Dispersion-sedimentation models

Due to density differences the particles have the tendency to settle. Thus, solid concentration profiles result which can be described on the basis of the sedimentation-dispersion model (78,79,80). This model involves two parameters, namely, the solids dispersion coefficient, E3, and the mean settling velocity, U5, of the particles in the swarm. Among others Kato et al. (81) determined 3 and U3 in bubble columns for glass beads 75 and 163 yum in diameter. The authors propose correlations for both parameters, E3 and U3. The equation for E3 almost completely agrees with the correlation of Kato and Nishiwaki (51) for the liquid phase dispersion coefficient. 

The mechanism by which solids are distributed throughout a vessel once they are suspended is different from that leading to suspension. It might be expected that the solids distribution would again be affected by the bulk flow pattern, i.e. the mean velocities throughout the vessel as well as the turbulence structure. These flows oppose the gravitational downwards force. As will be shown later, the measured vertical concentration profiles are very complicated, much more so than in solids transport in pipe flow, for example, where a steady decrease in concentration occurs from top to bottom. This concentration decay in pipes can be modelled rather easily by a one dimensional sedimentation-dispersion model. A similar model has been proposed for stirred vessels for the region above the impeller. 

In general, catalyst sedimentation has to be accounted for in slurry reactors. The distribution of the catalyst along the reactor can be computed using the sedimentation-dispersion model. As to the results of Kato et al. (73), the solid dispersion coefficients do not differ much from those of the liquid phase. From the data provided by Cova (74), Imafuku et al (75), and Kato et al. (73), the solids concentration profiles can be calculated. As in the FT process the catalyst particles are usually small, according to Kolbel and Ralek (35) the diameter should be less than 50 um, the catalyst profiles are not very pronounced, in accordance to the measurements of Cova (74). 

If catalyst particles of larger size and density are used and sedimentation might be important the correlations proposed by Kato et al. (73) are recommended to calculate the solids dispersion coefficient and the settling velocity of the particle. From this data the solids concentration profile can be computed with the sedimentation-dispersion model. 

The catalyst must not be uniformly distributed over the entire suspension volume which is considered by introducing the sedimentation-dispersion model (73-75). 

A theoretical analysis of the sedimentation-dispersion model by Jean et al. (1989) indicated that in a common expression of the sedimentation-dispersion model with U (defined as the cross-sectional averaged... 

Jean RH, Tang WT, Fan LS. The sedimentation-dispersion model for slurry bubble columns. AIChE J 35 662-665, 1989. 

A different approach which also starts from the characteristics of the emissions is able to deal with some of these difficulties. Aerosol properties can be described by means of distribution functions with respect to particle size and chemical composition. The distribution functions change with time and space as a result of various atmospheric processes, and the dynamics of the aerosol can be described mathematically by certain equations which take into account particle growth, coagulation and sedimentation (1, Chap. 10). These equations can be solved if the wind field, particle deposition velocity and rates of gas-to-particle conversion are known, to predict the properties of the aerosol downwind from emission sources. This approach is known as dispersion modeling. 

Cova (3 ) measured the solid concentration profiles of a Raney nickel catalyst with an average diameter of 15.7 ym in a h.6 cm id reactor, using water and acetone as the liquids. He developed a sedimentation diffusion model, assuming solid and liquid dispersion coefficients were equal, and slurry settling velocities were independent of solid concentration. The model was then applied to data for Raney nickel in 6.35 and kk.J cm id bubble columns, in both cocurrent and countercurrent flow. 

The limitations of the sedimentation-diffusion model for describing solid dispersion in these systems are discussed. 

The sedimentation diffusion model, when applied to the iron oxide system, gave solid settling velocities in agreement with theory. Solid dispersion coefficients were in the range predicted hy the Kato correlation, hut showed considerable experimental scatter. 

