Population balance classification

In this paper, three methods to transform the population balance into a set of ordinary differential equations will be discussed. Two of these methods were reported earlier in the crystallizer literature. However, these methods have limitations in their applicabilty to crystallizers with fines removal, product classification and size-dependent crystal growth, limitations in the choice of the elements of the process output vector y, t) that is used by the controller or result in high orders of the state space model which causes severe problems in the control system design. Therefore another approach is suggested. This approach is demonstrated and compared with the other methods in an example. 

The discussion of the transformation methods will be based on the population balance for a crystallizer with fines removal and product classification ... 

Several classification functions, C(L), are given in Figure 4.20. Here, the fraction of particles by mass reporting to the recycle stream is given as a function of particle size, L, for a screen and a cyclone. Several authors have used empirical classification functions instead of dassifier performance curves with reasonable results for the overall comminution-classification circuit control. The steady state (i.e., dmidt = 0) macroscopic population balance on a discrete mass basis over the grind-... 

This two step population balance for communition and classification can be rewritten in array form [21] ... 

In this chapter, the fundamentals of classification and comminution of ceramic powders have been described. Comminution is described by birth and death functions in a population balance. These birth and... 

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 ... 

For the plant constructors, the benefit of the population balance is mainly derived from the transparency of the interplay of the kinetic variables of nucleation rate and crystal growth rate, the mean retention time, the fines dissolving, and the classification of product crystals during withdrawal with respect to the resulting mean particle size of the crystallization process. Here, reference is made to the publications, such as that of Toyokura and Sakai [12]. 

The fact that the population balance is clearly able to determine the influences on particle size distribution is shown, for example, by the effects produced by the fines dissolution and the classification of product during withdrawal. 

The solute balance reduced to a constraint on growth rate. The model did not result, however, in the required state-space representation essential for multi-variable model based control. A finite difference approximation of the crystal size axis was used to convert population balance into a set of ODEs, which was then linearized at an operating point to obtain the required state-space representation. The need for a product classification step, whereby coarse crystals are removed at a finite size in addition to fines dissolution, was highlighted. Hydrocyclones were suggested for reducing CSD dispersion by using variable underflow discharge diameter as an additional input for control. 

Rousseau and Howell (1982) considered the merits of using different measurements for stabilizing low order cycling of CSD in a continuous crystallizer with both fines destruction and product classification. The analysis was carried out on a simulated process using population and mass balances along with kinetic equations and employed finite difference techniques to solve the system. The main advantages of using a finite difference method in comparison with a linearized form of analytical solution were cited as (a) no modifications to the models were necessary to accommodate different removal functions and (b) any form of nucleation kinetics could be used. 

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