Flocculation coefficient

Ks Smolukowski flocculation coefficient, meters3/sec Kw Proportionality factor in equation for force of adhesion by van der Waal s forces, N/meter K6 Proportionality factor, C/meter2 Ksg Proportionality factor, V/meter In Natural logarithm of L Distance between centers of two particles or bodies, meters Lm Migration distance, meters Ls Clearance between proximate surfaces of two bodies, meters... 

F is the flocculation coefficient, calculated from F = VI Vq, where V is the sediment volume and is the total initial volume of the suspension. F is some reference, frequently taken as the value of F for a nonflocculated or stable suspension. According to its definition, the greater the value of P the more unstable is the suspension. 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

Where a is the composite conductivity, a0 a proportionally coefficient, Vfc the percolation threshold and t an exponent that depends on the dimensionality of the system. For high aspect ratio nanofillers the percolation threshold is several orders of magnitude lower than for traditional fillers such as carbon black, and is in fact often lower than predictions using statistical percolation theory, this anomaly being usually attributed to flocculation [24] (Fig. 8.3). 

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... 

Laor, Y., and M. Rebhun. 1997. Compexation—flocculation A new method to determine binding coefficients of organic contaminants to dissolved humic substances. Environmental Science and Technology 31 3558. 

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). 

Equation 6.52 exhibits the inverse of the stability ratio playing the role of the sticking probability coefficient that reduces kmn below its value for pure transport control of flocculation whenever cluster interactions are repulsive. (Note that WM1I) 1 when V(r) vanishes, according to Eq. 6.51.) liquation 6.52 is a model for kni(1 known as Fuchsian kinetics. ... 

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. 

Depletion flocculation has also been induced in oil-in-water emulsions by adding different concentrations of a non-adsorbing biopolymer (xanthan) to the aqueous phase. At low frequencies, the attenuation coefficient of the emulsions decreased with increasing... 

The design of the flocculator of Figure 6.11 may be made by determining the power coefficients for laminar, transitional, and turbulent regime of flow field. We will, however, discuss its design in terms of the fundamental definition of power. Consider Fd as the drag by the water on the blade Fd is also the push of the blade upon the water. This push causes the water to move at a velocity Vp equal to the velocity of the blade. 

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