Collision source term model

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

The utility of the kinetic model is most evident when evaluating the collision source terms for the moments. As noted earlier, Eq. (6.109) is closed but results in complicated polynomial expressions for higher-order moments. In contrast, the kinetic model... 

In a fluid model the correct calculation of the source terms of electron impact collisions (e.g. ionization) is important. These source terms depend on the EEDF. In the 2D model described here, the source terms as well as the electron transport coefficients are related to the average electron energy and the composition of the gas by first calculating the EEDF for a number of values of the electric field (by solving the Boltzmann equation in the two-term approximation) and constructing a lookup table. 

Many disperse-phase systems involve collisions between particles, and the archetypical example is hard-sphere collisions. Thus, Chapter 6 is devoted to the topic of hard-sphere collision models in the context of QBMM. In particular, because the moment-transport equations for a GBPE with hard-sphere collisions contain a source term for the rate of change of the NDF during a collision, it is necessary to derive analytical expressions for these source terms (Fox Vedula, 2010). In Chapter 6, the exact source terms are derived... 

In order to account for variable particle numbers, we generalize the collision term iSi to include changes in IVp due to nucleation, aggregation, and breakage. These processes will also require models in order to close Eq. (4.39). This equation can be compared with Eq. (2.16) on page 37, and it can be observed that they have the same general form. However, it is now clear that the GPBE cannot be solved until mesoscale closures are provided for the conditional phase-space velocities Afp)i, (Ap)i, (Gp)i, source term 5i. Note that we have dropped the superscript on the conditional phase-space velocities in Eq. (4.39). Formally, this implies that the definition of (for example) [Pg.113]

The next step is to provide a closure for the pair correlation function appearing in the collision source and collisional-flux terms. For moderately dense flows, the collision frequency for finite-size particles is known to be larger than that found using the Boltzmann Stofizahlansatz (Carnahan Starling, 1969 Enksog, 1921). In order to account for this effect, the pair correlation function can be modeled as the product of two single-particle velocity distribution functions and a radial distribution function ... 

The other two collision source vectors, and can be evaluated using the definitions in Eqs. (6.104) and (6.106). As mentioned earlier, will be closed in terms of the moments of order two and lower, and their gradients. In contrast, C will not be closed in terms of any finite set of moments. Nevertheless, it can be approximated using quadrature-based moment methods as described in Section 6.5. In the fluid-particle limit d d2), neither CI2 i or C will contribute terms involving spatial gradients of the fluid properties (i.e. buoyancy, lift, etc.) to the fluid-phase momentum equation. As mentioned earlier, such terms result from the model for gapi i-n) and would appear, for example, on using the expression in Eq. (6.81). With the latter, Eq. (6.161) becomes... 

Keywords Atomization Chemical reactions Craiservation equations Constitutive equations Drop breakup Drop deformation Drop collisions Evaporation LES Newtonian fluids RANS Spray modeling Spray PDF Stochastic discrete particle method Source terms Turbulence... 

In (19.45), the drop collisions are accounted for via the source term,/cou- One of the most widely used collision models is the one developed by O Rourke [36]. In this model, the probability for a drop with index I to undergo n collisions with a drop of index 2 in a given volume V during the time interval At is given by the Poisson distribution... 

Refined cascade models have been incorporated in several modifications of the original EW-theory. Kostin and coworkers examined some of the basic relaxational assumptions using a mathematical model for a static collision density maintained at steady state by a constant source term (h5-45K ... 

In accordance with the work of Coulaloglou and Tavlarides [17] and Prince and Blanch [102], Luo [79] assumed that all the macroscopic source terms determining the death and birth rates could be defined as the product of a collision density and a probability. Thus modeling of bubble coalescence means modeling of a... 

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