Mass density disperse phase

For the case in which all particles have the same mass, Um = Un for the case in which all particles have the same material density, Um = Uy. Note that, like Un and Uy, the mass-average disperse-phase velocity will not usually be in closed form. In most applications, the mean particle velocity Up will be set to the mass-average disperse-phase velocity. Thus, unless noted otherwise, we will set Up = Um throughout the rest of this book. The particle-mass source terms are defined by... 

As material parameters, the densities and the viscosities of both phases as well as the interfacial tension a must be listed. We incorporate the material parameters of the dispersion phase and in the relevance list and note separately the material numbers p/pd and rj/rid- Additional material parameters are the (dimensionless) volume ratio of both phases (f) and the mass portion ci of the emulsifler (surfactant) (e.g., given in ppm). 

Efficient phase separation is critical, since cross-phase contamination has an inherently adverse effect on mass-transfer efficiency. In addition, carryover of solvent in aqueous effluent streams results in loss of solvent from the process, impacting process economics. Phase separation is affected by several physicochemical factors, including the viscosities and densities of the opposing bulk phases and the interfacial tension of the two-phase system. All of these properties contribute to the dimensionless dispersion number, which describes the tendency of two dispersed phases to separate... 

Industrial process streams are frequently treated as being single phase fluids, having simple properties of viscosity and density for calculations involving pumping, mass transfer, etc. In fact most industrial process streams occur as dispersions of two or more phases as discussed in earlier chapters. Dispersed phases introduce complications such that, in many cases, the viscosity is not expressed by a single number at constant temperature and pressure, but also depends upon whether the material is flowing, and even its recent history ... 

Note from Table 6.8 that the reduced viscosity gives the relative increase in the viscosity of the solution over the solvent, per unit of concentration. Since r/ is the limiting value of the reduced viscosity, it is a measure of the first increment of viscosity due to the dispersed particles and is therefore characteristic of the particles. Equation (6.33) predicts that the intrinsic viscosity should equal 2.5 for spherical particles. If the dispersed phase volume fraction is used to reflect the dry-weight concentration of particles that may become solvated when dispersed, then intrinsic viscosity measurements can be used to determine the extent of solvation as follows. Suppose the mass of colloidal solute in a solution is converted to the volume of unsolvated material using the dry density. If the particles are assumed to be uniformly solvated throughout the dispersion then the solvated particle volume exceeds that of the unsolvated particle volume by the factor 1 + (m], b/m2)(p2/Pi) where my, is the mass of bound solvent, m2 is the mass of the solute particle, p2 is the density of the particle and pi is the density of the solvent. Since 4>[Pg.185]

When two liquids are immiscible, the design parameters include droplet size distribution of the disperse phase, coalescence rate, power consumption for complete dispersion, and the mass-transfer coefficient at the liquid-liquid interface. The Sauter mean diameter, dsy, of the dispersed phase depends on the Reynolds, Froudes and Weber numbers, the ratios of density and viscosity of the dispersed and continuous phases, and the volume fraction of the dispersed phase. The most important parameters are the Weber number and the volume fraction of the dispersed phase. Specifically, dsy oc We 06(l + hip ), where b is a constant that depends on the stirrer and vessel geometry and the physical properties of the system. Both dsy and the interfacial area aL remain unaltered, if the same power per unit volume (P/V) is used in the scale-up. 

Static packing columns also have an improved mass transfer with less axial dispersion however, the dispersed phase should not wet the packings. These extraction towers, which depend on density differences, have fairly large throughput and are inexpensive extraction units. 

Under turbulent flow conditions, power dissipation is the controlling factor for mass transfer and phase dispersion. The power drawn by a single impeller in a liquid-gas system is typically lower than that drawn by the same impeller in liquid alone. The presence of the gas reduces the average density of the mixture, and the gas flow regime (e.g., flooding) may cause the impeller blades to be locally surrounded by a higher... 

P/ (Pc V) and the volume portion od the dispersed phase. A possible influence of the the mass density ratio of both phases was not considered. 

The dispersed phase volume fraction is related to the mass density by ... 

In an early attempt to calculate the phase fractions in an approximate implicit volume fraction-velocity-pressure correction procedure, Spalding [176, 177, 178, 180] calculated the phase fractions from the respective phase continuity equations. However, experience did show that it was difficult to conserve mass simultaneously for both phases when the algorithm mentioned above was used. For this reason, Spalding [179] suggested that the volume fraction of the dispersed phase may rather be calculated from a discrete equation that is derived from a combination of the two continuity equations. An alternative form of the latter volume fraction equation, particularly designed for fluids with large density differences, was later proposed by Carver [26]. In this method the continuity equations for each phase were normalized by a reference mass density to balance the weight of the error for each phase. 

If the disperse phase has particles with different volumes and different masses (or material densities), at least two internal coordinates are necessary to describe a particle state. In order to derive a transport equation of the disperse-phase mass density pp, we will let... 

In many applications, the particles will be composed of multiple chemical species. In such cases, it is necessary to introduce a vector of internal coordinates p whose components are the mass of each chemical species. Obviously, the sum of these internal coordinates is equal to the particle mass. By definition, if pa, is the mass of component a, then integration over phase space leads to a component disperse-phase mass density ... 

Consider again a system wherein all particles have the same volume and mass, and the disperse-phase momentum density is gp = ppOp. The transport equation for the disperse-phase momentum density for this case is ... 

When the elements of the disperse phase can be classified as equidimensional, namely they have nearly the same size or spread in multiple directions, and have constant material density, typically a single internal coordinate is used to identify the size of the elements. This could be particle mass (or volume), particle surface area or particle length. In fact, in the case of equidimensional particles these quantities are all related to each other. For example, in the trivial cases of spherical or cubic particles, particle volume and particle surface area can be easily written as Vp = k d and Ap = k d, or, in other words, as functions of a characteristic length, d (i.e. the diameter for the sphere and the edge for the cube), a volume shape factor, k, and a surface-area shape factor, k. For equidimensional objects the choice of the characteristic length is straightforward and the ratio between kp, and k is always equal to six. The approach can, however, be extended also to non-equidimensional objects. In this context, the extension turns out to be very useful only if... 

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