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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.

The atmospheric diffusion models that are widely used for regulatory purposes have been reviewed by [

The GPM has been expressed in terms of the horizontal and vertical diffusion parameters, σ y ( x ) and σ z ( x ) . Different formulae for σ y ( x ) and σ z ( x ) have been collected from different References and presented by [

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 σ y ( x ) and σ z ( x ) , and different stack heights. Different formulae of σ z ( x ) are used to derive an approximate formula for the downwind location of the maximum concentration that is used to predict 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 Gaussian plume model for estimating the concentration of pollutant released from a continuous point source at some point above the ground is given by [

C ( x , y , z , H ) = Q 2 π u σ y σ z e − y 2 / 2 σ y 2 [ e − ( z − H ) 2 / 2 σ z 2 + e − ( z + H ) 2 / 2 σ z 2 ] (1)

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 (1):

C ( x , 0 , 0 , H ) = Q π u σ y σ z exp [ − H 2 2 σ z 2 ] (2)

The variables σ y and σ z 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 [

σ z = H 2 (3)

Therefore, the rough estimation of the MGLC can be calculated from the following equation at x = x_{me}:

C m e ( x , 0 , 0 , H ) = 2 Q π e u H 2 ( σ z σ y ) (4)

The formula of x_{me} can be derived by solving Equation (3) using different systems of the vertical diffusion parameters [

The values of σ_{y} and σ_{z} as functions of distance for use with his suggested stability categories were suggested by [_{y} and σ_{z} for use with the original Pasquill stability categories were suggested by [_{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 [

σ y ( x ) = ( a 1 ln x + a 2 ) x (5)

σ z ( x ) = 1 2.15 exp ( b 1 + b 2 ln x + b 3 ln 2 x ) (6)

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

From Equation (6) and Equation (3) we find the formula of the estimated position of the MGLC as:

x m e = exp ( − b 2 ± b 2 2 − 4 b 3 [ b 1 − ln ( 2.15 H 2 ) ] 2 b 3 ) (7)

Coefficient | Stability | Categories | ||||
---|---|---|---|---|---|---|

A | B | C | D | E | F | |

a_{1} | −0.0234 | −0.0147 | −0.0117 | −0.0059 | −0.0059 | −0.0029 |

a_{2 } | 0.3500 | 0.2480 | 0.1750 | 0.1080 | 0.0880 | 0.0540 |

b_{1 } | 0.8800 | −0.9850 | −1.1860 | −1.3500 | −2.8800 | −3.8000 |

b_{2 } | 0.1520 | 0.8200 | 0.8500 | 0.7930 | 1.2550 | 1.4190 |

b_{3} | 0.1475 | 0.0168 | 0.0045 | 0.0420 | −0.0420 | −0.0550 |

where, the positive square root is selected to estimate x_{me} [

Specified a system of diffusion parameters that is applicable for short-term ground-level release over terrain with a low surface roughness [

σ y ( x ) = p y x q y (8)

σ z ( x ) = p z x q z (9)

where the coefficients p and q are specified in

x m e = ( H 2 P z ) 1 q z (10)

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 [

σ y = r x ( 1 + x / a ) p (11)

σ z = s x ( 1 + x / a ) q (12)

where r, s, a, p and q are constants depending on the atmospheric stability. Their values are given in

x m e = a q − 1 (13)

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)

Coefficient | Stability | Categories | ||||
---|---|---|---|---|---|---|

A | B | C | D | E | F | |

p_{y} | 0.4690 | 0.3060 | 0.2300 | 0.2190 | 0.2370 | 0.2730 |

q_{y } | 0.9030 | 0.8850 | 0.8550 | 0.7640 | 0.6910 | 0.5940 |

p_{z } | 0.0170 | 0.0720 | 0.0760 | 0.1400 | 0.2170 | 0.2620 |

q_{z } | 1.3800 | 1.0210 | 0.8790 | 0.7270 | 0.6100 | 0.5000 |

Atmospheric | Stability | Categories | |||
---|---|---|---|---|---|

Stability | r (m/km) | s (m/km) | a (km) | p | q |

A | 250 | 102 | 0.927 | 0.189 | −0.918 |

B_{ } | 202 | 96.2 | 0.37 | 0.162 | −0.101 |

C_{ } | 134 | 72.2 | 0.283 | 0.134 | 0.102 |

D_{ } | 78.7 | 47.5 | 0.707 | 0.135 | 0.465 |

E | 56.6 | 33.5 | 1.07 | 0.137 | 0.624 |

F | 37 | 22 | 1.17 | 0.134 | 0.70 |

are used to estimate the location of the MGLC (x_{me}). The values of x_{me} are used to estimate the values of the MGLC by using Equation (4). The maximum concentrations of pollutant and their downwind locations were calculated using Q = 3 g/s and u = 3 m/s for different effective heights (5 m, 45 m, 100 m, and 250 m) and different atmospheric stabilities. The comparison between the actual and estimated values is investigated through the calculation of the relative error:

Relative error = | actual value − estimated value actual value | (14)

The results of this study are presented in Tables 4-6.

