Population-balance method, description

The population balance approach is employed for the description of droplet dynamics in various flow fields. A significant advantage of the method is that a vehicle is provided to include the details of the breakage and coalescence processes in terms of the physical parameters and conditions of operation. A predictive multidimensional particle distribution theory is at hand which, in the case of well-defined droplet processes, can be employed for a priori prediction of the form and the magnitude of the particle size distribution. The physical parameters which affect the form... 

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 deterministic population balance equations governing the description of mass transfer with reaction in liquid-liquid dispersions present a framework for analysis. However, signiflcant difficulties exist in obtaining solutions for realistic problems. No analytical solutions are available for even the simplest cases of interest. Extension of the solution to multiple reactants for uniform drops is possible using a method of moments but the solution is limited to rate equations which are polynomials (E3). Solutions to the population balance equations for spatially nonhomogeneous dispersions were only treated for nonreacting dispersions (P4), and only a simple case was solved for a spray column (B19). Treatment of unmixed feeds presents a problem. 

The properties of microbial populations result from the characteristics of metabolism and Its control at the level of the Individual cell. Useful descriptions of population behavior may therefore be constructed based on understanding of single-cell metabolism, and, conversely, studies of population properties may be employed to extract Information about single-cell operation. Population balance equations provide the primary mathematical tool for these purposes, and flow cytometry allows experimental access to the distribution of cell states In the population. Application of these methods to bacteria and yeasts Is Illustrated using three exaaq>le systems. 

Kinetic approaches represent realistic and comprehensive description of the mechanism of network formation. Under this approach, reaction rates are proportional to the concentration of unreacted functional groups involved in a specific reaction times an associated proportionality constant (the kinetic rate constant). This method can be applied to the examination of different reactor types. It is based on population balances derived from a reaction scheme. An infinite set of mass balance equations will result, one for each polymer chain length present in the reaction system. This leads to ordinary differential or algebraic equations, depending on the reactor type under consideration. This set of equations must be solved to obtain the desired information on polymer distribution, and thus instantaneous and accumulated chain polymer properties can be calculated. In the introductory paragraphs of Section... 

Modeling is probably the tool of excellence for engineers (Chapter 9). It is used to simulate the reaction and the process system in order to shorten the time for development. It is based on models that can be physical or chemical, semi-empirical or empirical, descriptive or more fundamental. To describe the development of the molecular weight distribution upon reaction, moment methods or equations based on population balance are often used. 

The mathematical modeling of these polymers are based on either population balances [24-31] or Monte Carlo methods [32-35]. The output of these models includes a detailed description of the polymer architecture (sol MWD, gel fraction, cross-linking points, long chain branching). The models are rather complex and will not be discussed here. Nevertheless, for cases of practical interest, the average molecular weights can be calculated with a modest computational effort [36]. 

The general field of problems described above, except in some special areas, may be treated by the well-known methods and analytical models of mathematical physics. It has already been noted that the most general description of the neutron population usually starts with a neutron-balance relation of the Boltzmann type. The Boltzmann equation was developed in connection with the study of nonuniform gas mixtures, and the application to the neutron problem represents a considerable simplification of the general gas problem. (Whereas in gas problems all the particles are in motion, in reactor problems only the neutrons are in motion. ) The fundamental equation of reactor physics, then, is already a familiar one from the kinetic theory. Further, many of the most useful neutron models obtained from approximations to the Boltzmann equation reduce to familiar forms, such as the heat-conduction, Helmholtz, and telegraphist s equations. These simplifications result from the elimination of various independent variables in the... 

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