Single sphere

The first case is relevant in the discussion of colloid stability of section C2.6.5. It uses the potential around a single sphere in the case of a double layer that is thin compared to the particle, Ka 1. Furthennore, it is assumed that the surface separation is fairly large, such that exp(-K/f) 1, so the potential between two spheres can be calculated from the sum of single-sphere potentials. Under these conditions, is approximated by [42] ... 

F. Single spheres, forced =2.040.591 [S] Correlates large amount of data and compares [125]... 

UfQ = terminal velocity of a single sphere (infinite dilution) c = volume fraction sohd in the suspension n = function of Reynolds number Re = dpUto /[L as given Fig. 6-58... 

The terminal settling velocity for single spheres can he determined using the contrasts for the flovv regime. 

Temperature rise, centrifugal pump, 207-209 Terminal particle velocity, 228, 230 Particles, different densities, 238 Single spheres, 274 Solids in air, 237 Solids in water, 237 Test pressure, piping, 18 Thickeners and settleix/decanters, Decanter, 242... 

I I. Tayi.or, T.D. Physics of Fluids 6 (1963) 987. Heat transfer from single spheres in a low Reynolds number slip flow. 

Rowe. P.N.. Claxton, K.T., and Lewis, J.B. Trans. Inst. Chem. Eng. 41 (1965) T14. Heat and mass transfer from a single sphere ill an extensive flowing fluid. 

Fig. 1.—The arrangement of 45 spheres in icosahedral closest packing. At the left there is shown a single sphere, which constitutes the inner core. Next there is shown the layer of 12 spheres, at the corners of a regular icosahedron. The third model shows the core of 13 spheres with 20 added in the outer layer, each in a triangular pocket corresponding to a face of the icosahedron these 20 spheres lie at the corners of a pentagonal dodecahedron. The third layer is completed, as shown in the model at the right, by adding 12 spheres at corners of a large icosahedron the 32 spheres of the third layer lie at the corners of a rhombic triaconta-hedron. The fourth layer (not shown) contains 72 spheres.

In order to derive an expression for the interaction parameter T on the basis of elasticity theory, the elastic energy of a single sphere of volume F( is considered which is embedded in a spherical hole of volume Fq in the elastic medium ... 

In the theoretical section above, the nonlinear polarization induced by the fundamental wave incident on a planar interface for a system made of two centrosymmetrical materials in contact was described. However, if one considers small spheres of a centrosymmetrical material embedded in another centrosymmetrical material, like bubbles of a liquid in another liquid, the nonlinear polarization at the interface of a single sphere is a spherical sheet instead of the planar one obtained at planar surfaces. When the radius of curvature is much smaller than the wavelength of light, the electric field amplitude of the fundamental electromagnetic wave can be taken as constant over the whole sphere (see Fig. 7). Hence, one can always find for any infinitely small surface element of the surface... 

Basic research consists of exploratory studies into things for which an end use cannot be specified. It might include a study to determine the effect of chlorine molecules on the diffusivity of hydrocarbons or a study of the dissolution of single spheres in a flowing stream. The prospective dollar value of this research cannot be estimated. 

SK Friedlander. Mass and heat transfer to single spheres and cylinders at low Reynolds numbers. AIChE J 3 43-48, 1957. 

Having stated the above, it is absolutely essential to make some comments on the current package of measures regulating the Spanish pharmaceutical market. As we mentioned earlier, the expected effectiveness of tackling the problem of pharmaceutical consumption in a single sphere of intervention (supply, demand or wholesalers) is small. Equally, the study of the impact of any measure of this sort must therefore incorporate an integral approach to the problem (see Table 10.11). 

A sphere has a larger mass than a typical bar (L = 5D) resonating at the same frequency, and because it is equally sensitive for all directions and polarizations it has a cross-section (for the same material) that is about 75 times larger. A single sphere is also capable of determining the source direction and polarization. A spherical detector is the only detector for GWs with isotropic sky coverage and the capability of finding the location of the source. Both laser interferometers and bar detectors are unable to do this with just one detector six bar detectors would be needed to build an omni-directional observatory. 

