Particles of Arbitrary Shape

The following general statement was proved in [63] for the case of a uniform translational Stokes flow (Re -4 0) or a potential flow past a particle of an arbitrary shape the mean Sherwood number remains the same if the flow direction is changed to the opposite. 

Suppose that the axis of revolution makes an angle u with the translation flow velocity at infinity. In [358] the following approximate formula for the mean Sherwood number was obtained  

At low Peclet numbers, for the translational Stokes flow past an arbitrarily shaped body of revolution, formula (4.10.8) coincides with the exact asymptotic expression in the first three terms of the expansion [358], Since (4.10.8) holds identically for a spherical particle at all Peclet numbers, one can expect that for particles whose shape is nearly spherical, the approximate formula (4.10.8) will give good results for low as well as moderate or high Peclet numbers. 

For a steady-state viscous flow (without closed streamlines) past arbitrarily shaped smooth particles, one can calculate the mean Sherwood number by the approximate formula [359] 

The auxiliary variables Sho and Shoo in (4.10.9) are the leading terms of the asymptotic expansions of the mean Sherwood number at small and large Peclet numbers, respectively. (In (4.10.9), Sh, Sho, and Shoo are defined on the basis of the same characteristic length.) 

These results, which are expected to be reliable for Re 40, are correlated within 10% by the expression 

As shown in Chapter 4, the terminal velocity of a particle of arbitrary shape cannot be predicted with complete confidence, even at low Re. In this chapter, we have shown that the behavior of particles with well-characterized shapes is 

For calculating terminal velocities, it is convenient to use groups like those defined in Chapter 5  

Heywood s volumetric shape factor k, defined in Chapter 2, can be estimated rapidly, even for irregular particles, using Eq. (2-2). Table 6.3 gives values for regular shapes and some natural particles. Heywood (H2, H3) suggested that k be employed to correlate drag and terminal velocity, using dj and the projected 

Tetrahedron Cylinder with E = 1 viewed along axis viewed normal to axis Spheroids E = 0.5 E = 2 

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Gibbons R M 1969 Scaled particle theory for particles of arbitrary shape Mol. Phys. 17 81... 

For a particle of arbitrary shape a mass balance yields... 

No data are available for heat and mass transfer to or from disks or spheroids in free fall. When there is no secondary motion the correlations given above should apply to oblate spheroids and disks. For larger Re where secondary motion occurs, the equations given below for particles of arbitrary shape in free fall are recommended. 

Fig. 6.14 Ratio of terminal velocity of particle of arbitrary shape to that of sphere having the...

For particles of arbitrary shape held in a flow, Eqs. (6-34) and (6-37) should be used for Re > 1000. For particles in free fall the only data available (P2) show that the transfer is little affected by particle rotation with rotational velocities less than 50% of the particle velocity. The correlation for fixed particles was adequate provided that the equivalent diameter was used in place of L. For particles of arbitrary shape falling in the Newton s law regime, Eq. (6-35) should be used with replacing L and Sho taken as 2. 

The form factor for a particle of arbitrary shape can be calculated by numerical integration of (6.10). However, for certain regular geometrical shapes, it is possible to obtain analytical expressions for /. In this section we consider one such particle, a homogeneous sphere. 

Turner (1973) and McKellar (1976) applied RG theory to ensembles of randomly oriented particles of arbitrary shape the former author included spheres with anisotropic optical constants. Optically active particles have been treated within the framework of the RG approximation by Bohren (1977). 

However, (8.25) is not restricted to spheres but holds for particles of arbitrary shape. Thus, circular dichroism in particulate media includes a component that is the result of differential scattering, in contrast with circular dichroism in homogeneous media, which arises solely from differential absorption of left-circularly and right-circularly polarized light. 

The obvious advantage of the microwave experiment is that oriented single particles of arbitrary shape and, within limits, arbitrary refractive index, can be studied easily. Multilayered and other inhomogeneous particles pose no particular problems. 

McKellar, B. H. J., 1976. What property of a haze is determined by light scattering 2. Nonuniform particles of arbitrary shape, Appl. Opt., 15, 2464-2467. 

Brenner, H. (1963). Forced Convection Heat and Mass Transfer at Small Peclet Numbers from a Particle of Arbitrary Shape. Chem. Eng. Sci., 18,109. 

Bearing in mind the famous crystallographic correspondence principle between the outer shape of a crystal and its inner lattice structure, it is conceivable that it is impossible to form structurally ordered particles of arbitrary shapes so long as the reactions are carried out at or near the thermodynamic equilibrium. Inversely, special... 

Although this form accounts for the distribution of particles of arbitrary shape, the theory is well developed for spheres. In this case, one can also define the distribution function in terms of the particle radius (or diameter),... 

Analytical expression for the electrophoretic velocity of a sphere can be obtained for a thin but distorted double layer. Dukhin [6] first examined the effect of distortion of thin ion cloud on the electrophoresis of a sphere in a symmetric two-species electrolyte. Dukhin s approach was later simplified and extended by O Brien [7] for the case of a general electrolyte and a particle of arbitrary shape. Since 0(k 1) double layer thickness is much smaller than the characteristic particle size L, the ion cloud can be approximated as a structure composed of a charged plane interface and an adjacent diffuse cloud of ions. Within the double layer, the length scales for variation of quantities along the normal and tangential directions are k ] and L, respectively. 

