Population balance equations general case

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. 

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... 

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. 

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. 

Substituting into the general population balance equation yields for the micro-distributed case... 

We are now in a position to derive the population balance equation for the one-dimensional case. The reader interested in this may directly proceed to Section 2.7 since the next section prepares for derivation of the population balance equation for the general vectorial particle state space. 

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. 

The Population Balance Equation 2.7.3. Boundary Conditions for the General Case... 

In dealing with stochastic problems, it became clear from Section 7.3 that one is frequently faced with lack of closure, especially in situations where interaction occurs between particles or between particles and their environment. Such lack of closure arises because of the development of correlations between particle states promoted by preferential behavior between particle pairs of specific states or between the particle and its environment. The population balance equation, which generally comes about by making the crudest closure approximation, does not make accurate predictions in such cases of the average behavior of the system. The question naturally arises as to whether one can find other mean field descriptions by making more refined closure approximations on the unclosed product density equations. 

One-dimensional population balance models for both batch and continuous systems are described in this section as special cases of the generalized population balance model stated in General Population Balance Equations. ... 

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. 

For this case, we can start with the general population balance equation (6.2.51b) integrated over the volume of the vessel ... 

The population balance in equation 2.86 employs the local instantaneous values of the velocity and concentration. In turbulent flow, there are fluctuations of the particle velocity as well as fluctuations of species and concentrations (Pope, 1979, 1985, 2000). Baldyga and Orciuch (1997, 2001) provide the appropriate generalization of the moment transformation equation 2.93 for the case of homogeneous and non-homogeneous turbulent particle flow by Reynolds averaging... 

The population balance is a partial integro-differential equation that is normally solved by numerical methods, except for special simplified cases. Numerical solution of the population balance for the general case is not, therefore, entirely straightforward. Ramkrishna (1985) provides a comprehensive review. 

As is the case for all balance equations, we can generalize the population balance. The details of the derivation are presented elsewhere (Randolph and Larson 1988). 

The process dynamic model of a batch crystalliser is straightforward, fully described by the energy, mass and population balances. However, the dynamic of the crystal size distribution can be ignored if a batch is initially fed with seeds closely sized between two adjacent sieve sizes. General equations and constraints are developed for anhydrous salts. Additional equations are required to describe other transformations as in the case of hydrates and organic compounds. The subscript f and the superscript in the following equations denotes feed and saturation, respectively. The rate change in ... 

Performing the corresponding summations on the equations in Table 9.6, one obtains the (N, JVf)th moment formulation of Table 9.11. Some of the summation terms in these equations will not be evaluated for the general (N, M) case, but we will determine them by assigning values to N and M. Since we will not address branching, we take M = 0 here, but in principle this can be treated in a similar way. We will focus now on the TDB moment distributions and successively derive the model equations for the zeroth, first, and second moments, or N = 0,1, and 2. Solving the model thus essentially means solving the population balances of the real concentration distributions and P and the pseudo-distributions and... 

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