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###   # ISCST3 Tech Guide

Gaussian Plume Air Dispersion Model

## 6.1.5 The Dispersion Parameters

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Point Source Dispersion Parameters.

Equations that approximately fit the Pasquill-Gifford curves (Turner, 1970) are used to calculate Fy and Fz (in meters) for the rural mode. The equations used to calculate Fy are of the form: where: In Equations (1-32) and (1-33) the downwind distance x is in kilometers, and the coefficients c and d are listed in Table 1. The equation used to calculate Fz is of the form: where the downwind distance x is in kilometers and Fz is in meters. The coefficients a and b are given in Table 2. Tables 3 and 4 show the equations used to determine Fy and Fz for the urban option. These expressions were determined by Briggs as reported by Gifford (1976) and represent a best fit to urban vertical diffusion data reported by McElroy and Pooler (1968). While the Briggs functions are assumed to be valid for downwind distances less than 100m, the user is cautioned that concentrations at receptors less than 100m from a source may be suspect.

TABLE 1

Parameters used to calculate pasquill-gifford.

Fy = 465.11628 (x)tan(TH)

TH = 0.017453293 [c - d ln(x)]

 Pasquill Stability Category c d A 24.1670 2.5334 B 18.3330 1.8096 C 12.5000 1.0857 D 8.3330 0.72382 E 6.2500 0.54287 F 4.1667 0.36191

where Fy is in meters and x is in kilometers

TABLE 2

Parameters used to calculate pasquill-gifford

Fz(meters) = axb (x in km)

 Pasquill Stability Category x (km) a b .10 122.800 0.94470 0.10 - 0.15 158.080 1.05420 0.16 - 0.20 170.220 1.09320 0.21 - 0.25 179.520 1.12620 A* 0.26 - 0.30 217.410 1.26440 0.31 - 0.40 258.890 1.40940 0.41 - 0.50 346.750 1.72830 0.51 - 3.11 453.850 2.11660 3.11 ** ** .20 90.673 0.93198 B* 0.21 - 0.40 98.483 0.98332 0.40 109.300 1.09710 C* All 61.141 0.91465 .30 34.459 0.86974 0.31 - 1.00 32.093 0.81066 1.01 - 3.00 32.093 0.64403 D 3.01 - 10.00 33.504 0.60486 10.01 - 30.00 36.650 0.56589 30.00 44.053 0.51179 .10 24.260 0.83660 0.10 - 0.30 23.331 0.81956 0.31 - 1.00 21.628 0.75660 1.01 - 2.00 21.628 0.63077 E 2.01 - 4.00 22.534 0.57154 4.01 - 10.00 24.703 0.50527 10.01 - 20.00 26.970 0.46713 20.01 - 40.00 35.420 0.37615 40.00 47.618 0.29592 .20 15.209 0.81558 0.21 - 0.70 14.457 0.78407 0.71 - 1.00 13.953 0.68465 1.01 - 2.00 13.953 0.63227 2.01 - 3.00 14.823 0.54503 F 3.01 - 7.00 16.187 0.46490 7.01 - 15.00 17.836 0.41507 15.01 - 30.00 22.651 0.32681 30.01 - 60.00 27.074 0.27436 60.00 34.219 0.21716

* If the calculated value of Fz exceed 5000 m, Fz is set to 5000 m.

** Fz is equal to 5000 m.

Table 3

 Pasquill Stability Category Fy(meters)* A 0.32 x (1.0 + 0.0004 x)-1/2 B 0.32 x (1.0 + 0.0004 x)-1/2 C 0.22 x (1.0 + 0.0004 x)-1/2 D 0.16 x (1.0 + 0.0004 x)-1/2 E 0.11 x (1.0 + 0.0004 x)-1/2 F 0.11 x (1.0 + 0.0004 x)-1/2

* Where x is in meters

Table 4

 Pasquill Stability Category Fz(meters)* A 0.24 x (1.0 + 0.001 x)1/2 B 0.24 x (1.0 + 0.001 x)1/2 C 0.20 x D 0.14 x (1.0 + 0.0003 x)-1/2 E 0.08 x (1.0 + 0.0015 x)-1/2 F 0.08 x (1.0 + 0.0015 x)-1/2

* Where x is in meters.