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

From _{m}) and its location (x_{m}).

Stability class | C_{ma } g/m^{3 } | x_{ma } m | C_{me } g/m^{3}_{ } | x_{me } m | Relative C_{m } | Error% x_{m}_{ } | |
---|---|---|---|---|---|---|---|

H = 5 m | |||||||

A | 1.101E−2 | 9.9 | 1.099E−2 | 10.2 | 0.1 | 3.6 | |

B | 5.415E−3 | 31.3 | 5.415E−3 | 31.0 | 0.0 | 0.2 | |

C | 6.172E−3 | 40.4 | 6.171E−3 | 40.8 | 0.0 | 1.0 | |

D | 5.926E−3 | 64.6 | 5.910E−3 | 67.4 | 0.3 | 4.4 | |

E | 5.339E−3 | 101.1 | 5.338E−3 | 102.3 | 0.0 | 1.2 | |

F | 5.036E−3 | 164.5 | 5.032E−3 | 168.3 | 0.1 | 2.3 | |

H = 45 m | |||||||

A | 2.086E−4 | 77.2 | 2.094E−4 | 71.9 | 1.2 | 6.9 | |

B | 7.575E−5 | 304.1 | 7.565E−5 | 296.1 | 0.1 | 2.6 | |

C | 7.521E−5 | 478.8 | 7.521E−5 | 476.0 | 0.0 | 0.6 | |

D | 5.528E−5 | 959.9 | 5.520E−5 | 991.2 | 0.1 | 3.3 | |

E | 4.385E−5 | 1721.6 | 4.308E−5 | 1977.2 | 1.7 | 14.8 | |

F | 3.050E−5 | 3226 | 3.861E−5 | 4330.0 | 6.2 | 34.2 | |

H = 100 m | |||||||

A | 5.528E−5 | 132.6 | 5.684E−5 | 122.8 | 2.5 | 7.4 | |

B | 1.682E−5 | 667 | 1.678E−5 | 645.7 | 0.2 | 3.2 | |

C | 1.563E−5 | 1159.2 | 1.563E−5 | 1145.2 | 0.0 | 1.2 | |

D | 1.033E−5 | 2537.1 | 1.032E−5 | 2607.8 | 0.1 | 2.8 | |

E | 5.026E−6 | 6368.6 | 5.752E−6 | 8285.1 | 4.5 | 30.1 | |

F | 2.743E−6 | 14,699.4 | 2.013E−6 | 34,757.0 | 26.6 | 136.5 | |

H = 250 m | |||||||

A | 1.427E−5 | 231.7 | 1.382E−5 | 213.7 | 3.2 | 7.8 | |

B | 3.077E−6 | 1605.2 | 3.067E−6 | 1542.8 | 0.3 | 3.9 | |

C | 2.638E−6 | 3171 | 2.636E−6 | 3107.6 | 0.1 | 2.0 | |

D | 1.531E−6 | 7697.4 | 1.530E−6 | 7864.8 | 0.1 | 2.2 | |

E | 4.491E−7 | 41,722.4 | 3.828E−7 | 81,308.5 | 14.8 | 94.9 | |

F | 3.258E−8 | 122,425.9 | 5.544E−8 | 400,312.2 | 70.2 | 227.0 | |

Stability class | C_{ma } g/m^{3 } | x_{ma } m | C_{me } g/m^{3}_{ } | x_{me } m | Relative C_{m } | Error% x_{m } | |
---|---|---|---|---|---|---|---|

