The stability of dispersions of graphite in aqueous solution with ABDM (alkyl benzyl dimethyle ammonium chloride) as a surfactant was investigated. The adsorbed structures of the ABDM molecules on graphite particles were examined in relation to zeta potential and critical micelle concentration. It was found that the zeta potential seems to be strongly correlated with both ABDM monolayer and bilayer formations on the particle surfaces. The total potential energies resulting from the interaction of the ABDM adsorbed graphite particles in suspension were estimated using the DLVO theory, and highly dispersed systems were obtained at 20 kT and higher. The adsorption amount of ABDM, zeta potential, graphite powder contact angle, and particle sedimentation rate were measured to determine the dispersion stability. It was found that the high dispersible graphite suspension with settling time T 1/2 of 44.5 h at pH 10 and zeta potential of 22.5 mV was produced.

graphite dispersion adsorption surface modification zeta potential     

Optimizing the primary particle size distributions of pressurized metered dose inhalers by using inkjet spray drying for targeting desired regions of the lungs

Conventional suspension pressurized metered dose inhalers (pMDIs) suffer not only from delivering small amounts of a drug to the lungs, but also the inhaled dose scatters all over the lung regions. This results in much less of the desired dose being delivered to regions of the lungs. This study aimed to improve the aerosol performance of suspension pMDIs by producing primary particles with narrow size distributions. Inkjet spray drying was used to produce respirable particles of salbutamol sulfate. The Next Generation Impactor (NGI) was used to determine the aerosol particle size distribution and fine particle fraction (FPF). Furthermore, oropharyngeal models were used with the NGI to compare the aerosol performances of a pMDI with monodisperse primary particles and a conventional pMDI. Monodisperse primary particles in pMDIs showed significantly narrower aerosol particle size distributions than pMDIs containing polydisperse primary particles. Monodisperse pMDIs showed aerosol deposition on a single stage of the NGI as high as 41.75?±?5.76%, while this was 29.37?±?6.79% for a polydisperse pMDI. Narrow size distribution was crucial to achieve a high FPF (49.31?±?8.16%) for primary particles greater than 2?µm. Only small polydisperse primary particles with sizes such as 0.65?±?0.28?µm achieved a high FPF with (68.94?±?6.22%) or without (53.95?±?4.59%) a spacer. Oropharyngeal models also indicated a narrower aerosol particle size distribution for a pMDI containing monodisperse primary particles compared to a conventional pMDI. It is concluded that, pMDIs formulated with monodisperse primary particles show higher FPFs that may target desired regions of the lungs more effectively than polydisperse pMDIs.     

Hydrodynamic instability of a suspension of spherical particles through a branching network of circular tubes

A numerical method is presented for computing the unsteady flow of a monodisperse suspension of spherical particles through a branching network of circular tubes. The particle motion and interparticle spacing in each tube are computed by integrating in time a one-dimensional convection equation using a finite-difference method. The particle fraction entering a descendent tube at a divergent bifurcation is related to the local and instantaneous flow rates through a partitioning law proposed by Klitzman and Johnson involving a dimensionless exponent, q ≥ 1. When q = 1, the particle stream is divided in proportion to the flow rate; as q → ∞, the particles are channeled into the tube with the highest flow rate. The simulations reveal that when the network involves two or more generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous, self-sustained oscillations in the segment flow rates, pressure drop across the network, and particle spacing in each tube. A phase diagram is presented to establish conditions for unsteady flow. As found recently for blood flow in a capillary network, oscillations can be induced for a given network tree order by decreasing the ratio of the tube diameter from one generation to the next or by decreasing the diameter of the terminal segments. The instability is more prominent for rigid than deformable particles, such as drops, bubbles, and cells, due to strong lubrication forces between the tightly fitting particles and tube walls. Variations in the local particle spacing, therefore, have a more significant effect on the effective viscosity of the suspension in each tube and pressure drop required to drive a specified flow rate.     

Kinematics of Flow of a Slow Suspended-Particle Flux about a Sphere as Applied to the Sedimentation of a Bidisperse Suspension

A study is made of the kinematics of flow of fine particles about a coarse one in the process of sedimentation of the fine particles. An approximate analytical solution for the increase in the settling rate of fine particles, which is in good agreement with the approximation dependence obtained earlier as a result of numerical experiment, has been found. It has been shown that a relatively weak entrainment of most of the fine particles arriving at the peripheral region of the cell around the coarse particle is of primary importance for the effect of acceleration of sedimentation. A comparison with the available experimental data is made.     

用共沉降法制备组分连续变化的梯度材料

从最简单的沉降过程-单一粒径粉末的沉降开始，描述了颗粒沉降的特点，并简述了用共沉降法制备梯度材料的基本原理，着重阐述了用共沉降法制备梯度材料的基本工艺过程，介绍了用共沉降法制备梯度材料的现状和使具有不同烧结性能的组元同是致密化这一目前仍需解决的关键问题。     

Estimation of complex refractive index of polydisperse particulate systems from multiple-scattered ultraviolet-visible-near-infrared measurements

A method to extract the complex refractive index of spherical particles from a polydisperse suspension at concentrations where multiple light-scattering effects are significant is presented. The optical constants are estimated from total diffuse reflectance and transmittance measurements and inverting the measurements using the radiative transfer equation (RTE) and the Mie theory for scattering by polydisperse spherical particles. The method is tested by applying it to three different polydisperse polystyrene suspensions and extracting the optical constants of polystyrene particles in the wavelength range of 450-1200 nm. The effect of particle size, concentration, and polydispersity on the estimated values of the optical constants is also discussed.     

