Settling rate Stokes’ equation

It is worth elucidating mineral particle behavior in hydrothermal plumes in order to consider the formation mechanism of chimney and massive ores on the seafloor. Using the grain size data on sulfides and sulfates, the density of the fluids and of the minerals, the relationship between vertical settling rate and grain size of sulfides and sulfates can be derived based on the following Stokes equation ... 

All these ideas have been put into one equation, called Stokes law. Nothing against Sir Frederick Stokes, but vapor viscosities are almost always so small that they do not affect settling rates. Also, we never know the particle size distribution of the droplets. There is a more fruitful way to look at the settling tendency of droplets of liquid in an upflowing vapor stream, as shown in Fig. 26.1. The method states... 

Rate of Settling—At the present time two methods are widely used for computing settling rates of liquid suspensions. Rollason, cited by Egolf and McCabe (1937), proposed the use of Stokes equation in the form... 

The motion of a particle in a liquid is described by Stokes equation. If its diameter is d, the rate u at which it settles by gravity in a liquid of viscosity r] and density p is given by Eq. (21)... 

The sedimentation of particles—that is, their downward motion due to gravitational settling—follows Stokes s law. Turbulent diffusion represents an opposing force, and sedimentation equilibrium is established when both forces cancel. Assuming a constant production rate of particles at the earth surface for each size group and a balance of upward and downward fluxes in the atmosphere leads to the following equation ... 

Finally it should be pointed out that equations 2.44 and 2.46 have a limited application to small sizes (the minimum is 1 pm in a gravity field and much smaller in a centrifugal field depending on the speed of rotation) when Brownian motion effectively slows down settling rates and to large sizes when, for Reynolds numbers greater than about 0.2, increasing deviations from Stokes law occur. 

The stationary settling rate of a single, small, spherical particle in an infinite, stationary fluid under the influence of gravity can be calculated from Eqns. 1 and 2. It has been assumed in these equations that the flow around the particle is laminar and Stokes law of resistance may be applied. 

Sedimentation is another classical particle classification and sizing method for liquid-bom particles. Sedimentation methods are based on the rate of settling of particles in a liquid at rest under a gravitational or centrifugal field. The relationship between settling velocity and particle size is reduced to the Stokes equation at low Reynolds numbers ... 

Stokes law is rigorously applicable only for the ideal situation in which uniform and perfectly spherical particles in a very dilute suspension settle without turbulence, interparticle collisions, and without che-mical/physical attraction or affinity for the dispersion medium [79]. Obviously, the equation does not apply precisely to common pharmaceutical suspensions in which the above-mentioned assumptions are most often not completely fulfilled. However, the basic concept of the equation does provide a valid indication of the many important factors controlling the rate of particle sedimentation and, therefore, a guideline for possible adjustments that can be made to a suspension formulation. 

Stokes s law and the equations developed from it apply to spherical particles only, but the dispersed units in systems of actual interest often fail to meet this shape requirement. Equation (12) is sometimes used in these cases anyway. The lack of compliance of the system to the model is acknowledged by labeling the mass, calculated by Equation (12), as the mass of an equivalent sphere. As the name implies, this is a fictitious particle with the same density as the unsolvated particle that settles with the same velocity as the experimental system. If the actual settling particle is an unsolvated polyhedron, the equivalent sphere may be a fairly good model for it, and the mass of the equivalent sphere may be a reasonable approximation to the actual mass of the particle. The approximation clearly becomes poorer if the particle is asymmetrical, solvated, or both. Characterization of dispersed particles by their mass as equivalent spheres at least has the advantage of requiring only one experimental observation, the sedimentation rate, of the system. We see in sections below that the equivalent sphere calculations still play a useful role, even in systems for which supplementary diffusion studies have also been conducted. 

As the oil flows within its designated cross-section area through the horizontal vessel, free water droplets form and begin to drop at a terminal velocity rate. The Vessize program calculates this terminal velocity VTWO. As discussed previously in the oil dehydration section, the Stokes law settling equation is used for the water droplet fall rate VTWO. 

Droplets in the 3 to 100 micron (1 micron = 1/25,400 in.) range settle under the action of gravity at a rate given by Stokes law. The settling velocity is found by the equation... 

The so-caUed Sigma concept has been widely used in the field of centrifugal sedimentation ever since its first development by Ambler in 1952. It is a simplified relation between the machine performance in terms of X50, total volumetric flow rate Q and an index of the centrifuge size E. The cut size X50 is represented by its terminal settling velocity Vg in the given liquid under gravity so that from Stokes law (using equation 7.5b for the definition of K)... 

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