Sedimentation equations

Most probable settling velocity from sedimentation data Particle-size determination from sedimentation equation Sedimentation in an ultracentrifuge Solvation and ellipticity from sedimentation data Diffusion and Gaussian distribution Temperature-dependence of diffusion coefficients... 

Basic Sedimentation Equilibrium Equation. Sedimentation equilibrium experiments are performed at constant temperature. The condition for sedimentation equilibrium is that the total molar potential, m, for all components i be constant everywhere in the solution column of the ultracentrifuge cell. Mathematically this can be expressed as... 

Gravitational experiments can also use the sedimentation equilibrium principle (equating sedimentation and diffusion fluxes). The molecular weight is then given by the equation ... 

Piazza R, Bellini T and Degiorgio V 1993 Equilibrium sedimentation profiles of screened charged colloids a test of the hard-sphere equation of state Rhys. Rev. Lett. 71 4267-70... 

The 2eta potential (Fig. 8) is essentially the potential that can be measured at the surface of shear that forms if the sohd was to be moved relative to the surrounding ionic medium. Techniques for the measurement of the 2eta potentials of particles of various si2es are collectively known as electrokinetic potential measurement methods and include microelectrophoresis, streaming potential, sedimentation potential, and electro osmosis (19). A numerical value for 2eta potential from microelectrophoresis can be obtained to a first approximation from equation 2, where Tf = viscosity of the liquid, e = dielectric constant of the medium within the electrical double layer, = electrophoretic velocity, and E = electric field. 

The terminal velocity in the case of fine particles is approached so quickly that in practical engineering calculations the settling is taken as a constant velocity motion and the acceleration period is neglected. Equation 7 can also be appHed to nonspherical particles if the particle size x is the equivalent Stokes diameter as deterrnined by sedimentation or elutriation methods of particle-size measurement. 

Most authors who have studied the consohdation process of soflds in compression use the basic model of a porous medium having point contacts which yield a general equation of the mass-and-momentum balances. This must be supplemented by a model describing filtration and deformation properties. Probably the best model to date (ca 1996) uses two parameters to define characteristic behavior of suspensions (9). This model can be potentially appHed to sedimentation, thickening, cake filtration, and expression. 

Equation 8 provides the basis of comparison for the performance of various bottle centrifuges having the same material, and also, under certain circumstances, of other types of sedimentation centrifuges, if geometric dissimilarities are also considered. 

The capacity factor, Zg, defined by equation 7, is derived from a set of assumptions. An additional assumption is specific to the botde centrifuge. Namely, a particle is considered sedimented when it reaches the surface of the cake without contacting the tube wall. 

Equation 10 estimates the flow or throughput rate, above which particles of size d are less than 50% sedimented, and below which over 50% are mostly coUected. Equations 10 and 11 are also appHcable to the light particles rising in a heavy phase Hquid, provided that and are interchanged in equation 11. 

Equation 12 shows that 2 can be expressed as the product of a mean sedimentation area (2 7Z r l) and the G level (co r /s), and therefore reflects the increased sedimentation rate expected through a defined area having centrifugal acceleration instead of gravity. 

The relation of equation 21 for similar centrifuges requires identical sedimentation performance characteristics when operating on the same material. 

Other Sedimentation Seale-Up Equations. Some centrifuge supphers use an area-equivalent, description instead of S others use KQ or values. All of these ate in urhts of area. For a disk centrifuge,... 

Factors Influencing Centrifugal Sedimentation. The sedimentation velocity of a particle is defined by equations I and 2. Each of the terms therein effects separation. 

The particle size deterrnined by sedimentation techniques is an equivalent spherical diameter, also known as the equivalent settling diameter, defined as the diameter of a sphere of the same density as the irregularly shaped particle that exhibits an identical free-fall velocity. Thus it is an appropriate diameter upon which to base particle behavior in other fluid-flow situations. Variations in the particle size distribution can occur for nonspherical particles (43,44). The upper size limit for sedimentation methods is estabHshed by the value of the particle Reynolds number, given by equation 11 ... 