In solid-liquid mixing design problems, the main features to be determined are the flow patterns in the vessel, the impeller power draw, and the solid concentration profile versus the solid concentration. In principle, they could be readily obtained by resorting to the CFD (computational fluid dynamics) resolution of the appropriate multiphase fluid mechanics equations. Historically, simplified methods have first been proposed in the literature, which do not use numerical intensive computation. The most common approach is the dispersion-sedimentation phenomenological model. It postulates equilibrium between the particle flux due to sedimentation and the particle flux resuspended by the turbulent diffusion created by the rotating impeller. 

The design, scaleup and performance prediction of slurry reactors require models which must consider not only the hydrodynamic and mixing behavior of the three phases, but also the mass transfer between the phases along with the intrinsic kinetics. In the DCL and FTS processes, an axial dispersion model is applicable, with the solid phase assumed to follow sedimentation or dispersed flow model. However, in the CCC, where the solid particles take part in the reaction, dispersion model is no longer applicable. 

The design and development of a novel vibrating reed technique for on-line measurement of the sedimentation kinetics of two-phase dispersions is described. The technique has been tested in conjunction with a variety of solid/liquid and liquid/liquid dispersions with dense phase concentrations in the range 0-50 % v/v. Typical output include settling velocities, solids flux profiles as well as solids throughputs. Additionally, the performance of a number of sedimentation kinetics models proposed for dilute systems (0 - 2.81 % v/v) are evaluated by comparison with data obtained using the device. 

Some of the principles of free surface model proposed by Happel (H10, P7, P8) for studying the rate of sedimentation of solid particles may be adopted for studying heat and mass transfer in gas-liquid dispersions (R9). 

OIC Analytical instruments produce the fully computerized model 700 total organic carbon analyser. This is applicable to soils and sediments. Persulphate oxidation at 90-100°C non-dispersive infrared spectroscopy is... 

Stokes s law and the equations developed from it apply to spherical particles only, but the dispersed units in systems of actual interest often fail to meet this shape requirement. Equation (12) is sometimes used in these cases anyway. The lack of compliance of the system to the model is acknowledged by labeling the mass, calculated by Equation (12), as the mass of an equivalent sphere. As the name implies, this is a fictitious particle with the same density as the unsolvated particle that settles with the same velocity as the experimental system. If the actual settling particle is an unsolvated polyhedron, the equivalent sphere may be a fairly good model for it, and the mass of the equivalent sphere may be a reasonable approximation to the actual mass of the particle. The approximation clearly becomes poorer if the particle is asymmetrical, solvated, or both. Characterization of dispersed particles by their mass as equivalent spheres at least has the advantage of requiring only one experimental observation, the sedimentation rate, of the system. We see in sections below that the equivalent sphere calculations still play a useful role, even in systems for which supplementary diffusion studies have also been conducted. 

There are three chapters in this volume, two of which address the microscale. Ploehn and Russel address the Interactions Between Colloidal Particles and Soluble Polymers, which is motivated by advances in statistical mechanics and scaling theories, as well as by the importance of numerous polymeric flocculants, dispersants, surfactants, and thickeners. How do polymers thicken ketchup Adler, Nadim, and Brenner address Rheological Models of Suspensions, a closely related subject through fluid mechanics, statistical physics, and continuum theory. Their work is also inspired by industrial processes such as paint, pulp and paper, and concrete and by natural systems such as blood flow and the transportation of sediment in oceans and rivers. Why did doctors in the Middle Ages induce bleeding in their patients in order to thin their blood ... 

All the above formulas are one-parameter equations, i.e. they relate the dispersion viscosity only to the volume fraction of particles contained in it. This limits the range of applicability of the equations to not very high dispersion concentrations. To take account of the influence of the structure of concentrated dispersions on their rheological behavior, Robinson [12] suggested that the viscosity of dispersions is not only propertional to the volume fraction of solid phase, but is also inversely proportional to the fraction of voids in it. (At about the same time Mooney [40], who proceeded from a hydrodynamic model, arrived, using theoretical methods, at the same conclusion). Robinson s equation contains the relative sedimentation volume value — S, which depends on the particle size distribution of the dispersion... 

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