Point source Gaussian diffusion

TABLE 18.2 Point Source Gaussian Diffusion Formulas... 

Fig. 5. Diffusion of pollutants from a point source. PoUutant concentrations have separate Gaussian distributions in both the horizontal (j) and vertical directions. The spread is parameterized by the standard deviations ( O ) which are related to the diffusivity (fQ.

It is known that the vertical distribution of diffusing particles from an elevated point source is a function of the standard deviation of the vertical wind direction at the release point. The standard deviations of the vertical and horizontal wind directions are related to the standard deviations of particle concentrations in the vertical and horizontal directions within the plume itself. This is equivalent to saying that fluctuations in stack top conditions control the distribution of pollutant in the plume. Furthermore, it is known that the plume pollutant distributions follow a familiar Gaussian diffusion equation. 

RTDM (Rough Terrain Diffusion Model) is a sequential Gaussian plume model designed to estimate ground-level concentrations in rough (or flat) terrain in die vicinity of one or more co-located point sources. 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

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. 

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. 

SUMMARY OF GAUSSIAN POINT SOURCE DIFFUSION FORMULAS 923... 

Gaussian plume models These models use a time and spatially independent horizontal wind field, time-dependent point source, and no chemical reactions or loss mechanisms. A bell-shaped downwind distribution is assumed. The answer is a function of source strength, average wind velocity, and two diffusion parameters Easy to use Sanctioned by EPA for developing implementation plans for ambient air-quality standards Much experience with use Assumes wind field constant and uniform Limits use to 1 h and 10 km Not useful for reactive pollutants 114... 

Overcamp, T. J. 1976. A General Gaussian Diffusion-Deposition Model for Elevated Point Sources, Journal of Applied Meteorology, vol. 15, pp. 1167-1171. 

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. 

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. 

Gaussian model A dispersion model based on the concept that atmospheric diffusion is a random mixing process driven by turbulence in the atmosphere. The concentration at any point downwind of a release source is approximated by a Gaussian concentration profile in both the horizontal and vertical dimensions. 

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