Population balance classifier

At this point we must use the birth and death functions described in Section 4.2.4.4 and solve this equation for the population in the mill. Then using the population balance over the classifier equation (4.57), the product population, mp(L), can be determined from the population inside the grinding mill, m(L), as follows ... 

The width of the size distribution is often measured in terms of the coefficient of variation (c.v.) of the mass distribution. Randolph and Larson [98] have shown that the coefficient of variation d the mass distribution is constant at 50% for this type of precipitator. This coefficient of variation is usually too large for ceramic powders. Attempts to narrow the size distribution of particles generated in a CSTR can be made by classified product removal, as shown in Figure 6.24. The classification function, p(R), is similar to those discussed in Section 4.2 and can be easily added to the population balance as follows ... 

The mathematical modeling of polymerization reactions can be classified into three levels microscale, mesoscale, and macroscale. In microscale modeling, polymerization kinetics and mechanisms are modeled on a molecular scale. The microscale model is represented by component population balances or rate equations and molecular weight moment equations. In mesoscale modeling, interfacial mass and heat transfer... 

Tomiyama [148] and Tomiyama and Shimada [150] adopted a N + 1)-fluid model for the prediction of 3D unsteady turbulent bubbly flows with non-uniform bubble sizes. Among the N + l)-fluids, one fluid corresponds to the liquid phase and the N fluids to gas bubbles. To demonstrate the potential of the proposed method, unsteady bubble plumes in a water filled vessel were simulated using both (3 + l)-fluid and two-fluid models. The gas bubbles were classified and fixed in three groups only, thus a (3 + 1)- or four-fluid model was used. The dispersions investigated were very dilute thus the bubble coalescence and breakage phenomena were neglected, whereas the inertia terms were retained in the 3 bubble phase momentum equations. No population balance model was then needed, and the phase continuity equations were solved for all phases. It was confirmed that the (3 + l)-fluid model gave better predictions than the two-fluid model for bubble plumes with non-uniform bubble... 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (FIDE). Since analytical solutions of this equation are not available for most cases of practical interest, several numerical solution methods have been proposed during the last two decades as discussed by Williams and Loyalka [209] and Ramkrishna [151]. 

By the use of a generalized population balance the MSMPR modelf is extended to account for unsteady-state operation, classified product removal, crystals in the feed, crystal fracture, variation in magma volume, and time-dependent growth rate. These variations are not included in the following derivations. 

An even greater problem is the quantitative application of the population balance in the coarse grain crystallizers of DTB and fluidized bed design, which due to the cyclical behavior described above are almost impossible to describe with the MSCPR (mixed suspension classified product removal) and CSCPR (classified suspension classified product removal) approaches. 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (PIDE). In the PBE (12.308) the size property variable ranges from 0 to oo. In order to apply a numerical scheme for the solution of the equation a first modification is to fix a finite computational domain. The conventional approximation is to truncate the equation by substitution of the infinite integral limits by the finite limit value max- The function/(, r, t) denotes the exact solution of the exact equation. It might be assumed that the solution of the truncated PBE is sufficiently close to the exact equation so that the two solutions are practically equal. Hence, the solutions of both forms of the PBE are denoted by fiC, r, t). 

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