Kinetics flocculation

The foundations of the theory of flocculation kinetics were laid down early in this century by von Smoluchowski (33). He considered the rate of (irreversible) flocculation of a system of hard-sphere particles, i.e. in the absence of other interactions. With dispersions containing polymers, as we have seen, one is frequently dealing with reversible flocculation this is a much more difficult situation to analyse theoretically. Cowell and Vincent (34) have recently proposed the following semi-empirical equation for the effective flocculation rate constant, kg, ... 

With regard to reversible flocculation kinetics, the problem is even more challenging- Detailed models for the deflocculation process as well as the flocculation process are required computer simulation is probably going to be the only way forward here ... 

Hm has been shown to be always positive, which suggests that, in two-phase systems (such as oil-water), the particles will always be attracted to each other. This means that even air bubbles will attract each other, as is also found from experiments. A linear relation is found between H1216/n and yLD, as expected from Equation Experimental values of Am as determined from flocculation kinetics showed that this agreed with the theoretical relation. 

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... 

Flocculation kinetics. We have droplets in an emulsion with a density of one droplet per (10 /uu)3. Calculate the initial decrease of the concentration, assuming no energy barrier. 

Flocculation processes are complicated phenomena because of the varieties of both particle morphology and chemical reactions they encompass.34 A few concepts of a general nature have emerged, however, and they will be the focus of this chapter. From the perspective of kinetics, perhaps the most important of these broad generalizations is the distinction that can be made between transport-controlled and reaction-controlled flocculation, parallel to the classification of adsorption processes described in Section 4.5. Flocculation kinetics are said to exhibit transport control if the rate-limiting step is the movement of two (or more) particles toward one another prior to their close encounter and subsequent combination into a larger particle. Reaction control occurs if it is particle combination instead of particle movement (toward collision) that limits the rate of flocculation. 

Given that Eq. 6.1 (with D 2) applies to reaction-controlled flocculation kinetics, Eq. 6.54 implies that MM(t) [or MN(t)] must also exhibit an exponential growth with time. Therefore, by contrast with transport-controlled flocculation kinetics, a uniform value of the rate constant kmn cannot be introduced into the von Smoluchowski rate law, as in Eq. 6.17, to derive a mathematical model of the number density p,(t). Equations 6.22 and 6.24 indicate clearly that a uniform kinil leads to a linear time dependence in the... 

Direct particle counting of an initially monodisperse suspension was used to measure the time dependence of the q-moment M0, as given in the following table. Examine these data for conformity to either transport- or reaction-controlled flocculation kinetics and estimate the characteristic time scale, 2/kn p0, wherekn = kmn for m n 1. (Answer k n= 3.05 x 10 22 m3 s"1 = 2KD/Wmn, corresponding to Wmn = 4.07 X 104 for all m, n.)... 

Derrindinger, L. and Sposito, G., Flocculation kinetics and cluster morphology in illite/NaCl suspensions, J. Colloid Interface Sci., 222, 1, 2000. 

The subject of flocculation kinetics and the stabilization of dispersions has been dealt with in many recently published papers. Some of them are cited here (26-34.). 

Chen, L. A., G. A. Serad, and R. G. Carbonell (1998). Effect of mixing conditions on flocculation kinetics of wastewaters containing proteins and other biological compounds using fibrous materials and polyelectrolytes. Brazilian J. Chemical Eng. 15, 4, 358-368. 

Chen, W. J., D. P. Lin, and 1. P. Hsu (1998). Contrihution of electrostatic interaction to the dynamic stability coefficient for coagulation-flocculation kinetics of beta-iron oxy-hydroxides in polyelectrolyte solutions. J. Chem. Eng. Japan. 31, 5, 722-733. 

Particle counts are useful in pilot plant testing as one evaluation parameter in coiqunction with other water quality (e.g. turbidity) and operational (e.g. filter head loss) parameters. Information gained from particle counts can be used to follow flocculation kinetics and solid-removal effectiveness of unit processes. 

Gjnax < 5 IcT then flocculation will occur. Two types of flocculation kinetics may be distinguished (i) fast flocculation with no energy barrier and (ii) slow flocculation, where an energy barrier exists. 

The fast flocculation kinetics was investigated by Smoluchowski [12], who considered the process to be represented by second-order kinetics and the process to be simply diffusion-controlled. The number of particles n at any time t may be related to the initial number (at t = 0) by the following expression ... 

The slow flocculation kinetics was investigated by Fuchs [13], who related the rate constant k to the Smoluchowski rate by the stability constant W,... 

