Collision integrals

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... 

Wlien H has reached its minimum value this is the well known Maxwell-Boltzmaim distribution for a gas in themial equilibrium with a unifomi motion u. So, argues Boltzmaim, solutions of his equation for an isolated system approach an equilibrium state, just as real gases seem to do. Up to a negative factor (-/fg, in fact), differences in H are the same as differences in the themiodynamic entropy between initial and final equilibrium states. Boltzmaim thought that his //-tiieorem gave a foundation of the increase in entropy as a result of the collision integral, whose derivation was based on the Stosszahlansatz. 

When ions move under equilibrium conditions in a gas and an external electric field, the energy gained from the electric field E between collisions is lost to the gas upon collision so that the ions move with a constant drift speed v = KE. The mobility K of ions of charge e in a gas of density N is given in tenns of the collision integral by the Chapman-Enskog fomuila [2]... 

The values of s/k are less dran 600 K for most of the simple molecules which are found in high temperature systems, and hence the collision integral may be assumed to have a value of unity in tlrese systems. 

These representations -will be used to calculate collision integrals in Section 1.15. 

Coefficient Equations.—To determine the coefficients of the expansion, the distribution function, Eq. (1-72), is used in the Boltzmann equation the equation is then multiplied by any one of the polynomials, and integrated over velocity. This gives rise to an infinite set of coupled equations for the coefficients. Only a few of the coefficients appear on the left of each equation in general, however, all coefficients (and products) appear on the right side due to the nonlinearity of the collision integral. Methods of solving these equations approximately will be discussed in later sections. 

Bather than carrying out the calculation for the general case, which yields rather unwieldy expressions, only equations sufficient to obtain certain approximations will be developed. If we multiply the Boltzmann equation, Eq. (1-39), by 1 = i%( 2)3r )) (0.9>)> the resulting equation is simply the equation of conservation of mass, since integrating unity over the collision integral gives zero ... 

When we multiply Eq. (1-39) by Sff2 7 0), the integration again gives a collision integral term that goes to zero the resulting equation is ... 

The collision integral for Eq. (1-87) may be developed in the same manner and yields for Maxwell molecules ... 

Case of General Central Force Law.—The evaluation of the collision integrals for tire viscosity, Eq. (1-89), was simplified for... 

Consider electrons of mass m and velocity v, and atoms of mass M and velocity V we have mjM 1. The distribution function for the electrons will be denoted by /(v,<) (we assume no space dependence) that for the atoms, F( V), assumed Maxwellian as usual, in the collision integral, unprimed quantities refer to values before collision, while primed quantities are the values after collision. In general, we would have three Boltzmann equations (one each for the electrons, ions, and neutrals), each containing three collision terms (one for self-collisions, and one each for collisions with the other two species). We are interested only in the equation for the electron distribution function by the assumption of slight ionization, we neglect the electron-electron... 

To the lowest order of approximation, which is used to evaluate the collision integrals for the perturbation terms (vzfx and we take m/M — 0. There is, thus, no interchange of energy between the electrons and neutral atoms, so that... 

Although this lowest order approximation is used in determining the first order corrections to the distribution function, it is necessary to go to a higher order of approximation in determining the collision integral of Eq. (1-140). If we keep terms to first order in the small quantity m/M, the collision integral may be evaluated to give 28... 

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]) ... 

Maximum intrapellet temperatures, 25 305 effect of diffusion collision integral on, 25 301-303... 

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 collision integral has been evaluated and tabulated by Chapman and Cowling and Hirschfelder, Curtiss, and Bird. Bird, Stewart, and Lightfoot listed the values published by Hirschfelder et al. (1949), and they also tabulated values of ejj for common gases. Numerous correlations have been proposed for estimating Bjj and Ojj, and Ravindran et al. (1979) reviewed and compared the various methods. 

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