Equations for predicting growth or dissolution rates of spherical particles

Approximate analytical solutions to the transport equations describing the heat or mass transfer-controlled growth or dissolution of an isolated sphere are discussed and critically evaluated. The predictions of these equations are compared with finite-difference and similarity solutions. Recommendations are presented for using particular analytical expressions to evaluate the radius-time behavior of the spherical particle for various ranges of the two dimensionless groups of the problem.     

Lateral migration of spherical particles in porous flow channels: application to membrane filtration

Lateral migration of spherical rigid neutrally buoyant particles moving in a laminar flow field in a porous channel is induced by an inertial lift force (tubular-pinch effect) and by a permeation drag force due to convection into the porous walls. The analysis of Cox and Brenner [7], for the particle motion in a nonporous duct is extended to include the effect of the wall porosity. Criteria are established under which the inertial and permeation drag force in the lateral direction can be vectorially added. Particle trajectories and concentrations profiles are calculated for a plane Poiseuille flow with one porous wall. For particles with radius of 1 μm, inertial and permeation drag forces are of comparable size under flow conditions often met in ultra- and hyperfiltration of dilute suspensions. For smaller particles the permeation drag force dominates.     

Mass transfer to large particles in fluidised beds of smaller particles

A theory is proposed for predicting the transfer of a gas through a fluidised bed of small particles to a large particle. It is proposed that non steady-state mass transfer of the gas occurs by two mechanisms: (i) mass transfer of gas in clusters or packets of the smaller particles approaching the large particle; and (ii) gas convection. The theory developed enables prediction of the Sherwood number (N_sh, the dimensionless mass transfer coefficient) for a large particle, diameter d: N_sh=2ε_mf+$\text{4}\text{g3}\text{mfd}\text{(U}\text{mf}\text{ε}\text{mf}\text{+}\text{uinb}$)/π D_A${}^1{}^2$ where U_mf is the minimum fluidising velocity, εmf is the bed voidage at U_mf-0u_b is the mean bubble rise velocity and D_A is the gas diffusivity. This equation is shown to be in excellent agreement with Sherwood numbers determined from combustion experiments in which single large particles of petroleum coke were burned in air fluidised beds over a wide range of operating conditions. It is also shown that predictions using this expression are in close agreement with those from an empirical expression previously proposed by the autho     

Liquid fluidization of binary particle mixtures—I: Overall bed expansion

Semi-batch liquid fluidization of binary solid mixtures is investigated for bed expansion, for inversion of solid species and for stratification by size. This first of three papers deals with overall bed expansion, and it is shown that, even where there is total inter-mixing of the two solid species, the bed expands as if it were two single species beds in series. The superiority of this serial model over property-averaging models, and the inapplicability of relative velocity models, for predicting the expansion of semi-batch liquid-fluidized beds of multi-species solids are also demonstrated.     

A fully predictive criterion for the transition between particulate and aggregate fluidization

A simple general model for the interaction between a particle and the fluid in a fluidized suspension, enables a hydrodynamic criterion for the onset of bubbling in fluidized beds to be formulated in a compact and fully predictive form. Excellent agreement is found between the proposed criterion and experimental observations reported in the literature for liquid beds and gas fluidized systems operating under both ambient and elevated pressure condi     

Specular reflection in radiant heat transport across a spherical void

The emissivity factor for radiant heat transport across a spherical void with specular surfaces in a conducting solid is derived. To calculate the emissivity factor, the probability of all possible paths for radiation are formulated and summed. Emissivity factors for the diffuse and specular limits are compared. The resulting temperature dependence of void space radiation for several typical oxide materials is discussed.     

Dissolution kinetics: The nature of the particle attack of layered silicates in HF

The dissolution of layered silicates by hydrofluoric acid was studied with the aid of a slurry reactor. The nature of particle dissolution was delineat by measurements of reaction rates and by the examination of partially dissolved particles using a scanning electron microscope. The rates of particle d per unit mass were modeled as a function of the fraction x dissolved by an equation of the form [-r?(x)]/[-r?(x = 0)] = (1 - x) ^n?1. For the three layer silicates the parameter n takes on a value of $\text{1}\text{2}$, reflecting radial attack at the unit cell edges. The reactive edge surface area is considerably smaller than the total wetted particle surface area. In contrast, the two layer kaolinite structure dissolves at both the layer edges and the planar faces, and the parameter n varies from 0 to 1 as the particle dissolution proceeds to completion. Comparison of the rates of dissolution on the basis of the actual reactive surface areas show that layered silicates having the same octahedral sheet dissolve at the same rate. It is concluded that attack is preferential at surfaces where the octahedral sheets are exposed.     

Migration of suspended particles in plane stationary ultrasonic field

The time-averaged motion of microscopic particles in plane stationary sound fields is investigated. A generalized kinetic theory of particle migration in which the effect of particle diltusion is taken into account is formulated and the particle distributions at the rather early stage and after a sufficient sound irradiation are derived analytically from the theory. It is found that the nodal change of particle concentration of microscopic polystyrene latex dispersions is well predicted by the present theory, independently of the particle size, the sound frequency and the energy density of sound. The condition of formation of the Kundt's tube striation is also characterized theoretically.     

