Boltzmann Integral-Differential Equation
The Boltzmann equation for the frequency distribution, f (Gidaspow 1994), can be written as
where c and r were regarded as independent coordinates and where Newton’s law of motion was
For binary collisions of rigid particles, the right-hand side of the Boltzmann equation (2.23) assumes the form
where the primes indicate the quantities after particle interaction-collision and /2) is the product of the respective single-particle distributions. Hence, the Boltzmann equation is an integral-differential equation. Because of its nonlinearity, it must be solved by iteration. For the first approximation, one takes the Maxwellian distribution. The second approximation, as shown in detail in Chapman and Cowling (1970), will give rise to a Navier-Stokes-type equation. This is done efficiently using an altered form of the Boltzmann operator, as presented below,
Changing the coordinates from c to C,
Applying chain rule to Eqs. (2.25-2.27), the Boltzmann equation can be expressed as
A transport equation for a quantity у can be obtained starting with the Boltzmann equation by multiplying it by у (Eq. 2.29) and integrating over c (Eq. 2.30),
Now, we need to find the single-particle distribution function f(r, c, t) and the pair distribution function f(2)(rj_, ci r2, c2; t). Here, we take the Maxwellian velocity distribution function as the single-particle distribution (Savage and Jeffrey 1981; Jenkins and Savage 1983; Ding and Gidaspow 1990)
and the Enskog assumption for the pair distribution function is used next (Chapman and Cowling 1970; Lun et al. 1984; Ding and Gidaspow 1990). That is,
where go is the equilibrium radial distribution function (Savage and Jeffrey 1981).
Detailed derivation of the equations can be found elsewhere (Ding and Gidaspow 1990; Gidaspow 1994).