Collision frequency turbulent 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. 

In many, if not most, cases of practical interest, the fluid in which the particles are suspended is in turbulent motion. In Chapter 7. the effects of turbulence on the collision frequency function for coagulation were di.scussed. In the last chapter, nucleation in turbulent How was analy ,ed through certain scaling relations based on the form of the concentration and velocity fluctuations in the shear layer of a turbulent jet. In this section the GDE for turbulent flow is derived by making the Reynolds assumption that the fluid velocity and size distribution function cun be written as the sum of mean and fluctuating components ... 

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 mechanism of coagulation in a laminar flow has limited practical application since in the majority of applications the motion of liquid has a turbulent character. At turbulent motion, the collision frequency of particles increases very considerably in comparison with a quiescent environment or with laminar motion. Now consider the mechanism of particle coagulation in a turbulent flow, following the work [52]. 

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]. 

The principal shortcoming of the turbulent coagulation model offered by Levich [19] and rejected by many researchers is that it seriously overestimates the collision frequency of drops. Therefore the shear coagulation model [110] of particle coagulation in a turbulent flow has emerged as by far the most popular one. Since Levidfs model does not take into account the hydrodynamic interaction of particles, let us estimate the effect of this interaction on the collision frequency. 

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 mechanism of drop coagulation depends on the conditions of mixture flow. In laminar flow, the coagulation is caused by the approach of drops due to different velocities of their motion or in the non-uniform field of velocities of an external medium, or on sedimentation in the gravity field. In a turbulent flow, the approach of drops occurs due to chaotic turbulent pulsations. In comparison with the laminar flow, the number of collisions of drops in unit time increases. Any, even insignificant, mixing of the flow increases the collision frequency. 

The expressions for frequencies of bubble collision in laminar and turbulent flow which derived in the previous paragraphs make it possible to And the kernels of coagulation K co, V) and then proceed to solve the kinetic equation (25.1). Because the solution, generally speaking, presents significant mathematical difficulties, we shall only consider some simple special cases. 

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