Mean Concentration from Continuous Sources

Combining Eqs. (4.1) and (4.2), the mean concentration from the continuous source is expressed as... 

Equation (4.22) is the expression for the mean concentration from a continuous point source of strength q at the origin in an infinite fluid when the standard deviations of plume spread are different in the different coordinate directions and when the slender-plume approximation is invoked. 

We now turn to the case of a continuous source. The mean concentration from a continuous point source of strength q at height h above the (totally reflecting) earth is given by (it is conventional to let h denote the source height, and we do so henceforth)... 

TABLE 18.1 Expressions for the Mean Concentration from a Continuous Point Source in an Infinite Fluid in Stationary, Homogeneous Turbulence... 

It is assumed that the contaminant enters the water table or aquifer at a concentration near its solubility limit, although there is no practical means to verify this. This method is more favorable when the release occurred as a single, short-term episode. A long-term release from a continuing source would result in a date that more closely represents the last date upon which the contaminant entered the aquifer at or near its solubility limit. Should the contaminant enter the aquifer below its solubility limit, then a date earlier than the actual event would result. Conversely, should the contaminant enter the aquifer as NAPL for a period of time, a date in which all the NAPL dissolved in groundwater would result. If NAPL was present when measurements were obtained, then the zone of highest concentration would... 

Solutions were obtained in Section III for the mean concentration resulting from an instantaneous release of a quantity Q of material at the origin in an infinite fluid with stationary, homogeneous turbulence and a mean velocity in the x direction. We now wish to consider the case of a continuously emitting source under the same conditions. The source strength is specified as q (g sec )-... 

To illustrate the application of the Monte Carlo method, we consider the problem of simulating the dispersion of material emitted from a continuous line source located between the ground and an inversion layer. A similar case has been considered by Runca et al. (1981). We assume that the mean wind u is constant and that the slender-plume approximation holds. The line source is located at a height h between the ground (z = 0) and an inversion layer (z = Zi). If the ground is perfectly reflecting, the analytical expression for the mean concentration is found by integrating the last entry of Table II over y from -< to -Hoo. The result can be expressed as... 

The other method. of conducting the evaporation of vat liqiiors is by means of iron pans, with the fire and, heated air applied beneath. These—named fishing pans —are of considerable size, and shaped somewhat similar to a boat. They may bo heatad by a fire used solely for the purpose in which case to prevent injury to the pan, the arch of the furnace should be continued for some distance beneath, or the pan may be placed at the end of the black-ash furnace farthest from the fire, just as with the salting pan, except that the heated air is mads to pass under instead of over the pan. The evaporation is rapidly conducted by this method and when the lie becomes concentrated to a certain strength, small crystals of mo nob yd rated carbonate of soda— Na 0, COa, HO—constantly fall to the bottom, and as quickly as thoy fall are raked together to tho end of the pan farthest from the source of heat, and then Scooped or fished out by moans of perforated iron shovels. After being allowed some time to draiD, the salta so obtained are washed with fresh vat liquor, and then removed a reverberatory furnace, and all worked about until sufficiently dry. The deep-red mother liquor is either applied to the production of caustic soda or evaporated, and the residue heated with sawdust as before directed. 

This example can readily be generalized to three dimensions. If we continue to assume that there is a mean flow only in the x direction, then the expression for the mean concentration resulting from an instantaneous point source of unit strength at the origin is... 

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. 

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. 

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