Master density equation

Third, Section 7.3 goes into the direct derivation of stochastic equations of population balance. These equations are also obtainable from averaging the master density equations of Section 7.2, but are best obtained by using the methodology of Section 7.3. Some applications of stochastic analysis are shown in this section, which are of focal interest to the subject of this chapter. 

The Population Balance Equation via Averaging of the Master Density Equation... 

We have thus established the population balance equation and its boundary condition rigorously from the master density equation. It is possible in an entirely analogous manner to also derive equations in the higher order product densities by appropriately averaging Eq. (7.2.7) and thus facilitate the calculation of fluctuations. We do not take this route here because we shall derive the product density equations directly from their probability interpretations in Section 7.3. 

It is interesting to observe that Eq. (7.2.5) in the master density equation can be solved somewhat readily in a conceptual sense. The issue has been dealt with by the author at length in a publication referenced in footnote 7. We shall discuss only the broad features of this development here. 

We have thus derived the second-order product density equation for the pure breakage process based purely on probabilistic considerations. It could also have been derived by averaging the master density equation (7.2.5), using the definition (7.1.11). The boundary conditions for the second-order product density can be obtained in much the same manner as the first-order product density by accounting for the formation of nuclei. Thus, recognizing the symmetry of the product density, we have the boundary condition... 

Obviously, the preceding hierarchy of master density equations can also be closed at v = N. However, the product density equations may allow closure at a considerably lower value of r, which makes them much more attractive to solve than the master density equations. As pointed out earlier, even an analytical solution to the master density equation is not particularly valuable because of its combinatorial complexity. 

A discrete version of the master density equations (7.3.10), without particle growth, has been solved by Bayewitz et al (1974), and later by Williams (1979), to examine the dynamic average particle size distribution in an aggregating system with a constant kernel. When the population is small EN < 50) their predictions reveal significant variations from those predicted by the population balance equation. However, the solution of such master density equations is extremely difficult even for the small populations of interest for nonconstant kernels. It is from this point of view that a suitably closed set of product density equations presents a much better alternative for analysis of such aggregating systems. We take up this issue of closure again in Section 7.4. 

In deriving the product density equations, we shall take the route of first identifying the master density equation and obtain the former by averaging. We prefer this route to that of direct derivation of the product density equations in this case because the rate of change of the environmental variable is given by (7.3.11) which involves all the particles in the system. 

The author s derivation of the master density equation for a microbial population with environmental effects is available in Ramkrishna (1979). 

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