In x-ray sedimentation, a collimated beam of x-rays permits particle concentration detection as a function of mass. The relationship between the fraction of x-rays transmitted and the mass concentration of particles of atomic weight >12 is expressed as in equation 13 ... 

For many modeling purposes, Nhas been assumed to be 1 (42), resulting in a simplified equation, S = C, where is the linear distribution coefficient. This assumption usually works for hydrophobic polycycHc aromatic compounds sorbed on sediments, if the equdibrium solution concentration is <10 M (43). For many pesticides, the error introduced by the assumption of linearity depends on the deviation from linearity. 

Simulation of aerosol processes within an air quaUty model begins with the fundamental equation of aerosol dynamics which describes aerosol transport (term 2), growth (term 3), coagulation (terms 4 and 5), and sedimentation (term 6) ... 

Equations to calculate size distributions from sedimentation data are based on the assumption that the particles fall freely in the suspension. In order to ensure that particle-particle interactton does not prevent free fall, an upper-volume concentration hmit of around 0.2 percent is recommended. 

Centrifugal Sedimentation Methods These methods extend sedimentation methods into the submicron size range. Sizes are calculated from a modified version of Stokes equation ... 

Adsorption — An important physico-chemical phenomenon used in treatment of hazardous wastes or in predicting the behavior of hazardous materials in natural systems is adsorption. Adsorption is the concentration or accumulation of substances at a surface or interface between media. Hazardous materials are often removed from water or air by adsorption onto activated carbon. Adsorption of organic hazardous materials onto soils or sediments is an important factor affecting their mobility in the environment. Adsorption may be predicted by use of a number of equations most commonly relating the concentration of a chemical at the surface or interface to the concentration in air or in solution, at equilibrium. These equations may be solved graphically using laboratory data to plot "isotherms." The most common application of adsorption is for the removal of organic compounds from water by activated carbon. 

As follows for the filtration of incompressible sediment (at a constant rate), the pressure increases in a direct proportion to time. However, the above equation... 

Equating equations 20 and 21, we find that (K2/K1) is equal to 3. Hence, the resistance of the liquid relative to a spherical particle in the sedimentation process is... 

In sedimentation, the Eule number is often referred to as the resistance number. Multiplying and dividing the RHS of equation 52 by n/8, we obtain... 

Substituting the resistance force into equation 51 and expressing F and V in terms of d, the basic equation of sedimentation theory is obtained ... 

This expression represents the first form of the general dimensionless equation of sedimentation theory. As the desired value is the velocity of the particle, equation 64 is solved for the Reynolds number ... 

This is the second form of the dimensionless equation for sedimentation. The Reynolds number also may be calculated from this equation ... 

Therefore, the inertia forces have an insignificant influence on the sedimentation process in this regime. Theoretically, their influence is equal to zero. In contrast, the forces of viscous friction are at a maximum. Evaluating the coefficient B in equation 55 for a = 1 results in a value of 24. Hence, we have derived the expression for the drag coefficient of a sphere, = 24/Re. 

In applying this equation it is possible to determine the maximum size particle in laminar flow, taking into account the given conditions of sedimentation (p, Pp, n and a ). However, this equation does not determine what the flow regime is when d > d . 

The turbulent regime for Cq is characterized by the section of line almost parallel to the x-axis (at the Re" > 500). In this case, the exponent a is equal to zero. Consequently, viscosity vanishes from equation 46. This indicates that the friction forces are negligible in comparison to inertia forces. Recall that the resistance coefficient is nearly constant at a value of 0.44. Substituting for the critical Reynolds number, Re > 500, into equations 65 and 68, the second critical values of the sedimentation numbers are obtained ... 

Filters generally achieve a lower final moisture content than obtained by gravity sedimentation and are often fed from thickeners, as indicated in the schematic particulate process shown in Figure 9.2. In this chapter the principles of slurry filtration will be described and certain simplified filter design equations derived. For more complex derivations the reader is referred to specialist texts e.g. Coulson and Richardson (1991), Wakeman (1990a) and Purchas (1981). 

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