Generalized population balance equation moments

Not infrequently, practical needs can be fulfilled by calculating the (generally integral) moments of the number density function. The calculation of such moments can occasionally be accomplished by directly taking moments of the population balance equation producing a set of moment equations. 

Thus (4.4.8) shows how any term in the population balance equation involving the number density can be expressed purely in terms of the first M integral moments. While the use of generalized Laguerre polynomials with additional parameters provided some flexibility for the method of moments, Hulburt and Akiyama have reported difficulties with this method. 

Tab. 9.11. The general (N, M)th double moment formulation of the population balance equation set of Table 9.10, obtained by multiplying by the TDB number and branching number indices, / and and subsequent summation over these indices.

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

The population balance in equation 2.86 employs the local instantaneous values of the velocity and concentration. In turbulent flow, there are fluctuations of the particle velocity as well as fluctuations of species and concentrations (Pope, 1979, 1985, 2000). Baldyga and Orciuch (1997, 2001) provide the appropriate generalization of the moment transformation equation 2.93 for the case of homogeneous and non-homogeneous turbulent particle flow by Reynolds averaging... 

This problem has been introduced in the discussion of the classes approach. For reaction equations and a full set of population balances, see Tables 9.5 and 9.6. Here, we address the more general problem of more than one TDB per chain [9]. This occurs as a consequence of insertion of TDB chains created by disproportionation or of recombination termination. We start with the full 3D set of Table PVAc2 and then reduce it to a ID formulation by developing the TDB and branching moment expressions. The (N, M)th branching-TDB moments or pseudo distributions for living and dead chains are defined by ... 

Performing the corresponding summations on the equations in Table 9.6, one obtains the (N, JVf)th moment formulation of Table 9.11. Some of the summation terms in these equations will not be evaluated for the general (N, M) case, but we will determine them by assigning values to N and M. Since we will not address branching, we take M = 0 here, but in principle this can be treated in a similar way. We will focus now on the TDB moment distributions and successively derive the model equations for the zeroth, first, and second moments, or N = 0,1, and 2. Solving the model thus essentially means solving the population balances of the real concentration distributions and P and the pseudo-distributions and... 

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