Lateral and Vertical Virtual Distances

The equations in Tables 1 through 4 define the dispersion parameters for an ideal point source. However, volume sources have initial lateral and vertical dimensions. Also, as discussed below, building wake effects can enhance the initial growth of stack plumes. In these cases, lateral (xy) and vertical (xz) virtual distances are added by the ISC models to the actual downwind distance x for the Fy and Fz calculations. The lateral virtual distance in kilometers for the rural mode is given by: where the stability-dependent coefficients p and q are given in Table 1-5 and Fyo is the standard deviation in meters of the lateral concentration distribution at the source. Similarly, the vertical virtual distance in kilometers for the rural mode is given by: where the coefficients a and b are obtained form Table 1-2 and Fzo is the standard deviation in meters of the vertical concentration distribution at the source. It is important to note that the ISC model programs check to ensure that the xz used to calculate Fz at (x + xz) in the rural mode is the xz calculated using the coefficients a and b that correspond to the distance category specified by the quantity (x + xz).

To determine virtual distances for the urban mode, the functions displayed in Tables 3 and 4 are solved for x. The solutions are quadratic formulas for the lateral virtual distances; and for vertical virtual distances the solutions are cubic equations for stability classes A and B, a linear equation for stability class C, and quadratic equations for stability classes D, E, and F. The cubic equations are solved by iteration using Newton's method.

Table 5

FUNC {x_y~=~LEFT ( {{F_{yo}} over p} right ) ^{``1/q}}

 Pasquill Stability Category p q A 209.14 0.890 B 154.46 0.902 C 103.26 0.917 D 68.26 0.919 E 51.06 0.921 F 33.92 0.919

Procedures Used to Account for the Effects of Building Wakes on Effluent Dispersion

The procedures used by the ISC models to account for the effects of the aerodynamic wakes and eddies produced by plant buildings and structures on plume dispersion originally followed the suggestions of Huber (1977) and Snyder (1976). Their suggestions are principally based on the results of wind-tunnel experiments using a model building with a crosswind dimension double that of the building height. The atmospheric turbulence simulated in the wind-tunnel experiments was intermediate between the turbulence intensity associated with the slightly unstable Pasquill C category and the turbulence intensity associated with the neutral D category. Thus, the data reported by Huber and Snyder reflect a specific stability, building shape and building orientation with respect to the mean wind direction. It follows that the ISC wake-effects evaluation procedures may not be strictly applicable to all situations. The ISC models also provide for the revised treatment of building wake effects for certain sources, which uses modified plume rise algorithms, following the suggestions of Schulman and Hanna (1986). This treatment is largely based on the work of Scire and Schulman (1980). When the stack height is less than the building height plus half the lesser of the building height or width, the methods of Schulman and Scire are followed. Otherwise, the methods of Huber and Snyder are followed. In the ISC models, direction-specific building dimensions may be used with either the Huber-Snyder or Schulman-Scire downwash algorithms.

The wake-effects evaluation procedures may be applied by the user to any stack on or adjacent to a building. For regulatory application, a building is considered sufficiently close to a stack to cause wake effects when the distance between the stack and the nearest part of the building is less than or equal to five times the lesser of the height or the projected width of the building. For downwash analyses with direction-specific building dimensions, wake effects are assumed to occur if the stack is within a rectangle composed of two lines perpendicular to the wind direction, one at 5Lb downwind of the building and the other at 2Lb upwind of the building, and by two lines parallel to the wind direction, each at 0.5Lb away from each side of the building, as shown below: Lb is the lesser of the height and projected width of the building for the particular direction sector. For additional guidance on determining whether a more complex building configuration is likely to cause wake effects, the reader is referred to the Guideline for Determination of Good Engineering Practice Stack Height (Technical Support Document for the Stack Height Regulations) - Revised (EPA, 1985). In the following sections, the Huber and Snyder building downwash method is described followed by a description of the Schulman and Scire building downwash method.