Fig. 9. Velocity of a single sphere in a 3D LB gas. The black line is the data from LBM, which has the proper functional form v(t) = 11 (1 - exp(-grt/Hoo)). The grey line is theoretical terminal velocity, which is slightly higher than vx.

Henderson 575 presented a set of new correlations for drag coefficient of a single sphere in continuum and rarefied flows (Table 5.1). These correlations simplify in the limit to certain equations derived from theory and offer significantly improved agreement with experimental data. The flow regimes covered include continuum, slip, transition, and molecular flows at Mach numbers up to 6 and at Reynolds numbers up to the laminar-turbulent transition. The effect on drag of temperature difference between a sphere and gas is also incorporated. 

Table 5.1. Correlations for Drag Coefficient of a Single Sphere in Continuum and Rarefied Flows1575 ...

Correlations for heat transfer coefficient between a single sphere and surrounding gas have been proposed by many researchers (Table 5.2), for example, Whitaker,1584 and Ranz and Marshall,15051 among others. The correlation recommended by Whitaker is accurate to within 30% for the range of parameter values listed. All properties except jus should be evaluated at Tm. For freely falling liquid droplets, the Ranz-Marshall correlation 505 is often used. The correlations may be applied to mass transfer processes simply by replacing Nu and Pr with Sh and Sc, respectively, where Sh and Sc are the Sherwood number and Schmidt number, respectively. Modifications to the Ranz-Marshall correlation have been made by researchers to account... 

Table 5.2. Correlations for Heat Transfer Coefficient of a Single Sphere in Gas...

For a single sphere in a stagnant environment, i.e. where there is no convection, the limiting value of the Nusselt number can be shown (see, for example, Kay and Nedderman, 1985) to be... 

For heat transfer to a single sphere in turbulent flow, McAdams (1954) suggested... 

When heat is conducted away from a single sphere of diameter d through a large volume of stationary fluid, it can be shown that... 

The clonogenic assay is normally exploited to analyze the effect of compounds on the symmetry of division in NSCs/CSCs. In this assay, the number of secondary spheres generated from the dissociation of a single sphere reflects the frequency of NSC/CSCs present in the original primary clone. This analysis also returns an estimate of the relative frequencies between symmetric proliferative (two SCs generated at each cycle) and symmetric differentiative (two differentiated and/or dead cells generated after cell division). 

Study of the eflSciency of packed columns in liquid-liquid extraction has shown that spontaneous interfacial turbulence or emulsification can increase mass-transfer rates by as much as three times when, for example, acetone is extracted from water to an organic solvent (84, 85). Another factor which may be important for flow over packing has been studied by Ratcliff and Reid (86). In the transfer of benzene into water, studied with a laminar spherical film of water flowing over a single sphere immersed in benzene, they found that in experiments where the interface was clean... 

What is the largest number of spheres that can touch a single sphere (assume that each sphere has the same radius) For circles, we know the answer is six (Fig. 4.26). For spheres, the largest number is twelve, but this fact was not proved until 1874. In other words, the largest number of unit spheres that can touch another unit sphere is twelve. For hyperspheres, it is not yet known if the number is twenty-four, twenty-five, or twenty-six, nor is a solution known for higher dimensions, as far as I know. Mathematicians do know that it is possible for at least 306 equal spheres to touch another equal sphere in nine dimensions, and 500 can touch another in ten dimensions. But mathematicians are not sure if more can be packed ... 

To show the effect of increasing size dispersion on extinction,a series of calculations for water droplets is given in Fig. 11.6. The topmost curve reproduces the calculations of Fig. 11.5a for a single sphere the standard deviation a is increased in successively lower curves. 

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