Population balance, applications

The continuum form of the bubble population balance, applicable to flow of foams in porous media, can be obtained by volume averaging. Bubble generation, coalescence, mobilization, trapping, condensation, and evaporation are accounted for in the volume averaged transport equations of the flowing and stationary foam texture. 

Ramkrishna. D., 2000. Population balances. Theory and applications to particulate systems in engineering. New York Academic Press. 

Analytical solutions of the self-preserving distribution do exist for some coalescence kernels, and such behavior is sometimes seen in practice (see Fig. 40). For most practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the size range into discrete intervals and then solve the series of ordinary differential equations that result. A geometric discretization reduces the number of size intervals (and equations) that are required. Litster, Smit and Hounslow (1995) give a general discretized population balance for nucleation, growth and coalescence. Figure 41 illustrates the evolution of the size distribution for coalescence alone, based on the kernel of Ennis Adetayo (1994). 

Lo, S., Application of population balance to CFD modelling of gas-liquid reactors . Conference on Trends in Numerical and Physical Modelling for Industrial Multiphase Flows , Cargese, Corse 27-29 September (2000). 

Note that the RANS formulation used in (B.44) and (B.45) can easily be extended to the LES, as outlined in Section 5.10. Moreover, by following the same steps as outlined above, DQMOM can be used with the joint velocity, composition PDF transport equation. Finally, the reader can observe that the same methodology is applicable to more general distribution functions than probability density functions. Indeed, DQMOM can be applied to general population balance equations such as those used to describe multi-phase flows. 

A different approach in the use of orthogonal polynomials as a transformation method for the population balance is discussed in (8 2.) Here the error in Equation 11 is minimized by the Method of Weighted Residuals. This approach releases the restrictions on the growth rate and MSMPR operation, however, at the cost of the introduction of numerical integration of the integrals involved, which makes the method computationally unattractive. The applicability in determining state space models is presently investigated and results will be published elsewere. 

A population balance over a macroscopic region has many engineering applications. For this type of balance the general population balance developed in equation (3.7) can be simplified. Into a macroscopic volume, we can have an arbitrary number of inputs and outputs at flow rates . In addition, if we assume that the suspension is well mixed... 

Particle growth in a batdi environment is more difficult to predict because the steady state assumption previously used for the CSTR case is no loiter applicable. For a batch precipitator, the simplified population balance becomes... 

By application of this transformation under conditions of constant t, the dimensionless solution to the characteristic population balance for a batch reactor can be found to be... 

Herbst et al. [International J. Mineral Processing, 22, 273-296 (1988)] describe the software modules in an optimum controller for a grinding circuit. The process model can be an empirical model as some authors have used. A phenomenological model can give more accurate predictions, and can be extrapolated, for example from pilot-to full-scale application, if scale-up rules are known. Normally the model is a variant of the population balance equations given in the previous section. 

The SAXS/TGA approach has been demonstrated to be a useful technique for time-resolution of porosity development in carbons during activation processes. Qualitative interpretation of the data obtained thus far suggests that a population balance approach focusing on the rates of production and consumption of pores as a function of size may be a fruitful approach to the development of quantitative models of activation proces.ses. These then could become useful tools for the optimization of pore size distributions for particular applications by providing descriptions and predictions of how various activating agents and time-temperature histories affect resultant pore size distributions. 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

The work discussed in this section clearly delineates the role of droplet size distribution and coalescence and breakage phenomena in mass transfer with reaction. The population balance equations are shown to be applicable to these problems. However, as the models attempt to be more inclusive, meaningful solutions through these formulations become more elusive. For example, no work exists employing the population balance equations which accounts for the simultaneous affects of coalescence and breakage and size distribution on solute depletion in the dispersed phase when mass transfer accompanied by second-order reaction occurs in a continuous-flow vessel. Nevertheless, the population balance equation approach provides a rational framework to permit analysis of the importance of these individual phenomena. 

Numerical Solutions For many practical applications, numerical solutions to the population balance are necessary. Several numerical solution techniques have been proposed. It is usual to break the... 

Penlidis, A. Macgregor, J.F. Hamielec, A.E. Mathematical-modeling of emulsion polymerization reactors—a population balance approach and its applications. ACS Symp. Ser. 1986, 313, 219-240. 

Lo S (2000) Application of population balance to CFD modeling of gas-liquid reactors. Proc of Trends in numerical and physical modelling for industrial multiphase flows. Corse, 27-29 September... 

The application of similar advanced distribution functions in the context of population balance analysis of polymerization processes is familiar in reaction engineering [40, 97]. However, the microscopic balance equations used for this purpose are normally averaged over the whole reactor volume so that simplified macroscopic (global) reactor analysis of the chemical process behavior is generally performed [35]. 

Dorao CA (2006) High Order Methods for the Solution of the Population Balance Equation with Applications to Bubbly Flows. Dr ing thesis, Department of Chemical Engineering, The Norwegian University of Science and Technology, Trondheim... 

Ramkrishna D (2000) Population Balances Theory and Applications to Particulate Systems in Engineering. Academic Press, San Diego Randolph AD (1964) A Population Balance for Countable Entities. Can J Chem Eng 42(6) 280... 

Alexiadis, a., Vanni, M. Gardin, R 2004 Extension of the method of moments for population balances involving fractional moments and application to a typical agglomeration problem. Journal of Colloid and Interface Science 276, 106-112. 

Immanuel, C. D. Doyle, F. J. 2003 Computationally efficient solution of population balance models incorporating nucleation, growth and coagulation application to emulsion... 

The population balance equation is a framework for the modeling of particulate systems. These include dispersions involving solid particles, liquid drops, and gas bubbles spanning a variety of systems of chemical engineering interest. The detailed derivation of the population balance equation and its applications can be found in Ramkrishna (1985, 2000). Publications pioneering the general application of population balance are by Hulburt and Katz (1964), Randolph and Larson (1964), and Frederickson et al. (1967). 

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