Coagulation coefficient

When the particle sizes are much larger than the mean free path of the suspending medium, the Knudsen number (Kn = Agfa, where a and Ag are the radius of the particle and the mean free path of the medium, respectively) is extremely small. Under such conditions, the Brownian coagulation coefficient is well described by the Smoluchowski equation (1)... 

Smoluchowski s equation was extended up to Kn < 0.25, by introducing a slip correction factor C(Kri) in the expression for the diffusion coefficient. The coagulation coefficient is given for Kn < 0.25 by the expression... 

In the free molecular limit (Kn > 10), the coagulation coefficient is obtained from the kinetic theory of rarefied gases as (4)... 

For the transition region (0.1 < Kn < 10), Fuchs (5), assuming the sticking probability equal to unity, proposed the following semi-empirical expression for the coagulation coefficient... 

Sitarski and Seinfeld (6) were the first to provide a theoretical basis for Fuch s semi-empirical formula, by solving the Fokker-Plank equation by means of Grad s (7) 13-moment method. Their solution was further improved by Mork et al. (8). The Brownian coagulation coefficient predicted by these models agrees fairly well with the Fuchs interpolation formula. However, the model does not predict the proper free molecular limit. The validity of the Fuchs semiempirical formula was further reinforced, by the Monte Carlo simulations of Brownian coagulation, by Nowakowski and Sitarski (9). 

The coagulation coefficient is a function of the radius of the particle Rp, its mass m , the effective particle Knudsen number k, the temperature of the medium T, and the depth of the interaction potential well between two particles. Using the expression for the overall interaction potential given in Appendix I, the depth of the interaction potential well can be calculated from the knowledge of the Hamaker constant for the particle. The friction coefficient f is related to the diffusion coefficient of the particle, D, through the Einstein equation... 

The limiting behavior of the coagulation coefficient /3, in the continuum and the free molecular regimes can now be examined. In the continuum limit, where Krt — 0, one has... 

In order to calculate the coagulation coefficient ft, Eqs. [94J and [95] must be solved for the unknown constants A and B. It results that the constant A is given by... 

From Eqs. [80] to [83], the coagulation coefficient ft can be expressed in terms of the effective particle Knudsen number k and the diffusion coefficient of the particle D as... 

Therefore, the expression for the coagulation coefficient, which is valid for Iaige particles, reduces to the free molecular limit for the rate of collisions when Kw — cc. 

The coagulation coefficient given by Eq. [97] is referred to in what follows as the upper bound, even though, as explained later, it constitutes an upper bound only for sufficiently small particles. 

The lower bound for the coagulation coefficient, ft, is calculated from Eq. [81] using the values of the constant A determined from the numerical solution of Eqs. [74], [77], and [78]. The upper bound for the coagulation coefficient, ft, is obtained from Eq. [97]. In the above calculations, the Philips slip correction factor (Eq. [6]), is used for the calculation of the diffusion coefficient of particles. The results are expressed in terms of the dimensionless coagulation coefficient y defined as the ratio between the coagulation coefficient and the Smoluchowski coagulation coefficient... 

For particles of unit density, in air, at a pressure of 1 atm and temperature of 298°K, the upper and lower bounds of the dimensionless coagulation coefficients are plotted as a function of Knudsen number, for different Hamaker constants, in Fig. 3. Obviously, the upper bound for the coagulation coefficient is independent of the Hamaker constant. The upper and the lower bounds tend to the Smoluchowski expression for small Knudsen numbers. For large Knudsen numbers, the upper bound coincides with the free molecular limit, as can be seen from Fig. 3. The lower bound is found to decrease dramatically with a decrease in the Hamaker constant, for large Knudsen numbers. Both the lower and the upper bounds exhibit a maximum at intermediate values of the Knudsen number. 

Table 1 compares the dimensionless coagulation coefficient predicted by the present model with other models. Since the Hamaker constant for most of the aerosol systems is of the order of 10"12 eig, this value is used in the calculation of the lower bound. Particle diffusion coefficients based on Philips slip correction factor for an accommodation coefficient of unity are used for the calculation of the coagulation coefficients ft (the Fuchs interpolation formula) and fts (the Sitarski... 

Fio. 3. Dimensionless coagulation coefficients predicted by the upper and the lower bounds for equalsized particles of unit density, in air, at 1 atm and 298 K. 

Comparison of the Coagulation Coefficients Predicted by Different Models for Equal-Sized Particles,... 

Wagner and Kerker (12) and Chatteijee et al. (13) have measured the coagulation coefficients for monodispersed aerosols of diethyl hexyl sebacate. In their experiments, the... 

Experimental data of the coagulation coefficient for NaCI aerosols of different sizes obtained by Shon et al. (14) are compared with the upper bound, with the lower bound (for a Hamaker constant of 10 12 erg) and with the Fuchs interpolation formula in Fig. 9. The experimental data exhibit considerable scatter, the upper bound for the coagulation coefficient agrees somewhat better with the experimental data than the Fuchs interpolation formula. 

Experimental data regarding the coagulation coefficients of Pt aerosols, measured by Nolan and Keenan (15) are compared with different models in Table IV. The corrections for polydispersity in the above data, as reported by Mercer (16), are accounted for in the experimental values of the coagulation... 

The main result of the paper is shown in Fig. 3 in which the upper and lower bounds of the ratio between the coagulation coefficient and the Smoluchowski s coagulation coefficient are plotted as a function of the Knudsen number, for different Hamaker constants (for the lower bound), for particles of unit density. 

When the particle size is very small compared to the mean free path, i.e., Kn > 10, the particle can be regarded as a large spherical molecule undergoing independent binary collisions with the gas molecules and the coagulation coefficient is obtained from the kinetic theory of rarefied gases as (5)... 

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