In a systematic study of particle-particle interactions that also has often been overlooked, Cox and Brenner (1971) developed a comprehensive general theory for calculating the rheological properties of a suspension of particles of arbitrary shape to 0(02), including wall effects. They did not, however, attempt an explicit numerical calculation of ku and it is perhaps for this reason that their work has not received the thoughtful attention it deserves. 

For reliable application of the free volume concept of disperse systems one must have dependable methods of determination of the maximum packing fraction of the filler tpmax. Unfortunately, the possibility of a reliable theoretical calculation of its value, even for narrow filler fractions, seems to be problematic since there are practically no methods available for calculations for filler particles of arbitrary shape. The most reliable data are those obtained by computer simulation of the maximum packing fraction for spherical particles which give the value associated with possible particle aggregation, so that they are probable for fractions of small particle size. Deviations of particle shape is nearly cubic. At present the most reliable method of determination of [Pg.142]

One of the popular methods for evaluating effective diffusivities in heterogeneous catalysts is based on gas chromatography. A carrier gas, usually helium, which is not adsorbed, is passed continuously through a column packed with catalyst. A pulse of a diffusing component is injected into the inlet stream and the effluent pulse recorded. The main advantages of this transient method are its applicability to particles of arbitrary shapes, and that experiments can be carried out at elevated temperatures and pressures. Haynes [1] has given a comprehensive review of this method. 

Joshi (1987) and Clift (1993) have addressed this point in detail. Clift (1993) has provided the following systematic derivation. Consider a particle of arbitrary shape that is stationary in a fluid approaching with velocity Msup, where the fluid velocity need not be in the vertical direction (Fig. 49C). At any point on the surface of the particle, the fluid exerts a normal stress a and shear stress t. The force obtained by integrating t over the surface of the particle is the skin friction, and the component of this force parallel to u up is the skin friction drag. There is no argument over the formulation of t and hence skin friction. 

As stated before, the volume of catalyst per unit volume of reactor space, which is to be termed the fractional catalyst volume (equal to 1 minus the voidage) and the degree of utilization of this catalytic material are important factors in a high-pressure, relatively slow catalytic process, such as the hydroprocessing of oils. The effectiveness factor of catalyst particles of arbitrary shape can be correlated with a generalized Thiele modulus 4>gen, defined by... 

Briefly, these models assume that the orientation angles of differential sample surface elements in space are of equal probability. This assumption allows the use of geometric probability for deriving Che signal ratios for supported particles of different shape. It was found that, for truly random samples, the XPS signal of a supported phase which is present as equally sized but arbitrarily shsped convex particles is determined by their surface/volume ratio (and hence dispersion). In other words, the surface/volume ratios found for the supported compounds can be interpreted In terms of dimensions of particles of arbitrary shape.. Hence, the way in which XPS sees at siipported catalysts is similar to that in which the reactants do. 

It is important to note that Equation 5.365 is valid for particles of arbitrary shape and size, if the following requirement is fulfilled the dimensions of the particle and the local radii of curvature of the particle surface are much larger than the Debye screening length. 

Finally, linearity of the creeping-flow equations and boundary conditions allows a great a priori simplification in calculations of the force or torque on a body of fixed shape that moves in a Newtonian fluid. To illustrate this assertion, we consider a solid particle of arbitrary shape moving with translational velocity U(t) and angular velocity il(t) through an unbounded, quiescent viscous fluid in the creeping-flow limit Re 1 and Re/S < 1. The problem of calculating the force or torque on the particle requires a solution of... 

G. K. Youngren and A. Acrivos, Stokes flow past a particle of arbitrary shape A numerical method... 

In reviewing our analysis of (9-152) leading to (9-141), we may note that we have used the conditions (9-143) and (9-144) only on the velocity field. Thus, as stated earlier, the result (9-141) or (9-142) is valid in streaming flow for any heated body with a uniform surface temperature provided that these conditions are satisfied and that /V Higher-order terms in (9-142) will depend on the details of the flow, and thus on the Reynolds number Re, as well as the shape and orientation of the body relative to the free stream. However, in the creeping-flow limit, Brenner was able to extend (9-142) to one additional term for a particle of arbitrary shape,... 

H. Brenner, Forced convection heat and mass transfer at small Peclet numbers from a particle of arbitrary shape, Chem. Eng. Sci. 18, 109-22 (1963). 

Translational Stokes Flow Past Particles of Arbitrary Shape... 

Peclet numbers, the problem of mass exchange between a particle of arbitrary shape and a uniform translational flow were studied by the method of matched asymptotic expansions in [62]. The following expression was obtained for the mean Sherwood number up to first-order infinitesimals with respect to Pe ... 

For a particle of arbitrary shape in a translational flow, the first three terms of the asymptotic expansion of the dimensionless total diffusion flux as Pe — 0 have the form [62]... 

Particle of Arbitrary Shape in a Linear Shear Flow... 

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