Collision function

For sufficiently low density, g — 1, the collision function takes the form... 

For high densities, g cannot be set equal to one, and the collision function becomes much more complex and so is not given here. It turns out, however, that instead of using the full radial distribution function, it is sufficient to use the value at contact r — R, so that we define a new function ... 

Monodisperse hard-sphere collisions functions about spatial point x yields... 

These expressions for the collision function assume purely geometrical collisions and do not include electrostatic, van der Waals, or viscous forces. Corrections for these surface and fluid forces are available for Brownian coagulation and have been verifled experimentally by Lichten-belt et al. (12) in the absense of electrostatic forces for particles uniform in size. For shear coagulation, corrections have been computed for collisions between spheres of equal size, and experimental agreement with theory has been obtained only when electrostatic forces are absent (van de Ven and Mason (13) Zeichner and Schowalter (14)). Differential-sedimentation coagulation of hydrosols has not been examined theoretically or experimentally. 

The first simplification is to assume only one coagulation or sedimentation mechanism is dominant in a subrange of particle size. Figure 1 is a comparison of the collision functions for collision of an arbitrary particle with a particle of 1 /xm in diameter. Values of the collision functions were obtained directly from Equations 3, 4, and 6 for low fluid turbulence and low density particles. Because the collision functions plotted do not include the previously mentioned particle surface and fluid forces, this plot only approximates the dominance of a coagulation mechanism over a particle size interval. For particles less than 1 /xm. Brownian motion is the dominant collision mechanism, while particles from 1 to about 100 /xm collide because of fluid shearing. Collisions... 

Ph, Psh, Pds = collision functions for Brownian, shear, and diflFerential-sedimentation coagulation [L t" ] c = rate of turbulent energy dissipation... 

Pearson et al. (44) compared the collision functions of various collision mechanisms, shown in Table 1. It is interesting to note that, although all the mechanisms shown are dependent on the continuous-phase properties, the droplet sizes, and the flow conditions, only sedimentation and turbulent inertia are dependent on the density difference between the dispersed and continuous phases. These two mechanisms only occur where the droplets are of different size, and their collision functions tend to zero as the droplet sizes become closer. From the collision functions of these two mechanisms it is seen that turbulent inertia will only dominate sedimentary coalescence when the characteristic acceleration is greater than that of gravity ... 

Tabla 1 Collision Functions und Dimensional Parameters for Various Processes... 

Mechanism Collision function tfi) Source Dimensional parameter... 

As further aids, the designer can potentially utilize simulation programs, collision functions, computational modules for strength, etc. 

Equation (2.37) shows D varying almost as (although a more correct temperature variation is given by considering also the collision function of Fig. 2.5) and inversely as the pressure, which will serve for pressures up to abo,ut 1500 kN/m (15 atm) [19],... 

The functions and depend on the collision function model using gas density and temperature, and should satisfy the moment equation. The above-mentioned equation is substituted for the Boltzmann s equation, and a set of inhomogeneous linear equations is obtained by equating terms of equal order. The use of distribution functionsand so on leads to the determination of transport terms needed to close the continuum equations appropriate to the particular level of approximation. The continuum stress tensor and heat flux vector can be written in terms of the distribution function (f >). This can be further simplified in terms of macroscopic velocity and temperature derivatives. 

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