Coagulation kinetic equation

Filtration is analogous to coagulation in many respects. This is illustrated by juxtaposing the basic kinetic equations on particle removal ... 

Kinetic equations for reversible adsorption and reversible coagulation are established when the interaction potential has primary and secondary minima of comparable depths. The process is assumed to occur in two successive steps. First the particles move from the bulk of the fluid to the secondary minimum. A fraction of the particles which have arrived al the secondary minimum move further to the primary minimum. Quasi-steady state is assumed for each of the steps separately. Conditions are identified under which rates of reversible adsorption or coagulation at the primary minimum can be computed by neglecting the rate of accumulation at the secondary minimum. The interaction force boundary layer approach has been improved by introducing the tangential velocity of the particles near the surface of the collector into the kinetic equations. To account for reversibility a short-range repulsion term is included in the interaction potential. 

By splitting the quasi-steady-state assumption of diffusion of particles under the action of the interaction force field into two parts, kinetic equations which account for accumulation at both the primary and secondary minimum are formulated. Conditions are established under which, after a short transient, reversible adsorption or coagulation can be treated by neglecting accumulation at the secondary minimum. The effect of tangential velocity of particles on the rate of reversible adsorption is analyzed and a criterion established when the effect... 

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... 

The kinetic theory of fast irreversible coagulation was developed by von Smoluchowski. Later the theory was extended to the case of slow and reversible coagulation. In any case of coagulation (flocculation), the general set of kinetic equations reads... 

Consider the balance of particles of the sort k. Their number increases at collisions with particles of sort i and k — i, and decreases at collisions with particles of sorts k and i. The equation describing coagulation kinetics, has the form... 

The coagulation constant K figuring in the kinetic equation of coagulation (11.1), is, by its physical meaning, equal to the collision frequency of drops with unit concentration, therefore... 

Finally, we can write the expression for the coagulation kernel K(V,co) of the kinetic equation (11.1). Since the diffusion flux of drops of radius R2 toward the drop of radius Ri at the unit concentration ri2o = 1 has the meaning of the kernel of coagulation, the replacement of radiuses by volumes in (13.133) gives... 

Considering Eq. (15.22) in terms of volumes of drops rather than radii and taking into account that the core of the kinetic equation of coagulation K(V, to) is equal to the diffusion flux of drops with unit concentration, one obtains ... 

The temporal change of the size distributions of drops, n( V, t), under the assumption of a spatially uniform distribution and with regard to drop coagulations alone is described by the kinetic equation of coagulation, which follows from Eq. (15.1) ... 

The dispersiveness of a Uquid-gas mixture can be characterized by a continuous volume distribution of bubbles n(V, t, P) at the moment of time f at the point P of the considered volume. The distribution n satisfies the kinetic equation, which, with due regard of the diffusion growth and coagulation, and neglecting the breakup of bubbles, takes the form ... 

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. 

Let the process of coagulation progress in such a manner that the volume distribution of bubbles depends only on time t. In addition, we assume that the integration of bubbles occurs only due to their coagulation. Then the kinetic equation will take the form ... 

Ablation factor of separator Dimensionless parameter Kernel of kinetic equation (coagulation constant)... 

In principle, the kinetics of metal vapor condensation is analogous to the coagulation kinetics of an ideal suspension. There are unstable states, which can be described by the Smoluchowski equation for fast coagulation. 

Golovin, A. M., On the Kinetic Equation for Coagulating Cloud Droplets with Allowance for Condensation. Ill, Izv. Geophys. Ser. English Transl. Bull. Acad. Sci. USSR 9, 880-884, 10, 949-953 (1963). 

Galina,H. Mean-Field Kinetic Modeling of Polymerization The Smoluehowski Coagulation Equation, Vol. 137, pp. 135-172. 

All models described up to here belong to the class of equilibrium theories. They have the advantage of providing structural information on the material during the liquid-solid transition. Kinetic theories based on Smoluchowski s coagulation equation [45] have recently been applied more and more to describe the kinetics of gelation. The Smoluchowski equation describes the time evolution of the cluster size distribution N(k) ... 

Kinetic methods describing the evolution of distributions of molecules by systems of kinetic differential equations (obeying either the classic mass action law of chemical kinetics or the generalized Smoluchowski coagulation process). 

It Is to be remarked that the process described by the Infinite set of kinetic (coagulation) equations can be simulated by Monte-Carlo methods ( ). The Information on the number of molecules of the respective size Is stored In the computer memory and weighting for selection of molecules Is applied given by the number and reactivity of groups In the respective molecule. 

If the electrostatic barrier is removed either by specific ion adsorption or by addition of electrolyte, the rate of coagulation (often followed by measuring changes in turbidity) can be described fairly well from simple diffusion-controlled kinetics and the assumption that all collisions lead to adhesion and particle growth. Overbeek (1952) has derived a simple equation to relate the rate of coagulation to the magnitude of the repulsive barrier. The equation is written in terms of the stability ratio ... 

The first group, consisting of Sections 2.2-2.4, covers sedimentation. After some preliminaries, we discuss Stokes s law, a hydrodynamic equation that will appear again when we discuss electrokinetic phenomena in Chapter 12 and the kinetics of coagulation in Chapter 13. Stokes s law is a key relationship in understanding the rate of sedimentation and is used in the derivation of the sedimentation equation for spherical particles. Following this, the equation for the sedimentation coefficient, a... 

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