The Coagulation Equation

In the previous sections, we focused our attention on calculation of the instantaneous coagulation rate between two monodisperse aerosol populations. In this section, we will develop the overall expression describing the evolution of a polydisperse coagulating aerosol population. A good place to start is with a discrete aerosol distribution. 

The Discrete Coagulation Equation A spatially homogeneous aerosol of uniform chemical composition can be fully characterized by the number densities of particles of various monomer contents as a function of time, Afc(f). The dynamic equation governing 

Nk(t) can be developed as follows. We have seen that the rate of collision between particles of two types (sizes) is J 2 = K NxNj, expressed in units of collisions per cm3 of fluid per second. A fc-mer can be formed by collision of a (k - / )-mer with ay-mer. The overall rate of formation, J, of the fc-mer will be the sum of the rates that produce the fc-mer or 

To investigate the validity of this formula let us apply it to a trimer, which can be formed only by collision of a monomer with a dimer 

This expression appears to be incorrect, as we have counted the collisions twice. To correct for that we need to multiply the rate by a factor of so that collisions are not counted twice. Therefore 

FIGURE 12.10 Coagulation coefficient at 300 K, = I g cm, A/kT = 20 as a function of Knudsen number ( Kup) for particle radii ratios of 1,2, and 5 both in the presence and in the absence of interparticle forces. 

FIGURE 12.11 Enhancement over Brownian coagulation as a function of particle Knudsen number for A/kT = 20. In the continuum regime there is a retardation of coagulation because of the viscous forces. 

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Drake, R.L., 1972. A general mathematical survey of the coagulation equation. In Topics in current aerosol research, Pt. 2. Eds. G.M. Tidy and J.R. Brocks, New York Pergamon. 

For orthokinetic conditions the ratio of adsorption halftime to coagulation halftime can be calculated by integrating the adsorption Equation (6) and the coagulation Equation (2). 

To circumvent the problem of many equations, it is possible to represent the coagulation equation as a nonlinear integrodifferential equation of the form... 

Using K0 as the coagulation constant and neglecting the second term in Eq. 18.9 yield the usual form of the coagulation equation ... 

One recent development in astrophysical stochastic processes has been the widespread use of coagulation calculations for both nucleation phenomena and dust formation and processes that relate to the distribution function for masses and mass ratios in forming stellar systems. The use of the coagulation equation for the study of star and stellar system formation in particular has been quite recent and warrants a review. 

Perhaps the first studies to employ agglomeration were those related to the formation of the solar system. The use of the coagulation equation, essentially a macroscopic version of the master equation for systems which can be treated as being controlled by one independent variable and time, has been of some importance in recent simulations of the process of star formation. Employed for some time in the study of nucleation, " the results were first used extensively by Safronov and his collaborators in the modeling of planetary formation in the solar nebula. 

It is such considerations that have given rise to the application of analytic treatments of the coagulation equation. The first such treatment for clouds is that of Nakano, who assumed that all fragments initially have the Jeans mass. Taking several forms for the agglomeration coeflScient, which was assumed to be a simple mass-dependent quantity (or constant), the coagulation equation was numerically integrated. For a (/i, p ) constant or = + the resultant mass distribution is peaked between... 

For Brownian coagulation in the continuum range, the collision frequency function is given by (7.16). Substitution of the similarity form (7.69) reduces the coagulation equation for the continuous distribution (7.67) with (7.16) to the following form ... 

Figure 7.8 Sdt -prcscrving particle size distribution for Brownian coagulation, Tlie Ibnn is appaw-imatcly lognormal. The re.sult obtained by solution of the ordinary integrodiffereniial equation for the continuous spectrum is compared with the limiting solution of Hidy and Lilly (1965) for the discrete spectrum, calculated from the discrete form of the coagulation equation. Shown also are points calculated from analytical solutions for the lower and upper ends of the distribution (Friedlandcr and Wang. 1966).

DYNAMICS OF AEROSOL POPULATIONS 12.3.7 Solution of the Coagulation Equation... 

Solution of the Coagulation Equation by the Laplace Transform The Laplace transform of n v,t) is by definition... 

Galkin, V.A., On stability and stabilization of solutions of the coagulation equation. Diff. Urav. 14, (1978), 1863-1874. 

The existence of a self-similar solution for the coagulation equation has been addressed by the following Lushnikov (1973), Ziflf et al (1983), van Dongen and Ernst (1988). 

Drake, R. L., A General Mathematical Survey of the Coagulation Equation, in Topics in Current Aerosol Research, (Part 2), (G. M. Hidy and J. R. Brock, Eds.), p. 315, Pergamon Press, New York, 1970. 

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