A study on the states of the non isothermal tubular wall reactor

The thermal behaviour of a cooled tube wall reactor is analyzed.The main goal of this work is the characterization of the steady state. Three different steady state solutions, in which the control through the reactor differs from each other, are presented. The possibility of simultaneous existence of these states is analyzed as a function of the parameters of the model.For the zeroth order reaction here treated, it is possible to get isothermal operating conditions. Minimum thermal levels at the wall reactor for a given conversion are discussed and they result closely related to the isothermal conditions.An analytical solution is developed for the so called “highest steady state” (HISS). This expression makes possible to predict the “quenching” conditions of the reactor.     

Mass transfer in liquid fluidized beds of ion exchange particles

Mass transfer rate data in a shallow liquid fluidized bed of ion exchange resin particles have been obtained in the range 2 < Re_s, < 25. At low voidage, ε < 0.55–0.60, the rate of mass transfer is reduced and it is inferred that a fluidized bed tends to maintain an ordered axial structure. This is lost at high voidage due to increased flow perturbation caused by the distributor plate. A generalised correlation is given for fluidized, fixed and distended beds in the range 2 < Re_s, < 25.     

Suspending solid particles in an agitated tank with axial-type impellers

Experimental investigation of the critical impeller speed in systems solids-liquid agitated by axial-type turbine impellers was drawn to more general relations as regards the impact of two more important variables: angle of blade inclination and volume fraction of solids. Both variables manifest existence of separate hydrodynamic regions which are reflected in the respective shape of the theoretically derived correlation function.     

The effect of pressure on the flow of gas in fluidized beds of fine particles

The division of gas flow between the bubble and dense phases of fluidized beds of six different types of Group A powders has been studied at pressures of up to 20 bar using surface collapse and X-ray absorption measurements. It was found that with these fine powders as pressure increases at constant volumetric gas flowrate so the size and hold-up of bubbles decrease while their frequency increases. Contrary to previous measurements the average bubb velocity appears to decrease with increasing pressure. The dominant mode of bubble break-up in all the powders was found to be division from the rear, contrast to that observed with Group B powders at atmospheric pressure. Interstitial phase voidage was found to increase with increasing pressure.The results are interpreted in terms of a model which assumes a difference between the voidages, and hence the gas flow, of powder in the wakes behind     

Flow of generalized newtonian liquids through fixed beds of nonspherical particles

The flow of generalized Newtonian liquids through a random fixed bed of particles has been investigated and a universal method of calculation of the creeping flow was suggested. The usefulness of this method has been verified experimentally for the flow of power law, Ellis and Sutterby liquids through fixed beds of different nonspherical particles.     

Accelerated deposition of latex particles onto a rapidly dissolving steel surface

When immersed in a solution containing HF, H₂O₂ and latex, carbon steel rapidly dissolves without gas evolution at a rate limited by the diffusion of reactants or products of dissolution. The diffusion of ions through the boundary layer induces a local electric field of the order of 10 V/cm which attracts negatively charged latex particles toward the steel. Gravimetric measurements of the rate of accumulation of particles next to the steel correlate well with predictions of the electric-field strength over a range of concentration of acid, peroxide, latex and supporting electrolytes.     

Spatially patterned states in systems of interacting catalyst particles

Considering an assemblage of identical catalyst particles exposed to uniform surroundings, we show in this theoretical study that interparticle interactions give rise to spatially patterned steady states under some conditions. In such patterned states, the rate of reaction on one particle differs from the rate on all or some of the neighboring particles. Results, which are based on simple geometric configurations and on the assumption that a single nonisothermal reaction occurs, show that spatially patterned states of two types exist. The first occurs only if a single isolated particle has multiple steady states. The second type, not inferable from the behavior of a single isolated particle, is attributable to the instability of a spatially uniform state caused by particle-particle interactions. This type may occur in cases where the steady state of an isolated particle is unique and globally stable. Though the study is not extended here to complex assemblages, the results suggest that these phenomena may occur in fixed bed reactors and lead to insidious nonuniformities in reactor temperature and catalyst activity.     

Particle adhesion and removal in model systems—VIII: Deposition of rod-like β-FeOOH particles on steel

The adhesion of uniform rod-like β-FeOOH particles onto stainless steel was studied as a function of dispersion pH, ionic strength, and flow rate using the packed bed technique. The results showed the process to be essentially controlled by convective diffusion in the presence and in the absence of interaction forces. The behavior of particles of different shapes and sizes was compared and discussed.     

Reactions of solid particles with nonuinform distribution of solid reactant: The volume reaction model

The effect of non-uniform solid reactant distribution on conversion of solid particles in gas-solid reactions is analyzed based on the volume reaction model. Certain special features of such systems are pointed out. The possibility of ash layer formation in the kinetically controlled regime is discussed. Conditions leading to single or double ash layer formation, both at the center and surface of the particle, in the intermediate regime of diffusion with simultaneous reaction are described. Detailed mathematical equations which are useful for calculation of the conversion-time relationship for particles with non-uniform solid reactant distribution are presented. Comparison is made to reaction of uniform particles and differences in required reaction time for desired conversion are outlined.     

Further consideration on chemical absorption mechanism by aqueous slurries of sparingly soluble fine particles

The absorption rates of SO₂ into an aqueous slurry of Mg(OH)₂ and an aqueous slurry of CaCO₃ with adding MGSO₄, were theoretically derived in terms of the two-reaction-plane model. Experimental absorption rates of dilute S0₂ into above slurries can be predicted by the proposed model in which there is no particle suspending in between the gas-liquid interface and the primary reaction plane.