Settling radius

The diameter of a sedimenting species determined from Stokes law assuming a spherical shape. Also referred to as the Stokes diameter or (divided by a factor of 2) the settling radius . 

The Stokes settling radius of colloidal particles can be obtained from their sedimentation rate. An ultracentrifuge is used to increase the sedimentation rate, since colloidal particles settle very slowly under the influence of gravity alone. 

Here Ps and p, are the densities of solid particles and liquid respectively, g is the acceleration due to gravity, R is the particle radius (in this context called the Stokes settling radius) and r] is the viscosity of the medium. Thus... 

The particle can be assumed to be spherical, in which case M/N can be replaced by (4/3)ttR P2, and f by 671770R- In this case the radius can be evaluated from the sedimentation coefficient s = 2R (p2 - p)/9t7o. Then, working in reverse, we can evaluate M and f from R. These quantities are called, respectively, the mass, friction factor, and radius of an equivalent sphere, a hypothetical spherical particle which settles at the same rate as the actual molecule. 

Under these circumstances, the settling motion of the particles and the axial motion of the Hquid phase are combined to determine the settling trajectory of these particles. The trajectory of particles just reaching the bowl wall near the point of Hquid discharge defines a minimum particle size that starts from an initial radial location and is separated in the centrifuge. A radius ris chosen to divide the Hquid annulus in the bowl into two equal volumes initially containing the same number of particles. Half the particles of size i present in the suspension are separated the other half escape. This is referred to as a 50% cutoff. 

Here again an equation is estabUshed (2) to describe the trajectory of a particle under the combined effect of the Hquid transport velocity acting in the x-direction and the centrifugal settling velocity in thejy-direction. Equation 13 determines the minimum particle size which originates from a position on the outer radius, and the midpoint of the space, between two adjacent disks, and just reaches the upper disk at the inner radius, r. Particles of this size initially located above the midpoint of space a are all collected on the underside of the upper disk those particles initially located below the midpoint escape capture. This condition defines the throughput, for which a 50% recovery of the entering particles is achieved. That is,... 

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

Radius, a, nm D /D tr ions Distance diffused, x, nm Distance settled, x/... 

Equation (7.63) results in a polar diagram in the z-plane as shown in Figure 7.16. Figure 7.17 shows mapping of lines of constant a (i.e. constant settling time) from the. V to the z-plane. From Figure 7.17 it can be seen that the left-hand side (stable) of the. v-plane corresponds to a region within a circle of unity radius (the unit circle) in the z-plane. 

Lest I leave the erroneous impression here that colloid science, in spite of the impossibility of defining it, is not a vigorous branch of research, I shall conclude by explaining that in the last few years, an entire subspeciality has sprung up around the topic of colloidal (pseudo-) crystals. These are regular arrays that are formed when a suspension (sol) of polymeric (e.g., latex) spheres around half a micrometre in diameter is allowed to settle out under gravity. The suspension can include spheres of one size only, or there may be two populations of different sizes, and the radius ratio as well as the quantity proportions of the two sizes are both controllable variables. Crystals such as AB2, AB4 and AB13 can form (Bartlett et al. 1992, Bartlett and van... 

For a very dilute suspension, i.e., e = 1 and (e) = 1, the settling veloeity will be equal to the free-fall veloeity. As no valid theoretieal expression for the funetion (e) is available, eommon praetiee is to rely on experimental data. Note that a unit volume of thiekened sludge eontains e volume of liquid and (1 - e) volume of solid phase, i.e., a unit volume of partieles of sludge eontains e/(l - e) volume of liquid. Denoting a as the ratio of partiele surfaee area to volume, we obtain the hydraulie radius as the ratio of this volume, e/(l - e), to the surfaee, a, when both values are related to the same volume of partieles ... 

Because the radius of the solid discharge port is ususally less than the radius of the liquid overflow at the broader end of the bowl, part of the settled solids is submerged in the pond. 

The settling capacity for a given size of particles is a function of R, C and u, which itself is proportional to R. In general, for the sedimentation of heavy particles in a suspension it is sufficient that the radial component of Uf be less than at a radius greater than Rj. 

Increasing the radius of the suspended particles, Brownian motion becomes less important and sedimentation becomes more dominant. These larger particles therefore settle gradually under gravitational forces. The basic equation describing the sedimentation of spherical, monodisperse particles in a suspension is Stokes law. It states that the velocity of sedimentation, v, can be calculated as follows ... 

Sedimentation of particles follows the principle outlined above [Eq. (1)] in which particles in the Stokes regime of flow have attained terminal settling velocity. In the airways this phenomenon occurs under the influence of gravity. The angle of inclination, t /, of the tube of radius R, on which particles might impact, must be considered in any theoretical assessment of sedimentation [14,19]. Landahl s expression for the probability, S, of deposition by sedimentation took the form ... 

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