Equivalent spherical particle

As an additional guide, the values are correlated with the equivalent spherical particle diameter by Stokes law, as in equation 1. A density... 

As an additional guide, the Q0/Z values are correlated with the equivalent spherical particle diameter by Stokes law, as in equation 1. A density difference A8 of 1.0 g/cm3 and a viscosity of 1 mPa-sf =cP) are assumed, thus conversion to other physical characteristics of the system requires that the particle size scale be adjusted to equate a particle of 1.0 pm diameter to its j20/ - i 1 cm/s, according to the relationship Q0 /E = 10 7 x 1.09 AS/p, for A8 in g/cm3, and viscosity p in Pas. For interpretation of the particle sizes, the scale refers to the 50% cutoff particle size, and under actual centrifugation conditions the value of Z, determined from Figure 8, must be increased by efficiency factors to give the theoretical value of Z. 

To use the previous equations for non-spherical particles, the diameter d, must be the diameter of the equivalent spherical particle. The volume of the equivalent spherical particle Vs = r( ) must be equal to the volume of the non-spherical particle Vp = jidl, where is a volume shape factor. Expressing the equality and solving for the equivalent spherical diameter d produces... 

Note, again, that 4 is for a spherical particle. Eor nonspherical particles, the sieve diameter dp must be converted into its equivalent spherical particle by the equation mentioned in a previous paragraph. 

For comparison, the radius of an equivalent spherical particle, f (sphere)> can be calculated by equating the volume-to-surface ratios ... 

Hydrite Properties well-crystallized, axial ratio 10 1, narrow particle size distribution, equivalent spherical particle size 550 nm [2263],... 

The equivalent spherical particle diameter of an aggregate of irregularly shaped particles can be found by studying the inertial motion of particles in a medium. This inertial motion behavior is used in many applications, such as sedimentation vessels, electrostatic separators and precipitators, and particle collectors. The various forces that affect particle motion, shown in Figure 4, are briefly discussed below. 

Classification of a particle cloud into discrete sizes using cascade impaction may be interpreted as measuring aerodynamic (equivalent spherical particle) diameter. Several impaction stages (cascade impactor) are used in the classification of a poly disperse cloud (see Figure 35). 

Since equation 12 is valid when Darcy s law is applicable, the equivalent spherical particle diameter, ds, and the intrinsic permeability, k, can be related by... 

We can estimate a, the radius of the particle, from a knowledge of the diffusion coefficient. If the particle is not spherical, the frictional coefficient is larger than that given by the Stokes s law expression the nonspherical particles exert a frictional effect larger than that exerted by an equivalent spherical particle. 

Fig. 16 Variation of diffusivity with carbon number for linear alkanes in silicalite at 300 K showing comparison between self-diffusivities and corrected transport diffusivities obtained by different techniques, o MD simulation [74] hierarchical simulation [75] + QENS [78] A PFG NMR [76] V single crystal membrane [65] A ZLC [77]. The ZLC values were calculated based on the assumption of isotropic diffusion in an equivalent spherical particle. The present figure has been modified by the addition of further experimental data from a figure originally presented by Jobic [78]...

In this equation, z is the coordinate along the reactor axis, Dax is the axial diffusion coefficient and vint the interstitial velocity, which can be calculated from the gas flow speed Vsup using vint = Vsup/e. The axial diffusivity can be calculated from the molecular diffusion coefficient of the component. For Rc, the radius of the crystals, the equivalent spherical particle radius is taken, defined as the radius of the sphere having the same external surface area to volume ratio [3]. We have estimated a value of 25 pm for the zeolite crystals in the current study. 

From the emission intensity, an equivalent spherical particle size can be calculated for sample particulates. One material which is available as pure, spherical particles of well-controlled particle size is sihca, used for chromatography column packing. A micrograph of 3 p.m diameter Si02 beads collected on a polycarbonate filter is shown in Fig. 7.51(a). While relatively uniform in diameter (monodisperse), you can see that not all the beads are perfect single spheres. The silica particles are aspirated into the He... 

