Collision rate coagulating particles

Let us now consider coagulation of particles in the absence of any repulsive barrier. In addition, we assume that, although there are no interparticle forces that contribute to the transport of particles toward each other, there is sufficient attraction between the particles on contact for them to form a permanent bond. As early as 1917, Smoluchowski formulated the equations for the collision rate for particles transported by diffusion alone (Smoluchowski 1917), and we develop the same idea here. 

In summary, polymeric flocculants generally increase peri-kinetic flocculation rates compared with perikinetic coagulation rates. This is not necessarily true for orthokinetic flocculation, and experimental results in the literature are seemingly in conflict. Collision rate theory predicts that the polymer adsorption step may become rate limiting in orthokinetic flocculation. The present study was designed to elucidate the relationship between polymer adsorption rates and particle flocculation rates under orthokinetic conditions. 

Darling and van Hooydonk (1981) also considered how to reduce the diffusional collision rate to obtain slow coagulation and used the classical approach of Fuchs (Reerink and Overbeek, 1954), whereby an activation energy is computed from the pair interaction free energy of the aggregating particles. The reaction kernel is given by Eq. (6) divided by the stability ratio W,... 

The rate of coagulation is considered to be dominated by a binary process involving collisions between two particles. The rate is given by bn,nj, where nl is the number of particles of z th size and b a collision parameter. For collision between i - and / -sized particles during Brownian motion, the physicist M. Smoluchowski derived the relation ... 

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-... 

Let us consider a system of monodispersed aerosol particles. Because of the collisions between the particles, they will coagulate to form doublets. The rate of formation of a doublet depends, of course, on the particle radius, the particle density, the viscosity and temperature of the suspending medium, the Hamaker constant, and the particle number concentration. If the rate of formation of a doublet is itnuch less than its rate of dissociation, the doublets are unstable. These unstable doublets attain... 

Conceptually similar results were demonstrated by Krutzer et al. [14], who measured the orthokinetic coagulation rate under laminar Couette flow and isotropic turbulent flow (as well as other flow conditions). Despite equal particle collision rates, significance differences were observed in the overall rates indicating different collision efficiencies (higher collision efficiencies were found under a turbulent flow regime). Thus, identical chemical properties of a dispersion do not determine a single collision efficiency the collision efficiency is indeed dependent upon the physical transport occurring in the system. 

This condition is defined as rapid coagulation in which the rate of disappearance of primary particles, Jq, is equal to the frequency of collision between the particles ... 

In this equation, is the total interaction energy between the two colliding particles defined in the previous section. The stability ratio, W, for the system gives the ratio of rapid coagulation, Jp, to slow coagulation, J[= J W], DQi) is the position-dependent diflusion equation. This diffusion coefficient ratio is a factor that decreases the collision rate because of the difficulty in draining the liquid between the two solid surfaces. This diffiision coefficient ratio is given by [60,61]... 

For shear coagulation in laminar flow, the collision rate for N, particles of size a, with Nj particles of size Cj is given by [3, p. 298]... 

The rate at which two particles with masses mj and m- and concentrations nt and tij collide is given by np., where (3 is the coagulation kernel (34, 35). New particles of mass (m, + m() are formed at a rate of anj fifty, where a is the stickiness coefficient. If all aggregates are composed of unit particles of the same size, then m, = i m and (m, + m/ = mi+j = (i + j)m where m, is the mass of the unit particle. If no new unit particles are produced and there is no nonaggregation process making particles, the change in concentration of particles of size i is the difference between the rate at which the particles are formed by collision of smaller particles and the rate at which they are lost to formation of larger particles. 

Particle collision and coagulation lead to a reduction in the total number of particles and an increase in the average size. An expression for the time rate of change of the particle size distribution function can be derived as follows. 

Hence for colliding particles of the same diameter the effect of (he van der Waats forces on collision rate does not depend on the size but only on A/kT. The integral (7.35) has been evaluated and the result is shown in Fig, 7.3. The determination of the effect of the van der Waals forces on the coagulation rate thus reduces to the evaluation of i4/A 7 for particles of equal size. 

The collision rate we have derived is the rate, expressed as the number of 2 particles per second, at which the 2 particles collide with a single 1 (primary) particle. When there are Nu) 1 particles, the total collision rate between 1 and 2 particles per volume of fluid is equal to the collision rate derived above multiplied by Niq. Thus the steady-state coagulation rate (cm-3 s 1) between 1 and 2 particles is... 

To calculate the overall coagulation rate we need to add to this rate the collision rate of red particles among themselves and blue particles among themselves. Let us now paint one-half the red particles as green and one-half the blue particles as yellow. The rate of coagulation within each of the initial particle categories is... 

Typical measured values of (8 /v) 2 are on the order of 10 s-1, so turbulent shear coagulation is significantly slower than Brownian for submicrometer particles, and the two rates become approximately equal for particles of about 5 pm in diameter (Figure 13.A.2). The calculations indicate that coagulation by Brownian motion dominates the collisions of submicrometer particles in the atmosphere. Turbulent shear contributes to the coagulation of large particles under conditions characterized by intense turbulence. 

In the absence of a barrier to coagulation, and if the primary minimum is deep, every collision between a particle and a floe will lead to the growth of the floe. The rate of coagulation is then controlled entirely by the kinetics of the diffusion process leading to particle-particle collision. The theory of fast coagulation was developed originally by Smoluchowski (1918) and elaborated by Muller (1926). The rate equation has the same form as that for a bimolecular reaction ... 

A second factor reducing the rale of collision arises from the fact that, as two particles approach closely, liquid has to flow out from the region between them. Experimental evidence based on measurements of rapid coagulation (which is similarly affected by this hydrodynamic effect) indicates that the collision rate is reduced to about half its expected value. 

For a quantitative description of slow coagulation, Smoluchowski has suggested to formally introduce into the expression for the particle collision frequency a factor a < 1, describing the share of collisions resulting in formation of aggregates. Introduction of this factor is equivalent to increase of the characteristic coagulation time by a factor 1/a. The coagulation rate is characterized by stability factor W. It is the ratio of the partides collision rates without and with the force of electrostatic repulsion [58]... 

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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