Perimeter-equivalent factor

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... 

The graphical correlation is presented in terms of the perimeter equivalent factor Yj sed in Figs. 4.9 and 4.10. The points have been calculated for axisymmetric shapes. 

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. 

Figure 4.1. Shape factor ratio against perimeter-equivalent factor for particles of various shape in a stagnant medium 1, circular cylinder 2, oblate ellipsoid of revolution 3, prolate ellipsoid of revolution 4, cube...

E perimeter-equivalent factor, S. /(47raj ) rs friction stress on wall... 

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

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

Equation 2.73 is another way of writing equation 2.13 where, in this case, the pressure drop is expressed in height of fluid instead of in force per unit area. In equation 2.73, de is the equivalent diameter defined as four times the cross-sectional flow area divided by the appropriate flow perimeter,/is the Fanning friction factor for flow in an open channel and u is the mean velocity. Combining equations 2.72 and 2.73, and solving for u gives... 

In these equations, fy is the friction factor for longitudinally finned tubes, and D, is equivalent diameter (= 4ac/P, where ac is the cross-sectional flow area and P is the wetted perimeter). 

Schafer, [136] in a discussion on the accuracy of image analysis systems, states that the accuracy of area and equivalent diameter measurements is sufficient for most practical purposes. On the other hand he suggests improvements in the determination of perimeters and shape factors which he found to be unsatisfactory. 

The CO-H2 synthesis properties of metal/titania catalysts have been found in several studies to be essentially unaffected by the temperature of reduction, in contrast to the strong effect this factor has on chemisorption properties. This problem has focused attention on the special nature of the metal-titania contact perimeter. Reduction of titania, undoubtedly through hydrogen spillover, begins there. It is important to note that Ti J+ cations are produced by reduction temperatures as low as 473K, as shown by 02 adsorption/H20 decomposition measurements (36) or by temperature programmed reduction. In the latter study, the amount of-TiJ+ produced at temperatures below 503K was equivalent to a TiJ+/Pt atom ratio of 0.6 (37). 

The drainage equivalency of wicks and sand drains is often computed by equating void volumes or perimeters. If void volumes are used, assuming 85% for wicks and 30% for sand, a wick is shown to be equivalent of about a 2-to 3-inch diameter sand drain (depending upon whatever efficiency factor may seem appropriate). Based on experience, however, several wick manufacturer s literature states, Each wick drain can provide a greater vertical discharge capacity than a 6-inch-diameter sand drain (quote courtesy of U.S. Wickdrain, Inc). 

FRICTION FACTOR IN FLOW THROUGH CHANNELS OF NONQRCULAR CROSS SECTION. The friction in long straight channels of constant noncircular cross section can be estimated by using the equations for circular pipes if the diameter in the Reynolds number and in the definition of the friction factor is taken as an equivalent diameter, defined as four times the hydraulic radius. The hydraulic radius is denoted by r and in turn is defined as the ratio of the cross-sectional area of the channel to the wetted perimeter of the channel ... 

There is a confusing array of measurement options in modem digital image analyzers. Besides area and perimeter, it is possible to obtain measurements of geometrical features such as equivalent circular diameter, aspect ratio, shape factors, Feret diameters, Martin diameters, to name just a few. In most instruments, the operator has the flexibility to design his or her own geometrical property using the built-in measurements in some mathematical combination defined by the... 

Fanning friction factor (1/4 of the Darcy factor) d = diameter of the full, circular pipe equivalent to the channel of wetted perimeter R... 

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