page 0 Prepared by Associate Prof. Dr. Mohamad Wijayanuddin Ali Chemical Engineering Department Universiti Teknologi Malaysia

page 1 The applicable conditions are - - Constant mass release rate, Q m = constant, - No wind, = 0, -Steady state,  /  t = 0, and -Constant eddy diffusivity, K j = K * in all directions. For this case, Equation 9 reduces to the form, (10)

page 2 Equation 10 is more tractable by defining a radius as r² = x² + y² + z². Transforming Equation 10 in terms of r yields (11) For a continuous, steady state release, the concentration flux at any point, r, from the origin must equal the release rate, Q m (with units of mass/time). This is represented mathematically by the following flux boundary condition. (12) The remaining boundary condition is (13)

page 3 Equation 12 is separated and integrated between any point r and r = . (14) Solving Equation 14 for yields, (15) It is easy to verify by substitution that Equation 15 is also a solution to Equation 11 and thus a solution to this case. Equation 15 is transformed to rectangular coordinates to yield, (16)

page 4 The applicable conditions are - -Puff release, instantaneous release of a fixed mass of material, Q m * (with units of mass), -No wind, = 0, and -Constant eddy diffusivity, K j = K *, in all directions. Equation 9 reduces, for this case, to (17)

page 5 The initial condition required to solve Equation 17 is (18) The solution to Equation 17 in spherical coordinates is (19) and in rectangular coordinates is (20)

page 6 The applicable conditions are - Constant mass release rate, Q m = constant, - No wind, = 0, and - Constant eddy diffusivity, K j = K * in all directions. For this case, Equation 9 reduces to Equation 17 with initial condition, Equation 18, and boundary condition, Equation 13. The solution is found by integrating the instantaneous solution, Equation 19 or 20 with respect to time. The result in spherical coordinates is (21)

page 7 and in rectangular coordinates is (22) As t , Equations 21 and 22 reduce to the corresponding steady state solutions, Equations 15 and 16.

page 8 This case is shown in Figure 7. The applicable conditions are - Continuous release, Q m = constant, - Wind blowing in x direction only, = = u = constant, and - Constant eddy diffusivity, K j = K * in all directions. For this case, Equation 9 reduces to (23)

page 9 Equation 23 is solved together with boundary conditions, Equation12 and 13. The solution for the average concentration at any point is (24) If a slender plume is assumed (the plume is long and slender and is not far removed from the x-axis), (25) and, using, Equation 24 is simplified to (26) Along the centreline of this plume, y = z = 0 and (27)

page 10 This is the same as Case 2, but with eddy diffusivity a function of direction. The applicable conditions are - - Puff release, Q m * = constant, - No wind, = 0, and - Each coordinate direction has a different, but constant eddy diffusivity, K x, K y and K z. Equation 9 reduces to the following equation for this case. (28) The solution is (29)

page 11 This is the same as Case 4, but with eddy diffusivity a function of direction. The applicable conditions are - - Puff release, Qm* Qm* = constant, - Steady state,  /  t = o, - Wind blowing in x direction only, = = u = constant, - Each coordinate direction has a different, but constant eddy diffusivity, K x, Ky Ky and K z, and - Slender plume approximation, Equation 25.

page 12 Equation 9 reduces to the following equation for this case. (30) The solution is (31) Along the centreline of this plume, y = z = 0 and the average concentration is given by (32)

page 13 This is the same as Case 5, but with wind. Figure 8 shows the geometry. The applicable conditions are - - Puff release, Qm* Qm* = constant, - Wind blowing in x direction only, = = u = constant, and - Each coordinate direction has a different, but constant eddy diffusivity, K x, Ky Ky and K z,. The solution to this problem is found by a simple transformation of coordinates. The solution to Case 5 represents a puff fixed around the release point.

page 14 If the puff moves with the wind along the x-axis, the solution to this case is found by replacing the existing coordinate x by a new coordinate system, x - ut, that moves with the wind velocity. The variable t is the time since the release of the puff, and u is the wind velocity. The solution is simply Equation 29, transformed into this new coordinate system. (33)

page 15 This is the same as Case 5, but with the source on the ground. The ground represents an impervious boundary. As a result, the concentration is twice the concentration as for Case 5. The solution is 2 times Equation 29. (34)

page 16 This is the same as Case 6, but with the release source on the ground, as shown in Figure 9. The ground represents an impervious boundary. As a result, the concentration is twice the concentration as for Case 6. The solution is 2 times Equation 31. (35)

page 17 Figure 9 Steady-state plume with source at ground level. The concentration is twice the concentration of a plume without the ground.

page 18 For this case the ground acts as an impervious boundary at a distance H from the source. The solution is (36)

page 19 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.

page 20 with similar relations given for  y and  z. The dispersion coefficients,  x,  y, and z z represent the standard deviations of the concentration in the downwind, crosswind, and vertical (x,y,z) directions, respectively. 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.

page 21 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.

page 22 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

page 23 Figure 10 Horizontal dispersion coefficient for Pasquill-Gifford plume model. The dispersion coefficient is a function of distance downwind and the atmospheric stability class.

page 24 Figure 11 Vertical dispersion coefficient for Pasquill-Gifford plume model. The dispersion coefficient is a function of distance downwind and the atmospheric stability class.

page 25 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.

page 26 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.

page 27 Table 3 Equations and data for Pasquill-Gifford Dispersion Coefficients

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page 30 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)

page 31 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)

page 32 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.

page 33 For a specified concentration, *, the isopleths at ground level are determined by dividing the equation for the centreline concentration, Equation 40, by the equation for the general ground level concentration, Equation 39. This equation is solved directly for y. (45) The procedure is 1.Specify *, u, and t. 2.Determine the concentrations, (x,0,0,t), along the x-axis using Equation40. Define the boundary of the cloud along the x-axis. 3.Set (x,y,0,t) = * in Equation 45 and determine the values of y at each centreline point determined in step 2. The procedure is repeated for each value of t required.

page 34 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)

page 35 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.