The Framework of Population Balance

We are concerned with systems consisting of particles dispersed in an environmental phase, which we shall refer to as the continuous phase. The particles may interact between themselves as well as with the continuous phase. Such behavior may vary from particle to particle depending upon a number of properties that may be associated with the particle. The variables representing such properties may be either discrete or continuous. The discreteness or continuity of the property pertains to its variation from particle to particle. 

Continuous variables may be encountered more frequently in population balance analysis. They often arise as a natural solution to dealing with indefinite or variable discreteness. For example, a particle-splitting process where the products of splitting could conceivably have any size smaller than the parent particle is most naturally handled by assigning particle size as a continuous variable. The external coordinates denoting the position vector of (the centroid of) a particle describing continuous motion through space represent continuous variables. The temperature of a particle in a fluidized bed is another example of a continuous variable. 

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Gimbun, j., Nagy, Z. K. Rielly, C. D. 2009 Simultaneous quadrature method of moments for the solution of population balance equations, using a differential algebraic equation framework. Industrial U Engineering Chemistry Research 48, 7798-7812. 

The number density function, along with the environmental phase variables, completely determines the evolution of all properties of the dispersed phase system. The population balance framework is thus an indispensable tool for dealing with dispersed phase systems. This book seeks to address the various aspects of the methodology of population balance necessary for its successful application. Thus Chapter 2 develops the mathematical framework leading to the population balance equation. It... 

As observed earlier, in a bacterial population the life spans of the mother and daughter cells are negatively correlated, while those of sister cells are positively correlated. We show next that the negative correlation between mother and daughter is easily accounted for with the usual framework of population balance. The positive correlation between siblings, however, requires the framework just described in Section 7.5.1. We consider first the formulation to account for the negative correlation between mother and daughter. 

The development of the differential equations which describe the evolution of particle size and molecular weight properties during the course of the polymerization is based on the so-called "population balance" approach, a quite general model framework which will be described shortly. Symbols which will be used in the subsections to follow are all defined in the nomenclature. 

The main contribution from the work of Luo [95, 96] was a closure model for binary breakage of fluid particles in fully developed turbulence flows based on isotropic turbulence - and probability theories. The author(s) also claimed that this model contains no adjustable parameters, a better phrase may be no additional adjustable parameters as both the isotropic turbulence - and the probability theories involved contain adjustable parameters and distribution functions. Hagesaether et al [49, 50, 51, 52] continued the population balance model development of Luo within the framework of an idealized plug flow model, whereas Bertola et al [13] combined the extended population balance module with a 2D algebraic slip mixture model for the flow pattern. Bertola et al [13] studied the effect of the bubble size distribution on the flow fields in bubble columns. An extended k-e model was used describing turbulence of the mixture flow. Two sets of simulations were performed, i.e., both with and without the population balance involved. Four different superficial gas velocities, i.e., 2,4,6 and 8 (cm/s) were used, and the superficial liquid velocity was set to 1 (cm/s) in all the cases. The population balance contained six prescribed bubble classes with diameters set to = 0.0038 (m), d = 0.0048 (m), di = 0.0060 (m), di = 0.0076 (m), di = 0.0095 (m) and di = 0.0120 (m). 

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). 

In conclusion to this section, research in the RTD area is always active and the initial concepts of Danckwerts are gradually being completed and extended. The population balance approach provides a theoretical framework for this generalization. However, in spite of the efforts of several authors, simple procedures, easy to use by practitioners, would still be welcome in the field of unsteady state systems (variable volumes and flow rates), multiple inlet/outlet reactors, variable density mixtures, systems in which the mass-flowrate is not conserved, etc... On the other hand, the promising "generalized reaction time distribution" approach could be developed if suitable experimental methods were available for its determination. 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

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. 

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. 

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