Number density, local

Single-component PDA equipment is similar to LDA, but two detectors (not one) are installed with different detection angles. By means of simultaneous processing of signals supplied by the two detectors, information on the velocity and on the size of the scattering objects can be acquired. Therefore, velocity distribution, size distribution, and number density (local concentration)... 

Equation 2.101 can be used now with the understanding that P (z) is the pressure given by the EoS model for given temperature and number densities (local density and composition). Alternatively, Equation 2.102 can be used, but, now, the EoS model will be used to provide with the local Helmholtz free-energy density To(z) for given temperature, local density, and composition. 

For homogeneous systems, the average number density is n = (N) / V=v Let us define a local number... 

GL 25] ]R 8] [P 27] After irradiation for only 5 s, high conversions (> < obtained [21]. This is explained as being due to both the high absorption coefficient of the photosensitizer Bengal Rose and the high local number density of photons within the micro channel as a result of the large specific surface area. 

The atomic PDF is related to the probability to find a spherical shell around a generic atom (scattering center) in the material - it is defined as G(r) = Anp[p r)-p(, where p r) and po are, respectively, the local and average atomic number densities and r the radial distance. G(r) is the Fourier transform of the total structure factor Sid). ... 

Using Eqs. (36)-(39), one obtains the following expressions for the local Sherwood number and local limiting current density on the speherical surface... 

At equilibrium, the chemical potential for a given molecular species is constant throughout the system. The two terms on the right-hand side of (11.4) can vary in space, however, so as to add up to a constant. In an inhomogeneous system, the number density and excess chemical potential adjust so as to yield the same constant chemical potential. Due to the local nature of the excess chemical potential, it is reasonable to define an excess chemical potential at a single point in space and/or for a single molecular conformation [29]. That excess chemical potential then determines... 

For particles of equal mass, we thus have esps = mn with n the local number density of particles. From the KTGF, the time evolution of the granular temperature is given by... 

In this section, we will only discuss the basic principles of kinetic theory, where for detailed derivations we refer to the classic textbook by Chapman and Cowling (1970), and a more recent book by Liboff (1998). Of central importance in the kinetic theory is the single particle distribution function /s(r, v), which can be defined as the number density of the solid particles in the 6D coordinate and velocity space. That is, /s(r, v, t) dv dr is the average number of particles to be found in a 6D volume dv dr around r, v. This means that the local density and velocity of the solid phase in the continuous description are given by... 

In the case of droplets and bubbles, particle size and number density may respond to variations in shear or energy dissipation rate. Such variations are abundantly present in turbulent-stirred vessels. In fact, the explicit role of the revolving impeller is to produce small bubbles or drops, while in substantial parts of the vessel bubble or drop size may increase again due to locally lower turbulence levels. Particle size distributions and their spatial variations are therefore commonplace and unavoidable in industrial mixing equipment. This seriously limits the applicability of common Euler-Euler models exploiting just a single value for particle size. A way out is to adopt a multifluid or multiphase approach in which various particle size classes are distinguished, with mutual transition paths due to particle break-up and coalescence. Such models will be discussed further on. 

Thus, the source terms for each environment S(c) and Sk ((/)) will be closed. Of particular interest are the local nucleation rates /(c ). As discussed in Wang and Fox (2004), due to poor micromixing the local nucleation rates can be much larger than those predicted by the average concentrations /((c)). This results in a rapid increase in the local particle number density mo due to the creation of a very large number of nuclei. As discussed below, this will have significant consequences on the local rate of aggregation. 

In our Galaxy, as well as in others, populations of stars aggregate from place to place in the so-called globular clusters. There the local number density of stars is far bigger than in the rest of the Galaxy except perhaps in its core. 

If an atomic transition is optically pumped by a beam of laser radiation having the appropriate frequency, the population in the upper state can be considerably enhanced along the path of the beam. This causes an intensification of the spontaneous emission from this state, which contains information about the conditions within the pumped region, since the exponential decay time for the intensified emission depends upon both the electron number density and the electron temperature. The latter can be obtained from the intensity ratio of the fluorescence excited from two different lower levels, if local thermal equilibrium is assumed. This method has been dis-... 

A single-determinant wave-function of closed shell molecular systems is invariant against any unitary transformation of the molecular orbitals apart from a phase factor. The transformation can be chosen in order to obtain LMOs. Starting from CMOs a number of localization procedures have been proposed to get LMOs the most commonly used methods are as given by the authors of (Edmiston et ah, 1963) and (Boys, 1966), while the procedures provided by (Pipek etal, 1989) and (Saebo etal., 1993) are also of interest. It could be stated that all the methods yield comparable results. Each LMO densities are found to be relatively concentrated in some spatial region. They are, furthermore, expected to be determined mainly by that part of the molecule which occupies that given region and its nearby environment rather than by the whole system. 

Here is the spectral flux and n is the total hydrogen number density. Although we have strictly assumed an exponential density profile p(r), if the photosphere is sharp only the local value of AR(r), as defined by equation (1), is important. 

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