Collision polydisperse

The frequency of collisions is also expected to be greater in a polydisperse system than in a monodisperse system by the same logic as presented in item 1. 

According to this kinetic model the collision efficiency factor p can be evaluated from experimentally determined coagulation rate constants (Equation 2) when the transport parameters, KBT, rj are known (Equation 3). It has been shown recently that more complex rate laws, similarly corresponding to second order reactions, can be derived for the coagulation rate of polydisperse suspensions. When used to describe only the effects in the total number of particles of a heterodisperse suspension, Equations 2 and 3 are valid approximations (4). 

Note that the preceding equation is for ideal cases, in which the particles are monodis-persed, spherical, and totally elastic, and the contact surface is clean. In practice, the particles are usually nonspherical and polydispersed the collision could have involved some heat loss, plastic deformation, or even breakup and the contact surface may have impurities or contaminants. In these cases, a correction factor tj is introduced to account for the effects of these nonideal factors. The applicable form of the electric current through the ball probe is, thus, given by... 

The Fundamentals of Acoustic Agglomeration of Small Particulates. Let us consider a polydisperse aerosol consisting of submicrometer and micron sized particles. The mean separation distance between particles would typically be about 100 micrometers. Brownian movement of the particles is caused by the collision of the thermally agitated air molecules with the particles. Also any convection currents or turbulence in the carrier gas will of course cause the particles to be partially entrained and moved in the air. If we next impose an acoustic field of acoustic pressure p, the acoustic velocity u will be given by... 

Most aerosols are polydisperse when formed, some more than others. For example, an examination of sawdust would reveal particles of various sizes, as would that of any material formed by attrition. Since raindrops could grow by condensation or by a series of collisions with other drops, they would also be expected to be polydisperse. In fact, monodisperse aerosols are very rare in nature, and when they do appear, generally they do not last very long. Some high-altitude clouds are monodisperse, as are some materials formed by condensation. Sometimes it is satisfactory to represent all the particle sizes by only a single size. Other times more information is needed about the distribution of all particle sizes. Of course, a simple plot of particle frequency versus size gives a picture of the sizes present in the aerosol, but this may not be enough for a complete quantitative analysis. 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

The rest of this chapter is organized as follows. First, in Section 6.1, we consider the collision term for monodisperse hard-sphere collisions both for elastic and for inelastic particles. We introduce the kinetic closures due to Boltzmann (1872) and Enksog (1921) for the pair correlation function, and then derive the exact source terms for the velocity moments of arbitrary order and then for integer moments. Second, in Section 6.2, we consider the exact source terms for polydisperse hard-sphere collisions, deriving exact expressions for arbitrary and integer-order moments. Next, in Section 6.3, we consider simplified kinetic models for monodisperse and polydisperse systems that are derived from the exact collision source terms, and discuss their properties vis-d-vis the hard-sphere collision models. In Section 6.4, we discuss properties of the moment-transport equations derived from Eq. (6.1) with the hard-sphere collision models. Finally, in Section 6.5 we briefly describe how quadrature-based moment methods are applied to close the collision source terms for the velocity moments. 

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. 

As noted earlier, because the collisions are binary, the extension to polydisperse systems is straightforward and simply adds collision terms for every possible collision partner in the system. 

Freret, L., Laurent, F, de Chaisemarun, S. et al. 2008 Turbulent combustion of polydisperse evaporating sprays with droplet crossing Eulerian modeling of collisions at finite Knudsen and validation, in Proceedings of the 2008 CTR Summer Program, Stanford (CA) Center for Turbulence Research, pp. 277-288. 

If the polydispersity of bubbles generated in air-dissolved flotation or electroflotation is high, there is no need for additional introduction of centimicron bubbles. Optimal flow of two-stage flotation corresponds to the maximum attainable degree of monodispersity of bubbles. In this case the ratio between volume fractions of micro- and macrobubbles and collision efficiencies of the processes of particle capture by small bubbles and bubble coagulation must be such that the particle capture process outweighs the process of coalescence. 

Consider a polydisperse emulsion, assuming a spatially homogeneous case, with low volume concentrations of the disperse phase. Assume further that it is possible to limit ourselves to consideration of pair interactions of drops. The dynamics of enlargement (integration) of drops due to their collision and coalescence is then described by the following kinetic equation... 

Thus, the coalescence rate in a polydisperse emulsion is almost two times smaller than in a monodisperse one. It is explained by the fact that in an electric field, the collision frequency is higher for drops of commensurable sizes than for drops whose sizes differ by a lot (see Fig. 13.4). 

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