Complete Uniform Distribution Function

For the complete uniform distribution function, the external excitation is a vector plane wave propagating along the Z-axis of the global coordinate system and the scattering plane is the XZ-plane. The code computes the following orientation-averaged quantities  

The scattering angles, at which the scattering matrix is evaluated, are uniformly spaced in the interval (6 min,RND, max.RNo)- The elements of the scattering matrix are expressed in terms of the ten average quantities 

The orientation-averaged extinction matrix K) is computed by using (1.125) and (1.126), and note that for macroscopically isotropic and mirror-symmetric media the off-diagonal elements are zero and the diagonal elements are equal to the orientation-averaged extinction cross-section per particle. 

For macroscopically isotropic and mirror-symmetric media, the orientation-averaged scattering and extinction cross-sections (Cscat) = (C scat)i and (C ext) = (C ext)i are calculated by using (1.124) and (1.122), respectively, while for macroscopically isotropic media, the code additionally computes (Cscat)v accordingly to (1.133), and (Cext)v as (C xt)v = Ku). 

The physical correctness of the computed results is tested by using the inequalities (1.138) given by Hovenier and van der Mee [103], The message that the test is not satisfied means that the computed results may be wrong. 

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T-matrix Program 185 3.1.1 Complete Uniform Distribution Function... 

Distribution functions measure the (average) value of a property as a function of an independent variable. A typical example is the radial distribution function g(r) that measures the probability of finding a particle as a function of distance from a typical particle relative to that expected from a completely uniform distribution (i.e. an ideal gas with density N V). The radial distribution function is defined in eq. (14.38). 

The scattering characteristics depend on the type of the orientation distribution function. By convention, the uniform distribution function is called complete if the Euler angles Op, / p and 7p are uniformly distributed in the intervals (0,360°), (0,180°) and (0,360°), respectively. The normalization constant is 47t for axisymmetric particles and for nonaxisymmetric particles. The uniform distribution function is called incomplete if the Euler angles Op, /3p and 7p are uniformly distributed in the intervals (apmin, pmax), (/ pmin,/ pmax), and (7pmin, 7pmax)> respectively. For axisymmetric particles, the orientational average is performed over Op and / p, and the normalization constant is... 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

In a homogeneously catalyzed reaction the determination of the kinetic factors for the process is usually straightforward. In a solution, reactants and the soluble catalysts are uniformly distributed throughout the reaction medium and the reaction rate can be expressed as a function of the concentrations of these substances. A heterogeneously catalyzed process is more complex because the catalyst is not uniformly distributed throughout the reaction medium. Consider a two phase system, either vapor/solid or liquid/solid, with the solid phase the catalyst. In such a system several steps are needed to complete the catalytic cycle ... 

Nearly Uniform Suspension Regime This is the state of suspension at which particle concentration and particle size distribution are roughly uniform throughout the vessel. Any further increase in agitation speed or power does not appreciably enhance the solids distribution in the fluid. A coefficient of variation of the solid concentration of about 0.05 (i.e., a uniformity of 95%) is often considered adequate for most process applications. In practice, a concentration gradient as a function of vertical position will always exist. Complete uniformity is theoretically unattainable and impractical to achieve because a thin clear fluid layer always exists at the air-liquid interface, as the axial lift velocity, and, hence, the particle concentration approaches zero near the liquid surface. Nearly uniform suspension is often the desired process result for operations where a representative sample of solids is required or a uniform concentration of solids must... 

We conclude by mentioning a recent development, although we cannot go into details. In the logistic attractor of Example 11.5.2, the scaling varies from place to place, unlike in the middle-thirds Cantor set, where there is a uniform scaling by everywhere. Thus we cannot completely characterize the logistic attractor by its dimension, or any other single number-—we need some kind of distribution function that tells us how the dimension varies across the attractor. Sets of this type are called multifractals. 

According to the equation log m is a linear function of log m and the constant z can be found from the slope. The constant z is a measure of the heterogeneity of the membranes, becoming zero for a perfectly uniform membrane, as demanded by the approximate form of the Donnan equation for a 1 1 electrolyte. The z of various commercial ion exchange membranes has been measured using salt and acid solutions.111112 The complete distribution function is given by... 

Diffusion provides another good example. A small impermeable sack containing a solute is placed in an infinite container full of solvent. At t = 0 the sack is broken (the constraint is removed) and the solute diffuses through the solvent until a new state of equilibrium is reached in which the solute is uniformly distributed throughout the solution. The property that is measured is the concentration of solute as a function of position and time. Initially the solute is found only in the sack so that the sack geometry defines the initial concentration distribution 0. This experiment is completely described by the diffusion equation the solution of which gives4... 

With Higbie s distribution function all elements at the surface have the same age. Such a situation could be encountered with a quiescent liquid or with completely laminar flow. In that case is simply given by Eq. 6.4.b-4 in which t takes a definite value I, the uniform time of exposure. With Danckwert s age distribution function Eq. 5.4-1 the average rate of absorption per unit surface, is given by ... 

Almost all thermal destruction and incineration problems of today involve multiphase situations. In two-phase flows, accurate size and velocity measurements of particles are important for a broad spectrum of applications. Solid particles are not spherical in shape. Their shape is qnite complex and far from being uniform. Complete characterization of local properties includes such attributes as particle size and number density distribntion, a velocity distribution function related to particles of different size, mean velocity of the gas phase, local gas and particle temperatures, composition of both the gas and particle phases, turbulence properties, and the Uke. In particle-laden flows the influence of turbulence on the particles and vice versa is important. The reUable and precise measurement of any one of the above properties is a nontrivial task. 

Equations 3.62 through 3.74 form a closed set of equations that can be solved for the functions < r,z),CH+ r,z),co2(r,z). Using these functions, the faradaic current density y>(z) can be obtained. This function can be employed to calculate the effectiveness factor of Pt utilization of the pore, defined as the total current produced by the pore, normalized by an ideal current that would be obtained, if reactant and potential distributions were completely uniform, with 4> (z) = , H+... 

NMR in Fig. 1 (top and middle). If the sample is partially ordered, like a partially aligned liquid crystal polymer, the NMR line shape reflects the orientational distribution function of the coupling tensors [28, 29]. Figure 1 (bottom) depicts the special case of a completely aligned sample. Such a sample with macroscopically uniform alignment, for nematic phases usually induced by the magnetic field, gives rise to a simple doublet. 

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