Particle arbitrary shape

Here B is an optical constant, or is the total polarizability of the particle, and n is the number of components in each particle. The indexes i and j refer to components of the same particle. If the assumption of independent particles was not made, then the indexes could refer to components of any two particles, and the autocorrelation expression could not be written as a simple sum of contributions from individual particles. The spatial vector r(r) refers to the center of mass of the particle. R(r). In the case of a nonspherical particle (arbitrary shape), Eq. (I0) would describe the coupled motion of the center of mass and the relative arrangement of the components of the particle. For spherical particles, translational and rotational motion arc uncoupled and we have a simplified expression for the electric field time correlation function ... 

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. 

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... 

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. 

Here we concentrate on cylindrical containing walls, although there is some work on particles near plane boundaries and surfaces of arbitrary shape. Most of the work on rigid particles refers to spheres, and it is then convenient to use the diameter ratio... 

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). 

A discussion of some theoretical approaches to scattering by randomly inhomogeneous particles is followed in the final section by an outline of recent progress in constructing solutions to problems of scattering by nonspherical particles, including those of arbitrary shape. 

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. 

Purcell and Pennypacker calculated scattering and absorption by various rectangular particles with different refractive indices for size parameters up to about two. More recently, Kattawar and Humphreys (1980) used the Purcell-Pennypacker method to investigate scattering by two spheres as a function of separation. Arbitrary shapes can be treated by this method but excessive computation time appears to preclude its use for large size parameters. 

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. 

The above construction and accordingly the conclusion, obviously hold for an arbitrary electrolyte in 1Zn. 772 and symmetry of the electrolyte, resulting in the sinh p nonlinearity instead of a general sum of exponents, are not essential for the derivation above and were assumed for the sake of brevity of presentation only. The appropriate result—force saturation—is most likely true for particles of an arbitrary shape, although the corresponding generalization still has to be carried out. Another relevant task would be an actual computation of the limiting repulsion force as a function of, say, particle separation. 

Proof of boundedness of the force of interaction between two charged particles of an arbitrary shape in H3, held at a given distance from each other in an electrolyte solution, upon an infinite increase of the particle s charge. (It was shown in 2.2 that the repulsion force between parallel symmetrically charged cylinders saturates upon an infinite increase of the particle s charge. This is also true for infinite parallel charged plane interaction [9]. The appropriate result is expected to be true for particles of an arbitrary shape.)... 

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

This expression determines the kinetics of the excitation accumulation from IV (0) = 0 up to the stationary value N, after instantaneous switching the permanent illumination at t 0. This is a particular case of the more general convolution recipe derived in Ref. 201 for pulses of arbitrary shape. Hence, for equal diffusion coefficients, the convolution formula follows from IET as well as from the many-particle theory employed in Ref. 201. The generality of the latter allows us to use in the convolution formula the system response P(t), calculated with any available theory. The same is valid for the stationary equation (3.458), used above. Although P(t) obtained with different theories is different, as well as N (t), the relationship between N and P remains the same. 

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),... 

If the electrical potential is low ( 25 mV), a Poisson-Boltzmann equation can be approximated satisfactorily by a linear expression, which is more readily solvable. Various problems have been solved in the literature. These include, for example, planar [24], spherical [25], spheroidal [26], and arbitrary shaped particles [27]. The result for a linear Poisson-Boltzmann equation will not be discussed here. 

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. 

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