Collision integrals reduced

In order to reduce the complexity of the problem, several approximation schemes have been developed. In the BGK model, the collision integral is replaced by a simple local term ensuring that the well-known Maxwell distribution is reached at thermal equilibrium [16]. The linearization method assumes that the phase space distribution is given by a small perturbation h on top of a (local) Maxwell distribu-tion/o (see, e.g., [17, 18]) ... 

Here P is the total pressure of the system, and ly is the collision integral, which is a function of the reduced temperature T = k T/Cij. The molecular... 

The reduced collision integrals, which depend on the particular form of the potential-energy function, are usually found tabulated as a function of the reduced temperature. However, an approximate fit to the reduced collision integral is given as... 

For a Lennard-Jones 6-12 potential (nonpolar interactions), the reduced collision integral may be approximated by a fit as... 

In practice, most often the expressions for transport properties are written in terms of reduced collision integrals... 

Finally, the reduced collision integral is usually expressed in terms of a reduced temperature T. If the intermolecular potential energy function can be expressed in the form [178]... 

Using this nondimensionalization, we can write the reduced collision integrals as a function of T alone ... 

Thus the reduced collision integrals can be done once and for all and tabulated as a function of T for a given intermolecular potential function... 

The remaining part of the collision integral can be calculated analytically by identifying the different cases, thereby reducing the discretized KE to a form similar to Eq. (7.52). A detailed description can be found in Aristov (2001). 

The function ffjl is derived analytically from the hard-sphere-collision integral, and readers interested in the exact forms are referred to Tables 6.1-6.3 of Chapter 6. One crucial issue is the description of the equilibrium distribution with QBMM. In fact, since the nonlinear collision source terms that drive the NDF and its moments to the Maxwellian equilibrium are approximated, the equilibrium is generally not perfectly described. The error involved is generally very small, and is reduced when the number of nodes is increased, but can be easily overcome by using some simple corrections. Details on these corrections for the isotropic Boltzmann equation test case are reported in Icardi et al. (2012). 

Usually, the so-called model kinetic equations are applied in practical calculations. They maintain the main properties of the exact collision integral and, at the same time, they reduce significantly the computational efforts. The most usual model kinetic equation was proposed by Bhatnagar, Gross, and Krook (BGK) which reads [4]... 

The term Qab is the collision integral, depending in a complicated way on temperature and the interaction energy of the colliding molecules, ab- ab values as a function of the reduced temperature T = kT/eab where k is the Boltzmann constant, have been tabulated. " The main disadvantage of the HBS equation is the difficulty encountered in evaluating [Pg.598]

In Figure 14.1, a comparison is presented between three older correlations for (Hanley 1974 Kestin et al. 1984 Rabinovich et al. 1988) and a correlation recently derived from the work of Bich etal. 990). The latter correlation serves as the baseline in the temperature range 100-2000 K. In each of the four cases the correlations, which originated from equation (14.4) for were not expressed in terms of the reduced effective collision cross section 6j[ but instead in terms of a reduced collision integral S2(2,2) Furthermore, for the higher-order correction the second-order Kihara approximation (Maitland et al. 1987) was applied. As detailed in the Appendix to Chapter 4, the conversion from collision integrals to effective cross sections is simple for isotropic interatomic potentials, namely. 

The collision integral term accounts for the effects of intermolecular interactions on the diffusion process and is a function of the reduced temperature... 

Once the reduced temperature has been determined, the collision integral can be read off Figure 5.1 or calculated using the equation shown in the upper portion of the plot. 

FIGURE 5.1 Plot of the Leonard-Jones reduced temperature and diffusion collision integral. 

To determine the collision integral using Figure 5.1, the reduced temperature must be calculated (Equation 5.6), which requires the determination of the interaction energies for naphthalene (Equation 5.8 or 5.9) and the binary air-naphthalene system (Equation 5.7, e- =1.34 x 10 " ergs) ... 

For the elastic-collision case the quantities in Eqs. (3.46) are again available in tables as a function of the reduced temperature T. From the point of view of computation, it is convenient also to have the collision integrals and their ratios expressed as polynomials in T. The appropriate polynomial coefficients for a fourth-order fit of and second-order fits of B, and C over limited ranges of T are given in Table 2. 

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