The Population Balance Equation

Using a readily established generalization of the Reynolds transport theorem in three-dimensional space to general vector spaces we may write 

Equation (2.6.1) is crucial to the development of the population balance equation for the general case, which is treated in the next section. 

Although we are ready for the derivation of the general population balance equation, we shall begin for the sake of simplicity with the one-dimensional case. 

5 For an elegant derivation of the Reynolds transport theorem, see Serrin (1959). 

If we denote the number density by / (x, t the rate of change in the number of particles in [a, h] caused by this traffic at a and b is given by 

The chemical engineering community began the first efforts that can be associated with the concepts of the population balance in the early 1960s. 

Jakobsen, Chemical Reactor Modeling, doi 10.1007/978-3-540-68622-4 9, Springer-Verlag Berlin Heidelberg 2008 

The population balance concept was first presented by Hulburt and Katz [37]. Rather than adopting the standard continuum mechanical framework, the model derivation was based on the alternative Boltzmann-t q)e equation familiar from classical statistical mechanics. The main problems investigated stem from solid particle nucleation, growth, and agglomeration. 

Randolph [95] and Randolph and Larson [96], on the other hand, formulated a generic population balance model based on the generalized continuum mechanical framework. Their main concern was solid particle crystallization, nucleation, growth, agglomeration/aggregation and breakage. 

Ramkrishna [93, 94] adopted the concepts of Randolph and Larson to investigate biological populations. An outline of the population balance model derivation from the continuum mechanics point of view was discussed. 

Similar PBE modeling approaches have also been used in the theory of aerosols in which the gas is the continuous phase [27, 136], in chemical-, mechanical- and 

Carrica et al. [11] investigated compressible bubbly two-phase flow around a surface ship. They developed a population balance from kinetic theory using the particle mass as internal coordinate or property, whereas most earlier work on solid particle analysis used particle volume (or diameter). In flows where compressibility effects in the gas are important (as in the case of laboratory bubble colunms operated 

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Given expressions for the crystallization kinetics and solubility of the system, the population balance (equation 2.4) can, in principle, be solved to predict the performance of both batch and of continuous crystallizers, at either steady- or unsteady-state... 

Hence, the population balance equation for a restricted-in-space agglomera tion process can be written as... 

Tavlarides presents a sophisticated model for representing coalescence and breakage of droplets in liquid-liquid dispersions. The model relies on the population balance equation and still requires the adjustment of 6 parameters. The solution of such equations is difficult and requires the use of Monte-Carlo methods... 

Use of Saturation. Because of the potential for simplification of the population balance equations, much recent work has concentrated on studying saturation phenomena. First proposed by Piepmeier (9), and elaborated on by Daily (10), saturation in atomic species can lead to complete elimination of the need to know any collisional rates, and in molecular species may provide substantial simplification of the balance equation analysis. 

If random scission occurs together with end-chain scission (but at a different rate) then defining x by dx/dt = k(i the population balance equations (PBEs) for these combined processes are... 

The evolution of the size distribution of aerosols due to coagulation can be described by the population balance equation... 

For a batch miU, the summation of the left-hand size of the above equation is zero. In this case, an analytical solution can be obtain to the population balance (equation 4.29) ... 

The solution to the population balance equation is shown in Figure... 

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 population balance equations are very general and may be applied to batch, semicontinuous, and continuous emulsion polymerizations. Furthermore, both seeded and ab initio polymerizations are comprehended by Eq. (5) in all (or part) of the three commonly considered polymerization intervals. The following sections show how the different possibilities are reflected in different functional forms of the elements of the matrices O and K and of the vector c. It should be remembered, however, that certain conceivable situations are not comprehended by Eq. (5) for example, if the monomer molecules are not freely exchanged between the latex particles so that the monomer concentration inside each latex particle is determined by its growth history. 

Min and Ray (1974) in their review article detailed many experimental studies on the PSDs of emulsion polymers. In this section, some more recent results that compare the predictions of the population balance equations [Eq. (5)] with experimental data will be discussed. 

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)... 

Often for engineering purposes the complete characterization of the dispersion by (x t) is not necessary and knowledge of average properties such as average size, surface area, or mass concentration is adequate for design purposes. It then may be expedient to reformulate the population balance equation in terms of the moments of the distribution. 

