Coagulation velocity gradient

Coagulation a Collision efficiency volumetric concentration of suspended particles G Velocity gradient t time i... 

The rate of coagulation of particles in a liquid depends on the frequency of collisions between particles due to their relative motion. When this motion is due to Brownian movement coagulation is termed perikinetic when the relative motion is caused by velocity gradients coagulation is termed orthokinetic. 

The presence of velocity gradients in the system may also increase the rate of coagulation above the value given by Equation (24) or (42). 

The ratio of the probability of a collision induced by a fluid velocity gradient (dv/dx) (i.e., orthokinetic coagulation) to the collision probability under the influence of Brownian motion (perikinetic coagulation—what we have considered so far) has been shown to be (Probstein 1994)... 

The rate constant k0 for orthokinetic coagulation is determined by physical parameters (velocity gradient du/dz, floe volume ratio of the dispersed phase, = sum over the product of particle number and volume), and the collision efficiency factor a0 observed under orthokinetic transport conditions ... 

Chemical parameters determine the surface characteristics of the suspended colloids, the concentration of the coagulant and its effects upon the surface properties of the destabilized particles, and the influence of other constituents of the ionic medium upon the coagulant and the colloids. The extent of the chemical and physical interactions between the colloidal phase and the solution phase determines the relative stability of the suspended colloids. One speaks of stable suspensions when all collisions between the colloids induced by Brownian motion or by velocity gradients are completely elastic the colloidal particles continue their... 

The rate of coagulation depends upon the collision frequency, which is controlled by physical parameters describing perikinetic or ortho-kinetic particle transport (temperature, velocity gradient, number concentration and dimension of colloidal particles), and the collision efficiency factor a measuring the extent of the particle destabilization which is primarily controlled by chemical parameters. 

The coagulation rate depends upon physical parameters (temperature, velocity gradient, number and dimension of colloid), determining the collision frequency and upon chemical parameters (pH, Al(III) dosage, surface concentration of dispersed phase S), affecting the collision efficiency factor a... 

Figure 6.4 Comparison of key variables that control coagulation in natural and artificial systems G = mean water velocity gradients (s-1), = particle concentration, and a = collision efficiency. (From Stumm and Morgan, 1996, with permission.)...

Figure 14.4. (a) Simulation results for a continuous initial particle size distribution, with /3 = 4 and size range from 1 nm to 100 /xm initial total mass concentration, 3 mg liter" = 1.5 g cm temperature = 15 C coagulation with a velocity gradient G = 10 s" and a collision efficiency factor a = 0.05 (see Sections 14.4 and 14.8 for the definition of these terms), (b, c) Simulation results for a continuous initial particle size distribution with /3 = 4 and size ranging from 1 nm to 100 pm after 2 days with different initial mass concentrations ranging from 0.01 to 10 mg liter" (p = 2.0 g cm" temperature = 15°C G = 0.5 a. = 0.05). (b) Evolution of particle size versus concentration with ordinates expressed as percentage of initial value for each size class, (c) Evolution of mean size value with concentration. (Adapted from Filella and Buffle, 1993.)... 

Illustrative Cases. Three cases are illustrated in Figure 9, marked by the circles labeled A, B, and C. Case A refers to classical experiments by Swift and Friedlander (27) on the coagulation of monodisperse latex particles (diameter = 0.871 pm) in shear flow and in the absence of repulsive chemical interactions. Considering a velocity gradient of 20 s 1, HA is 0.0535, log HA is — 1.27, and dfdj is 1.0 for these experimental conditions. The circle labeled A in Figure 9 marks these conditions and indicates that the hydrodynamic corrections to Smoluchowsla s model predict a reduction of about 40% in the aggregation rate by fluid shear. The experimental measurements by Swift and Friedlander showed a reduction of 64%. This observed reduction from Smoluchowski s rectilinear model was therefore primarily physical or hydro-dynamic and consistent with the curvilinear model. 

Particles in a fluid in which a velocity gradient du/dy exists have a relative motion that may bring them into contact and cause coagulation (Figure 13.A.1). Smoluchowski in 1916 first studied this coagulation type assuming a uniform shear flow, no fluid dynamic interactions between the particles, and no Brownian motion. This is a simplification of the actual physics since the particles affect the shear flow and the streamlines have a curvature around the particles. 

Smoluchowski showed that for a velocity gradient V = du/dy perpendicular to the x axis the coagulation rate is... 

Hydrodynamic Forces Fluid mechanical interactions between particles arise because a particle in motion in a fluid induces velocity gradients in the fluid that influence the motion of other particles when they approach its vicinity. Because the fluid resists being squeezed out from between the approaching particles, the effect of so-called viscous forces is to retard the coagulation rate from that in their absence. 

An important effect in Equation (3.34) is that the collision radius enters as Rl and since we have approximated Rc = 2R it becomes 8R consequently this indicates that shearing is very sensitive to particle size. For this reason small particles are rather insensitive to shearing forces, whereas larger particles, for instance with R > 0.5 pm, can often be flocculated by stirring or shaking particularly at electrolyte concentrations close to the ccc. The sensitized coagulation which occurs in the presence of a velocity gradient is known as orthokinetic flocculation. 

It is a general observation that gentle stirring promotes coagulation. The reason for this is that velocity gradients in the flow field create relative particle movements and therefore an increased collision frequency. The simplest case to treat is that of a uniform shear field. 

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