Population balance equations, solution numerical

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. 

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. 

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

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. 

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]. 

Alopaeus, V., Laakkonen, M. Aittamaa, J. 2006 Numerical solution of moment-transformed population balance equation with fixed quadrature points. Chemical Engineering Science 61, 4919-4929. 

This book provides a consistent treatment of these issues that is based on a general theoretical framework. This, in turn, stems from the generalized population-balance equation (GPBE), which includes as special cases all the other governing equations previously mentioned (e.g. PBE and BE). After discussing how this equation originates, the different computational models for its numerical solution are presented. The book is structured as follows. 

When the particle densities are not small, the coupled equations (2.11.1) and (2.11.2) must be solved simultaneously. Generally, such solutions can only be obtained numerically. The solution of population balance equations is of concern in Chapter 4. 

A particularly attractive approach that has evolved more recently is that of discretizing population balance equations and solving the discrete equations numerically. The effectiveness of this technique lies in rapid solutions of selected properties of the population that may be of interest to a specific application. 

Since we are interested in dynamic analysis starting from some initial conditions which can be arbitrary, it is clear that one cannot associate a self-similar solution from the very beginning of the process except for an initial condition that happens to be compatible with the self-similar solution. Thus, the question of a self-similar solution basically arises when the system has evolved away from the initial condition. There is thus a sense of independence of the self-similar solution from the initial condition. This independence may, however, apply only for a class of initial conditions outside of which no self-similar solution may be attained. Questions in regard to the conditions under which a self-similar solution exists for a population balance equation, and the class of initial conditions for which the solution can approach such a self-similar solution, are indeed mathematically very deep and cannot be answered within the scope of this treatment. On the other hand, numerical solutions can be examined for their approach to self-similarity. 

The same authors also presented an example of the use of the population balance equation (PBE) (distribution of biomass m) coimected to the multi-zone/CFD model. This example is in several respects relevant for the assessment of the modeling approach. The coupling of the integro-differential equation of the population balance is a numerical challenge, which can nowadays be tackled within the environment of a CFD approach, albeit without consensus on the proper closure assumptions. Still, the computational effort for the numerical solution of the population balance embedded in the multizonal model is extensive, and it is difficult to extend this approach to multiple state variables necessary for dynamic metabolic models. This is an important argument to favor the alternative method of an agent-based Lagrange-Euler approach discussed in Section 3.5. 

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 are available for particular cases. 

In mathematical terms the population balance equation (PBE) is classified as a non-linear partial integro-differential equation (PIDE). In the PBE (12.308) the size property variable ranges from 0 to oo. In order to apply a numerical scheme for the solution of the equation a first modification is to fix a finite computational domain. The conventional approximation is to truncate the equation by substitution of the infinite integral limits by the finite limit value max- The function/(, r, t) denotes the exact solution of the exact equation. It might be assumed that the solution of the truncated PBE is sufficiently close to the exact equation so that the two solutions are practically equal. Hence, the solutions of both forms of the PBE are denoted by fiC, r, t). 

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