Arbitrary size

As an idealization of the classified-fines removal operation, assume that two streams are withdrawn from the crystallizer, one corresponding to the product stream and the other a fines removal stream. Such an arrangement is shown schematically in Figure 14. The flow rate of the clear solution in the product stream is designated and the flow rate of the clear solution in the fines removal stream is set as (R — 1) - Furthermore, assume that the device used to separate fines from larger crystals functions so that only crystals below an arbitrary size are in the fines removal stream and that all crystals below size have an equal probabiHty of being removed in the fines removal stream. Under these conditions, the crystal size distribution is characterized by two mean residence times, one for the fines and the other for crystals larger than These quantities are related by the equations... 

In batch classification, the removal of fines (particles less than any arbitrary size) can be correlated by treating as a second-order reaction K = (F/Q)[l/x(x — F)], where K = rate constant, F = fines removed in time 0, and x = original concentration of fines. 

Sometimes a product that does not meet these requirements must be adjusted by adding a specially crushed fraction. No crushing device available will give any arbitrary size distribution, and so crushing with a small reduction ratio and recycle of oversize is practiced when necessary. 

The generalization of a Coulson-Fischer type wave function to the molecular case with an arbitrary size basis set is known as Spin Coupled Valence Bond (SCVB) theory. ... 

Mirrors BBMCA rule (e), and its rotated equivalents, allows groups of particles to be built up to form stable configurations. Such configurations can then be used as mirrors to reflect balls, and thereby to act as signal routers. Figure 6.15, for example, shows the smallest possible fixed configuration consisting of four particles. Since adjacent squares remain uncoupled from one another, mirrors of arbitrary size can be built up from this basic four-particle mirror. 

Schematic representation of reciprocal rate curve for cascade of three arbitrary size CSTR s.

Consider a cylindrical region of arbitrary size and shape within a fluid, as shown in Fig. 4-1. We will apply a momentum balance to a slice of the... 

Given any data matrix A of arbitrary size (as rows x columns) the matrix A can be written or defined using the computation of Singular Value Decomposition [6-8] as... 

With these results, the general equations of Section 15.2.1 can be transformed into equations analogous to those for a constant-density BR. The analogy follows if we consider an element of fluid (of arbitrary size) flowing through a PFR as a closed system, that is, as a batch of fluid. Elapsed time (t) in a BR is equivalent to residence time (t) or space time (r) in a PFR for a constant-density system. For example, substituting into equation 15.2-1 [dWd/A - FAJ(-rA) = 0] for dV from equation 15.2-15 and for d/A from 15.2-13, we obtain, since FAo = cAoq0,... 

The performances of a BR and of a PFR may be compared in various ways, and are similar in many respects, as discussed in Section 15.2.2.1, since an element of fluid, of arbitrary size, acts as a closed system (i.e., a batch) in moving through a PFR. The residence time in a PFR, the same for all elements of fluid, corresponds to the reaction time in a BR, which is also the same for all elements of fluid. Depending on conditions, these quantities, and other performance characteristics, may be the same or different. 

Note that 5 and Sex relate to unit area. The generalization of this argument to randomly placed clusters of arbitrary size and shape is given in Ref. 4. 

Light Scattering Technique. Properties of the light scattered by a large number of droplets can be used to determine droplet size distribution. Dobbins et al. 694 first derived the theoretical formulation of scattering properties of particles of arbitrary sizes and refractive indices in polydispersions of finite optical depth. Based on... 

Several authors(22 30) have contributed to developing the formalism with which the effects of an interface on a dipole inside or near a particle can be treated. In the Rayleigh regime (/ > a), Gersten and Nitzan have made several contributions to the theory of molecular decay rates and energy transfer/22 24) Kerker et alP solved the boundary value problem for a dipole and a spherical particle of arbitrary size, and NcNulty et al.,(26) Ruppin,(27) Chew,(28) and Druger and co-workers(29,30) have used the solution to solve some of the problems of interest. 

A variety algorithms can be used to calculate the loadings and. scores for PCA. A comiEonly employed approach is the singular value decomposition CS T>) algoriiim (Golub and Van Loan, 1983, Chapter 2). A matrix of arbitrary size can be 5sftten as R = USV. The U matrix contains the coordinates of the... 

Both the amplitudes and relative phase of the field components parallel and perpendicular to the scattering plane can be changed upon scattering by a sphere of arbitrary size and composition. Symmetry, however, precludes any mechanism for transforming parallel to perpendicularly polarized light, and... 

At the present time the electromagnetic scattering theory for a sphere, which we have called Mie theory, provides the only practical method for calculating light-scattering properties of finite particles of arbitrary size and refractive index. Clearly, however, many particles of interest are not spheres. It is therefore of considerable importance to know the extent to which Mie theory is applicable to nonspherical particles. To determine this requires generalizing from a large amount of experimental data and calculations. We summarize... 

All the applications of light scattering that have been discussed so far have been restricted to very small particles and fairly small indices of refraction or to fairly small particles and very small indices of refraction. We have finally reached the point at which it seems appropriate to relax all these restrictions and consider the scattering by a particle of arbitrary size and index of refraction. 

A. Nir and A. Acrivos, On the Creeping Motion of Two Arbitrary-sized Touching Spheres in a Linear Shear Field, J. Fluid Mech., 59, 209-223 (1973). 

R. P. Messmer, From finite clusters of atoms to the infinite solid. I. solution of the eigenvalue problem of a simple tight binding model for clusters of arbitrary size, Phys. Rev. B15 1811-1816 (1977). 

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