Rate coefficient flocculation

The power-law relationship in Eq. 6.1 also has implications for measurements of floccule size and dimension during the flocculation process itself. If the principal contributor to floccule growth is collisional encounters between particle units of comparable size, the increase in N per encounter will be equal approximately to N itself. Moreover, if diffusionally mediated collisions are the cause of these encounters, the kinetics of collision will be described by a second-order rate coefficient 9... 

Under the assumption that the target and incoming floccules engage in Brownian motion independently,18 D = Dm + Dn, and the rate coefficient becomes... 

Equation 6.16d should be a good estimate of the floccule formation rate coefficient if D scales as R 1 and if collisions between floccules of approximately equal size dominate.20... 

Equation 6.18 is graphed in Fig. 6.6 for the cases q = 1, 2, 3. The number density of primary particles, pj(t), decreases monotonically with time as these particles are consumed in the formation of floccules. The number densities of the floccules, on the other hand, rise from zero to a maximum at t = (q - l)/2KDp0, and then decline. This mathematical behavior reflects creation of a floccule of given size from smaller floccules, followed by a period of dominance, and finally consumption to form yet larger particle units as time passes. Both experimental data and computer simulations, like that whose visualization appears in Fig. 6.1, are in excellent qualitative agreement with Eq. 6.18 when they are used to calculate the pq(t).13,14 Thus the von Smoluchowski rate law with a uniform rate coefficient appears to capture the essential features of diffusion-controlled flocculation processes. 

If the von Smoluchowski rate law (Eq. 6.10) is to be consistent with the formation of cluster fractals, then it must in some way also exhibit scaling properties. These properties, in turn, have to be exhibited by its second-order rate coefficient kmn since this parameter represents the flocculation mechanism, aside from the binary-encounter feature implicit in the sequential reaction in Eq. 6.8. The model expression for kmn in Eq. 6.16b, for example, should have a scaling property. Indeed, if the assumption is made that DJRm (m = 1, 2,. . . ) is constant, Eq. 6.16c applies, and if cluster fractals are formed, Eq. 6.1 can be used (with R replacing L) to put Eq. 6.16c into the form... 

Since z > 0 for an increasing cluster size with time (Eq. 6.44), 6 < 1 in Eq. 6.48. Thus the homogeneity condition satisfied by the second-order rate coefficient determines the rate of floccule growth and the corresponding decline in the number of floccules (Eqs. 6.42 and 6.44). 

Diffusionally mediated collisions between two floccules of equal size can be described by a second-order rate coefficient KD = 8irRD, where R is the radius and D is the diffusion coefficient of a floccule. Upon invoking the Stokes-Einstein relation, D = kBT/67ri7R, one derives Eq. 6.2. For an introductory discussion of the second-order rate law for particle collisions, see, for example, Chap. 11 in P. C. Hiemenz, Principles of Colloid and Surface Chemistry, Marcel Dekker, New York, 1986. 

We therefore find that under stagnant and neutral buoyancy conditions, the relative rate of diffusion controlled collision is dominated by the radius of the antifoam drops—the larger the drops, the slower the rate of collision because both the number concentration and the diffusion coefficients of drops are lower. Increasing viscosity of the detergent liquid and decreasing the volume fraction of the antifoam also both decrease the rate of flocculation. 

Flocculation kinetics can be described in different ways. Here we introduce a treatment first suggested by Smoluchowski [547], and described in Ref. [538], p. 417. The formalism can also be used to treat the aggregation of sols. A prerequisite for coalescence is that droplets encounter each other and collide. Smoluchowski calculated the rate of diffusional encounters between spherical droplets of radius R. The rate of diffusion-limited encounters is SttDRc2, where c is the concentration of droplets (number of droplets per unit volume). For the diffusion coefficient D we use the Stokes-Einstein relation D = kBT/finr/R. The rate of diffusion-limited encounters is, at the same time, the upper limit for the decrease in droplet concentration. Both rates are equal when each encounter leads to coalescence. Then the rate of encounters is given by... 

The attenuation coefficient in the flocculated emulsion is lower at low frequencies and higher at high frequencies than that of the nonflocculated emulsions. The decrease in attenuation at low frequencies on flocculation as a result of the thermal overlap effects mentioned earlier, whereas the increase at high frequencies results from increased scattering of ultrasound by the floes. The same ultrasonic spectroscopy technique has been used to study the disruption of floes in a shear field (38). As the emulsions are exposed to higher shear rates the floes become disrupted and their attenuation spectra become closer to that of nonflocculated droplets. 

FIGURE 10.13. In a cx)lloidal system, the rate of particle flocculation will depend on the rate of particle collision. That rate, in turn, wiU depend on the diffusion coefficients of the respective particles and their effective particle diameters (or collision cross sections). 

Hindered-settling systematics. The hindered-settling velocity of slurries may be expressed as a coefficient times the Stokes law settling rate. Table 4-12 summarizes [120] typical coefficients obtained in theoretical and empirical investigations. A plot of these coefficients versus porosity showed that they are substantially in agreement. Steinour [121] introduced the concept of immobilized water in his treatment of flocculated suspensions and showed that by defining... 

---