Volume-averaged population

The continuum form of the bubble population balance, applicable to flow of foams in porous media, can be obtained by volume averaging. Bubble generation, coalescence, mobilization, trapping, condensation, and evaporation are accounted for in the volume averaged transport equations of the flowing and stationary foam texture. 

The zeroth order moments of the volume averaged bubble population equations, i.e., the balances on the total bubble density in flowing and stationary foam, have the form of the usual transport equations and can be readily incorporated into a suitable reservoir simulator. 

The purpose of this Appendix is to volume-average the population balance of bubble number density... 

F1g 42 Test of the relationships between particle size distribution and column characteristics. In both plots the van Deemter C term and the permeability are plotted against the square of the average particle size. In Figure 4.2u the x axis is derived from the population-averaged particle size, and in Figure 4.2h the x axis is derived from the volume-averaged particle size. 

The chemically important Maximum Probability Domains for the same systems are shown in Figs. 7 and 8. For magnesium oxide, the cation and anion 10 electrons MPD have high probability 0.85 and 0.57 for cation and anion, respectively the volumes of two ions are very different, 19.4 and 90.5 cubic bohrs the MPD average net charges are, respectively, -i-l. 85 and -1.85 and these are very close to the formal ones. The two-electron MPD reported in Fig. 8 has a probability equal to 0.42, a volume of 56.0 cubic bohrs, and an average population of 2.04 electrons. 

The Euler-Euler model is based on the volume average form of the transport equation developed for multiphase systems [46] and is coupled with the population balance equations for the particle agglomeration [139]. The general model equations... 

Equation (3.2.15) becomes identical to (3.2.8). Thus the applicability of Equation (3.2.8) by using volume-averaged breakage functions for describing the evolution of drop size distributions in a stirred vessel depends upon the rapid circulation of the drop population through the recirculation zone. This discussion also points to the inadequacy of Equation (3.2.8) in describing the process in large stirred vessels where the assumption of uniform population density in the vessel may not be borne out. 

The diametei of average mass and surface area are quantities that involve the size raised to a power, sometimes referred to as the moment, which is descriptive of the fact that the surface area is proportional to the square of the diameter, and the mass or volume of a particle is proportional to the cube of its diameter. These averages represent means as calculated from the different powers of the diameter and mathematically converted back to units of diameter by taking the root of the moment. It is not unusual for a polydispersed particle population to exhibit a diameter of average mass as being one or two orders of magnitude larger than the arithmetic mean of the diameters. In any size distribution, the relation ia equation 4 always holds. 

This result can be useful for design purposes when the diffusivities, partition coefficients, feed-stream conditions, dispersed-system volume, gas-phase holdup (or average residence time), and the size distribution are known. When the size distribution is not known, but the Sauter-mean radius of the population is known, (293) can be approximated by... 

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