Collision integral properties

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

Monchick and Mason [289] have given tables of the collision integrals and transport properties for the Stockmayer potential. These table were calculated by integrating the potential over all orientations of the dipoles. The tables are actually presented as a function... 

All of the transport properties from the Chapman-Enskog theory depend on 2 collision integrals that describe the interactions between molecules. The values of the collision integrals themselves, discussed next, vary depending on the specified intermolecular potential (e.g., a hard-sphere potential or Lennard-Jones potential). However, the forms of the transport coefficients written in terms of the collision integrals, as in Eqs. 12.87 and 12.89, do not depend on the particular interaction potential function. 

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

The general properties of the Boltzmann collision integral are well known and will not be discussed here. 

Similarly, we may discuss the properties of the kinetic equations for atoms. For example, the collision integral hnab of Eq. (3.68) contains the following contributions for k = 3 ... 

Table B2 Collision integrals for predicting transport properties of gases at low densities ...

M. Sommerfeld and J. Kussin. Analysis of collision effects for turbulent gas-particle flow in a horizontal channel, part ii. integral properties and validation. Int. J. Multiphase Flow, 29(4) 701-718, 2003. 

The Lennard-Jones potential [4], [5], [9] (sixth-power attraction, twelfth-power repulsion) is quite realistic and appears to be the one most commonly used in practice. This potential contains two adjustable parameters (a size and a "strength ), which are defined and listed for various chemical compounds in [5], [6], and [9]. The collision integrals appearing in the first approximations to the transport properties are tabulated as functions of useful dimensionless forms of these two parameters in [5], [6], and [9]. Similar tabulations for other potentials may also be found in [5]. 

Gas transport properties are required to apply the theory given in Sections 3.3 and 3.4. Viscosities of pure nonpolar gases at low pressures are predicted from the Chapman-Enskog kinetic theory with a Lennard-Jones 12-6 potential. The collision integrals for viscosity and thermal conductivity with this potential are computed from the accurate curve-fits given by Neufeld et al. (1972). 

The above equation expresses the diffusion of vapor A through B medium, in which, Ma and Mb ate the molecular weight (kg/kmol), Po the absolute pressure, Oab and CIq are the molecule collision diameter and the molecule collision integral which are functions of substance properties and can be determined by empirical equations... 

The parameter the diffusion collision integral, is a function of k T/e, where is the Boltzmann constant and e is a molecular energy parameter. Values of tabulated as a function of k T/e, have been published (Hirschfelder et al., 1964 Bird et al., 1960). Neufeld et al., (1972) correlated using a simple eight parameter equation that is suitable for computer calculations (see, also, Danner and Daubert, 1983 Reid et al., 1987). Values of a and e/k (which has units of kelvin) can be found in the literature—for only a few species—or estimated from critical properties (Reid et al., 1987 Danner and Daubert, 1983). The mixture a is calculated as the arithmetic average of the pure component values. The mixture e is taken to be the geometric average of the pure component values. 

Although it is not strictly necessary for the monodisperse case, we employ a notation with subscripts 12 to denote the two particles involved in the colhsion. The reason for doing so is that in the polydisperse case, where particles 1 and 2 have different properties, it will be straightforward to modify the collision integrals with very little change in the notation. 

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]... 

Table B2 Collision Integrals for Predicting Transport Properties of Gases at Low Densities...

A further improvement appears to be necessary if at least one of the colliding species has a permanent dipole moment. Paul and Warnatz [86] investigated correction methods in the context of transport properties. Although we will not discuss details of these corrections, the basic effect is that permanent dipole moments decrease the collision diameter slightly, while the attractive energy is strongly increased. Because the value of the collision integral depends on lj, the overall effect at... 

Finally, we note an important property of the conservation equations, Eqs. (3.30) and (3.32), in the context of the Boltzmann collision integral. Letting primes denote postcollisional values and unprimed denote the precollisional values, as in the Boltzmann collision integral, the conservation equations, when applied to a collision, lead to... 

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