Radius sedimentation times

It is obvious from the shape of sedimentation curves (Fig. 5.19) that at rise of mixing rate they shift to the region of particles with smaller sizes (higher sedimentation times), i.e. under increase of turbulence level of mixing decrase of average particles radius in reaction zone occurs. 

Equivalent spherical radius Approximate size Sedimentation rate (time to settle 30 cm)... 

There are two essential consequences of this relation. Because larger droplets sediment or rise much faster (a 5-p.m drop rises 625 times faster than a 0.2-p.m droplet), the process is equal to shearing, leading to enhanced flocculation. The ratio between flocculation due to shear and to diffusion of droplets is proportional to the cube of the radius. Secondly, flocculation to droplet aggregates means an enhanced sedimentation rate. Sis drops ia an octahedral arrangement gives approximately four times the sedimentation rate. 

Consider the simple initial condition t = 0 where the sohd concentration (t),o is constant across the entire shiny domain ix < r < rb where / l and l b are, respectively, the radii of the shiny surface and the bowl. At a later time t > 0, three layers coexist the top clarified layer, a middle shiny layer, and a bottom sediment layer. The air-liquid interface remains stationaiy at radius / l, while the hqiiid-slurry interface with radius i expands radiaUy outward, with t with i given by ... 

Criticize or defend the following proposition The data give the time required for particles to fall 20 cm, making it easy to convert time to sedimentation velocity for each point. Equation (11) may then be used to convert the velocity into the radius of an equivalent sphere. The resulting graph of W versus radius is a cumulative distribution function similar to that shown in Figure 1.18b. 

The radial concentration scans obtained from the UV spectrophotometer of the analytical ultracentrifuge can be either converted to a radial derivative of the concentrations at a given instant of time (dc/dr)t or to the time derivative of the concentrations at fixed radial position (dc/dt)r (Stafford, 1992). The dcf dt method, as the name implies, uses the temporal derivative which results in elimination of time independent (random) sources of noise in the data, thereby greatly increasing the precision of sedimentation boundary analysis (Stafford, 1992). Numerically, this process is implemented by subtracting pairs of radial concentration scans obtained at uniformly and closely spaced time intervals c2 — G)/( 2 — h)]. The values are then plotted as a function of radius to obtain (dc/dt) f versus r curves (Stafford, 1994). It can be shown that the apparent sedimentation coefficient s ... 

We can use the same record of the trajectory of a particle to determine at the same time its hydrodynamic radius and its density. For sedimentation at low Reynolds number, the average vertical velocity is given by the balance between the body force and the friction force ... 

The time required to sediment particles depends on the rotor speed, the radius of the rotor, and the effective path length traveled by the sedimented particles, that is, the depth of the liquid in the tube. Duplication of conditions of centrifugation is often desirable. The following is a useful formula for calculating speed required of a rotor whose radius differs from the radius with which a prescribed RCF was originally defined ... 

To obtain the q(r) / Pmm curve from sedimentation curve, c (R, A/=const.), one can plot the relative concentration, c/c0, as a function of particle radius obtained from particle displacement, AR, that occurred over the time, At, using eq. (V.53). If the diffusion rate is negligibly small, the c = c (AR / AO curves match each other at all times, At. The latter allows one to separate sedimentation and diffusion in polydisperse systems as well. To... 

Collision efficiency was calculated by the method proposed for the first time by Dukhin Derjaguin (1958). To calculate the integral in Eq. (10.25) it is necessary to know the distribution of the radial velocity of particles whose centre are located at a distance equal to their radius from the bubble surface. The latter is presented as superposition of the rate of particle sedimentation on a bubble surface and radial components of liquid velocity calculated for the position of particle centres. Such an approximation is possibly true for moderate Reynolds numbers until the boundary hydrodynamic layer arises. At a particle size commensurable with the hydrodynamic layer thickness, the differential of the radial liquid velocity at a distance equal to the particle diameter is a double liquid velocity which corresponds to the position of the particle centre. Such a situation radically differs from the situation at Reynolds numbers of the order of unity and less when the velocity in the hydrodynamic field of a bubble varies at a distance of the order ab ap. At a distance of the order of the particle diameter it varies by less than about 10%. Just for such conditions the identification of particle velocity and liquid local velocity was proposed and seems to be sufficiently exact. In situations of commensurability of the size of particle and hydrodynamic boundary layer thickness at strongly retarded surface such identification leads to an error and nothing is known about its magnitude. 

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