Collisional-flux term

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 ... 

Likewise, the components of the collisional-flux term in Eq. (6.22) can be written as... 

The analytical expressions for will be used directly with quadrature-based moment methods to evaluate the collision source and collisional-flux terms for each integer moment. The numerical implementation of these terms in the context of quadrature is discussed in Section 6.5. 

We can now express the collision source term and collisional-flux term for the integer moments of order y as... 

It is noted that the collisional source and flux terms given by (28) and (30) in Jenkins and Mancini [70] are somewhat different from the corresponding terms (9.192) and (9.193) given by Gidaspow [49] and both of these formulations are different from (2.9) and (2.10) given by Chao [21]. To a first order approximation the collisional flux terms in these reports are almost identical. The only difference is the power of the equivalent particle diameter. The reason for this difference is that Jenkins and Mancini [70] were considering circular disks whereas Gidaspow [49] on the other hand considered spherical particles. A disk (sometimes spelled disc) is the projection of a sphere on a plane perpendicular to the sphere-radiant point line. For the spheres a three-dimensional description of the collisions is achieved... 

For a large particle in a fluid at liquid densities, there are collective hydro-dynamic contributions to the solvent viscosity r, such that the Stokes-Einstein friction at zero frequency is In Section III.E the model is extended to yield the frequency-dependent friction. At high bath densities the model gives the results in terms of the force power spectrum of two and three center interactions and the frequency-dependent flux across the transition state, and at low bath densities the binary collisional friction discussed in Section III C and D is recovered. However, at sufficiently high frequencies, the binary collisional friction term is recovered. In Section III G the mass dependence of diffusion is studied, and the encounter theory at high density exhibits the weak mass dependence. 

On the left-hand side of this equation, the collisional flux, deflned by Eq. (6.70), appears. On the right-hand side, the collision source terms, deflned by Eqs. (6.68) and (6.69), appear. The kinetic fluxes (or free transport) correspond to the moments. ... 

As with the first-order moments, this expression has contributions due to the kinetic and collisional fluxes on the left-hand side, and due to the collision source terms on the right-hand side. The contributions due to like-particle collisions (Oi,200,11 and C2oo,ii) have the same forms as in the case of monodisperse particles described above. We will thus look briefly at the terms due to unlike-particle collisions. 

The corresponding peculiar velocity dependent flux term 4>( ) is computed from a modified form of (4.26) and the corresponding source term f2( ) is calculated from a modified form of (4.28). The flux- and source term formulas are modified by exchanging the microscopic particle velocity c with the peculiar velocity C in accordance with the collisional operator definition (4.99). Further details on the derivation of the source and flux relations are given in Sect. 4.1.5. 

This equation is a Maxwell-Enskog type of equation for multiple particle mixtures with an interstitial gas. The first term on the RHS, f2g.j ipi), denotes a collisional source. The second term, (ipi), denotes a collisional flux. In these two terms, the notation j denotes any particle type, so the two terms have both contributions from inner-collisions among particles of type i and inter-collisions between particles of type i and any other particles of type j. Detailed expressions for the two terms were derived by Gidaspow [49] for mono-particle systems, and by Manger [105] and Jenkins and Mancini [70] for binary particle systems. 

Generalized source term in granular kinetic theory Generalized flux term in granular kinetic theory Dense gas collisional transfer flux Dense gas kinetic transfer flux... 

The collisional flux is not present in the kinetic theory of dilute gases, and the third term on the right-hand side of the transport equation must be included when the particle property is a function of C rather than c. 

Thus, with knowledge of the factor S/XB, which S = a ve), X = cr xgve), and B = T, the photon fluxes can be converted into particle fluxes. In general terms the excitation rate X can be replaced by a population rate provided by a collisional-radiative model, which treats the excitation and de-excitation from all possible levels into the emitting one. The same holds for all possible ionization processes. In the case of molecules not only ionization but also dissociation will play a significant role, and, therefore, the ionization rate S... 

Several other related aspects of TCFs can be mentioned, but will not be covered here to concentrate instead on calculational methods and applications of collisional TCFs. An earlier alternative approach in terms of superoperators [18] suggests ways of extending the formalism to include phenomena where the total energy is not conserved due to interactions with external fields or media. It has led to different TCFs which however have not been used in calculations. Information-theory concepts can be combined with TCFs [10] to develop useful expressions for collisional problems [19]. Collisional TCFs can also be expressed as overlaps of time-dependent transition amplitude functions that satisfy differential equations and behave like wavepackets. This approach to the calculation of TCFs was developed for Raman scattering [20] and has more recently been extended using collisional TCFs for general interactions of photons with molecules [21] and for systems coupled to an environment [22-25]. This approach has so far been only applied to the interaction of photons with molecular systems. Flux-flux TCFs [26-28] have been applied to reactive collision and molecular dynamics problems, but their connection to collisional TCFs have not yet been studied. 

Collisional Rate of Change Derivate Flux and Source Terms... 

In this section the dynamics of inelastic binary particle collisions are examined. The theory represents a semi-empirical extension of the binary collision theory of elastic particles described in Sect. 2.4.2. The aim is to determine expressions for the total change in the first and second moments of the particle velocity to be used deriving expressions for the collisional source (4.27) and flux (4.26) terms. 

Further details of the derivation of the closure relations for the energy dissipation source term, 7, and the kinetic and collisional energy fluxes, and are given in... 

The physical interpretations of the terms in (4.105) can be outlined as follows. The terms on the LHS denotes the time dependent term and the convective term. The first two terms on the RHS represent the transport of energy by particle velocity fluctuations and by particle collisions, respectively. The term q = adpd CC )m denotes the kinetic heat flux and q ° = represents the collisional heat... 

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