brownian coagulation model

8/14/2019 Brownian Coagulation Model

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This is an author version of the paper: Celada A. T., Salcido A. (2003). Simple Model for Brownian Coagulation of

Atmospheric Polydisperse Aerosols. Journal of Aerosol Science, vol.1:S337-S338. European Aerosol Conference 2003.

This version has been reviewed but does not include the final publisher corrections and published layout.

SIMPLE MODEL FOR BROWIA COAGULATIO

OF ATMOSPHERIC POLYDISPERSE AEROSOLS

A.T. CELADA and A. SALCIDOInstituto de Investigaciones Elctricas, Divisin de Energas Alternas, Reforma No. 113, Col. Palmira,

Cuernavaca, Morelos, C.P. 62490, [email protected]

Keywords: ATMOSPHERIC AEROSOL, BROWNIAN COAGULATION, AEROSOL DYNAMICS,

POLYDISPERSE AEROSOLS.

INTRODUCTION

A simple probabilistic model for brownian coagulation of polydispersed aerosols is proposed. This model

(called COAG) is based on a discrete balance equation that gives the rate of change of the number density of

particles with diameters within a given range, in terms of the rates of formation and loss of particles in allother diameter ranges. The model does not consider the particles monomeric structure such as it is proposed inthe Smoluchowsky theory, instead it uses a probabilistic estimate of formation or loss of particles dependentof the diameters ranges of the colliding particles. To test the model, it was used to try to reproduce the results

of three aerosol coagulation experiments carried out by Rooker and Davies (1979). The computer simulationresults were found in good agreement with the experimental data.

METHODS

Given the full diameter spectrum of the aerosol particles, it is divided in a given number M of bins, k, notnecessarily all with the same length. If it is assumed that only binary collisions occur between the aerosol

particles, a particle in kwill be created depending of three factors: (i) the probability that one collision occurs

in the system, (ii) the probability that the collision was between particles of bins i and j, and (iii) theprobability that the output of the collision ij be a particle of bin k. The first of these probabilities

depends on the dynamic regime of the aerosol motion. Then, if it is assumed brownian coagulation, theproduct of the first two probabilities may be expressed as follows:

= 4 + +

= 2

where ri and rj are, respectively, the mean radii of the particles in the i and j bins, Di and Dj are thecorresponding diffusion coefficients, ni and nj are the number densities of the particles in these bins, and n is

the aerosol number density. The last probability factor, denoted by Qijk

, may be estimated using the volumeconservation of the colliding particles. In the computational implementation of the model, this probability, for


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each bin k, is calculated numerically taking into account all possible sizes of the coagulating particles(polydisperse aerosol) and stored as a lookup table. Taking into account these observations, the basic balance

equation of the model can be written as

= 2 12

, (1 )

This equation gives the rate of change of the number density of the particles that belong to the bin k as resultof the brownian coagulation of the particles of all other bins. This equation does not include any reference toother loss mechanisms such as sedimentation or wall adherence. This equation may be solved by any of thewell-known numerical methods for systems of ordinary differential equations.

To test the model, it was considered a series of CaCO3 aerosol coagulation experiments carried out by Rooker

and Davies in 1978. In these experiments, they measured the total number density of the aerosol particles asfunction of time, and calculated the coagulation and wall adherence coefficients. The aerosol consisted of

compact roughly spherical particles with a range of radii between 0.0025-0.015 m. The particle sizedistribution was obtained from a series of micrographs taken across the deposit in a thermal precipitator, and it

was approximately lognormal with median radius of 0.005 m and = 1.5.

In the computer simulations we performed to test the model, it was considered a diameter range from 0.005 to

0.1 m divided in six bins: 0.005-0.01, 0.01-0.015, 0.015-0.020, 0.020-0.025, 0.025-0.030, and 0.030-0.1. Thenumber densities for each bin was simulated for a time interval of 1800 s with time step of 1s, and the totalnumber density was obtained as the sum of the bin number densities each time step. For comparison purposes,the total number density was corrected using the wall adherence coefficient reported by Rooker and Davies. InFigures 1, 2, and 3, the simulation results obtained for the experiments C50, C63 and C90, respectively, of the

series reported by Rooker and Davies are shown (solid line). In the same figures the experimental data areincluded (open circles).

Figure 1. Comparison of the experimental data and the estimates obtained with the model COAG for theRooker and Davies (1979) coagulation

experiment C50. The rate of change of total number density of particles for CaCO3 aerosol to 25C is shown.


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This is an author version of the paper: Celada A. T., Salcido A. (2003). Simple Model for Brownian Coagulation of

Atmospheric Polydisperse Aerosols. Journal of Aerosol Science, vol.1:S337-S338. European Aerosol Conference 2003.

This version has been reviewed but does not include the final publisher corrections and published layout.





CONCLUSIONS

A simple probabilistic model for brownian coagulation of polydisperse aerosols was presented. To test the

correctness of the model estimates, a comparison of the simulation results with experimental data fromRooker and Davies was made. It was found a very good agreement between the model estimates and theexperimental data. This agreement allows us to conclude that the physical and probabilistic basis of the model

reflect appropriately the brownian coagulation mechanism of polydisperse aerosols.

REFERENCES

Rooker, S. J., & Davies C. N. (1979) Measurement of the coagulation rate of a high Knudsen number aerosol

with allowance for wall losses.J. Aerosol Sci.10 139-150.

brownian coagulation model

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