Calculation of the hindered settling velocity of a bimodal mixture of solid particles of arbitrary shape in a Newtonian fluid

A method for calculating the settling velocity of a bimodal mixture of particles of arbitrary shape is proposed. The method uses the concept of effective particle diameter, which characterizes the particle shape, volume, and midsection area. The method takes into account the contact interaction between particles of different sizes due to their collisions. The calculation results are compared with experimental data on settling of bimodal mixtures of limestone particles of arbitrary shape.     

Calculation of the settling velocity of particles for the hindered sedimentation of polydisperse mixtures of arbitrarily shaped solid particles in a Newtonian liquid

The method developed earlier for calculating the hindered settling velocity of arbitrarily shaped particles is generalized for polydisperse particle-size distributions. The calculated results are in satisfactory agreement with the experimental data on the sedimentation of bi-and trimodal mixtures of spherical glass particles.     

Calculation of the Velocity of Free Settling of Solid Particles in a Newtonian Liquid

A cubic equation for the velocity of free settling of solid particles in a Newtonian liquid is derived on the basis of a simple approximate expression for the drag coefficient of spherical particles as a function of the Reynolds number using the concept of the effective diameter of a particle of arbitrary shape, which characterizes the surface, volume, and midsection of the particle. The exact and approximate solutions are compared, and some experimental data are analyzed.     

Calculation of the velocity of obstructed settling of monodisperse solid particles in a Newtonian liquid

A model for calculating the velocity of obstructed settling of monodisperse solid particles in a Newtonian liquid is proposed. The model is based on an experimental expression for the drag coefficient for flow past a single sphere and an empirical expression for the viscosity of a solid-liquid mixture as a function of the volume concentration of the solid phase that takes into account the obstruction to the flow in the interparticle space.Translated from Teoreticheskie Osnovy Khimicheskoi Tekhnologii, Vol. 38, No. 6, 2004, pp. 624–629.Original Russian Text Copyright © 2004 by Kondratev, Naumova.     

The use of the variation method to define the velocity of constrained turbulent motion of disperse particles in liquid

A method of calculating the velocity of the constrained turbulent motion of spherical solid particles was elaborated based on the variation principle of minimum energy dissipation intensity.     

Experimental study on the effects of big particles physical characteristics on the hydraulic transport inside a horizontal pipe

This paper presents an experimental study of the physical characteristic effects of large particles on hydraulic transport in a horizontal pipe. The particles are spherical and are large with respect to the diameter of the pipe (8%, 10%, 16%and 25%). Experiments were done to test the important parameters in solid transport (pressure, velocity, etc.). As a result, the relationship between the pressure gradient forces and the mixture velocity was sub-stantially different from the pure liquid flow. However, in a single-phase flow a monotonous behavior of the pres-sure drop curve is observed, and the curve of the solid particle flow attains its minimum at the critical velocity. The regimes are characterized with differential pressure measurements and visualizations.     

General method of measurement of velocity of laminar constrained motion of spherical solid and gas particles in liquids

The general method of calculating the velocity of the constrained laminar motion of spherical solid and gas particles has been devised based on the variation principle of the intensity of the dissipation of energy.     

Determination of the free settling parameters of spherical particles in power law fluids

A new generalized method for rapid determination of the terminal free settling parameters of spherical particles in shear-thinning power law fluids has been proposed. On the basis of experimental data published elsewhere and the authors' own results, general plots have been obtained. These plots, together with the equations presented, permit the computation of the unknown parameters, such as the terminal free settling velocity or settling diameter, which characterize the steady motion of spherical particles in the power law fluids.     

On the Settling Velocity in a Nonstationary Atmosphere

A general drag coefficient has been used in the equation of motion for solid spherical particles. The time constants, stopping times, and settling velocities in a still atmosphere are computed for a wide range of Reynolds numbers. The settling times are compared with the times calculated when a particle is falling in a fluctuating atmosphere. It is found that such particles will get significantly longer settling times owing to an enhancement in the drag coefficient caused by an increase of the relative velocity between the particle and the fluid. Surprisingly, this enhancement is present for a horizontal wind field due to a coupling between particle motion in different directions, but it is also present for a vertical field. The effect is most pronounced in the intermediate Reynolds number region, slightly above the Stokes range, where the increase in settling time can be more than 10% for certain fluctuation frequencies and amplitudes. This indicates that such particles must be carefully treated when they are falling in a nonstationary medium     

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.     

Impact of Aggregate Size and Structure on Biosolids Settleability

Settling is an important solid–liquid separation process in wastewater treatment. While solids concentration is an important factor influencing settling velocity of particles, other solids characteristics including particle size, shape, and structure also play an important role in settling rate. The “compactness” of bacterial aggregates in particular is recognized to exert a great influence on solid phase dynamic behavior since it has a substantial effect on fluid flow through the aggregate, which, in turn, affects the particles buoyancy. With the recognition that biosolids structure can be described by fractal methods, we now have a convenient means of parameterizing aggregate “compactness.” In this article, we examine the fractal nature of bacterial aggregates (which are the main component of the solids in wastewater treatment processes) using small angle light-scattering methods and assess the impact of aggregate compactness (as described by fractal dimension) on settling velocity of both single aggregates and large aggregate clusters. The results indicate that settling velocity is strongly dependent on both size and aggregate structure, with the larger and less compact flocs settling more quickly as a result of the significant extent of flow through the bacterial assemblages.     

