Particle flux convective

On the other hand the analysis by a detailed diffusion/convection model leads to a mass-balance equation in a slab adsorbent particle which includes terms relating to diffusive flux, convective flux and accumulation ... 

This equation can be exploited for determining the particle diffusion coefficient by measiuing experimentally the number of irreversibly adsorbed particle as a function of time. This is done under the diffusion transport conditions by eliminating all natural and forced convection ciurents [2,9,10,16]. However, due to the fact that particle flux decreases gradually with the time, diffusion-controlled adsorption becomes very inefficient for long times. Indeed, in these experiments, adsorption times reached tens of hours. 

The particle flux density (JJ can be due to the combination of convection, diffusion, and magnetic force given as... 

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

For the region near the attachment point, Mullis found a strong effect of axial position on flux, but no satisfactory general correlation for this effect. In addition, he found no quantitative relation for the heat-transfer characteristics of jets directed toward the propellant surface. Under most conditions studied by Mullis, the radiation contribution is approximately 10% of the convective flux. The effects of solid-particle impingement were not investigated. 

The data of Fig. 20 also point out an interesting phenomenon—while the heat transfer coefficients at bed wall and bed centerline both correlate with suspension density, their correlations are quantitatively different. This strongly suggests that the cross-sectional solid concentration is an important, but not primary parameter. Dou et al. speculated that the difference may be attributed to variations in the local solid concentration across the diameter of the fast fluidized bed. They show that when the cross-sectional averaged density is modified by an empirical radial distribution to obtain local suspension densities, the heat transfer coefficient indeed than correlates as a single function with local suspension density. This is shown in Fig. 21 where the two sets of data for different radial positions now correlate as a single function with local mixture density. The conclusion is That the convective heat transfer coefficient for surfaces in a fast fluidized bed is determined primarily by the local two-phase mixture density (solid concentration) at the location of that surface, for any given type of particle. The early observed parametric effects of elevation, gas velocity, solid mass flux, and radial position are all secondary to this primary functional dependence. 

The left-hand sideofEq. (40)isthe accumulationofparticlesofagivensize. The terms on the right-hand side are, in turn, the bulk flow into and out of the control volume, the convective flux along the size axis due to layering and attrition, the birth of new particles due to nucleation, and birth and death of granules due to coalescence. 

Cross-flow filtration systems utilize high liquid axial velocities to generate shear at the liquid-membrane interface. Shear is necessary to maintain acceptable permeate fluxes, especially with concentrated catalyst slurries. The degree of catalyst deposition on the filter membrane or membrane fouling is a function of the shear stress at the surface and particle convection with the permeate flow.16 Membrane surface fouling also depends on many application-specific variables, such as particle size in the retentate, viscosity of the permeate, axial velocity, and the transmembrane pressure. All of these variables can influence the degree of deposition of particles within the filter membrane, and thus decrease the effective pore size of the membrane. 

Examine data taken under a natural convection lateral vertical surface spread for a particle board as shown in Figures 8.12(a) and (b) [17]. The correlation of these data based on Equation (8.29) suggests that the critical flux for ignition is... 

When the voltage is critical, regime b), there is no concentration polarization because the electrophoretic transport is equal to the convective transport. Any build up of species on the membrane will be dissipated due to diffusion driven by the concentration difference. In this regime, increasing the tangential velocity is expected to have no influence on the flux because fluid shear can only improve the transport of particles down a concentration gradient. In this case, there is no concentration gradient. 

In Figure 50, the lower curve for E=3.9 v/cm shows a transition in slope. The flux decreases with decreasing Reynolds numbers until a point is reached where the convective transport of particles toward the membrane is just equal to the electrophoretic migration away from the membrane-i.e. the voltage is now the critical voltage. Further decreases in the Reynolds number will not decrease the flux as there is now no concentration polarization. 

The concept of critical flux ( Jcrit) was introduced by Field et al. [3] and is based on the notion that foulants experience convection and back-transport mechanisms and that there is a flux below which the net transport to the membrane, and the fouling, is negligible. As the back transport depends on particle size and crossflow conditions the Jcrit is species and operation dependent. It is a useful concept as it highlights the... 

Since the particle residence time near the tube walls is small relative to their thermal relaxation times (see Table 1), the temperature of the particles near the wall does not depart appreciably from the mean bed temperature. The heat transfer coefficient from the particles is then governed by conduction across the gap separating the particles from the wall. The sum of the conductive and convective fluxes can be calculated from the expression derived by Decker and Glicksman (7 ) ... 

Boundary layer formulation. Many membrane processes are operated in cross-flow mode, in which the pressurised process feed is circulated at high velocity parallel to the surface of the membrane, thus limiting the accumulation of solutes (or particles) on the membrane surface to a layer which is thin compared to the height of the filtration module [2]. The decline in permeate flux due to the hydraulic resistance of this concentrated layer can thus be limited. A boundary layer formulation of the convective diffusion equation can give predictions for concentration polarisation in cross-flow filtration and, therefore, predict the flux for different operating conditions. Interparticle force calculations are used in two ways in this approach. Firstly, they allow the direct calculation of the osmotic pressure at the membrane. This removes the need for difficult and extensive experi-... 

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