Shape factors circularity

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)... 

Where parameter c is known as the Kozeny constant, which is interpreted as a shape factor that is assigned different values depending on the configuration of the capillary (as a point of reference, c = 0.5 for a circular capillary). S is the specific surface area of the chaimels. For other than circular capillaries, a shape factor is included ... 

The critical pressure, pc, for water is 3,206 psia (22 MPa). The product of constants CaC is 0.23, which was evaluated from existing water DNB data for circular tubes. As Eq. (5-20) was developed from a uniform heat flux distribution, a shape factor Fc (Tong et al., 1966a) should be applied to the correlation in a case with nonuniform heat flux distribution. 

Further, it is known that real-world capillaries or pores are not always circular shaped. In fact, in oil reservoirs, the pores are more triangular shaped or square shaped than circular. In this case, the rise in capillaries of other shapes, such as rectangular or triangular (Birdi et al 1988 Birdi, 1997, 2002) can be measured. These studies have much significance in oil recovery or water treatment systems. In any system in which the fluid flows through porous material, it would be expected that capillary forces would be one of the most dominant factors. 

Since the spherieity and circularity are so difficult to determine for irregular partieles, Wadell (Wl) proposed that ij/ and be approximated by operational shape factors ... 

The area defining is projected parallel to the direction of motion. The modified circularity % is related to Heywood s shape factor (see Chapter 2) by... 

Before closing this chapter, we feel that it is useful to list in tabular form some isothermal pressure-flow relationships commonly used in die flow simulations. Tables 12.1 and 12.2 deal with flow relationships for the parallel-plate and circular tube channels using Newtonian (N), Power Law (P), and Ellis (E) model fluids. Table 12.3 covers concentric annular channels using Newtonian and Power Law model fluids. Table 12.4 contains volumetric flow rate-pressure drop (die characteristic) relationships only, which are arrived at by numerical solutions, for Newtonian fluid flow in eccentric annular, elliptical, equilateral, isosceles triangular, semicircular, and circular sector and conical channels. In addition, Q versus AP relationships for rectangular and square channels for Newtonian model fluids are given. Finally, Fig. 12.51 presents shape factors for Newtonian fluids flowing in various common shape channels. The shape factor Mq is based on parallel-plate pressure flow, namely,... 

The shape factor for a rubber block, for example a circular disk of diameter d and height h , is usually defined as the ratio of one bounded surface to the free surface area. Hence, one obtains with Eq. (50) ... 

Particle sizes combined with shape factors have been the subject of many of the recent studies regarding flow of solids. Sphericity, circularity, surface-shape coefficient, volume-shape coefficient, and surface-volume-shape coefficient are some of the most commonly used shape factors. It is generally accepted that the flowability of powders decreases as the shapes of particles become more irregular. Efforts to relate various shape factors to powder bulk behavior have become more successful recently, primarily because of the fact that shape characterization techniques and methods for physically sorting particles of different shapes are... 

This section will discuss some of the commonly used and cited shape factors in the pharmaceutical industry the shape factors discussed in this section are Wadelfs true sphericity and circularity, rugosity coefficient, correction factor, Dallavalle s shape factor, Heywood s shape factor, Schneiderhohn aspect ratio, one plane critical stability (OPCS), and Podczeck s two- and three-dimensional factor. There are also many other shape factors but they are beyond the scope of this chapter (11,14,26-31). 

Though Podczeck et al. claim that this method is able to detect small deviations from circularity and differentiates the degree of elliptical figures, they later developed an improved three-dimensional shape factor based on this two-dimensional model ... 

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...

The capacity and/or the capacitance of an isopotential ellipsoid are presented in several texts and handbooks such as those by Flugge [22], Jeans [40], Kellogg [43], Mason and Weaver [62], Morse and Feshbach [68, 69], Smythe [98], and Stratton [111]. The results presented in these texts are used to develop expressions for the shape factors of several bodies spheres, oblate and prolate spheroids (see Fig. 3.3), circular and elliptical disks, and ellipsoids. The shape factor for the ellipsoid is general it reduces to the shape factor for the other bodies. 