By combining the Rayleigh theory with the Smoluchowski-Fuchs theory of flocculation kinetics [7, 8], the following expression can be obtained for the variation of turbidity with time. 

Birkner, F. B. Morgan, J. J. Polymer Flocculation Kinetics of Dilute... 

Here k is Boltzmann s constant, T is the absolute temperature, /x is the fluid viscosity, Vi and Vj are the volumes of particles with sizes i and /, respectively, and G is the velocity gradient of the fluid. Equations 14 and 15 often are written in terms of the diameters of the particles rather than in terms of their volumes. However, when two particles collide, it is useful to consider that their total volume is conserved (coalesced-sphere assumption) therefore, the equations for flocculation kinetics are expressed in terms of particle volumes throughout this work. 

Most analyses of flocculation kinetics assume a homogeneous suspension (that is, a suspension containing particles of only one size) at the onset of flocculation. To compare such suspensions with the heterogeneous suspensions evaluated here, an influent containing 132 mg/L (50ppm) of particles having the same volume-average diameter (0.688 /xm) as the standard case was assumed. Such a suspension also has the... 

Further reviews of coalescence and flocculation kinetics were reported by Becher (211), Tadros and Vincent (90), and Hartland (217). For all practical purposes the above treatments usually suffice in crude-oil studies. Extensive treatments of coalescence and flocculation kinetics were modeled as required for various other emulsion applications. Borwanker et al. (218) developed a mathematical model to account for flocculation and coalescence kinetics occurring simultaneously. They modified Van den Tempel s treatment for coalescence to include coalescence occurring in small floes. They showed how the rate-controlling mechanism could change from coalescence-rate controlling to flocculation-rate controlling during an emulsion hfetime. They further extended the model for concentrated emulsions. 

Einarson and Berg (1993) have attempted to explain the data on flocculation kinetics of latex particles with a block copolymer adsorbed on them. The polymer was polyethylene oxide (PEO)/polypropylene oxide (PPO). PPO is water insoluble and forms the part that adsorbs on the latex PEO forms streaming tails into water. Some charge effects remain after the polymer adsorption. The total potential is DLVO plus elastic plus osmotic effects. After fitting the model to the experimental data, they were able to calculate the value of 6, which they called the adlayer thickness. Their data on the stability ratio of latex with and without the polymer and as a fimction of NaCl concentration are shown in Figure 3.23. Note that the polymer stabilizes the colloid by almost one order of magnimde in NaQ concentration. That is, polymers may be necessary to maintain stability in aqueous media containing substantial electrolyte. 

In this chapter we examine some issues in mass transfer. The reader has already been introduced to some of the key aspects. In Chapter 3 (Section 7), flocculation kinetics of colloidal particles is considered. It shows the importance of diffusivity in the rate process, and in Equation 3.72, the Stokes-Einstein equation, the effect of particle size on diffusivity is observed, leading to the need to study sizes, shapes, and charges on colloidal particles, which is taken up in Chapter 3 (Section 4). Similarly some of the key studies in mass transfe in surfactant systems— dynamic surface tension, smface elasticity, contacting and solubilization kinetics—are considered in Chapter 6 (Sections 6, 7, 10, and 12 with some related issues considered in Sections 11 and 13). These emphasize the roles played by different phases, which are characterized by molecular aggregation of different kinds. In anticipation of this, the microstructures are discussed in detail in Chapter 4 (Sections 2,4, and 7). Section 2 also includes some discussion on micellization-demicellization kinetics. 

As mentioned above, two-component flocculants often present advantages over a single-component flocculant, such as better control of flocculation kinetics and improved floe strength. Most dual-component flocculants consist of two polyelectrolytes, two pol5miers, or a polyelectrolyte and a nanocolloid. Usually, one of the components adsorbs on the surface of the particles to be flocculated and the second component bridges these polymer-coated particles. Therefore, this combination of patching and bridging is believed to be responsible for excellent results, as described for instance for retention systems (Fig. 5) [10]. 

Couette flocculators are basic research apparatuses of flocculation kinetics, and can be used with particle counters (e.g. Coulter counter) as a standard for other laboratory flocculators which have ill-defined velocity gradients. 

Ga is seen to increase very sharply with decreasing h when the latter reaches small values. In the absence of repulsion between the particles or droplets, the latter will aggregate (flocculate) by simple diffusion through the medium. This leads to fast flocculation kinetics and the rate constant for the process ko has been calculated using the Smolulokowski equation,... 

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