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300             19. Transport and Dispersion of Air Pollutants

       C. Determination of Dispersion Parameters

       1, By Direct Measurements of Wind Fluctuations
         Hay and Pasquill (5) and Cramer (6, 7) have suggested the use of fluctua-
       tion statistics from fixed wind systems to estimate the dispersion taking
       place within pollutant plumes over finite release times. The equation used
       for calculating the variance of the bearings (azimuth) from the point of
       release of the particles, o-p, at a particular downwind location is



       where cr^ is the variance of the azimuth angles of a wind vane over the
       sampling period T calculated from average wind directions averaged over
       averaging periods of duration s; s equals 17/3, where T is the travel time to
       the downwind location; T is equivalent to xlu, where x is the downwind
       distance from the source and u is the transport wind speed. Here ft is
       the ratio of the time scale of the turbulence moving with the air stream
        (Lagrangian) to the time scale of the turbulence at a fixed point (Eulerian).
        Although ft has considerable variation (from about 1 to 9), a reasonable fit
        to field data has been found using a value of 4 for ft.
         A similar equation can be written for vertical spread from an elevated
        source. The standard deviation of the vertical distribution of pollutants at
        the downwind distance x is given by



       where a z is in meters and <r e is the standard deviation of the elevation
        angle, in radians, over the sampling period r calculated from averaged
       elevation angles over averaging periods s. Here, as before, s equals 17/3
       where T is travel time, and ft can be approximated as equal to 4; x in Eq.
        (19-9) is in meters. In application, a values can be calculated over several
        set averaging periods s. The distances to which each or applies are then
       given by x = ftus,
         To calculate plume dispersion directly from fluctuation measurements,
       Draxler (8) used equations in the form




       He analyzed dispersion data from 11 field experiments in order to determine
       the form of the functions f y and / 2, including release height effects. Irwin
        (9) has used simplified expressions for these functions where both / y and
       f z have the form


       where travel time T is xlu; T 0 is 1000 for f y; T 0 is 500 for / 2 for unstable
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