Point source diffusion formulas

V. Point Source Diffusion Formulas Based on a Gaussian Distribution. 233... 

The presumption of a Gaussian distribution for the mean concentration from a point source, although demonstrated only in the case of stationary, homogeneous turbulence, has been made widely and, in fact, is the basis for many of the atmospheric diffusion formulas in common use. Based on the developments of Section IV, we present in this section the Gaussian point source diffusion formulas that have been used for practical calculations. 

Summary of Gaussian Point Source Diffusion Formulas... 

The various point source diffusion formulas we have derived are summarized in Table 18.2. 

SUMMARY OF GAUSSIAN POINT SOURCE DIFFUSION FORMULAS 923... 

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. 

Dispersion Models Based on Inert Pollutants. Atmospheric spreading of inert gaseous contaminant that is not absorbed at the ground has been described by the various Gaussian plume formulas. Many of the equations for concentration estimates originated with the work of Sutton (3). Subsequent applications of the formulas for point and line sources state the Gaussian plume as an assumption, but it has been rigorously shown to be an approximate solution to the transport equation with a constant diffusion coefficient and with certain boundary conditions (4). These restrictive conditions occur only for certain special situations in the atmosphere thus, these approximate solutions must be applied carefully. 

In a tristimulus colorimeter, three or four filters duplicate the response of the standard observer. The filters, which correspond to the three primary colours in the spectrum (red, green and blue), can be combined to match most colours. The more sophisticated instruments carry a fourth filter to simulate the blue part of the CIE [[X bar]] function shown in Fig. 5.1. The other essential parts of a tristimulus colorimeter are a white light source, an array of photometers and, nowadays, a computer or an interface to one, as shown in Fig. 5.2a. The computer can collect responses as well as carry out data transformations between CIE and other colour scale systems or between different standard white light sources or white diffusers. Hunter and Harold (1987) give a good summary of the transformation formulae employed. Data from a colorimeter is given as a three-point output, commonly CIELAB, HunterLab or 7, x, y. 

Up to this point in this chapter we have developed the common theories of turbulent diffusion in a purely formal manner. We have done this so that the relationship of the approximate models for turbulent diffusion, such as the K theory and the Gaussian formulas, to the basic underlying theory is clearly evident. When such relationships are clear, the limitations inherent in each model can be appreciated. We have in a few cases applied the models obtained to the prediction of the mean concentration resulting from an instantaneous or continuous source in idealized stationary, homogeneous turbulence. In Section 18.7.1 we explore further the physical processes responsible for the dispersion of a puff or plume of material. Section 18.7.2 can be omitted on a first reading of this chapter that section goes more deeply into the statistical properties of atmospheric dispersion, such as the variances a (r), which are needed in the actual use of the Gaussian dispersion formulas. 

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