Discharge Characteristics of Polydisperse Powders through Conical Hoppers. Part 1: Predictions for Fine, Granular, Free Flowing Powders

Discharge characteristics of fine polydisperse granular powders of equal-solid density and near-spherical particle shape through a conical hopper were investigated by measuring solid discharge rates of powders. Effects of orifice size of hopper and size distribution of powders on discharge rate were determined by means of experiments conducted for six different sizes of hopper orifice and three different powder types under gravity flow conditions. A new effective mean diameter characterizing polydisperse powders is first introduced and determined from the particle size versus weight fraction distribution of a powder as the size corresponding to 50% cumulative weight fraction. This effective mean diameter was efficiently used in two modified forms of the Beverloo equation to predict discharge rates of polydisperse powders through hopper orifices.     

DEM Applications to Aeolian Sediment Transport and Impact Process in Saltation

Discharge characteristics of fine polydisperse granular powders of equal-solid density and near-spherical particle shape through a conical hopper were investigated by measuring solid discharge rates of powders. Effects of orifice size of hopper and size distribution of powders on discharge rate were determined by means of experiments conducted for six different sizes of hopper orifice and three different powder types under gravity flow conditions. A new effective mean diameter characterizing polydisperse powders is first introduced and determined from the particle size versus weight fraction distribution of a powder as the size corresponding to 50% cumulative weight fraction. This effective mean diameter was efficiently used in two modified forms of the Beverloo equation to predict discharge rates of polydisperse powders through hopper orifices.     

Two-dimensional viscous flow of a suspension due to particle settling in a vertical vessel

Summary This paper investigates the influence of the viscosity of the suspension together with wall friction on sedimentation in vessels with vertical walls. The settling process is described by four basic equations, i.e. two continuity and two momentum equations. Assuming a very dilute suspension, an asymptotic expansion for small particle concentration is carried out. To first order, the well-known one-dimensional solution is recovered. The motion of the liquid, however, is a second-order effect governed by the biharmonic equation. In order to find a solution satisfying the boundary conditions, harmonic expansions are used. Thus the problem is reduced to a system of linear algebraic equations. Solutions for the two-dimensional flow of the liquid and the motion of the particles are given, and the influence on the shape of the interface between the suspension and the clear liquid is discussed.With 7 Figures     

Sedimentation kinetics of polydisperse second-phase inclusions in molten selenium

The behavior of polydisperse second-phase inclusions in molten selenium during settling is modeled using solid carbon particles as an example. A theoretical limit of the particle concentration in the surface layer of the melt is evaluated as a function of particle size, and the particle concentration is determined as a function of time, melt viscosity, and surface layer thickness. The calculated histograms of the particle size distribution in the surface and bottom layers of molten selenium are in satisfactory agreement with experimental data.     

The sedimentation velocity of a particle in a wide range of Reynolds numbers in the application to the analysis of the separation curve

A simple interpolation formula for the sedimentation velocity of a particle, including the limiting cases of small and large Reynolds numbers is proposed. A comparison with other known similar formulas is given.It is shown that the adjustment of sedimentation law is important to describe the hydrocyclones separation curve for large particles.A way of representing the separation curve, which allows justify the assumptions underlying the model of Schubert–Neesse is shown. It is shown that the correction of formula for the sedimentation velocity significantly affects the estimate of the effective coefficient of turbulent diffusion in the apparatus.     

Sedimentation and suspension flows: Historical perspective and some recent developments

Sedimentation and suspension flows play an important role in modern technology. This special issue joins nine recent contributions to the mathematics of these processes. The Guest Editors provide a concise account of the contributions to research in sedimentation and thickening that were made during the 20th century with a focus on the different steps of progress that were made in understanding batch sedimentation and continuous thickening processes in mineral processing. A major breakthrough was Kynch's kinematic sedimentation theory published in 1952. Mathematically, this theory gives rise to a nonlinear first-order scalar conservation law for the local solids concentration. Extensions of this theory to continuous sedimentation, flocculent and polydisperse suspensions, vessels with varying cross-section, centrifuges and several space dimensions, as well as its current applications are reviewed.     

Aggregate formation in metallic ferromagnetic liquids

Ferromagnetic liquids containing 45A diameter tin or antimony coated iron particles dispersed in mercury have been prepared electrolytically. The fluids are found to have a magnetisation which is time dependent when an applied field is abruptly reduced. The characteristic decay time which is of the order of 60s cannot be attributed to magnetisation changes due to small noninteracting particles of 45A diameter but it is consistent with the presence of particle aggregates of 10⁴A diameter. Measurements of sedimentation rates and viscosity are also consistent with the presence of aggregates of 10⁴A diameter. Measurements of the sedimentation rate in a gravitational field show that if the particles are coated with tin and particularly antimony fluid stability is improved and aggregate size reduced. It is suggested that particle aggregation may be prevented by ensuring that the liquid contains a distribution of particles with a median diameter appreciably smaller than 45A and a small standard deviation.     

Sediments of soft spheres arranged by effective density

Colloidal sedimentation has been studied for decades from both thermodynamic and dynamic perspectives. In the present work, binary mixtures of colloidal spheres are observed to separate spontaneously into two distinct layers on sedimentation. Both layers have a high volume fraction and contain distinct compositions of particles. Although predicting these compositions using settling dynamics is challenging, here we show that the compositions are readily predicted thermodynamically by minimizing the gravitational energy of the system. As the random packing fraction of a mixture of spheres exceeds that of monodisperse spheres of either type, the mixture produces a denser suspension that forms the bottom phase. Experimentally, the use of charged particles and low-ionic-strength solutions provides interparticle repulsions that keep the packed particles mobile, avoiding a glassy state that would prevent particles from reaching their equilibrium configuration. We extend this work beyond binary systems, showing similar separated layers for a five-component mixture of particles.