Coagulation Brownian

Consider a problem on definition of collision frequency of small spherical particles executing Brownian motion in a quiescent liquid. In Section 8.2, Brownian motion was considered as diffusion with a effective diffusion factor. It was supposed that suspension is sufficiently diluted, so it is possible to consider only the pair interactions of particles. To simplify the problem, consider a bi-disperse system of particles, that is, a suspension consisting of particles of two types particles of radius ai and particles of radius a2. In this formulation, the problem was first considered by Smolukhowski [59]. 

In the course of Brownian motion particles are in state of chaotic motion and randomly collide with each other. Choose one particle of radius ai called the test particle with a system of coordinates connected to this particle, with the origin at the particle s center (Fig. 10.4). 

The adopted diffusion model of Brownian motion allows to us consider the collision frequency of particles of radius U2 with the test particle of radius ai as a diffusion flux of particles U2 toward the particle a. Assume the surface of the particle ai to be ideally absorbing. It means that as soon as the particle U2 will come into contact with the particle ai, it will be absorbed by this particle. In other words, absorption occurs as soon as the center of the particle U2 reaches the surface of a sphere of radius Rc = a U2. The quantity Rc is called the coagulation radius. Hence, the concentration of particles a2 should be equal to zero at 

The characteristic time of Brownian diffusion is estimated by the expression 

For particles of radius a 0.1 mm in a water solution under normal conditions, this time is 5 s. Hence, if we consider times greater than thrown, the process of diffusion of particles U2 can be regarded as stationary. The number concentration of these particles n is described by the diffusion equation 

Aerosol particles suspended in a fluid may come into contact because of their Brownian motion or as a result of their motion produced by hydrodynamic, electrical, gravitational, or other forces. Brownian coagulation is often referred to as thermal coagulation. 

The theory of coagulation will be presented in two steps. At first, we are going to develop an expression describing the rate of collisions between two monodisperse particle populations consisting of N particles with diameter Dp and N2 with diameter Dp2. In the next step a differential equation describing the rate of change of a full coagulating aerosol size distribution will be derived. 

At this point, we need to describe mathematically the distribution of aerosols around the fixed particle. We need to consider, as with mass transfer to a particle, the continuum, free molecular, and transition regimes. 

Continuum Regime Assuming that the distribution of particles around our fixed particle can be described by the continuum diffusion equation then that distribution N r,t) satisfies 

The initial condition assumes that the particles are initially homogeneously distributed in space with a number concentration Aq. The first boundary condition requires that the number concentration of particles infinitely far from the particle absorbing sphere is not influenced by it. Finally, the boundary condition at r = 2Rp expresses the assumption that the fixed particle is a perfect absorber, that is, that particles adhere at every collision. 

FIGURE 13.4 The collision of two particles of radii Rp is equivalent geometrically to collisions of point particles with an absorbing sphere of radius 2Rp. Also shown in the corresponding coordinate system. 

The collision rate is initially extremely fast (actually it starts at infinity) but for t 4Rp/nD, it approaches a steady-state value of /coi = 8nRp DNq. Physically, at t — 0, other particles in the vicinity of the absorbing one collide with it, immediately resulting in a mathematically infinite collision rate. However, these particles are soon absorbed by the stationary particle and the concentration profile around our particle relaxes to its steady-state profile with a steady-state collision rate. One can easily calculate, given the Brownian diffusivities in Table 9.5, that such a system reaches steady state in 10-4 s for particles of diameter 0.1 pm and in roughly 0.1 s for 1 pm particles. Therefore neglecting the transition to this steady state is a good assumption for atmospheric applications. 

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Higashitani K and Matsuno Y 1979 Rapid Brownian coagulation of colloidal dispersions J. Chem. Eng. Japan 12 460-5... 

Spielman, L. A., Viscous interactions in Brownian coagulation. J. Colloid Interface Sci. 33, 562-571 (1970). 

G. Narsimhan, E. Ruckenstein The Brownian Coagulation of Aerosols over the Entire Range of Knud-sen Numbers Connection between the Sticking Probability and. the Interaction Forced JOURNAL OF COLLOID AND INTERFACE SCIENCE 104 2 (1985) 344-369. 

G. Narsimhan, E. Ruckenstein Monte Carlo Simulation of Brownian Coagulation over the Entire Range of Particle Sizes from Near Molecular to Colloidal Connection between Collision Efficiency and Interparticle ForcesJOURNAL OF COLLOID AND INTERFACE SCIENCE 107 1 (1985) 174-193. 

Very small aerosol particles can be generated via physical or chemical nucleation, and their subsequent growth is due either to condensation on the surface of the particles or to the Brownian coagulation of the particles themselves. If... 

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)... 

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). 

A model for Brownian coagulation of equal-sized electrically neutral aerosol particles is proposed. The model accounts for the van der Waals attraction and Born repulsion in the calculation of the rate of collisions and subsequent coagulation. In this model, the relative motion between two particles is considered to be free molecular in the neighborhood of the sphere of influence. The thickness of this region is taken to be equal to the correlation length of the relative Brownian motion. The relative motion of the particles outside this region is described... 

The effect of interparticle forces on Brownian coagulation of aerosols is usually accounted for through a phenomenological sticking probability, i.e., the probability of coagulation upon collision. 

Dispersed systems consisting of particles suspended in gases are frequently encountered in nature and in industrial surroundings. Brownian coagulation is an important mechanism by which coarsening of particle sizes in such aerocolloidal systems occurs. It plays... 

The rate of the Brownian coagulation between two particles of specified sizes can be predicted from (i) the rate of collisions between the particles, obtained through the analysis of their relative motion and (ii) the probability that a collision will lead to coagulation. It is customary to ignore the inter-... 

MONTE CARLO COMPUTER SIMULATION OF BROWNIAN COAGULATION... 

The Monte Carlo simulation of Brownian coagulation involves the evaluation of the ensemble average of the coagulation rate over a large number of particle pairs, through the generation of particle trajectories. The inter-particle forces due to the van der Waals attraction and Born repulsion are accounted for in the description of the relative motion [40] two Particles. The relative Brownian motion of two particles is described by the... 

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