Distribution functions dispersion

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

Aerosol, monodisperse An aerosol with a size-distribution function described by a geometrical standard deviation less than 1.15. If the deviation is between 1.15 and 1.5, it is classified as a quasi-mono-disperse aerosol. 

From the axial dispersion viewpoint alone there is no doubt that the experimental reduction of dispersion or its conation would be preferable to assuming it negligible. Either of these options require a simple method for the assessment of axial dispersion which does not depend upon absolute molecular weight averages or assumption of distribution functions (5, Such a method will be shown in Section 3 of this report. However, first the problem of copolymer analysis which led to this method as a byproduct win be examined. 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

For both foam systems, the calculated distribution functions of the cell size from the SEM images are presented in Figure 9.20. The nanocomposite foams nicely obeyed a Gaussian distribution. In the case of PLA/ODA foamed at 150 °C under a high pressure of 24 MPa, we can see that the width of the distribution peaks, which indicates the dispersity for cell size, became narrow, accompanied by a finer dispersion of silicate particles. 

Two numerical methods have been used for the solution of the spray equation. In the first method, i.e., the full spray equation method 543 544 the full distribution function / is found approximately by subdividing the domain of coordinates accessible to the droplets, including their physical positions, velocities, sizes, and temperatures, into computational cells and keeping a value of / in each cell. The computational cells are fixed in time as in an Eulerian fluid dynamics calculation, and derivatives off are approximated by taking finite differences of the cell values. This approach suffersfrom two principal drawbacks (a) large numerical diffusion and dispersion... 

One good example of noncovalent functionalization for subsequent hybridization is the use of benzyl alcohol (BA) [118]. n-n interactions between the aromatic ring of BA and the CNT sidewalls result in a good dispersibility in ethanol. Furthermore, BA offers a well-ordered and well-distributed functionalization [119] of hydroxyl groups on the sidewalls of the CNTs that can be used to hybridize the material with a large number of metal oxides using conventional chemical methods [60]. 

At high temperatures above Tb 617 K PMN behaves Hke all other simple perovskites. The dynamics of the system is determined by the soft transverse optical (TO) phonon which exhibits a normal dispersion and is imderdamped at all wave vectors. Below Tb, in addition to the soft mode—which becomes overdamped—a new dielectric dispersion mechanism appears at lower frequencies which can be described by a correlation time distribution function /(t). 

Fig. 10 Relaxation time distribution function/(r) describing the dielectric dispersion in relaxor PMN. The short timescale maximnm describes the glassy-type dynamics, whereas the long timescale part refers to the polar clnster dynamics. The same featnres are obtained in PMN, PLZT, and SEN relaxors...

We shall first discuss the dispersion and backmixing models which adequately characterize flow in tubular and packed-bed systems then we shall consider combined models which are used for more complex situations. In connection with the various applications, the direct use of the age-distribution function for linear kinetics will also be illustrated. 

Db R) Radial dispersion coefficient, general dispersion model in cylindrical coordinates Molecular diffusivity Exit age distribution function, defined in Section I... 

A different approach which also starts from the characteristics of the emissions is able to deal with some of these difficulties. Aerosol properties can be described by means of distribution functions with respect to particle size and chemical composition. The distribution functions change with time and space as a result of various atmospheric processes, and the dynamics of the aerosol can be described mathematically by certain equations which take into account particle growth, coagulation and sedimentation (1, Chap. 10). These equations can be solved if the wind field, particle deposition velocity and rates of gas-to-particle conversion are known, to predict the properties of the aerosol downwind from emission sources. This approach is known as dispersion modeling. 

The fate of the pollutant moving in the aquifer along the streamlines is determined by the advection-dispersion equation, Eq. 25-10 or 25-18. For Pe 1, that is, for locations x dis / if, the concentration cloud can be envisioned to originate from an infinitely short input atx = 0of total mass (a so-called5 input) that by dispersion is turned into a normal distribution function along the x-axis with growing standard deviation. Since the arrival of the main pollution cloud at some distance x is determined... 

The interaction parameters for the water molecules were taken from nonempirical configuration interaction calculations for water dimers (41) that have been shown to give good agreement between experimental radial distribution functions and simulations at low sorbate densities. The potential terms for the water-ferrierite interaction consisted of repulsion, dispersion, and electrostatic terms. The first two of these terms are the components of the 6-12 Lennard-Jones function, and the electrostatic term accounts for long-range contributions and is evaluated by an Ewald summation. The... 

While the benefits cf flotation in oil-water separators are -til known, the high dispersion in the cell (as shown by the distribution function) results in very poor separator.. A good deal of the material bret- s through very early by force of rr.ix. g and thus is... 

Walstra, P. 1969B. Studies on milk fat dispersion. III. The distribution function of globule size in cow s milk and the process of milk fat formation. Neth. Milk Dairy J. 23, 111-123. 

We conclude therefore that, for small values of the dispersion number (D/uL < 0.01), the C-curve for a pulse input of tracer into a pipe is symmetrical and corresponds exactly to equation 2.20 for a Gaussian distribution function ... 

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