Gaussian Concentration Distributions

We have seen that under certain idealized conditions the mean concentration of a species emitted from a point source has a Gaussian distribution. This fact, although strictly true only in the case of stationary, homogeneous turbulence, serves as the basis for a large class of atmospheric diffusion formulas in common use. The collection of Gaussian-based formulas is sufficiently important in practical application that we devote a portion of this chapter to them. The focus of these formulas is the expression for the mean concentration of a species emitted from a continuous, elevated point source, the so-called Gaussian plume equation. 

The transition probability density Q expresses physically the probability that a tracer particle that is at Jt, y, z/ at f will be at x,y,z at t. We showed that under conditions of stationary, homogenous turbulence Q has a Gaussian form. For example, in the case of a 

Up to this point we have considered an infinite domain. For atmospheric applications a boundary at z = 0, the Earth, is present. Because of the barrier to diffusion at z = 0 it is necessary to modify the z dependence of Q to account for this fact. We can separate out the z dependence in (18.83) by writing 

Type of interaction between the diffusing material and the surface 

Total Reflection at z = 0 We assume that the presence of the surface at z = 0 can be accounted for by adding the concentration resulting from a hypothetical source at z = -z to that from the source at z -- z in the region z 0. Then Qr assumes the form 

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Using the Gaussian plume model and the other relations presented, it is possible to compute ground level concentrations C, at any receptor point (Xq, in the region resulting from each of the isolated sources in the emission inventory. Since Equation (2) is linear for zero or linear decay terms, superposition of solutions applies. The concentration distribution is available by computing the values of C, at various receptors and summing over all sources. 

The assumptions made in tlie development of Eq. 12.6.1 are (1) tlie plume spretid lias a Gaussian distribution in both tlie horizontal and vertical planes witli standard deviations of plume concentration distribution in the horizontal and vertical of Oy and respectively (2) tlie emission rate of pollutants Q is uniform (3) total reflection of tlie plume takes place at tlie eartli s surface and (4) tlie plume moves downwind with mean wind speed u. Altliough any consistent set of units may be used, tlie cgs system is preferred. 

Fig. 5 Sedimentation concentration distribution plots for guar gum using SEDFIT. a g (s) vs. s b c(s) vs. s. A Gaussian fit to the data lighter line) is also shown in (a). Rotor speed was 40000 rpm at 20.0 °C, concentration was 0.75 mg/ml in 0.02% NaNs. The guar had been heated at 160 °C for 10 min at a pressure of 3bar. From [49]...

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/... 

We note the similarity of Eqs. (3.27) and (3.15). In particular, if we define o- = 2X t, o-y = 2Kyyt, and = 2K t, the two expressions are identical when y = w = 0. Thus, we see that the mean concentration from an instantaneous point source in an infinite fluid with stationary, homogeneous turbulence has a Gaussian form, with the variances of the concentration distribution related to the variances of the wind velocity fluctuations or to constant eddy difihisivities. 

If it is now assumed that the crosswind and vertical concentration distributions in each disk are Gaussian, i.e.. 

Of the two standard deviations a-y and tr, more is known about cry. First, most of the experiments from which cr, and a- values are inferred involve ground-level measurements. Such measurements provide an adequate indication of a-y, whereas vertical concentration distributions are needed to determine a. Also, the Gaussian expression for vertical con-... 

Intermolecular collisions do not cause large deviations from the ideal gas law at STP for molecules such as N2 or He, which are well above their boiling points, but they do dramatically decrease the average distance molecules travel to a number which is far less than would be predicted from the average molecular speed. Collisions randomize the velocity vector many times in the nominal round trip time, leading to diffusional effects as discussed in Chapter 4. If all of the molecules start at time t = 0 at the position x = 0, the concentration distribution C(x,t) at later times is a Gaussian ... 

Modern HPLC is a routine tool in any analytical laboratory. Standard HPLC system represents a separation output in the form of chromatogram (typical modern chromatogram is shown in Figure 1-6). Each specific analyte in the chromatogram is represented by a peak. In the absence of the strong specific analyte interactions with the stationary phase and at relatively low analyte concentration, peaks are symmetrical and resemble a typical Gaussian (normal distribution) curve. 

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