Product density equations

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. 

Equation (7.3.2), however, provides only the average behavior of the system. It is of interest to observe here that the average behavior of the system could be obtained by dealing only with the first-order product density, viz., the expected population density. A truly stochastic formulation must consider, however, the higher order densities in order to calculate the average fluctuations about the mean behavior. The calculation of fluctuations was the subject of Sections 7.1.1.1 and 7.1.1.2. Since the higher order densities were the basic implements of this calculation, it will be our objective to first formulate the second-order product density equation for the breakage process under discussion. 

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... 

In the next section, we consider the derivation of product density equations for an aggregation process. 

Thus, the product density equations are clearly unclosed, making it impossible to solve them without a suitable closure hypothesis. If it is known that there are no more than particles in the system initially, the closure is automatic at r = AT, since in the equation for /n /no+i — identify equations in the master density, using probabilistic arguments as in the breakage process. The resulting hierarchy of equations is slightly different from (7.3.9) ... 

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. 

Product Density Equations for the Environment-Dependent Case... 

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. 

Equations (7.3.12) and (7.3.13) may now be averaged to obtain the product density equations. The first density function of interest is / y(y, which is the only true probability density in the set of product densities. Using the definition for this density, Eq. (7.3.13) may be directly integrated with respect to all the particle coordinates, divided by v and summer over all V to yield... 

We address applications here in which closure problems are not encountered. Thus, the average behavior of the population can be obtained from solving the first-order product density equation, and average fluctuations (of any order) about the mean can be calculated progressively by solving higher-order product density equations. In order to elucidate the nature of what can be obtained from such a theory, we shall consider a simple enough example for which analytical answers can be found. It is followed by a second example which has potential application to the study of cell death kinetics and hence to sterilization processes. 

The product density equations can be solved by Lapalace transform with respect to age. Thus, defining... 

We thus have the first two moments of the population by solving the first-order and second product density equations for the given process. The coefficient of variation of the total population, denoted COVN, which is obtained by calculating /VN/EN, is seen to be... 

In dealing with stochastic problems, it became clear from Section 7.3 that one is frequently faced with lack of closure, especially in situations where interaction occurs between particles or between particles and their environment. Such lack of closure arises because of the development of correlations between particle states promoted by preferential behavior between particle pairs of specific states or between the particle and its environment. The population balance equation, which generally comes about by making the crudest closure approximation, does not make accurate predictions in such cases of the average behavior of the system. The question naturally arises as to whether one can find other mean field descriptions by making more refined closure approximations on the unclosed product density equations. 

The foregoing condition may be used to determine t) in (7.4.5). Equations (7.4.5) and (7.4.6) then consist of the population balance model for aggregating systems with a local volume of mixing as defined in this section. It is our objective next to compare the prediction of this population balance model with Monte Carlo simulations using the interval of quiescence in Section 4.6 as well as with predictions made from the product density equations using various closure approximations. The Monte Carlo approach provides one... 

What is left in the sequel is the identification of the product density equations for the process above. Following Sampson and Ramkrishna (1986), we directly write the equations in the product densities integrated over the volume so that the densities are only in particle size coordinate but not in spatial volume and recognize them with a hat on top. This implies that we must use the redefined aggregation frequency a x, x ) in describing the aggregation process. The expected population density or the first-order product density must satisfy... 

The closed set of product density equations is given by Eqs. (7.4.7) through (7.4.9) in which if is any one of the set if, if2, if3, if4, if 5. The solution of the set of product density equations is, however, most efficiently done by recognizing a set of integral constraints that arise naturally. Taking... 

Multiplying the second-order product density equation (7.4.8) by xx and integrating over the semi-infinite interval with respect to both x and x we get on recognition of the conservation of mass (which eliminates the aggregation terms) the following integral constraint ... 

Sampson, K. J. and D. Ramkrishna, Particle Size Correlations in Brownian Agglomeration. Closure Hypotheses for Product Density Equations, J. Colloid Inter/ Sci. 110, 410-423 (1986). 

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