Model population balance

The basis of a population balance model is that the number of entities with some property in a system is a balanceable quantity. Properties include, among others, size, mass, and age. 

The population balance, including these effects, can be written as follows  

In this equation, the term on the left-hand side represents the rate of change of the number concentration of agglomerates of size k (vahd for any size k). This is the result of the processes accounted for on the right-hand side of the equation. The first term accounts for the formation of agglomerates of size k due to collisions of two smaller 

Initial suspension After 3 minutes After 6 minutes After 9 minutes After 12 minutes After 15 minutes 

Rgure 3.4. Change in floe size distribution during a flocculation process (Pelin, K., Licentiate thesis, Chalmers University of Technology 1999). 

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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-... 

Gertlauer, A., Mitrovic, A., Motz, S. and Gilles, E.-D., 2001. A population balance model for crystallization processes using two independent particles properties. Chemical Engineering Science, 56(7), 2553-2565. 

Leblanc and Fogler developed a population balance model for the dissolution of polydisperse solids that included both reaction controlled and diffusion-controlled dissolution. This model allows for the handling of continuous particle size distributions. The following population balance was used to develop this model. 

SE LeBlanc, HS Fogler. Population balance modeling of the dissolution of polydis-perse solids Rate limiting regimes. AIChE J 33 54-63, 1987. 

Symbols G = general polymer system, PB = population balance model, ADA = age... 

Figure 35. Granule growth mechanisms for population balance modeling. (From Litster and Ennis, 1994.)...

The authors would like to acknowledge that the section dealing with population balance modeling of granulation processes is an abbreviated version of material prepared by Dr. J. D. Litster, University of Queensland, for a joint short course given by Dr. B. J. Ennis and Dr. J. D. Litster. 

Adetayo, A. A., Litster, J. D., Pratsinis, S. E., and Ennis, B. J., Population Balance Modelling of Dram Granulation of Materials with Wide Size Distributions, Powder Tech, 82 37-49 (1995)... 

G. Fevotte, C. Alexandre and S.O. Nida, A population balance model of the solution-mediated phase transition of citric acid, AIChE J., 53, 2578-2589 (2007). 

Tavare and Garside ( ) developed a method to employ the time evolution of the CSD in a seeded isothermal batch crystallizer to estimate both growth and nucleation kinetics. In this method, a distinction is made between the seed (S) crystals and those which have nucleated (N crystals). The moment transformation of the population balance model is used to represent the N crystals. A supersaturation balance is written in terms of both the N and S crystals. Experimental size distribution data is used along with a parameter estimation technique to obtain the kinetic constants. The parameter estimation involves a Laplace transform of the experimentally determined size distribution data followed a linear least square analysis. Depending on the form of the nucleation equation employed four, six or eight parameters will be estimated. A nonlinear method of parameter estimation employing desupersaturation curve data has been developed by Witkowki et al (S5). 

In some areas, e.g. aerosol physics and crystallisation, population balance models are used in situations when a number balance equation is required as well as conventional mass and energy balances. Randolph and Larson review this theory as it applies specifically to particulate systems [15], whilst Froment and Bischoff [16] present population balance equations in the context of an extension of classical RTD theory. 

Fig. 7. Comparison of mean particle size obtained in a wet milling of a-alumina (cm = 0.2) with the results of a population balance model.

Equation (A12) is widely used in RE, but it does not account for the specific interactions of the dispersed phase. In this respect current research is focused on drop population balance models, which account for the different rising velocities of the different-size droplets and their interactions, such as droplet breakup and coalescence (173-180). 

The rate of cooling, or evaporation, or addition of diluent required to maintain specified conditions in a batch crystallizer often can be determined from a population-balance model. Moments of the population density function are used in the development of equations relating the control variable to time. As defined earlier, the moments are... 

The unstructured and segregated model considers the cell culture as a heterogeneous population, with individual cells characterized by age, mass, or size. The variation of properties follows a statistical distribution. The most concise way to deal with such population heterogeneity is through a population balance model (PBM). These models can become mathematically complex because the equations that describe the population have to consider that the properties vary in the population and... 

The representation of a property variation (mass, age, volume, etc.) within a cellular population is made using population balance models (PBM). These models allow the calculation of the variation of the rate of a population characteristic considering the rates of increase and disappearance of the specific characteristic within the cell population. Thus, they allow a calculation of one special feature in the population. Mathematical models that were originally unable to consider the diversity of character-... 

As an example, the single-variable population balance model, single-staged, mass structured is indicated by Equation 93. 

S. A. Schmidt, M. Simon, L Lagar Garcia, et ah, Estimation of Populance Parameters in Liquid-Liquid Extraction Columns., in 2nd Int. Conf. Population Balance Modelling. 2004, Valencia, Spain. 

S. E. Pratsinis, et al., Population balance modeling of drum granulation of materials with wide size distribution, Powder Technol., 1995, 82, 37-49. 

FIGURE 18.4 Comparison of population balance model with Monte Carlo model. 

Staggs, J. E. J. Population balance models for the thermal degradation of PMMA. Polymer 2007 48 3868. 

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... 

Haseltine, E. L. Systems Analysis of Stochastic and Population Balance Models for Chemical Reactions . Department of Chemical Engineering, Madison, University of Wisconsin (2005). Haseltine, E. L., and Rawlings, J. B. J. Chem. Phys. 117(15), 6959 (2002). 

Martin, P.J., Chin, N.L., et al. (2004). Aeration during bread dough mixing. II. A population balance model of aeration. Food Bioproducts Process. 82(C4), 268-281. 

To use the nucleation rate with population balance models, it is necessary to divide the nucleation rate J = dN(f )ldt by the critical size of the nuclei, r, giving... 

This expression for the rate of change in the population density of nuclei will be used later in the population balance model as an initial condition. Katz has shown that classical nucleation theory predicts well the dependence of the supersatvuntion ratio [10] on the nucleation rate. 

For different values of n ing(i ) = i2", other kinetic expressions can be developed. Figure 8.10 [18] shows the type of powder produced on spray diydng a solution that consists of metal salts of barium and iron in the ratio 1 12 (i.e., barium ferrite). Here we see the remains of the spherical droplets with a surface that consists of the metal salt precipitates, which form a narrow size distribution of platelet crystals (see Figure 8.10(a) and (b)). This narrow crystal size distribution is predicted by the population balance model if nudeation takes place over a short period of time. When these particles are spray roasted (in a plasma gun), the particles are highly sintered into spherical particles (see Figure 8.10(c)). 

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