Collision frequency function

For Brownian motion, the collision frequency function is based on Fick s first law with the particle s diffusion coefficient given by the Stokes-Einstein equation. The Stokes-Einstein relation states that... 

The lognormal distribution function can be interpreted physically as the result of a process of breakup of larger particles at rates that are normally distributed with respect to particle size (Aitchison and Brown, 1957). Approximately lognormal distributions also result when the aerosol size distribution is controlled by coagulation (Chapter 7). In this case the value of the standard deviation is determined by the form of the particle collision frequency function. Multimodal aerosols may result when particles from several different types of sources are mixed. Such distributions are often approximated by adding lognormal distributions, each of which corresponds to a mode in the observed distribution and to a particular type of source. 

Aerosols are unstable with respect to coagulation. The reduction in surface area that accompanie.s coalescence corresponds to a reduction in the Gibbs free energy under conditions of constant temperature and pressure. The prediction of aerosol coagulation rates is a two-step process. The first is the derivation of a mathematical expression that keeps count of particle collisions as a function of particle size it incorporates a general expression for tlie collision frequency function. An expression for the collision frequency based on a physical model is then introduced into the equation Chat keep.s count of collisions. The collision mechanisms include Brownian motion, laminar shear, and turbulence. There may be interacting force fields between the particles. The processes are basically nonlinear, and this lead.s to formidable difficulties in the mathematical theory. 

The quantity XjXj = 0 because the motion of the two particles is independent. The collision frequency function first derived by Smoluchowski is then obtained by substitution in (7.13) ... 

A simple solution to the kinetic equation for Brownian coagulation can be obtained for nearly monodisperse systems. Setting u/ = Vj in (7.16). the collision frequency function is given by... 

Figure 7.] Varialicin of collision frequency function (i(a. 02) with particle size ratio a ju2 for air at 23°C and 1 atm based on Fuchs (1964, p. 294). The value of f i, 2) is. smallest for particles of equal size (01/02 = I) and the spread in value with particle size is smallesl. Forrii/o =. P go< s through a weak maximum for Knudsen number near 5. The value of (0. 03) highest for interacting panicles of very different sizes (large 01/02). The lowest curves correspond to the continuum regime.

To derive an expression for the collision frequency function, refer to Fig. 7.4a, which shows a single panicle in the shear field with radius a, interacting with particles of radius Qj (Fig. 7.4b). The velocity of the particles normal to the surface of the page, relative to the particle shown, is xUlu/dx). Hence the flow of particles into the shaded portion of the strip dx is... 

The collision frequency function for coagulation by laminar shear is, therefore,... 

The various collision mechanisms are compared in Fig. 7.7 which shows the collision frequency function for l- m particles interacting with particles of other sizes. Under conditions corresponding to turbulence in the open atmosphere ( j ss 5cm /sec- ), either Brownian motion or differential sedimentation plays a dominant role. Brownian motion controls for particles smaller than 1 jam. At lower altitudes in the atmosphere and in turbulent pipe flows, shear becomes important. 

The determination of the form of ijf is carried out in two steps. First, the special form of the distribution function (7.69) is tested by substitution in the equation of coagulation for the continuous distribution function (7.67) with the appropriate collision frequency function, if the transformarion is consistent with the equation, an ordinary integrodifferential equation for as a function of t) is obtained. The next step is to find a solution of this equation subject to the integral constraints (7.70) and (7,71) and also find the limits on n(u). For some collision kernels, solutions for (tj) that satisfy these constraints may not exist. 

Consider the flow of an aerosol through a 4-in. duct at a velocity of 50 ft/sec. Compare the coagulation rate by Brownian motion and laminar shear in the viscous sublayer, near the wall. Present your results by plotting the collision frequency function for particles with dp = 1 /im colliding with particle.s of other sizes. Assume a temperature of 20 C. Hint In the viscous. sublayer, the velocity distribution is given by the relation... 

The contribution of the coagulation term [dtft/di]co3g vanishes identically no matter what the form of the collision frequency function. The coagulation mechanism only shifts matter up the distribution function from small to large sizes and does not change the local volumetric concentration of aerosol. 

In many, if not most, cases of practical interest, the fluid in which the particles are suspended is in turbulent motion. In Chapter 7. the effects of turbulence on the collision frequency function for coagulation were di.scussed. In the last chapter, nucleation in turbulent How was analy ,ed through certain scaling relations based on the form of the concentration and velocity fluctuations in the shear layer of a turbulent jet. In this section the GDE for turbulent flow is derived by making the Reynolds assumption that the fluid velocity and size distribution function cun be written as the sum of mean and fluctuating components ... 

This is the equation that is usually solved in calculating simultaneous coagulation and settling in a well-mixed chamber. Numerical solutions for special values of the collision frequency function have been obtained by Lindauer and Castleman (1971). They report results for the decay in the mass concentration as a function of chamber height and lime. 

Here rif, n, and are the number concentrations of particles with sizes i, /, and k respectively, t is time, and p(i,j) is a collision frequency function that depends on the mode of interparticle contact. 

For the flocculation tank, the significant transport mechanisms leading to interparticle collisions are assumed to be Brownian diffusion and fluid shear. Expressions for the collision frequency functions for these mechanisms were derived by Smoluchowski and are as follows ... 

Assumptions made in modeling the flocculation tank include (1) ideal or plug flow exists throughout the tank (2) the velocity gradient is identical everywhere in the tank (3) flocculation occurs simultaneously by Brownian diffusion and fluid shear, and the collision frequency functions are simply additive (4) the particles are distributed uniformly... 

Flocculation by Brownian diffusion is characterized by the collision frequency function presented in Equation 14. For flocculation by differential settling, the corresponding function is ... 

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