The range of dimensionless shape factor for the oblate spheroids is 8 < St 4ji. The highest and lowest values correspond to the sphere and the circular disk, respectively. The radii of the disk and sphere are set to one unit. 

The dimensionless shape factor range for the elliptical disks is approximately 3.4 < S < 8. The highest value corresponds to a circular disk of unit radius, and the lowest value corresponds to an elliptical disk with a 10 to 1 aspect ratio. 

Circular Toroid. The circular toroid is characterized by the ring diameter d and the toroid diameter D (Fig. 3.4a). The analytical solution [94] for the shape factor is written as an infinite series in which each term consists of toroidal or ring functions [1] ... 

Square Toroid. The accurate numerical values of shape factors for square toroids, which are characterized by the inner and outer diameters D, and D , respectively (Fig. 3.4b), were reported by Wang [122] for a wide range of the diameter ratio DJD . The dimensionless results were found to be in close agreement with the analytical results for the equivalent circular toroid defined by... 

Finite Circular Cylinder. The dimensionless shape factor for an isothermal right circular cylinder of length L and diameter d (Fig. 3.4c) was obtained from the analytical solution for the capacitance [96,97], Using the square root of the total surface area, the result is recast as... 

The dimensionless shape factor for the right circular cylinder is in very close agreement with the values for the oblate spheroid in the range 0 < Lid < 1 and with the values for the prolate spheroid in the range 1 < Lid < 8. The difference when Ud = 1 is less than 1 percent. This shows that the results for the sphere and a finite circular cylinder of unit aspect ratio are very close. The simple expression obtained from the Smythe solution can be used to estimate the shape factors of circular disks, oblate spheroids, and prolate spheroids in the range 0 < Ltd < 8. For Ud > 8, the prolate spheroid asymptotic result can be used to provide accurate results for long circular cylinders and other equivalent bodies. 

Finite Square Cylinder. The dimensionless shape factors for finite square cylinders of length L and side dimension W (Fig. 3.4d) can be calculated using the finite circular expression by means of the equivalent aspect ratio... 

Circular and Rectangular Annulus. The dimensionless shape factors of isothermal circular and rectangular annuli are presented next. 

Circular annulus. The circular annulus has inner and outer radii a and b, respectively. The two capacitance analytical solutions of Smythe [95] are recast into the following two expressions, which relate the dimensionless shape factor to the radii ratio e = alb ... 

P. A. A. Laura and G. Sanchez Sarmiento, Heat Flow Shape Factors for Circular Rods with Regular Polygonal Concentric Inner Bore, Nuclear Engineering and Design (47) 227-229,1978. 

G. K. Lewis, Shape Factors in Conduction Heat Flow for Circular Bars and Slabs with Internal Geometries, Int. J. Heat Mass Transfer, Vol. 11, pp. 985-992,1968. 

Wheeler, A., Advances in Catalysis 3, 249 (1951) introduced a roughness factor in a similar way a shape factor, taking account of the non-circular character of the cross-section seems to be more logical (see J. J. Steggerda, Thesis, Delft, 1955). 

Once an image has been segmented, the microstructural characteristics can be measured, for example ice crystal and air bubbles size. Figure 6.1 shows that the ice crystals and air bubbles are not spherical. Therefore, several different measurements of size can be made, for example the maximum diameter, the minimum diameter, or the equivalent circular diameter i.e. the diameter of a circle with the same area). The shape factor (defined by equation 6.3) is a measure of how far a particle is from being circular it takes a value of one for a sphere and becomes smaller as the particle becomes less spherical. 

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... 

Sieving by dehydrated zeolite crystals is based on the size and shape diffierences between the crystal apertures and the adsorbate molecule. In some instances, the aperture is circular, such as in zeolite A. In others, it may take the form of an ellipse such as in dehydrated chabazite. In this case, subtle differences in the adsorption of various molecules result from this shape factor. 

In this e, the shape factor, has a value of 1 for circular, less than 1 for flat or elliptical and greater than 1 for hollow fiber. Although no definitive data are available, its value should also be more than 1 for trilobal and multi-lobal fibers with circular symmetry. is a constant whose value depends on the units in which E, d and p are expressed. 

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