Univariate PBE

Expressing the drift, diffusion, and point processes as a unique source term (Ramkrishna, 2000) yields 

Typically, different assumptions regarding the functional form for the NDF within a single finite interval are employed, and, in general, these can be written as n(t, ) = Ni t)ki f) for f fi+, where i = 1. M, and fc ( ) is a polynomial that has to fulfill the property 

In the case of a zeroth-order polynomial, a (constant) approximation is employed within each class (Greenberg et al, 1986, 1993)  

Very often the discretization is written not in terms of the number of particles belonging to a single class, but rather in terms of the volume (or mass) of the particles belonging to a specific class. If pp is the density of the particles and Vp( ) is the functional relationship 

the case of constant approximation of the NDF within each interval results in 

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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. 

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... 

As with class and sectional methods for univariate PBE, the particle-velocity magnitude space is uniformly discretized over M intervals of size A < 2 < 3 < < m-i <... 

Consider a simple homogeneous system with a continuous inlet and outlet. The corresponding univariate PBE is... 

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. 

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. 

The MOM is also applicable for bivariate PBEs, as explained by Hulburt Katz (1964). In this case the moments of the NDF are defined as wjk = / n d, where and are the two internal coordinates and k = k, k2) is the exponent vector containing the order of the moment with respect to the two internal coordinates. Moments can have a particular physical meaning and, as for the univariate case, mo,o is the total particle number density. Likewise, the ratio of different moments can be used to characterize integral properties of the population, for example mi,o/mo,o is the number-average value of the first internal coordinate whereas mo,i/mo,o is the number-average value of the second one 2- For details and examples on the relationship between bivariate moments and measurable quantities, readers are referred, for example, to the work of Rosner et al. (2003). As for the univariate case, one has to solve the evolution equation for the moments. 

Insofar as the possibility of directly using Eq. (7.104) is concerned, as for univariate cases, the main limitation resides in the difficulty of finding analytical solutions and developing simple and stable closures. Analytical solutions are very rare and exist only for rates of continuous change and for kernels independent of the internal coordinates. Under these simplification hypotheses, for the case of nucleation, growth, diffusion, aggregation, and breakage (with additive internal coordinates), the bivariate PBE becomes... 

For simplicity, only the ID case is discussed here, and we denote the univariate NDF by n t, X, ), where is a passive internal coordinate. The PBE with only the advection term is... 

In this appendix we discuss the probiem of moment conservation with the DQMOM, and present a variation caiied DQMOM fuiiy conservative (DQMOM-FC) that can be empioyed to partiaiiy overcome it. To iiiustrate the issue, we consider a one-dimensional spatially inhomogeneous PBE for the univariate NDF n(t, x, with constant velocity and constant diffusion coefficient as described in Section 8.3. However, as will become clearer below, the extension to other, more complex, problems is straightforward. 

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