Diameter equivalent

Zanker s article, reprinted here, gives a way to estimate the elusive equivalent diameter of solids. 

Particle diameter is a primary variable important to many chemical engineering calculations, including settling, slurry flow, fluidized beds, packed reactors, and packed distillation towers. Unfortunately, this dimension is usually difficult or impossible to measure, because the particles are small or irregular. Consequently, chemical engineers have become familiar with the notion of equivalent diameter of a particle, which is the diameter of a sphere that has a volume equal to that of the particle. 

The equivalent diameter can be calculated from the dimensions of regular particles, such as cubes, pyramids. 

8 = fractional free volume, dimensionless s = specific surface = particle surface per unit volume of bed 

Examples. What is the equivalent diameter of a rectangular pyramid with base sides 1 and 2 in., and a height 

Fusain fibers -Flint sarrd, jagged flakes Flue dust, fused, aggregates 

Flint sand, jagged /, Glass, crushed, jagged 

Mica flakes Bari saddles Raschig rings 

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The limits of pore size corresponding to each process will, of course, depend both on the pore geometry and the size of the adsorbate molecule. For slit-shaped pores the primary process will be expected to be limited to widths below la, and the secondary to widths between 2a and 5ff. For more complicated shapes such as interstices between small spheres, the equivalent diameter will be somewhat higher, because of the more effective overlap of adsorption fields from neighbouring parts of the pore walls. The tertiary process—the reversible capillary condensation—will not be able to occur at all in slits if the walls are exactly parallel in other pores, this condensation will take place in the region between 5hysteresis loop and in a pore system containing a variety of pore shapes, reversible capillary condensation occurs in such pores as have a suitable shape alongside the irreversible condensation in the main body of pores. 

D, Equivalent diameter of a cross section, usually 4 times free area divided by wetted perimeter D, for equivalent diameter of window m ft... 

Laminar Flow Normally, laminar flow occurs in closed ducts when Nrc < 2100 (based on equivalent diameter = 4 X free area -i-perimeter). Laminar-flow heat transfer has been subjected to extensive theoretical study. The energy equation has been solved for a variety of boundaiy conditions and geometrical configurations. However, true laminar-flow heat transfer veiy rarely occurs. Natural-convecdion effects are almost always present, so that the assumption that molecular conduction alone occurs is not vahd. Therefore, empirically derived equations are most rehable. 

Limiting Nusselt numbers for laminar flow in annuli have been calculated by Dwyer [Nucl. Set. Eng., 17, 336 (1963)]. In addition, theoretical analyses of laminar-flow heat transfer in concentric and eccentric annuh have been published by Reynolds, Lundberg, and McCuen [Jnt. J. Heat Ma.s.s Tran.sfer, 6, 483, 495 (1963)]. Lee fnt. J. Heat Ma.s.s Tran.sfer, 11,509 (1968)] presented an analysis of turbulent heat transfer in entrance regions of concentric annuh. Fully developed local Nusselt numbers were generally attained within a region of 30 equivalent diameters for 0.1 < Np < 30, lO < < 2 X 10, 1.01 <... 

For rectangular ducts Kays and Clark (Stanford Univ, Dept. Mech. Eng. Tech. Rep. 14, Aug. 6, 1953) published relationships for headng and cooling of air in rectangular ducts of various aspect rados. For most noncircular ducts Eqs. (5-39) and (5-40) may be used if the equivalent diameter (= 4 X free area/wetted perimeter) is used as the characteristic length. See also Kays and London, Compact Heat Exchangers, 3d ed., McGraw-Hill, New York, 1984. 

Noncircular Ducts Equations (5-50 ) and (5-50/ ) may be employed for noncircular ducts by using the equivalent diameter D = 4 X free area per wetted perimeter. Kays and London (Compact Heat Exchangers, 3rd ed., McGraw-HiU, New York, 1984) give charts for various noncircular duels encountered in compact heat exchangers. 

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 ... 

Re using the equivalent diameters defined in the following. This situation is, by arbitrary definition, opposite to that for the hydraulic diameter used for turbulent flow. 

Equivalent diameter of window D [required only if laminar flow, defined as (Vrc), 100, exists]... 

Most efficient performance is obtained with plates having open areas equal to 2 to 3 percent of the total heat-transfer area. The plate should be located at a distance equal to four to six hole (or equivalent) diameters from the heat-transfer surface. 

Predic tions from Eq. (14-201) ahgn well with the Tatterson data. For example, for a velocity of 43 iti/s (140 ft/s) in a 0.05-m (1.8-inch) equivalent diameter channel, Eq. (14-201) predicts D32 of 490 microns, compared to the measured 460 to 480 microns. 

Cross-sectional Equivalent diameter Nominal thickness Nominal overall Maximum resistance Current rating d.c. or... 

These two nomographs provide a convenient means of estimating the equivalent diameter of almost any type of particle Figure 1 of regular particles from their dimensions, and Figure 2 of irregular particles from fractional free volume, specific surface, and shape. 

Also, in cases where the dimensions of a regular particle vary throughout a bed of such particles or are not known, but where the fractional free volume and specific surface can be measured or calculated, the shape factor can be calculated and the equivalent diameter of the regular particle determined from Figure 2. 

Figure 1. Equivalent diameters of regular-shaped bodies.

Figure 2. Equivalent diameters of irregular and complicated bodies.

In Figure 1, the equivalent diameter is related to regular shapes through equivalent volumes by the following formulas ... 

What is the equivalent diameter of crushed glass (t ) = 0.65) with a fractional free volume of 0.55 and a specific... 

Zanker, Adam, Estimating Equivalent Diameters of Solids. Chemical Engineering, July 5, 1976, p. 101. 

Airborne partieulate matter may eomprise liquid (aerosols, mists or fogs) or solids (dust, fumes). Refer to Figure 5.2. Some eauses of dust and aerosol formation are listed in Table 4.3. In either ease dispersion, by spraying or fragmentation, will result in a eonsiderable inerease in the surfaee area of the ehemieal. This inereases the reaetivity, e.g. to render some ehemieals pyrophorie, explosive or prone to spontaneous eombustion it also inereases the ease of entry into the body. The behaviour of an airborne partiele depends upon its size (e.g. equivalent diameter), shape and density. The effeet of partiele diameter on terminal settling veloeity is shown in Table 4.4. As a result ... 

Note - In designing a system based on the settling velocity of nonspherical particles, the linear size in the Reynolds number definition is taken to be the equivalent diameter of a sphere, d, which is equal to a sphere diameter having the same volume as the particle. 

Consider again the simple motion of a sphere. In this case, the equivalent diameter of a sphere, d, is equal to its geometric diameter, d. Equating the above expressions and replacing 5 by d (and denoting the Euler umber, Eu, by Y), we obtain an expression for the resistance force ... 

Credit for additional height of the flame center for multiple valve installations may be taken by clustering the safety valve discharge pipes to the atmosphere. The following procedure should be used for determining equivalent diameter and exit velocity to be used in the flame center calculation. Diameter and velocity are based on the total acmal area of the clustered vents. 

Djj. The Grashof number Nq, = Dj pgpAto/p" were is equivalent diameter, g is acceleration due to gravity, p is coefficient of volumetric expansion, p is viscosity, p is density, and Atg is the difference between the temperature at the wall and that in the bulk fluid. Nq, must be calculated from fluid properties at the bulk temperature. 

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