Perimeter-equivalent sphere

Fig. 4.13 Correlation for conductance factor of axisymmetric particles in stagnant media (based on perimeter-equivalent sphere).

For axisymmetric bodies of an arbitrary shape, we introduce the notion of a perimeter-equivalent sphere. To this end, we project all points of the surface of the body on a plane perpendicular to its axis. The projection is a circle of radius a . The perimeter-equivalent sphere has the same radius. 

This dependence is shown in Figure 2.7 by solid line. The values of the axial drag of an axisymmetric body relative to the drag of the perimeter-equivalent sphere are plotted on the ordinate. The values of the perimeter-equivalent factor E equal to the ratio of the surface area of the particle to the surface area of the perimeter-equivalent sphere are plotted on the abscissa. 

Bowen and Masliyah examined the axial resistance of cylinders with flat, hemispherical and conical ends, and of double-headed cones and cones with hemispherical caps, together with the established results for spheroids. Widely used shape factors (including sphericity) did not give good correlations, while Eqs. (4-26) and (4-27) were found to be inapplicable to particles other than cylinders and spheroids. The best correlation was provided by the perimeter-equivalent factor Yj defined in Chapter 2. With this parameter, the equivalent sphere has the same perimeter as the particle viewed normal to the axis. Based on their numerical results, Bowen and Masliyah obtained the correlation... 

An irregular particle can be described by a number of sizes. There are three groups of definitions the equivalent sphere diameters, the equivalent circle diameters and the statistical diameters. In the first group are the diameters of a sphere which would have the same property as the particle itself (e.g. the same volume, the same settling velocity, etc.) in the second group are the diameters of a circle that would have the same property as the projected outline of the particles (e.g. projected area or perimeter). The third group of sizes are obtained when a linear dimension is measured (usually by microscopy) parallel to a fixed direction. 

It is of interest to note that, by judicious definition of the characteristic diameter of nonspherical bodies, good agreement with the equations for spherical solids was obtained. A diameter defined by the total surface area of the body, divided by the perimeter normal to flow, was successfully used for spheres, hemispheres, cubes, prisms, and cylinders (PI), yielding a = 0 b - 0.692 m = 0.514 and n = [Eq. (4)]. Similar results were obtained for spheroids (S14), namely a = 0 6 = 0.74 w = 0.5 and n =. The commonly used equivalent diameter of a sphere of the same volume as the body yields transfer coefficients increasing with eccentricity (SI4). 

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