Huber and Snyder building downwash procedures

The first step in the wake-effects evaluation procedures used by the ISC model programs is to calculate the gradual plume rise due to momentum alone at a distance of two building heights using Equation (1-23) or Equation (1-25). If the plume height, he, given by the sum of the stack height (with no stack-tip downwash adjustment) and the momentum rise is greater than either 2.5 building heights (2.5 hb) or the sum of the building height and 1.5 times the building width (hb + 1.5 hw), the plume is assumed to be unaffected by the building wake. Otherwise the plume is assumed to be affected by the building wake.

The ISC model programs account for the effects of building wakes by modifying both Fy and Fz for plumes with plume height to building height ratios less than or equal to 1.2 and by modifying only Fz for plumes from stacks with plume height to building height ratios greater than 1.2 (but less than 2.5). The plume height used in the plume height to stack height ratios is the same plume height used to determine if the plume is affected by the building wake. The ISC models define buildings as squat (hw \$ hb) or tall (hw < hb). The ISC models include a general procedure for modifying Fz and Fy at distances greater than or equal to 3hb for squat buildings or 3hw for tall buildings. The air flow in the building cavity region is both highly turbulent and generally recirculating. The ISC models are not appropriate for estimating concentrations within such regions. The ISC assumption that this recirculating cavity region extends to a downwind distance of 3hb for a squat building or 3hw for a tall building is most appropriate for a building whose width is not much greater than its height. The ISC user is cautioned that, for other types of buildings, receptors located at downwind distances of 3hb (squat buildings) or 3hw (tall buildings) may be within the recirculating region.

The modified Fz equation for a squat building is given by: where the building height hb is in meters. For a tall building, Huber (1977) suggests that the width scale hw replace hb in Equation (1-37). The modified Fz equation for a tall building is then given by: where hw is in meters. It is important to note that Fz´ is not permitted to be less than the point source value given in Tables 1-2 or 1-4, a condition that may occur.

The vertical virtual distance, xz, is added to the actual downwind distance x at downwind distances beyond 10hb for squat buildings or beyond 10hw for tall buildings, in order to account for the enhanced initial plume growth caused by the building wake. The virtual distance is calculated from solutions to the equations for rural or urban sigmas provided earlier.

As an example for the rural options, Equations (1-34) and (1-37) can be combined to derive the vertical virtual distance xz for a squat building. First, it follows from Equation (1-37) that the enhanced Fz is equal to 1.2hb at a downwind distance of 10hb in meters or 0.01hb in kilometers. Thus, xz for a squat building is obtained from Equation (1-34) as follows: where the stability-dependent constants a and b are given in Table 2. Similarly, the vertical virtual distance for tall buildings is given by: For the urban option, xz is calculated from solutions to the equations in Table 4 for Fz = 1.2hb or Fz = 1.2 hw for tall or squat buildings, respectively.

For a squat building with a building width to building height ratio (hw/hb) less than or equal to 5, the modified Fy equation is given by: The lateral virtual distance is then calculated for this value of Fy.

For a building that is much wider than it is tall (hw/hb greater than 5), the presently available data are insufficient to provide general equations for Fy. For a stack located toward the center of such a building (i.e., away form either end), only the height scale is considered to be significant. The modified Fy equation for a very squat building is then given by: For hw/hb greater than 5, and a stack located laterally within about 2.5 hb of the end of the building, lateral plume spread is affected by the flow around the end of the building. With end effects, the enhancement in the initial lateral spread is assumed not to exceed that given by Equation (1-42) with hw replaced by 5 hb. The modified Fy equation is given by: The upper and lower bounds of the concentrations that can be expected to occur near a building are determined respectively using Equations (1-43) and (1-44). The user must specify whether Equation (1-43) or Equation (1-44) is to be used in the model calculations. In the absence of user instructions, the ISC models use Equation (1-43) if the building width to building height ratio hw/hb exceeds 5.

