Collisional Rate of Change

In this section the derivation of the collision operators for dilute and dense suspensions are examined. Introductory the established formulas for dilute and dense gases consisting of elastic particles are outlined. Thereafter the kinetic theory of inelastic particles are considered. 

It is explained in chap 2 that the Boltzmann equation is an equation of motion for the one-particle distribution function and is appropriate to a rare gas [86, 50]. In this particular case appropriate expressions for the collision 

To elucidate the extended theory we refer to the derivation of the dilute gas collision operator and explain the important modifications that have to be introduced for dense gases. Basically, in order to calculate the collision rate of change we adopt similar principles as outlined for dilute gases in sects 2.4.3 and 2.4.2. However, a modified expression for the collision frequency is required. 

To deduce the formula for the dense gas collision frequency a modified relation for the volume of the collision cylinder is required. As mentioned in chap 2, it is customary to consider the motion of particles 2 relative to the center of particles 1 (see Fig 2.2). For a binary molecular collision to occur the center of particle 2 must lie on the sphere of influence with radius di2 about the center of particle 1, see Fig 2.7. The radius of the sphere of influence is defined by (2.152). Besides, since the solid angle dk centered about the apse line k is conveniently used in these calculations in which the billiard ball molecular model is adopted, it is also necessary to specify the direction of the line connecting the centers of the two particles at the instant of contact [86]. The two angles 6 and 4 are required for this purpose. Moreover, when the direction of the apse line lies in the range of 0, 4 and 6 - - dO, 4 + d4 , at the instant of collision, the center of particle 2 must lie on the small rectangle da cut out on the sphere of influence of particle 1 by the angles dO and d . The area of this rectangle is  

This relation is consistent with (2.167), after dividing the latter relation by dt, and denotes the basis for the dense gas extensions made by Enskog as explained in sect 2.11. 

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To calculate the collisional rate of change for dense suspensions Gidaspow [22] adopted the dense gas collision operator (4.13), but in this case the particles are inelastic so g2i is related to g2i through the empirical relation (2.123). The kinetic fluxes for dilute suspensions were determined adopting the dilute gas collision operator (4.10), valid for elastic particles, instead. For dense suspensions the kinetic fluxes were approximated by those deduced for dilute suspensions. 

The collisional rate of change Z (V )coiiision of any particle property tjj is the integral over all possible binary collisions of the change in 0 in a particular collision multiplied by the probability frequency of such a collision. Hence, particle 1 gains of the microscopic property tjj during the collision... 

A similar expression for the collisional rate of change for particle 2 can be obtained. In this case we utilize the collision s unmetry properties, so this relation is achieved by interchanging the labels 1 and 2 in (4.15) and replacing k by —k. As distinct from the previous analysis, to determine this probability frequency at the instant of a collision between particles labeled 1 and 2 we now take the center of the second particle to be located at position r and the center of particle 1 to be at r — di2k. This approach represents a collision dynamically identical but statistically different from the previous one [31] [49] [32]. The result is ... 

Each of the integrals equals the collisional rate of change (V )coiiision, but a more symmetric expression of /i(V )Collision might be achieved taking one half of the sum of (4.15) and (4.16) [31, 32]. 

To complete the reformulation of the Boltzmann equation replacing the microscopic particle velocity with the peculiar velocity, the collisional rate of change term has to be modified accordingly. Jenkins and Richman [32] proposed the following approximate formula ... 

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

To ensure that the spatial shifting is correct Lun et al. [102] introduced a new position vector without label, defined such that r = ri and r2 = r -F dn. After the new position vector was introduced in (4.15), an expression similar to (4.15) for the collisional rate of change of solid particle 2 was derived. To solve this task Lun et al. [102] did utilize the collision symmetry properties such that this relation is... 

It is also expected that reactive collisions may diminish the effects of collisional damping of the z-oscillation. An unreactive collision removes energy from the z-mode oscillation so that the ion contributes more signal current at its original cyclotron frequency whereas a reactive collision removes an ion from a reactant population giving a true indication of the loss from the original population. The loss rate from the reactant population for ions of z-oscillation, Az, is proportional to the density of reactant ions of amplitude Az. Thus, for very reactive ions, no change in sensitivity due to collisional relaxation is expected. 

Other possible solvent effects depend on the heavy-atom effect, which may favor intersystem crossing, and on the viscosity, which may change the rate of diffusion and hence the collisional triplet-triplet energy transfer. These effects have already been mentioned in previous sections. 

If dissociation is investigated without dilution by inert carrier gas, the rate constant may change with time at low pressures during reaction. As A dissociates the composition of collisional partners will change and so hence the effective value of k will change with time. 

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