Relation between the Actual and Estimated Maximum Ground Level Concentration of Air Pollutant and Its Downwind Locations

In this work we used the Gaussian plume model to calculate the actual 
maximum ground level concentration (MGLC) of air pollutant and its downwind 
location by using different systems of dispersion parameters and for different 
stack heights. An approximate formula for the prediction of downwind position 
that produces the MGLC of a pollutant based on the Gaussian formula was derived 
for different diffusion parameters. The derived formula was used to calculate 
the approximate MGLC. The actual and estimated values are presented in tables. 
The comparison between the actual and estimated values was investigated through 
the calculation of the relative errors. The values of the relative errors 
between the actual and estimated MGLC lie in the range from: 0 to 70.2 and 0 to 
1.6 for Pasquill Gifford system and Klug system respectively. The errors 
between the actual and estimated location of the MGLC lies in the range from: 
0.2 to 227 and 0.7 to 9.4 for Pasquill Gifford system and Klug system 
respectively.

have been collected from different References and presented by [7].
The concentration of a pollutant is a function of a number of variables, such as the emission rate, the location of the receptor from the source and the atmospheric conditions as: wind speed, wind direction and the vertical temperature changes in the local atmosphere.
The major purpose of the present study is to derive an approximate formula for estimating the downwind location of the maximum concentration that is used to find the MGLC of the released pollutant based on the Gaussian model.
The actual MGLC of air pollutant and its location was calculated by using the Gaussian formula for different dispersion parameters The actual and estimated values are presented in tables. The comparison between the actual and estimated values was investigated through the calculation of the relative errors.

Model Description
The Gaussian plume model for estimating the concentration of pollutant released from a continuous point source at some point above the ground is given by [8] where, C = Concentration of pollutant in air (g•m −3 ), Q = Emission rate (g•s −1 ), u = Wind speed at the effective release height (m•s −1 ), x = Downwind distance from the source (m), y = Lateral distance from the plume center line (m), z = Vertical height above ground (m), H = Effective release height above the ground (m), σ y and σ z are the lateral and vertical dispersion parameters (m).
The second exponential term within the brackets is the term due to reflection at the ground surface. The ground level concentration for an elevated release below the centerline of the plume is obtained by setting y and z = 0 in Equation The variables y σ and z σ depend on the downwind distance from the source (x), and the atmospheric stability. The actual MGLC below the plume centerline (C ma ) and its location (x ma ) can be calculated by using Equation (2). The rough estimation of the position of the MGLC (x me ) is calculated [10]: Therefore, the rough estimation of the MGLC can be calculated from the following equation at x = x me : The formula of x me can be derived by solving Equation (3) using different systems of the vertical diffusion parameters [11] as follows:

Pasquill-Gifford System
The values of σ y and σ z as functions of distance for use with his suggested stability categories were suggested by [12]. The modified values of σ y and σ z for use with the original Pasquill stability categories were suggested by [13]. The combination of Pasquill and Gifford parameters is called P-G scheme. In this scheme σ y and σ z are obtained from graphs as a function of downwind distance, x, for each stability class. These curves can be approximated by the following equations [14] ( ) ( ) where the constants a 1 , a 2 , b 1 , b 2 , and b 3 depend on the atmospheric stability and their values are presented in Table 1.
From Equation (6) and Equation (3) we find the formula of the estimated position of the MGLC as:

Power Law Method (Klug System)
Specified a system of diffusion parameters that is applicable for short-term ground-level release over terrain with a low surface roughness [15] [16]. Klug does not exceed source distances of 2 or 3 km. In this range the diffusion parameters can be described by power law functions as: where the coefficients p and q are specified in Table 2. The formula of the estimated location of the MGLC will be as:

Standard Scheme
In this scheme, the crosswind dispersion parameter σ y (x) and the vertical dispersion parameter σ z (x) for various stability classes can be analytically expressed based on Pasquill-Gifford (P-G) curves as follows [17]: where r, s, a, p and q are constants depending on the atmospheric stability. Their values are given in Table 3 (Green et al., 1980). The derived formula of the estimated location of the MGLC is: The actual MGLC (C ma ) and its location (x ma ) were calculated by using Equation (2) by differentiating it with respect to "x" and equate the result with zero, then find the value of maximum downwind distance and substituting in Equation (2) to find the actual MGLC. Equation (7), Equation (10) and Equation (13) The results of this study are presented in Tables 4-6.

Results and Discussion
The actual and estimated maximum ground level concentrations of pollutant and their downwind locations were calculated using "emission rate" Q = 3 g/s and "wind speed" u = 3 m/s for different effective source heights (5 m, 45 m, 100 m, and 250 m) and for different atmospheric stabilities. We see from Equation (7), Equation (10) and Equation (13) that the formula for x me derived by using Pasquill-Gifford system and Power law method is dependent on the stability of air and the effective source height (H), while the derived formula using the Standard scheme was found to be dependent on the atmospheric stability only. The results of this study are presented in tables. The comparisons between the actual and estimated values are investigated through the calculation of the relative error.
Tables 4-6 reveal that for each effective stack height (H) as the atmospheric stability tends to be stable both the actual and estimated position of the MGLC tends to be far from the stack except for stability class B at H = 5 m in Table 5.
Also, the increase in the effective source height gives the same trend. These tables clarify that for each stability class the higher H gives the lower and the more distant concentration.

Conclusions
From Table 4 for Pasquill Gifford system, one can see that higher H gives higher error for the estimation of maximum concentration (C m ) and its location (x m ). Open Journal of Air Pollution  Table 6 shows that the values of x me estimated by Equation (13)