Although Equation (1-43) provides the highest concentration estimates for squat buildings with building width to building height ratios (hw/hb) greater than 5, the equation is applicable only to a stack located near the center of the building when the wind direction is perpendicular to the long side of the building (i.e., when the air flow over the portion of the building containing the source is two dimensional). Thus, Equation (1-44) generally is more appropriate then Equation (1-43). It is believed that Equations (1-43) and (1-44) provide reasonable limits on the extent of the lateral enhancement of dispersion and that these equations are adequate until additional data are available to evaluate the flow near very wide buildings.

The modified Fy equation for a tall building is given by: The ISC models print a message and do not calculate concentrations for any source-receptor combination where the source-receptor separation is less than 1 meter, and also for distances less than 3 hb for a squat building or 3 hw for a tall building under building wake effects. It should be noted that, for certain combinations of stability and building height and/or width, the vertical and/or lateral plume dimensions indicated for a point source by the dispersion curves at a downwind distance of ten building heights or widths can exceed the values given by Equation (1-37) or (1-38) and by Equation (1-42) or (1-43). Consequently, the ISC models do not permit the virtual distances xy and xz to be less than zero.

Schulman and Scire refined building downwash procedures

The procedures for treating building wake effects include the use of the Schulman and Scire downwash method. The building wake procedures only use the Schulman and Scire method when the physical stack height is less than hb + 0.5 LB, where hb is the building height and LB is the lesser of the building height or width. In regulatory applications, the maximum projected width is used. The features of the Schulman and Scire method are: (1) reduced plume rise due to initial plume dilution, (2) enhanced vertical plume spread as a linear function of the effective plume height, and (3) specification of building dimensions as a function of wind direction. The reduced plume rise equations were previously described in Section 1.1.4.11.

When the Schulman and Scire method is used, the ISC dispersion models specify a linear decay factor, to be included in the Fz's calculated using Equations (1-37) and (1-38), as follows: where Fz´ is from either Equation (1-37) or (1-38) and A is the linear decay factor determined as follows: where the plume height, he, is the height due to gradual momentum rise at 2 hb used to check for wake effects. The effect of the linear decay factor is illustrated in Figure 1-1. For Schulman-Scire downwash cases, the linear decay term is also used in calculating the vertical virtual distances with Equations (1-40) to (1-41).

When the Schulman and Scire building downwash method is used the ISC models require direction specific building heights and projected widths for the downwash calculations. The ISC models also accept direction specific building dimensions for Huber-Snyder downwash cases. The user inputs the building height and projected widths of the building tier associated with the greatest height of wake effects for each ten degrees of wind direction. These building heights and projected widths are the same as are used for GEP stack height calculations. The user is referred to EPA (1986) for calculating the appropriate building heights and projected widths for each direction. Figure 1-2 shows an example of a two tiered building with different tiers controlling the height that is appropriate for use for different wind directions. For an east or west wind the lower tier defines the appropriate height and width, while for a north or south wind the upper tier defines the appropriate values for height and width.

Procedures Used to Account for Buoyancy-Induced Dispersion.

The method of Pasquill (1976) is used to account for the initial dispersion of plumes caused by turbulent motion of the plume and turbulent entrainment of ambient air. With this method, the effective vertical dispersion Fze is calculated as follows: where Fz is the vertical dispersion due to ambient turbulence and )h is the plume rise due to momentum and/or buoyancy. The lateral plume spread is parameterized using a similar expression: where Fy is the lateral dispersion due to ambient turbulence. It should be noted that )h is the distance-dependent plume rise if the receptor is located between the source and the distance to final rise, and final plume rise if the receptor is located beyond the distance to final rise. Thus, if the user elects to use final plume rise at all receptors the distance-dependent plume rise is used in the calculation of buoyancy-induced dispersion and the final plume rise is used in the concentration equations. It should also be noted that buoyancy-induced dispersion is not used when the Schulman-Scire downwash option is in effect.