The population balance equation is employed to describe the temporal and steady-state behavior of the droplet size distribution for physically equilibrated liquid-liquid dispersions undergoing breakage and/or coalescence. These analyses also permit evaluation of the various proposed coalescence and breakage functions described in Sections III,B and C. When the dispersion is spatially homogeneous it becomes convenient to describe particle interaction on a total number basis as opposed to number concentration. To be consistent with the notation employed by previous investigators, the number concentration is replaced as n i,t)d i = NA( i t)dXi, where N is the total number of particles per unit volume of the dispersion, and A(xj t) dXi is the fraction of drops in increment X, to X( + dxi- For spatially homogeneous dispersions such as in a well-mixed vessel, continuous flow of dispersions, no density changes, and isothermal conditions Eq. (102) becomes... 

Delichatsios and Probstein (D4-7) have analyzed the processes of drop breakup and coagulation/coalescence in isotropic turbulent dispersions. Models were developed for breakup and coalescence rates based on turbulence theory as discussed in Section III and were formulated in terms of Eq. (107). They applied these results in an attempt to show that the increase of drop sizes with holdup fraction in agitated dispersions cannot be attributed entirely to turbulence dampening caused by the dispersed phase. These conclusions are determined after an approximate analysis of the population balance equation, assuming the size distribution is approximately Gaussian. 

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. 

Analysis of Mass Transfer or Reaction in Dispersions with the Population Balance Equation... 

A simplified homogeneous dispersed-phase mixing model was proposed by Curl (C16). Uniform drops are assumed, coalescence occurs at random and redispersion occurs immediately to yield equal-size drops of the same concentration, and the dispersion is assumed to be homogeneous. Irreversible reaction of general order s was assumed to occur in the drops. The population balance equations of total number over species concentration in the drop were derived for the discrete and continuous cases for a continuous-fiow well-mixed vessel. The population balance equation could be obtained from Eq. (102) by taking the internal coordinate to be drop concentration and writing the population balance equation in terms of number to yield... 

Hulbert and Akiyama (HI8) analyzed the population balance equation to determine coalescence and redispersion effects on extent of conversion, deriving a set of equations for the moments of the distribution. The results agreed with those of Shain (S18). The authors claim the moment method reduces computation effort although there is a loss of information on the distribution. 

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. 

Considering a single growth direction with one characteristic length L, and a well-mixed crystallizer with growth and nucleation as the only dominating phenomena the population balance equation (PBE) has the form... 

The main challenge in formulating these equations is related to the definition of the collision operator. So far this approach has been restricted to the formulation of the population balance equation. That is, in most cases a general transport equation which is complemented with postulated source term formulations for the particle behavior is used. Randolph [80] and Randolph and Larson [81] used this approach deriving a microscopic population balance equation for the purpose of describing the behavior of particulate systems. Ramkrishna [79] provides further details on this approach considering also fluid particle systems. 

In a series of papers Lathouwers and Bellan [43, 44, 45, 46] presented a kinetic theory model for multicomponent reactive granular flows. The model considers polydisersed particle suspensions to take into account that the physical properties (e.g., diameter, density) and thermo-chemistry (reactive versus inert) of the particles may differ in their case. Separate transport equations are constructed for each of the particle types, based on similar principles as used formulating the population balance equations [61]. 

Venneker et al [118] made an off-line simulation of the underlying flow and the local gas fractions and bubble size distributions for turbulent gas dispersions in a stirred vessel. The transport of bubbles throughout the vessel was estimated from a single-phase steady-state flow fleld, whereas literature kernels for coalescence and breakage were adopted to close the population balance equation predicting the gas fractions and bubble size distributions. 

Several extensions of the two-fluid model have been developed and reported in the literature. Generally, the two-fluid model solve the continuity and momentum equations for the continuous liquid phase and one single dispersed gas phase. In order to describe the local size distribution of the bubbles, the population balance equations for the different size groups are solved. The coalescence and breakage processes are frequently modeled in accordance with the work of Luo and Svendsen [74] and Prince and Blanch [92]. 

The fundamental derivation of the population balance equation is considered general and not limited to describe gas-liquid dispersions. However, to employ the general population balance framework to model other particulate systems like solid particles and droplets appropriate kernels are required for the particle growth, agglomeration/aggregation/coalescence and breakage processes. Many droplet and solid particle closures are presented elsewhere (e.g., [96, 122, 25, 117, 75, 76, 46]). 

The formulations of the population balance equation based on the continuum mechanical approach can be split into two categories, the macroscopic- and the microscopic population balance equation formulations. The macroscopic approach consists in describing the evolution in time and space of several groups or classes of the dispersed phase properties. The microscopic approach considers a continuum representation of a particle density function. 

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