Population balance equations, solution

Bove [16] proposed a different approach to solve the multi-fluid model equations in the in-house code FLOTRACS. To solve the unsteady multifluid model together with a population balance equation for the dispersed phases size distribution, a time splitting strategy was adopted for the population balance equation. The transport operator (convection) of the equation was solved separately from the source terms in the inner iteration loop. In this way the convection operator which coincides with the continuity equation can be employed constructing the pressure-correction equation. The population balance source terms were solved In a separate step as part of the outer iteration loop. The complete population balance equation solution provides the... 

The general population balance equation requires numerical methods for its solution and several have been proposed (e.g. Gelbard and Seinfeld, 1978 Hounslow, 1990a,b Hounslow etai, 1988, 1990), of which more later. Fortunately, however, some analytic solutions for simplified cases also exist. 

In principle, given expressions for the crystallization kinetics and solubility of the system, equation 9.1 can be solved (along with its auxiliary equations -Chapter 3) to predict the performance of continuous crystallizers, at either steady- or unsteady-state (Chapter 7). As is evident, however, the general population balance equations are complex and thus numerical methods are required for their general solution. Nevertheless, some useful analytic solutions for design purposes are available for particular cases. 

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... 

N. M. Faqir, Numerical solution of the two dimensional population balance equation describing the hydrodynamics of interacting liquid-liquid dispersions. Chem. Eng. Sci., 2004, 59 (12), 2567-2592. 

Madras, G. and McCoy, B. Numerical and similarity solutions for reversible population balance equations with size-dependent rates. Journal of Colloid and Interface Science 2002 246 356. 

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... 

A number of numerical approaches have been applied to the solution of the general population balance equation ... 

The population balance simulator has been developed for three-dimensional porous media. It is based on the integrated experimental and theoretical studies of the Shell group (38,39,41,74,75). As described above, experiments have shown that dispersion mobility is dominated by droplet size and that droplet sizes in turn are sensitive to flow through porous media. Hence, the Shell model seeks to incorporate all mechanisms of formation, division, destruction, and transport of lamellae to obtain the steady-state distribution of droplet sizes for the dispersed phase when the various "forward and backward mechanisms become balanced. For incorporation in a reservoir simulator, the resulting equations are coupled to the flow equations found in a conventional simulator by means of the mobility in Darcy s Law. A simplified one-dimensional transient solution to the bubble population balance equations for capillary snap-off was presented and experimentally verified earlier. Patzek s chapter (Chapter 16) generalizes and extends this method to obtain the population balance averaged over the volume of mobile and stationary dispersions. The resulting equations are reduced by a series expansion to a simplified form for direct incorporation into reservoir simulators. 

A simplified one-dimensional transient solution of the bubble population balance equations, verified by experiments, has been presented elsewhere (5) for a special case of bubble generation by capillary snap-off. 

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. 

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. 

For a polydisperse system of particles undergoing both creaming and gravity-induced flocculation, the time evolution of the particle size distribution, and hence the time rate of change of the total particle concentration, can only be determined from the solution of the governing population balance equations (10,11). For the special case of two different sized particles, this system of balance equations reduces to ... 

Sha et al [130, 131] developed a similar multifluid model for the simulation of gas-liquid bubbly flow. To guarantee the conservation of mass the population balance part of the model was solved by the discrete solution method presented by Hagesaether et al [52]. The 3D transient simulations of a rectangular column with dimensions 150 x 30 x 2000 (mm) and the gas evenly distributed at the bottom were run using the commercial software CFX4.4. For the same bubble size distribution and feed rate at the inlet, the simulations were carried out as two, three, six and eleven phase flows. The number of population balance equations solved was 10 in all the simulations. It was stated that the higher the number of phases used, the more accurate are the results. 

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... 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (FIDE). Since analytical solutions of this equation are not available for most cases of practical interest, several numerical solution methods have been proposed during the last two decades as discussed by Williams and Loyalka [209] and Ramkrishna [151]. 

The remaining chapters in this book are organized as follows. Chapter 2 provides a brief introduction to the mesoscale description of polydisperse systems. There, the mathematical definition of a number-density function (NDF) formulated in terms of different choices for the internal coordinates is described, followed by an introduction to population-balance equations (PBE) in their various forms. Chapter 2 concludes with a short discussion on the differences between the moment-transport equations associated with the PBE and those arising due to ensemble averaging in turbulence theory. This difference is very important, and the reader should keep in mind that at the mesoscale level the microscale turbulence appears in the form of correlations for fluid drag, mass transfer, etc., and thus the mesoscale models can have non-turbulent solutions even when the microscale flow is turbulent (i.e. turbulent wakes behind individual particles). Thus, when dealing with turbulence models for mesoscale flows, a separate ensemble-averaging procedure must be applied to the moment-transport equations of the PBE (or to the PBE itself). In this book, we are primarily... 

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