Models population balance and

Macroscopic Reactor Modeling - Population Balances and the Method of Moments... 

Macroscopic Reactor Modeling - Population Balances and the Method of Moments 411 Taking the first derivative of Eq. (68), one obtains Eq. (69). 

Schreiner etal. (2001) modelled the precipitation process of CaC03 in the SFTR via direct solution of the coupled mass and population balances and CFD in order to predict flow regimes, induction times and powder quality. The fluid dynamic conditions in the mixer-segmenter were predicted using CFX 4.3 (Flarwell, UK). 

Coating Mass Uniformity and Distribution. Basically, there have been two approaches to model the accumulation of mass (coating material) on the surface of bed particles (i) the use of population balances and (ii) the probabilistic modelling of the spray-particle interaction. We will look at each of these approaches and see how it may be possible to combine these methods to give a fuller picture of coating performance. 

Himmelblau [32] and Himmelblau and Bischoff [33] have considered three types of model which are useful in process analysis, i.e. empirical models, population balance models and transport phenomena models. Empirical models involve mathematical relationships between dependent and independent variables, which are postulated either entirely a priori, or by considering the nature of the experimental data, or by analogies, etc. On the other hand, transport phenomena models are based on the laws of... 

For such characterized CH seed, the authors were able to obtain a simple solution to the batch crystallization population balance, and offered a relatively simple technique to model the behavior of a batch crystallizer. 

Table 2.4 Simplified homopolymerization mechanism for a preliminary model using population balances and the method of moments...

We will start with a very simple homopolymerization model that includes only initiation, propagation, transfer to hydrogen, -hydride elimination and imimolecular catalyst deactivation, as depicted in Table 2.4. From our previous discussion of the standard model for polymerization with coordination catalysts, it is known that several steps are not included in Table 2.4. It will be shown, however, that general expressions for population balances and the methods of moments starting with this simplified mechanism can be developed and later they can be extended, rather easily, to include more polymerization steps. 

The following sections develop in detail some of the important aspects of granulation process modeling, through the use of population balances and alternative approaches. 

In the following model development, we will use the polymerization kinetics mechanism described by Eqs. (1)-(14) to derive the population balances and moment equations for homopolymerization with a catalyst containing only one site type. Catalysts containing two or more site types are handled similarly by defining a set of equations with distinct polymerization kinetic constants for each different site type. 

Population balances and the method of moments can also be combined with the multigrain model and other polymer particle growth models. In this case, the population balances are defined for each position in the particle to obtain the radial profiles of chain length averages [36, 51-60]. 

Pigou, M. and Morchain, J. (2015) Investigating the interactions between physical and biological heterogeneities in bioreactors using compartment, population balance and metabolic models. Chem. Eng. Sci., 126, 267 - 282. 

Correlations of nucleation rates with crystallizer variables have been developed for a variety of systems. Although the correlations are empirical, a mechanistic hypothesis regarding nucleation can be helpful in selecting operating variables for inclusion in the model. Two examples are (/) the effect of slurry circulation rate on nucleation has been used to develop a correlation for nucleation rate based on the tip speed of the impeller (16) and (2) the scaleup of nucleation kinetics for sodium chloride crystalliza tion provided an analysis of the role of mixing and mixer characteristics in contact nucleation (17). Pubhshed kinetic correlations have been reviewed through about 1979 (18). In a later section on population balances, simple power-law expressions are used to correlate nucleation rate data and describe the effect of nucleation on crystal size distribution. 

Although evidence exists for both mechanisms of growth rate dispersion, separate mathematical models were developed for incorporating the two mechanisms into descriptions of crystal populations random growth rate fluctuations (36) and growth rate distributions (33,40). Both mechanisms can be included in a population balance to show the relative effects of the two mechanisms on crystal size distributions from batch and continuous crystallizers (41). 

The energy laws of Bond, Kick, and Rittinger relate to grinding from some average feed size to some product size but do not take into account the behavior of different sizes of particles in the mill. Computer simulation, based on population-balance models [Bass, Z. Angew. Math. Phys., 5(4), 283 (1954)], traces the breakage of each size of particle as a function of grinding time. Furthermore, the simu-... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

The general form of the population balance including aggregation and rupture terms was solved numerically to model the experimental particle size distributions. While excellent agreement was obtained using semi-empirical two-particle aggregation and disruption models (see Figure 6.15), PSD predictions of theoretical models based on laminar and turbulent flow considerations... 

---