Sphere surface area

Figure 5.10 When a point rf is in region IS, the conditional probability P12 that a point r" (with r = r — r" ) is in region 2S is equal to the fraction of the sphere surface area above the boundary plane.

Now, consider a real liquid, where the molecules tend to cluster around each other rather than spreading themselves randomly. In that case, GiR) is the fraction of the spheres surface area that intersects other molecules (Fig. 12.2), and pG(R) gives the probability per unit volume of finding a molecule at a distance R from the reference molecule. The number of neighboring molecules N R) within a radius R in that case is... 

The PCM algorithm is as follows. First, the cavity siuface is determined from the van der Waals radii of the atoms. That fraction of each atom s van der Waals sphere which contributes to the cavity is then divided into a nmnber of small surface elements of calculable surface area. The simplest way to to this is to define a local polar coordinate frame at tlie centre of each atom s van der Waals sphere and to use fixed increments of AO and A(p to give rectangular surface elements (Figure 11.22). The surface can also be divided using tessellation methods [Paschual-Ahuir d al. 1987]. An initial value of the point charge for each surface element is then calculated from the electric field gradient due to the solute alone ... 

The most direct test is to compare the BET area with the geometrical area of the solid. Unfortunately, comparisons of this kind are relatively rare on account of experimental difficulties. The choices are to work with, say, single crystals having a well defined surface, when techniques of quite extraordinary sensitivity will be needed for measurement of the adsorption or, to obtain a larger surface area by use of thin sheets, narrow rods or small spheres, and run the risk that the surface will not be truly smooth so that the actual area will exceed the geometrical area. 

Since the diffusion coefficient is constant for a given material, Eq. (2.63) shows that the time required for a displacement increases with the square of the distance traveled. This can be understood by thinking that the displacement criterion would be met by finding the diffused particle anywhere on the surface of a sphere of radius x after time t if it started at the origin. The surface area of a sphere is proportional to the square of its radius. 

Because of the diversity of filler particle shapes, it is difficult to clearly express particle size values in terms of a particle dimension such as length or diameter. Therefore, the particle size of fillers is usually expressed as a theoretical dimension, the equivalent spherical diameter (esd), ie, the diameter of a sphere having the same volume as the particle. An estimate of regularity may be made by comparing the surface area of the equivalent sphere to the actual measured surface area of the particle. The greater the deviation, the more irregular the particle. 

The external surface area of the filler can be estimated from a psd by summing the area of all of the equivalent spheres. This method does not take into account the morphology of the surface. It usually yields low results which provide Htde information on the actual area of the filler that induences physical and chemical processes in compounded systems. In practice, surface area is usually determined (5) from the measured quantity of nitrogen gas that adsorbs in a monolayer at the particle surface according to the BET theory. From this monolayer capacity value the specific surface area can be determined (6), which is an area per unit mass, usually expressed in m /g. 

The surface mean diameter is the diameter of a sphere of the same surface area-to-volume ratio as the actual particle, which is usually not a perfect sphere. The surface mean diameter, which is sometimes referred to as the Sauter mean diameter, is the most useful particle size correlation, because hydrodynamic forces in the fluid bed act on the outside surface of the particle. The surface mean diameter is directly obtained from automated laser light diffraction devices, which are commonly used to measure particle sizes from 0.5 to 600 p.m. X-ray diffraction is commonly used to measure smaller particles (see Size TffiASURETffiNT OF PARTICLES). 

Sphericity. Sphericity, /, is a shape factor defined as the ratio of the surface area of a sphere the volume of which is equal to that of the particle, divided by the actual surface area of the particle. 

This development has been generalized. Results for zero- and second-order irreversible reactions are shown in Figure 10. Results are given elsewhere (48) for more complex kinetics, nonisothermal reactions, and particle shapes other than spheres. For nonspherical particles, the equivalent spherical radius, three times the particle volume/surface area, can be used for R to a good approximation. 

E] Compared napthalene sublimination to aqueous absorption to obtain fBerl saddles, = diameter of sphere with same surface area as pacldug piece. Lc — operating void space = e —, where e =... 

E] Gas absorption aud desorption from water aud organics plus vaporization of pure liquids for Raschig riugs, saddles, spheres, aud rods, dp = nominal pacldug size, Cp = dry pacldug surface area/volume, = wetted pacldug surface area/volume. Equations are dimensionally consistent, so any set of consistent units can be used. <3 = surface tension, dynes/cm. 

Based on diameter of a sphere having the same surface area as the particle. 

X = ratio of surface area of particle to surface area of a sphere with diameter D,... 

Irregular-shaped particles exhibit greater surface area than regular-shapea cubes and spheres, the amount of this increase being possibly 25 percent. The effect of particle size and size distribution on effective surface, in a shaft employed for calcination of limestone, is shown in... 

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