Product particle size distribution population balance

Simplify the macroscopic population balance to describe the particle size distribution in a continuous constant volume isothermal well-mixed crystallizer with mixed product removal operating at steady state. Assume the crystallizer feed streams are free of suspended particles, that the crystallizer operates with ne igible breakage, and that agglomeration and crystallization cause no change in the volume of the system. 

These models require information about mean velocity and the turbulence field within the stirred vessels. Computational flow models can be developed to provide such fluid dynamic information required by the reactor models. Although in principle, it is possible to solve the population balance model equations within the CFM framework, a simplified compartment-mixing model may be adequate to simulate an industrial reactor. In this approach, a CFD model is developed to establish the relationship between reactor hardware and the resulting fluid dynamics. This information is used by a relatively simple, compartment-mixing model coupled with a population balance model (Vivaldo-Lima et al., 1998). The approach is shown schematically in Fig. 9.2. Detailed polymerization kinetics can be included. Vivaldo-Lima et a/. (1998) have successfully used such an approach to predict particle size distribution (PSD) of the product polymer. Their two-compartment model was able to capture the bi-modal behavior observed in the experimental PSD data. After adequate validation, such a computational model can be used to optimize reactor configuration and operation to enhance reactor performance. 

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. 

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. 

Particulate processes are characterized by properties like the paxticle shape, size, surface area, mass, and product purity. In crystallization the particle size and total number of crystals vary with time. Thus, determining particle size distribution (PSD) is important in crystallization. A population balance formulation describes the process of crystal size distribution with time most effectively. Thus, modeling of a batch crystallizer involves use of population balances to model the crystal size... 

Crystal nucleation and growth in a crystalliser cannot be considered in isolation because they interact with one another and with other system parameters in a complex manner. For a complete description of the crystal size distribution of the product in a continuously operated crystalliser, both the nucleation and the growth processes must be quantified, and the laws of conservation of mass, energy, and crystal population must be applied. The importance of population balance, in which all particles are accounted for, was first stressed in the pioneering work of Randolph and Larson1371. ... 

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 ... 

Particulate products, such as those from comminution, crystallization, precipitation etc., are distinguished by distributions of the state characteristics of the system, which are not only function of time and space but also some properties of states themselves known as internal variables. Internal variables could include size and shape if particles are formed or diameter for liquid droplets. The mathematical description encompassing internal co-ordinate inevitably results in an integro-partial differential equation called the population balance which has to be solved along with mass and energy balances to describe such processes. 

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