On motion of bubbles and drops

Dynamically interacting effects of surfactant impurities and neighbor particles on the transfer mechanism of momentum and terminal velocities of spherical bubbles, drops, or solid particle assemblages at low Reynolds numbers are analysed and formulated. The steady-state equations of viscous fluid motion and continuity (one set for each phase) have been solved simultaneously with a conservation equation for diffusing surfactant impurities. By employing a cell model with zero vorticity at its outer boundary, and modified interfacial conditions the terminal relative velocity of an ensemble of fluid particles is evaluated and compared with some available experimental data. The analysis of solid-fluid mixture is shown to become a special case of the analysis of fluid-fluid mixtures where adventitious surfactant impurities stop internal circulation thereby causing drop and bubble assemblages to behave as suspension of solid particles.     

A note on the aerodynamic diameter and the mobility of non-spherical aerosol particles

The aerodynamic diameter of a non-spherical aerosol particle is primarily related to the final settling velocity of the particle. The aerodynamic diameter is not obtained directly from mobility measurements by formally calculating a sphere diameter from the mobility equation for a spherical particle. Instead, a correction factor involving the dynamic shape factor of the non-spherical particle must be applied.     

Measurement and modeling of the settling velocity of isometric particles

The sedimentation of solid particles of simple form and complex form has been studied. The applied data in the case of isometric particles have been compared to those of spherical particles of same volume. This comparison allows highlighting an equivalent sedimentation diameter concept.Dimensionless factor depending on the extended equivalent diameter, the geometric aspect of the particle and the flow have been defined. The use of this factor allows finding an interrelationship with another dimensionless factor which is the Archimedes number. This last depends on the physical parameter of the particle and the carrying fluid. The numerical treatment of this interrelationship permitted to deduct an empirical model for the settling velocity of particles. This model is applied with success to the data of the considered measures and well compared to the results obtained by other models.     

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.     

Calculation of the parameters of an ensemble of particles in a well-stirred fluidized-bed reactor

Consideration is made of the problem of determining the parameters (the mass, the particle size distribution, and the entrainment, settling, and drain flows) of a fixed fluidized bed of reacting particles in a continuous reactor in the approximation that the solid phase is well stirred. A closed set of equations for solving this problem is derived. Special cases are studied, namely, beds of constant-size particles or particles contracting or growing during processing in a reactor with or without overflow devices. Formulas are obtained for the fluidized-bed parameters at different particle size distribution functions in the feed flow and also different size dependence of the particle contraction or growth velocity.     

A Model for the Relative Velocity of a Particle in an Impingement Dryer: Application to Heat Transfer

A simplified mathematical model for the relative gas-particle motion in a confined jet impingement dryer is developed. Model predictions based on an unsteady momentum balance are in good agreement with the observed cycling motion of a spherical particle. The model is applied to coriander seeds submerged in a flow field of superheated steam. It is found that relative motion occurs in unsteady turbulent regime, and that steady settling velocity of particles is never achieved. Model results are applied to correlate experimental heat transfer data of an impingement dryer. Experimental Nu numbers compare fairly well with correlations for fluidized systems.     

Mathematical modelling of a solid particle motion in a re‐circulatory fluidised bed unit

Two fundamental properties of the particles are their velocity and consequent displacement versus time history during the granulation process. Knowledge of the particle velocity might supply much information about the sub‐processes. In this paper, an analytical model of a solid particle motion in an internal re‐circulatory fluidised bed unit is developed and validated against experimentally obtained data. The model predictions show good correspondence with the experimental results for the spherical particles. For the case of non‐spherical granules, the agreement between the model and the experiments is not equally convincing but still adequate.     

Distribution of solid particles in liquid fluidized beds

A method is proposed for calculating the steady-state distribution of mixtures of solid particles and of the axial bed porosity in liquid fluidized beds. The extent of bed stratification is assumed to be determined by the differences among the settling velocities of the fluidized particles and the random motion of these particles that is a result of fluidization. Predictions of particle distribution based on the present method were found to agree reasonably well with experimental data.     

Particulate character, inertial effects and diffusion effects in concentrated suspensions

In the sedimentation of suspensions, the particles modify their settling velocity in accordance with the solids concentration. Kynch's theory assumes that in each layer, the settling velocity only depends on the solids concentration, and neglects the acceleration and deceleration processes when the particles descend with a velocity distinct from that corresponding to the solids concentration at this layer. The Kynch theory also assumes that there are so many small particles that the solid mass can be considered homogeneously distributed in the suspension, so the particulate character of the suspension is ignored. Kynch's theory also neglects the diffusion effects. An analysis of the particulate character of the suspension, the inertial effects and the diffusion effects are discussed in this paper, deducing a critical value for considering the particulate character negligible and a critical size for considering the diffusion effect also negligible. Finally, several data of different flocculated suspensions are satisfactorily analysed in view of the conclusions obtained previously.