Collision frequency Brownian coagulation

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. 

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... 

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... 

Brownian Coagulation Dynamics of Discrete Distribution for an Initially Monodisperse Aerosol 192 Brownian Coagulation Effect of Particle Force Fields 196 Effect of van der Waals Forces 197 Effect of Coulomb Forces 200 Collision Frequency for Laminar Shear 200 Simultaneous Laminar Shear and Brownian Motion 202 Turbulent Coagulation 204... 

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... 

For particles whose size exceeds 0.1 pm, we get P 2)t iPn)br-The expressions given in this section for collision frequencies in the processes of Brownian, shear, and turbulent coagulation are derived with no account taken of hydrodynamic, molecular, and electrostatic interactions of particles. Taking them into account considerably complicates the problem. In particular, in the fac-... 

The use of turbulent emulsion flow regime to facilitate integration of drops is justifled by the substantial increase of collision frequency that is achieved in a turbulent flow as compared to the collision frequency during the sedimentation of drops in a quiescent liquid or in a laminar flow. Particles suspended in the liquid are entrained by turbulent pulsations and move chaotically inside the volume in a pattern similar to Brownian motion. Therefore this pulsation motion of particles can be characterized by the effective factor of turbulent diffusion Dj, and the problem reduces to the determination of collision frequency of particles in the framework of the diffusion problem, as it was first done by Smoluchowsld for Brownian motion [18]. A similar approach was first proposed and realized in [19] for the problem of coagulation of non-interacting particles. The result was that the obtained frequency of collisions turned out to be much greater than the frequency found in experiments on turbulent flow of emulsion in pipes and agitators [20, 21]. 

For Browrtian diffusion of small particles, the influence of hydrodynamic interaction on the collision frequency was studied in works [28, 29], which also mention the decrease in the collision frequency by a factor of 1.5-2. This decrease is not as large as in the case of turbulent coagulation. There are two reasons why the effect of hydrodynamic interaction on the collision frequency of particles differs so substantially in the cases of turbulent flow and Brownian motion. First, the particle size is different in these two cases (the characteristic size of particles participating in Brownian motion is smaller than that of particles in a turbulent emulsion flow). Second, the hydrodynamic force behaves differently (the factor of Browrtian diffusion is inversely proportional to the first power of the hydrodynamic resistance factor h, and the factor of turbulent diffusion - to the second power of h). 

The rate of coagulation of particles in a liquid depends on the frequency of collisions between particles due to their relative motion. When this motion is due to Brownian movement coagulation is termed perikinetic when the relative motion is caused by velocity gradients coagulation is termed orthokinetic. 

Assuming an infinite repulsion potential, the particles would be stable for ever however, since in reality, repulsion potentials are finite there is always the probability of particle aggregation due to thermal fluctuations. The rate of particle coagulation is a function of the frequency of particles encounters, and of the probability of coagulation at this state [65]. Without repulsion coagulation will proceed very rapidly, even in fairly dilute dispersions, with the particles aggregating at the same rate at which they become encountered, by diffusion through the continuous phase. This rate is termed the Brownian collision rate or the... 

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