As can be seen from these data, irregularly shaped particles with an equivalent size (diameter) of 70-110 /zm show an increase in average force of adhesion with increasing particle size the values obtained for the average force of adhesion with irregularly shaped particles are smaller than the values obtained for equivalent spherical particles. As the particle size increases from 70 to 110 fim, the difference in adhesive force between spherical and irregular particles becomes less pronounced. 

We will express the equivalent diameter of cyUndrical particles in accordance with Eq. (III. 15). Under these conditions, the adhesion of cylindrical particles will correspond to the adhesion of equivalent spherical particles. When we apply Eq. (III. 15) to cylindrical particles, we find that the equivalent diameter is given by... 

Let us remember that the quantity d J is understood to be the diameter of a spherical particle equivalent in contact area to the cylindrical particle. Since the adhesion in a liquid medium is determined to a great degree by the contact area between particles and surface, the introduction of the equivalent diameter means that the adhesive interaction of the cylindrical and equivalent spherical particle wiU be exactly the same. 

The adhesive-force distribution of adherent particles with irregular and spherical shapes is shown in Fig. VI. 16. With respect to equivalent diameter, the adhesive-force distribution of irregular particles also follows a log-normal law. The adhesion of irregular particles in the aqueous medium is greater than that of the equivalent spherical particles. This higher level of adhesion for the irregular particles is observed over the entire range of values of ap. 

The median force of adhesion for the irregular particles is L5-2 times that of the equivalent spherical particles. 

In conclusion, we will present comparative data on the median force of adhesion for particles of different shapes and sizes on unpainted and painted metal surfaces (Fig. VI. 17). In all cases, the median force of adhesion for the irregularly shaped particles is greater than the force of adhesion for equivalent spherical particles. 

Thus we see that, in a liquid medium, the adhesive-force distribution for irregularly shaped particles also follows a log-normal law. This enables us to determine the median force of adhesion and to calculate the average force of adhesion. The values found for the median force of adhesion of irregularly shaped particles are generally greater than those for the equivalent spherical particles. In a liquid medium, in all cases, for particles of different shapes, we find that the adhesive force varies directly with the particle size. 

The mercury porosimetry results presented in Figure 5.6 illustrate the influence of particle size and size distribution on the size and size distribution of the porosity in green ceramic filled glass (CFG) composites consisting of 65 vol % of 0.4—1.5-)lm median particle size alumina and 35% borosilicate glass. The porosimetry results reveal that a relatively broad size distribution of pores exist within the green powder compacts and that pore size distribution and mean pore radius decreases with the substitution of fine alumina for the coarse in the CFG composites. The mean equivalent cylindrical pore radius, r, and surface-volume mean equivalent spherical particle radius, of a powder compact consisting of >l- J,m particles are proportional to one another and related by the fractional porosity, e, of the powder compact by... 

Another concept of critical interest in dealing with solid particles is that of shape. The weight or volume (or some other characteristic dimension) of particles of irregular shape may sometimes be expressed in terms of equivalent spherical particles. This procedure is useful, because it simplifies calculations, but it tends to lead the novice to believe that particles are more often spherical than not. Of course, this is not the case. 

Figure 9 Two-dimensional illustration of a non-spherical particle packing system (solid line) and its equivalent spherical particle packing system (broken line).

Although most modem particle characterization methods are developed, validated and presumably used for spherical particles or equivalent spherical particles, real particles are rarely such ideal. In many instances, particle shape affects powder packing, bulk density, and many other macroscopic properties. Shape characterization of particulate systems only scatters in the literature [60], since there are hardly any universal methodologies available. Several methods exist that use shape coefficients, shape factors, Fourier analysis, or fractal analysis to semi-quantitatively describe shape [Ij. 

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