Cases 1 through 10 above all depend on the specification of a value for the eddy diffusivity, K j. In general, K j changes with position, time, wind velocity, and prevailing weather conditions. While the eddy diffusivity approach is useful theoretically, it is not convenient experimentally and does not provide a useful framework for correlation. Sutton solved this difficulty by proposing the following definition for a dispersion coefficient. (37) with similar relations given for  y and  z. The dispersion coefficients,  x,  y, and  z represent the standard deviations of the concentration in the downwind, crosswind and vertical (x, y, z) direction, respectively.

A plume dispersing in a normal distribution along two axes - distance crosswind and distance vertically

Values for the dispersion coefficients are much easier to obtain experimentally than eddy diffusivities. The dispersion coefficients are a function of atmospheric conditions and the distance downwind from the release. The atmospheric conditions are classified according to 6 different stability classes shown in Table 2. The stability classes depend on wind speed and quantity of sunlight. During the day, increased wind speed results in greater atmospheric stability, while at night the reverse is true. This is due to a change in vertical temperature profiles from day to night.

The dispersion coefficients, y y and z z for a continuous source were developed by Gifford and given in Figures 10 and 11, with the corresponding correlation given in Table 3. Values for x x are not provided since it is reasonable to assume x x =  y. The dispersion coefficients y y and z z for a puff release are given in Figures 12 and 13. The puff dispersion coefficients are based on limited data (shown in Table 3) and should not be considered precise. The equations for Cases 1 through 10 were rederived by Pasquill using relations of the form of Equation 37. These equations, along with the correlation for the dispersion coefficients are known as the Pasquill- Gifford model.

Table 2 Atmospheric Stability Classes for Use with the Pasquill-Gifford Dispersion Model Stability class for puff model : A,B : unstable C,D : neutral E,F : stable

Figure 10 Horizontal dispersion coefficient for Pasquill-Gifford plume model. The dispersion coefficient is a function of distance downwind and the atmospheric stability class.

Figure 11 Vertical dispersion coefficient for Pasquill-Gifford plume model. The dispersion coefficient is a function of distance downwind and the atmospheric stability class.

Figure 12 Horizontal dispersion coefficient for puff model. This data is based only on the data points shown and should not be considered reliable at other distances.

Figure 13 Vertical dispersion coefficient for puff model. This data is based only on the data points shown and should not be considered reliable at other distances.

Table 3 Equations and data for Pasquill-Gifford Dispersion Coefficients

This case is identical to Case 7. The solution has a form similar to Equation 33. (38) The ground level concentration is given at z = 0. (39)

The ground level concentration along the x-axis is given at y = z= 0. (40) The centre of the cloud is found at coordinates (ut,0,0). The concentration at the centre of this moving cloud is given by (41) The total integrated dose, D tid received by an individual standing at fixed coordinates (x,y,z) is the time integral of the concentration. (42)

The total integrated dose at ground level is found by integrating Equation 39 according to Equation 42. The result is - (43) The total integrated dose along the x-axis on the ground is (44) Frequently the cloud boundary defined by a fixed concentration is required. The line connecting points of equal concentration around the cloud boundary is called an isopleth.

This case is identical to Case 9. The solution has a form similar to Equation 35. (46) The ground level concentration is given at z = 0. (47)

The concentration along the centreline of the plume directly downwind is given at y = z= 0. (48) The isopleths are found using a procedure identical to the isopleth procedure used for Case 1. For continuous ground level releases the maximum concentration occurs at the release point.

This case is identical to Case 10. The solution has a form similar to Equation 36. (49)

The ground level concentration is found by setting z = 0. (50) The ground centreline concentrations are found by setting y = z= 0. (51)

The maximum ground level concentration along the x-axis, max, is found using. (52) The distance downwind at which the maximum ground level concentration occurs is found from (53) The procedure for finding the maximum concentration and the downwind distance is to use Equation 53 to determine the distance followed by Equation 52 to determine the maximum concentration.

For this case the centre of the puff is found at x = ut. The average concentration is given by (54)

The time dependence is achieved through the dispersion coefficients, since their values change as the puff moves downwind from the release point. If wind is absent (u = 0), Equation 54 will not predict the correct result. At ground level, z = 0, and the concentration is computed using (55)

The concentration along the ground at the centreline is given at any y = z = 0, (56) The total integrated dose at ground level is found by application of Equation 42 to Equation 55. The result is (57)

For this case, the result is obtained using a transformation of coordinates similar to the transformation used for Case 7. The result is (58) where t is the time since the release of the puff.

The plume model describes the steady state behaviour of material ejected from a continuous source. The puff model is not steady-state and follows the cloud of material as it moves with the wind. As a result, only the puff model is capable of providing a time dependence for the release. The puff model is also used for continuous releases by representing the release as a succession of puffs. For leaks from pipes and vessels, if t p is the time to form one puff, then the number of puffs formed, n, is given by (59)

where t is the duration of the spill. The time to form one puff, t p, is determined by defining an effective leak height, H eff. Then, (60) where u is the wind speed. Empirical results show that the best H eff to use is (61) For a continuous leak, (62)

and for instantaneous release divided into a number of smaller puffs, (63) where (Q m * ) total is the release amount. This approach works for liquid spills, but not for vapor releases. For vapor releases a single puff is suggested. The puff model is also used to represent changes in wind speed and direction.

On an overcast day, a stack with an effective height of 60 meters is releasing sulfur dioxide at the rate of 80 grams per second. The wind speed is 6 meters per second. The stack is located in rural area.Determine: a.a.The mean concentration of SO 2 on the ground 500 meters downwind. b.b.The mean concentration on the ground 500 meters downwind and 50 meters crosswind. c.c.The location and value of the maximum mean concentration on ground level directly downwind.

a.This is a continuous release. The ground concentration directly downwind is given by Equation 51. (51) From Table 2, the stability class is D. the dispersion coefficients are obtained from Figures 10 and 11. The resulting values are  y = 39 meters and  z = 22.7 meters. Substituting into Equation 51

b.The mean concentration 50 meters crosswind is found using Equation 50 and setting y = 50. The results from part a are applied directly,

c.The location of the maximum concentration is found from Equation 53, From Figure 11, the dispersion coefficient has this value at x = 1200 m. At x = 1200 m, from Figure 10,  y = 88 m. The maximum concentration is determined using Equation 52,