Bivariate and multivariate PBE

Readers interested in a detailed discussion concerning the application of the different CM to this equation are referred to the specialized literature (Immanuel Doyle, 2005 Lau-renzi et al 2002 Qamar Warnecke, 2007 Vale McKenna, 2005 Xiong Pratsinis, 1991, 1993). In what follows, only a brief summary is reported. Let us now discuss a simple geometric extension to bivariate problems of the discretization presented for univariate PBE. Given the interval of the internal coordinate delimited by and 

Also in this case it is possible to define pivotal points and 4 resulting 

Class methods have been applied to bivariate PBE only for simple problems, and generally their extension to multivariate problems is quite complicated. In what follows selected examples are discussed to illustrate these difficulties. 

By applying the integration over the and 4 intervals for the two internal coordinates and by applying the definition in Eq. (7.40), we obtain 

As for the univariate case, when a particle is formed after an aggregation event, it has to be assigned to the neighboring cells (four in a bivariate case). When the fixed-pivot technique is used to assign the particles to cells, the following expression is obtained  

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Depending on the number of internal coordinates (univariate versus multivariate) and depending on whether the particle velocity is part of the internal-coordinate vector, very different solution methods have been developed. As a consequence, in this chapter the methods for cases with and without particle velocity are discussed separately. First, the methods developed for the solution of a univariate PBE (i.e. one internal coordinate) are discussed. Second, the approaches for the solution of bivariate and multivariate PBE (i.e. 

As introduced in the previous section, class and sectional methods are based on a discretization of the internal coordinate so that the GPBE becomes a set of macroscopic balances in state space. Indeed, the fineness of the discretization will be dictated by the accuracy needed in the approximation of the integrals and derivative terms appearing in the GPBE. As has already been anticipated, the methods differ according to the number of internal coordinates used in the description and depend on the nature of the internal coordinates. Therefore, in what follows, we will discuss separately the univariate, bivariate, and multivariate PBE, and the use of these methods for the solution of the KE. 

Equation (2.14) must be coupled with initial conditions given for the starting time and with boundaries conditions in physical space O and in phase space O. Analytical solutions to Eq. (2.14) are available for a few special cases and only under conditions specified by some very simple hypotheses. However, numerical methods can be used to solve this equation and will be presented in Chapters 7 and 8. The numerical solution of Eq. (2.14) provides knowledge of the NDE for each time instant and at every physical point in the computational domain, as well as at every point in phase space. As has already been mentioned, sometimes the population of particles is described by just one internal coordinate, for example particle length (i.e. f = L), and the PBE is said to be univariate. When two internal coordinates are needed, for example particle volume and surface area (i.e. = (v, a)), the PBE is said to be bivariate. More generally, higher-dimensional cases are referred to as multivariate PBEs. Another important case occurs when part of the internal-coordinate vector is equal to the particle-velocity vector (i.e. when the particles are characterized not by a unique velocity field but by their own velocity distribution). In that case, the PBE becomes the GPBE, as described next. 

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