Spherical coefficient

Table 4.10 Mechanical strength and spherical coefficient of the spherical catalyst prepared by spraying method...

For the spherical particle, y>p = l and for the non-spherical particle, Shape coefficient presents the extent of difference between particle and sphere, sometimes also called as spherical coefficient. The Eqs. (7.33) and (7.34) are suitable to particles whose inner spaces are non-hollow, but not to ring-columnar hollow particles. 

Corner J 1948 The second virial coefficient of a gas of non-spherical molecules Proc. R. Soc. A 192 275... 

Finally, the assumed spherical synnnetry of the interactions implies that the volume element r 2 is dri2- For angularly-dependent potentials, the second virial coefficient... 

The nth virial coefficient can be written as sums of products of Mayer/-fiinctions integrated over the coordinates and orientations of n particles. The third virial coefficient for spherically syimnetric potentials is... 

Let us compare computations of the effectiveness factor, using each of the three approximations we have described, with exact values from the complete dusty gas model. The calculations are performed for a first order reaction of the form A lOB in a spherical pellet. The stoichiometric coefficient 10 for the product is unrealistically large, but is chosen to emphasize any differences between the different approaches. 

The spherical geometry assumed in the Stokes and Einstein derivations gives the highly symmetrical boundary conditions favored by theoreticians. For ellipsoids of revolution having an axial ratio a/b, friction factors have been derived by F. Perrin, and the coefficient of the first-order term in Eq. (9.9) has been derived by Simha. In both cases the calculated quantities increase as the axial ratio increases above unity. For spheres, a/b = 1. 

The particle can be assumed to be spherical, in which case M/N can be replaced by (4/3)ttR P2, and f by 671770R- In this case the radius can be evaluated from the sedimentation coefficient s = 2R (p2 - p)/9t7o. Then, working in reverse, we can evaluate M and f from R. These quantities are called, respectively, the mass, friction factor, and radius of an equivalent sphere, a hypothetical spherical particle which settles at the same rate as the actual molecule. 

Assuming spherical particles, the drag coefficient, in the laminar, the Stokes flow regime is... 

Fig. 1. Diag coefficient vs particle Reynolds number for spherical particles where (-) corresponds to the theoretical value of = 24/Re (eq. 4).

In addition, dimensional analysis can be used in the design of scale experiments. For example, if a spherical storage tank of diameter dis to be constmcted, the problem is to determine windload at a velocity p. Equations 34 and 36 indicate that, once the drag coefficient Cg is known, the drag can be calculated from Cg immediately. But Cg is uniquely determined by the value of the Reynolds number Ke. Thus, a scale model can be set up to simulate the Reynolds number of the spherical tank. To this end, let a sphere of diameter tC be immersed in a fluid of density p and viscosity ]1 and towed at the speed of p o. Requiting that this model experiment have the same Reynolds number as the spherical storage tank gives... 

Figure 5 shows conduction heat transfer as a function of the projected radius of a 6-mm diameter sphere. Assuming an accommodation coefficient of 0.8, h 0) = 3370 W/(m -K) the average coefficient for the entire sphere is 72 W/(m -K). This variation in heat transfer over the spherical surface causes extreme non-uniformities in local vaporization rates and if contact time is too long, wet spherical surface near the contact point dries. The temperature profile penetrates the sphere and it becomes a continuum to which Fourier s law of nonsteady-state conduction appfies. 

The Stokes-Einstein equation has already been presented. It was noted that its vahdity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

G. 5-25-E, continuous phase coefficient, stagnant drops, spherical... 

The drag force is exerted in a direction parallel to the fluid velocity. Equation (6-227) defines the drag coefficient. For some sohd bodies, such as aerofoils, a hft force component perpendicular to the liquid velocity is also exerted. For free-falling particles, hft forces are generally not important. However, even spherical particles experience lift forces in shear flows near solid surfaces. 

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... 

Equations (6-236) to (6-239) are based on experiments on cube-oc tahedrons, octahedrons, cubes, and tetrahedrons for which the sphericity f ranges from 0.906 to 0.670, respectively. See also Chft, Grace, and Weber. A graph of drag coefficient vs. Reynolds number with y as a parameter may be found in Brown, et al. (Unit Operations, Whey, New York, 1950) and in Govier and Aziz. 

PasquiU Atmo.spheric Diffusion, Van Nostrand, 1962) recast Eq, (26-60) in terms of the dispersion coefficients and developed a number of useful solutions based on either continuous (plume) or instantaneous (puff) releases, Gifford Nuclear Safety, vol, 2, no, 4, 1961, p, 47) developed a set of correlations for the dispersion coefficients based on available data (see Table 26-29 and Figs, 26-54 to 26-57), The resulting model has become known as the Pasquill-Gifford model. 

The bubbles shapes in gas purging vary from small spherical bubbles, of radius less than one centimen e, to larger spherical-cap bubbles. The mass transfer coefficient to these larger bubbles may be calculated according to the equation... 

Let us first assume that we have a spherical particle with a radius of 5 p.m similar to an idealized toner particle, which is comprised of polystyrene, in contact with an electrically conducting substrate. A typical electric charge on a toner particle of that size is of the order of 10" " C. The Hamaker coefficient (Eq. 15) for such as system would be about 1.5 eV. 

By virtue of its chemical and thermal resistances, borosilicate glass has superior resistance to thermal stresses and shocks, and is used in the manufacture of a variety of items for process plants. Examples are pipe up to 60 cm in diameter and 300 cm long with wall tliicknesses of 2-10 mm, pipe fittings, valves, distillation column sections, spherical and cylindrical vessels up 400-liter capacity, centrifugal pumps with capacities up to 20,000 liters/hr, tubular heat exchangers with heat transfer areas up to 8 m, maximum working pressure up to 275 kN/m, and heat transfer coefficients of 270 kcal/hz/m C [48,49]. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

Consider spherical molecules A and B having radii and Tb and diffusion coefficients Da and Db- First, suppose that B is fixed and that the rate of reaction is limited by the rate at which A molecules diffuse to the B molecule. We calculate the flux 7(A- B) of A molecules to one B molecule. Let a and b be the concentrations (in molecules/cm ) of A and B in the bulk, and let r be the radius of a sphere centered at the B molecule. The surface area of this sphere is Aitr, so by Pick s first law we obtain... 

C = Overall Drag Coefficient, Dimensionless Dp = Diameter of Spherical Particle,ft. 

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