Moment-transport equation numerical solution

For all other cases, it will be necessary to solve the moment-transport equations derived from the GPBE as described in Chapter 4. In Chapter 8 the numerical algorithms used to find approximation solutions to the GPBE using quadrature-based moment methods are presented in detail. 

The MOM was introduced for particulate systems by Hulburt Katz (1964). In their pioneering work these authors showed how it is possible to solve the PBF in terms of the moments of the NDF and to derive the corresponding transport equations. A similar approach can be used for the solution of the KF, and a detailed discussion on the derivation of the moment-transport equations can be found in the works of Struchtrup (2005) and Truesdell Muncaster (1980). The main issue with this technique is in the closure problem, namely the impossibility of writing transport equations for the lower-order moments of the NDF involving only the lower-order moments. Since the work of Hulburt Katz (1964) much progress has been made (Frenklach, 2002 Frenklach Harris, 1987 Kazakov Frenklach, 1998), and different numerical closures have been proposed (Alexiadis et al, 2004 Kostoglou Karabelas, 2004 Strumendo Arastoopour, 2008). The basis... 

The numerical solution of the moment-transport equation is again done by using operator splitting to update first for the spatial fluxes and then for the drag. The realizable scheme... 

General solution of the population balance is complex and normally requires numerical methods. Using the moment transformation of the population balance, however, it is possible to reduce the dimensionality of the population balance to that of the transport equations. It should also be noted, however, that although the mathematical effort to solve the population balance may therefore decrease considerably by use of a moment transformation, it always leads to a loss of information about the distribution of the variables with the particle size or any other internal co-ordinate. Full crystal size distribution (CSD) information can be recovered by numerical inversion of the leading moments (Pope, 1979 Randolph and Larson, 1988), but often just mean values suffice. 

If time-dependent functions Q(l ) are inserted into Equation (7-2), the integral defies explicit solution. In that case, a numerical solution will be necessary. The equation will become simpler the moment we are dealing with a short-term emission only (e.g., a fl evaporation) or whenever the emission period T is small in comparison to the period of time necessary to transport the pollutants to the receptor. Assuming an emission level of h as well as a short-term emission of the entire toxic mass of Q, the integral in Equation (7-2) can be solved, and the result is a concentration on the ground of... 

Brenner (1980) has explored the subject of solute dispersion in spatially periodic porous media in considerable detail. Brenner s analysis makes use of the method of moments developed by Aris (1956) and later extended by Horn (1971). Carbonell and Whitaker (1983) and Koch et al. (1989) have addressed the same problem using the method of volume averaging, whereby mesoscopic transport coefficients are derived by averaging the basic conservation equations over a single unit cell. Numerical simulations of solute dispersion, based on lattice scale calculations of the Navier-Stokes velocity fields in spatially periodic structures, have also been performed (Eidsath et al., 1983 Edwards et al., 1991 Salles et al., 1993). These simulations are discussed in detail in the Emerging Areas section. 

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