Particles Stokes law

For a solid particle, Stokes Law gives the terminal velocity (i ,g) as... 

Biological particles. Stokes law applies to spherical particles, which are large in comparison with the molecules that comprise the liquid medium, and are present at a concentration low enough to avoid modification of the liquid viscosity. Most biological particles are not spherical, and Strokes law must be modified to take this into account. One approach to this problem is to consider that the biological particles shapes could be approximated by ellipsoids of revolution , or spheroids with one major and one minor axis. Calculations show that the frictional force over these ellipsoids is greater than that expected for spherical particles of the same volume.3... 

The sedimentation technique is reliable for particle size determination when rf is in a size range of 2-50 pm. The falling rate of smaller particles is affected by Brownian motion resulting from collisions with the molecules of the liquid and other interactions between particles. Stokes law is valid only for laminar or streamline flow (i.e., when there is no turbulence). The Reynolds number (Re) is a measure of when the process transitions from turbulent to laminar flow ... 

The frictional coefficient / contains information about the size and shape of the particle. For spherical particles, Stokes law, given in Equation 34-10, holds for laminar-flow conditions... 

This result is often called the Stokes-Einstein formula for the difflision of a Brownian particle, and the Stokes law friction coefficient 6iiq is used for... 

A physical value for 7 for each particle can be chosen according to Stokes law (with stick boundary conditions) ... 

A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

In the derivation of both Eqs. (9.4) and (9.9), the disturbance of the flow streamlines is assumed to be produced by a single particle. This is the origin of the limitation to dilute solutions in the Einstein theory, where the net effect of an array of spheres is treated as the sum of the individual nonoverlapping disturbances. When more than one sphere is involved, the same limitation applies to Stokes law also. In both cases contributions from the walls of the container are also assumed to be absent. 

Rigid, unsolvated spheres. Stokes law, Eq. (9.5), provides a relationship between f and the radius of the particle. Since this structure is a reasonable model for some protein molecules, experimental D values can be interpreted, via f, to yield values of R for such systems. Note that this application can also yield a value for M, since M = N pj [(4/3)ttR ], where pj is the density of the unsolvated material. 

Since f is a measurable quantity for, say, a protein, and since the latter can be considered to fail into category (3) in general, the friction factor provides some information regarding the eilipticity and/or solvation of the molecule. In the following discussion we attach the subscript 0 to both the friction factor and the associated radius of a nonsolvated spherical particle and use f and R without subscripts to signify these quantities in the general case. Because of Stokes law, we write... 

Eig. 5. Target efficiency of spheres, cylinders, and ribbons. The curves apply for conditions where Stokes law holds for the motion of the particle (see also N j ia Table 5). Langmuir and Blodgett have presented similar relationships for cases where Stokes law is not vaUd (149,150). Intercepts for ribbon or... 

Fig. 14. Drag coefficient for terminal settling velocity correlation (single particle) where A represents Stokes law B, intermediate law and C, Newton s...

In particle-size measurement, gravity sedimentation at low soHds concentrations (<0.5% by vol) is used to determine particle-size distributions of equivalent Stokes diameters ia the range from 2 to 80 pm. Particle size is deduced from the height and time of fall usiag Stokes law, whereas the corresponding fractions are measured gravimetrically, by light, or by x-rays. Some commercial instmments measure particles coarser than 80 pm by sedimentation when Stokes law cannot be appHed. 

As an additional guide, the values are correlated with the equivalent spherical particle diameter by Stokes law, as in equation 1. A density... 

A. Solid particles suspended in agitated vessel containing vertical baffles, continuous phase coefficient -2 + 0.6Wi f,.Wi D Replace Osi p with Vj = terminal velocity. Calculate Stokes law terminal velocity [S] Use log mean concentration difference. Modified Frossling equation K, -< T.d,P. [97] [146] p.220... 

The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Re = d pii/ where [L = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes Law gives 24... 

Wall Effects When the diameter of a setthng particle is significant compared to the diameter of the container, the settling velocity is reduced. For rigid spherical particles settling with Re < 1, the correction given in Table 6-9 may be used. The factor k is multiplied by the settling velocity obtained from Stokes law to obtain the corrected set-... 

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