Suspensions continuity equation

One-dimensional flow models are adopted in the early stages of model development for predicting the solids holdup and pressure drop in the riser. These models consider the steady flow of a uniform suspension. Four differential equations, including the gas continuity equation, solids phase continuity equation, gas-solid mixture momentum equation, and solids phase momentum equation, are used to describe the flow dynamics. The formulation of the solids phase momentum equation varies with the models employed [e.g., Arastoopour and Gidaspow, 1979 Gidaspow, 1994], The one-dimensional model does not simulate the prevailing characteristics of radial nonhomogeneity in the riser. Thus, two- or three-dimensional models are required. 

In a sheared suspension, the effects are two-fold. First, the expression for bulk stress itself must be modified. Second, the probability density is affected since the continuity equation for the latter must be replaced by a convection-diffusion equation. As a consequence, the distinction between open and closed trajectories loses some of its meaning. Batchelor (1977) gives the equivalent viscosity of a sheared suspension subject to strong Brownian motion as... 

In the case of multiparticle blockage, as the suspension flows through the medium, the capillary walls of the pores are gradually covered by a uniform layer of particles. This particle layer continues to build up due to mechanical impaction, particle interception and physical adsorption of particles. As the process continues, the available flow area of the pores decreases. Denoting as the ratio of accumulated cake on the inside pore walls to the volume of filtrate recovered, and applying the Hagen-Poiseuille equation, the rate of filtration (per unit area of filter medium) at the start of the process is ... 

A number of analytical solutions have been developed since that of von Smoluchowski, all of which contain some assumptions and constraints. Friedlander [33] and Swift and Friedlander [34] developed an approach relaxing the above constraint of an initially monodisperse suspension. Using a continuous particle size distribution function, a nonlinear partial integro-differential equation (with no known solution) results from Eq. (5). Friedlander [35] demonstrated the utility of a similarity transformation for representation of experimental particle size distributions. Swift and Friedlander [34] employed this transformation to reduce the partial integro-differential equation to a total integro-differential equation, and dem-... 

Consider first the effect of a dispersed phase, of volume fraction continuous phase of viscosity D0 and dispersed particles (droplets) which do not attract. At low volume fractions the Einstein equation should apply to a suspension of solid particles at constant temperature,... 

Pritchett et al. (1978) were the first to report numerical solutions of the nonlinear equations of change for fluidized suspensions. With their computer code, for the first time, bubbles issuing from a jet with continuous gas through-flow, could be calculated theoretically. Figure 14 shows as an illustration the computed mo-... 

It is frequently desirable to be able to describe emulsion viscosity in terms of the viscosity of the continuous phase tjq) and the amount of emulsified material. A very large number of equations have been advanced for estimating suspension (or emulsion, etc.) viscosities. Most of these are empirical extensions of Einstein s equation for a dilute suspension of spheres ... 

As the size ratio of the sand particle to the oil droplets d d increases to about 2, there is less dependence on the oil concentration, as shown in Figure 21b. When the size ratio increases to about 3, as shown in Figure 21c, the relative viscosity becomes independent of the oil concentration this result indicates that the emulsions act as a continuous phase toward the solids. Under this condition, the solids and the droplets behave independently, and no interparticle interaction occurs between the solids and the droplets. Yan et al. (64) showed that when the emulsions behave as a continuous phase toward the solids, the viscosity of the mixtures can be predicted quite accurately from the viscosity data of the pure emulsions and the pure solids suspensions. The viscosity of an emulsion-solids mixture having an oil concentration of Pq (solids-free basis) and a solids volume fraction of 0s (based on the total volume) can be calculated from the following equation ... 

For a given cross section of the dilute phase (this phase includes a part of the transition zone) the particles contact the ascending gas in a complicated way. Swarms of suspended particles may show a mass-transfer resistance for gas-particle contact, whereas a very dilute suspension of catalyst will not involve any appreciable mass-transfer resistance. Accordingly, one can assume that the fracticm directly ccmtacts the reactant gas the rest of the catalyst Ced in dense clusters, indirectly ccmtacts the gas with the overall mass-transfer coefficient k Oc- Under the simplifying assumption of ideal plug flow of gas through the dilute phase, the equations of continuity are given as follows ... 

A mathematical model for styrene polymerization, based on free-radical kinetics, accounts for changes in termination coefficient with increasing conversion by an empirical function of viscosity at the polymerization temperature. Solution of the differential equations results in an expression that calculates the weight fraction of polymer of selected chain lengths. Conversions, and number, weight, and Z molecular-weight averages are also predicted as a function of time. The model was tested on peroxide-initiated suspension polymerizations and also on batch and continuous thermally initiated bulk polymerizations. 

The basis for interpreting the transport data is mixture theory, which relates the transport properties of the bulk suspension to those of the continuous and dispersed phases. Of the many mixture relations that have been proposed, we employ those of Maxwell and Hanai (Equations 1 and 2, respectively) ... 

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