Analytical Properties of the Gaussian Plume Equation

APPENDIX 18.2 ANALYTICAL PROPERTIES OF THE GAUSSIAN PLUME EQUATION 

When material is emitted from a single elevated stack, the resulting ground-level concentration exhibits maxima with respect to both downwind distance and windspeed. Both directly below the stack, where the plume has not yet touched the ground, and far downwind, where the plume has become very dilute, the concentrations approach zero therefore a maximum ground-level concentration occurs at some intermediate distance. Both at very high wind speeds, when the plume is rapidly diluted, and at very low 

To investigate the properties of the maximum ground-level concentration with respect to distance from the source and windspeed, we begin with the Gaussian plume equation evaluated along the plume centerline (y = 0) at the ground (z = 0)  

We want to calculate the location xm of the maximum ground-level concentration for any given wind speed. By differentiating (18.117) with respect to x and setting the resulting equation equal to zero, we find 

Now we consider the effect of wind speed u on the maximum ground-level concentration. The highest concentration at any downwind distance can be determined as a function of u. Differentiating (18.A. 11) with respect to u, we obtain 

TABLE 18.4 Relationship Between Pasquill-GifTord Stability Classes and Temperature Stratification 

Stability Class Ambient Temperature Gradient a7 /az(X/100m) Potential Temperature Gradient 30/3z(X/lOOm) 

We shall deal here with buoyant and forced plumes only. Our interest is in predicting the rise of both buoyant and forced plumes in calm and windy, thermally stratified atmospheres. 

Chracterization of plume rise in terms of the exhaust gas properties and the ambient atmospheric state is a complex problem. The most detailed approach involves solving the coupled mass, momentum, and energy conservation equations. This approach is generally not used in routine calculations because of its complexity. An alternate approach, introduced by Morton et al. (1956), is to consider the integrated form of the conservation equations across a section normal to the plume trajectory (e.g., see Fischer et al., 1979 and Schatzmann, 1979). 

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