Shape factors, for particles

Example 18. Shape Factors for Particles in Packed Bed Exchange... 

In Leva s correlation for pressure drop in packed beds of particles, lower pressure drops are predicted for larger shape factors. For particles of the same volume per particle, the pressure drop per unit length is proportional to < , to the — 3 power where a increases from I to 2 as the flow increases from the laminar to the turbulent regime. 

This can be done via Eqs. (29) through (32). From the dissolution data, the coefficient B2 is obtained and through the results from microscopy the moments r2 and p3 can be evaluated. By knowing N, the initial number of particles, and the density of the solid, the average initial volume shape factor for a polydisperse powder can be estimated. 

The slip correction factors are important for particles smaller than 1 pm in diameter, which is rarely the case for pharmaceutical aerosols. Slip correction is required for the Stokes equation to remain predictive of particle behavior for these small particles. Therefore, assuming the absence of shape effects for particles in the Stokes regime of flow, Eq. (1) collapses into the following expression ... 

The range of specific surface area can vary widely depending upon the particle s size and shape and also the porosity.t The influence of pores can often overwhelm the size and external shape factors. For example, a powder consisting of spherical particles exhibits a total surface area, S, as described by equation (1.6) ... 

R. Aris, On shape factors for irregular particles-I. The steady state problem. Diffusion and reaction, Chem. Eng. Sci. 6 (1957) 262. 

The term in brackets is also a measure of particle-shape and may be termed the specific shape-factor of a particle. For spheres (a, = t, a, = 7t/6), we have a = 4.85, and for a cube a = 6. The specific shape-factor for irregular particles is usually greater than 6.0. 

L = actual depth of methanator bed in ft plus 3 ft more added on to account for AP due to nozzles, distributor, and supports e = voidage (fractional free volume of packing) = 0.40 for this case h = exponent = a function of the Reynolds number = 2.0 for this case D = average catalyst particle diameter, ft = 0.0238 ft for this case gc = dimensional constant = 32.17 (lb mass) (ft)/(lb forceXs2) p = fluid density, lb/ft3, based on average temperature and entering pressure = 0.464 lb/ft3 for this case (j)s = shape factor for the solid catalyst particles = 1.0 for this case... 

Sphericity shape factor Circularity shape factor 1 W where a0 = shape factor for equidimensional particle and thus represents part of av which is due to geometric shape only av = volume shape coefficient m = flakiness ratio, or breadth/thickness n = elongation ratio, or length/breadth Sphericity = (surface area of sphere having same volume as particle) / (surface area of actual particle) Circularity = (perimeter of particle outline)2 / 4tr(cross-sectional or projection area of particle outline)... 

Church [39] proposed the use of the ratio of the expected values of Martin s and Feret s diameters as a shape factor for a population of elliptical particles. Cole [40] introduced an image-analyzing computer (the Quantimet 720) to compare longest chord, perimeter and area for a large number of particles. Other parameters have been proposed by Pahl el. al [7,41], Beddow [42] and Laird [29]. 

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...

R. Aris, Shape Factors for Irregular Particles. 1. The Steady-State Problem. Diffusion and Re-... 

B shape the shape factors for the interstitial energy transport by radiation and molecular flow, respectively. The values of the relative particle to particle contact surface area, and the three shape factors, and must be... 

Equation 2.65 defines the useful quantity known as the specific surface area, i.e. the surface area per unit mass of solid. The constant F( = fsjfv = /31a) may be called the overall, surface-volume or specific surface shape factor. For spheres and cubes, F = 6. For other shapes F > 6. For an octahedron fy = J 2) 3,fs = 2 /3 and F = 7.35. Values of F 10 are frequently encountered in comminuted solids, and much higher values may be found for flakes and plate-like crystals. If the particles are elongated or needle-shaped, their volume and surface area may be calculated on the assumption that they are... 

Shape factors. Many particles in packed beds are often irregular in shape. The equivalent diameter of a particle is defined as the diameter of a sphere having the same volume as this particle. The sphericity shape factor 5 of a particle is the ratio of the surface area of this sphere having the same volume as the particle to the actual surface area of the particle. For a sphere, the surface area Sp = irDp and the volume is Vp irDpl6. Hence, for any particle, 4>s =n-DpSp, where Sp is the actual surface area of the particle and Dp is the diameter (equivalent diameter) of the sphere having the same volume as the particle. Then... 

In the equations, according to Larson and Garside [10], j denotes the exponent for the suspension density M, i is the quotient of the exponents m (for the supersaturation s at the nucleation rate and n (for the supersaturation s at the crystal growth rate G), Ip is the dominant particle size of the population balance, fey is the volume shape factor for the crystallized mass produced, and is the quotient of the proportionality constants of the secondary nucleation rate B and k "of the crystal growth rate G. 

Particle shape factor. The particle shape factor is a dimensionless parameter. For non-hollow particles, the shape factor is defined as the ratio of the external surface area 1S5 of a spherical particle with the same particle volume to the actual surface area Sp of the particle. This is equivalent to the ratio of the particle diameter dp to the diameter dp of a spherical particle with the same volume, that is, = Ss/Sp = ds/dp, where d = 6Vp/Sp. The shape factor indicates how much a particle differs from a spherical one. For spherical particles, 1 for non-spherical particles, is less than 1. The particle shape factor can be calculated from the volume and external surface area. The volume and external surface area for particles with... 

Thomas (1961) defined a shape factor for nonspherical particles as... 

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