H = 5 m | |||||||

A | 2.184E−3 | 51.2 | 2.150E−3 | 47.8 | 1.6 | 6.6 | |

B | 3.713E−3 | 48.9 | 3.705E−3 | 45.3 | 0.2 | 3.3 | |

C | 3.440E−3 | 79.5 | 3.439E−3 | 78.9 | 0.0 | 0.7 |
---|---|---|---|---|---|---|

D | 5.085E−3 | 83.4 | 5.084E−3 | 84.9 | 0.0 | 1.8 |

E | 5.937E−3 | 92 | 5.924E−3 | 97.0 | 0.2 | 5.5 |

F | 5.539E−3 | 166.5 | 5.515E−3 | 182.1 | 0.4 | 9.4 |

H = 45 m | ||||||

A | 5.763E−5 | 251.8 | 5.672E−5 | 235.1 | 1.6 | 6.6 |

B | 6.143E−5 | 403.5 | 6.129E−5 | 389.9 | 0.2 | 3.4 |

C | 4.509E−5 | 968.5 | 4.509E−5 | 962.2 | 0.0 | 0.8 |

D | 5.614E−5 | 1714 | 5.612E−5 | 1743.8 | 0.0 | 1.7 |

E | 5.475E−5 | 3375.5 | 5.463E−5 | 3558.0 | 0.2 | 5.4 |

F | 4.524E−5 | 13,482.5 | 4.505E−5 | 14,750.0 | 0.4 | 9.4 |

H = 100 m | ||||||

A | 1.538E−5 | 449.1 | 1.514E−5 | 419.3 | 1.6 | 6.6 |

B | 1.383E−5 | 881.6 | 1.380E−5 | 852.3 | 0.2 | 3.3 |

C | 9.332E−6 | 2402.8 | 9.331E−6 | 2384.1 | 0.0 | 0.8 |

D | 1.092E−5 | 5140.6 | 1.091E−5 | 5230.1 | 0.0 | 1.7 |

E | 9.971E−6 | 12,497.7 | 9.950E−6 | 13,137.9 | 0.2 | 5.4 |

F | 7.884E−6 | 66,581 | 7.850E−6 | 75,839.6 | 0.4 | 9.4 |

H = 250 m | ||||||

A | 3.377E−6 | 872 | 3.324E−6 | 814.4 | 1.5 | 6.6 |

B | 2.501E−6 | 2162.9 | 2.495E−6 | 2091.1 | 0.2 | 3.3 |

C | 1.531E−6 | 6814.6 | 1.531E−6 | 6761.6 | 0.0 | 0.8 |

D | 1.667E−6 | 18,129.1 | 1.666E−6 | 18,445.2 | 0.0 | 1.7 |

E | 1.413E−6 | 56,129.5 | 1.410E−6 | 59,166.3 | 0.2 | 5.4 |

F | 1.062E−6 | 416,131 | 1.057E−6 | 455,247.4 | 0.4 | 9.4 |

Stability class | C_{ma } g/m^{3 } | x_{ma } m | x_{me } m | Relative error% x_{m}_{ } |
---|---|---|---|---|

H = 5 m | ||||

A | 4.11E−3 | 33 | −317.7 | 10.6 |

B | 4.575E−3 | 36.5 | −336.1 | 10.2 |

C | 5.076E−3 | 50 | −315.1 | 7.3 |

D | 5.465E−3 | 77.5 | −1321.5 | 18.05 |

E | 5.286E−3 | 111 | −2845.7 | 26.6 |

F | 5.149E−3 | 173.5 | −3900 | 23.5 |

H = 45 m |

A | 7.350E−5 | 223.5 | −317.7 | 2.4 |
---|---|---|---|---|

B | 6.481E−5 | 320.1 | −336.1 | 2.05 |

C | 6.438E−5 | 491 | −315.1 | 1.64 |

D | 5.251E−5 | 932.9 | −1321.5 | 2.41 |

E | 4.395E−5 | 1476.5 | −2845.7 | 2.92 |

F | 3.166E−5 | 2939.7 | −3900 | 1.24 |

H = 100 m | ||||

A | 1.968E−5 | 393.3 | −317.7 | 1.8 |

B | 1.467E−5 | 689.2 | −336.1 | 1.48 |

C | 1.330E−5 | 1165 | −315.1 | 1.27 |

D | 8.116E−6 | 2877.2 | −1321.5 | 1.45 |

E | 5.349E−6 | 5469.5 | −2845.7 | 1.52 |

F | 2.535E−6 | 15,718.2 | −3900 | 1.24 |

H = 250 m | ||||

A | 4.752E−6 | 688.5 | −317.7 | 1.46 |

B | 2.769E−6 | 1632.4 | −336.1 | 1.2 |

C | 2.188E−6 | 3187.7 | −315.1 | 1.09 |

D | 8.209E−6 | 13,099.8 | −1321.5 | 1.10 |

E | 3.236E−7 | 41,423.9 | −2845.7 | 1.06 |

F | 8.250E−8 | 243,818.1 | −3900 | 1.02 |

_{me} estimated by Equation (13) are negative values so the approximation σ z = H / 2 is not suitable for the standard scheme.

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.

From this discussion we conclude that the approximation σ z = H / 2 is most suitable for the Klug system and Pasquill Gifford system.

The authors declare no conflicts of interest regarding the publication of this paper.

Essa, K.S.M., Etman, S.M. and El-Otaify, M.S. (2020) Relation between the Actual and Estimated Maximum Ground Level Concentration of Air Pollutant and Its Downwind Locations Open Journal of Air Pollution, 9, 27-35. https://doi.org/10.4236/ojap.2020.92003