Dispersion Gaussian distribution

At the effective stack height, the dispersion of the pollutants are assumed to spread out as a Gaussian distribution. The basic dispersion equation considers... 

If we consider an absorption band showing a normal (Gaussian) distribution [Fig. 17.13(a)], we find [Figs. (b) and (d)] that the first- and third-derivative plots are disperse functions that are unlike the original curve, but they can be used to fix accurately the wavelength of maximum absorption, Amax (point M in the diagram). 

Each oil-dispersant combination shows a unique threshold or onset of dispersion [589]. A statistic analysis showed that the principal factors involved are the oil composition, dispersant formulation, sea surface turbulence, and dispersant quantity [588]. The composition of the oil is very important. The effectiveness of the dispersant formulation correlates strongly with the amount of the saturate components in the oil. The other components of the oil (i.e., asphaltenes, resins, or polar substances and aromatic fractions) show a negative correlation with the dispersant effectiveness. The viscosity of the oil is determined by the composition of the oil. Therefore viscosity and composition are responsible for the effectiveness of a dispersant. The dispersant composition is significant and interacts with the oil composition. Sea turbulence strongly affects dispersant effectiveness. The effectiveness rises with increasing turbulence to a maximal value. The effectiveness for commercial dispersants is a Gaussian distribution around a certain salinity value. 

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. 

When considering the composition inhomogeneity of Markovian copolymers, the finiteness of the chemical size of macromolecules cannot be ignored, because fractional composition distribution W(/ f) in the limit / -> oo turns out to be equal to the Dirac delta function 5(f - X). For macromolecules of finite size f2> 1 the function W(/ f) is the Gaussian distribution whose center and dispersion (Eq. 2) are described by relationships (Eq. 8) and the following one... 

The center and dispersion (Eq. 2) of the Gaussian distribution describing the composition inhomogeneity of a random copolymer comprising macromolecules whose length is 12> 1 have the very simple appearance... 

Problems arise to get informations about the diffusion coeffients Ky and Kz. If equation (3.4) is interpreted as Gaussian distribution, a lot of available dispersion data can be taken into consideration because they are expressed in terms of standard deviations of the concentration distribution. Though there is no theoretical justification the Gaussian plume formula is converted to the K-theory expression by the transformation /11/... 

The results just obtained for < y) are, however, rarely used in applications because (v ) and T are generally not known. The Gaussian dispersion parameters aj and al are, in a sense, generalizations of (Cj) and particle displacement variances o-y and a-] are not calculated by Eq. (8.8). Rather, they are treated as empirical dispersion coefficients the functional forms of which are determined by matching the Gaussian solution to data. In that way, the empirically determined a-y and deviations from stationary, homogeneous conditions which are inherent in the assumed Gaussian distribution. 

It is interesting to note that Figure 6.6 approaches a Gaussian distribution as more tanks are placed in the series. In fact, equation (6.12) has a dimensionless variance, which will prove useful in comparisons of tanks-in-series and plug flow with dispersion (which will be discussed later in this chapter). 

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 ... 

The exact formulation of the inlet and outlet boundary conditions becomes important only if the dispersion number (DjuL) is large (> 0.01). Fortunately, when DjuL is small (< 0.01) and the C-curve approximates to a normal Gaussian distribution, differences in behaviour between open and closed types of boundary condition are not significant. Also, for small dispersion numbers DjuL it has been shown rather surprisingly that we do not need to have ideal pulse injection in order to obtain dispersion coefficients from C-curves. A tracer pulse of any arbitrary shape is introduced at any convenient point upstream and the concentration measured over a period of time at both inlet and outlet of a reaction vessel whose dispersion characteristics are to be determined, as in Fig. 2.18. The means 7in and fout and the variances and out for each of the C-curves are found. 

GRBs emit photons in pulses containing photons with a combination of different wavelengths, whose sources are believed to be ultrarelativistic shocks with Lorentz factor y = (9(100) [28]. Let us consider a wavepacket of photons emitted with a Gaussian distribution in x at the time t = 0. One has to find out how such a pulse would be modified at the observation point at a subsequent time t, because of the propagation through the spacetime foam, as a result of the refractive index. This is similar to the motion of a wavepacket in a conventional dispersive medium. The Gaussian wavepacket may be expressed at t 0 as the real part of... 

In order to evaluate the influence of a dispersion of formal potentials or redox constants (through different tunneling distances) in the CV peak broadening, a Gaussian distribution of both these parameters can be assumed with a mean p and standard deviation a. The current can be obtained as a weighted sum ... 

Comparing the solution of this equation to a Gaussian distribution shows that an estimate of the Pedet number can be obtained from the variance of the N distribution Pe = 2L2/a2. To obtain estimates of Peclet numbers in the bed alone the assumption was made that the mixing at the ends of the bed and the dispersion within the bed are independent in the sense that the variances of the N profiles are additive. Figure 3 shows data both with and without the bed, and the result of subtracting the variance due to end mixing from the total variance. 

---