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ISSN: 2472-114X
International Journal of Accounting Research
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Theoretical Solution of the Diffusion Equation in Unstable Case

Essa KSM1*, Mina AN2, Hamdy HS2 and Khalifa AA2

1Department of Mathematics and Theoretical Physics, Nuclear Research Centre, Atomic energy Authority, Cairo, Egypt

2Faculty of science, Beni -Suef University, Egypt

*Corresponding Author:
Essa KSM
Department of Mathematics and Theoretical Physics
Nuclear Research Centre, Atomic Energy Authority, Cairo, Egypt
Tel: 202-296-4810
E-mail: [email protected]

Received Date: January 18, 2016; Accepted Date: April 27, 2016; Published Date: May 02, 2016

Citation: Essa KSM, Mina AN, Hamdy HS, Khalifa AA (2016) Theoretical Solution of the Diffusion Equation in Unstable Case. Int J Account Res 4:129. doi:10.4172/2472-114X.1000129

Copyright: © 2016 Essa KSM, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

The diffusion equation is solved in two dimensions to obtain the concentration by using separation of variables under the variation of eddy diffusivity which depend on the vertical height in unstable case. Comparing between the predicted and the observed concentrations data of Sulfur hexafluoride (SF6) taken on the Copenhagen in Denmark is done. The statistical method is used to know the best model. One finds that there is agreement between the present, Laplace and separation predicted normalized crosswind integrated concentrations with the observed normalized crosswind integrated concentrations than the predicted Gaussian model.

Keywords

Diffusion equation; Separation of variables; Laplace technique; Gaussian model; Eddy diffusivity

Introduction

The analytical solution of the atmospheric diffusion equation contains different depending on Gaussian and non–Gaussian solutions. An analytical solution with power law of the wind speed and eddy diffusivity with realistic assumption is derived by Demuth [1] and Essa [2]. Most of the fundamental theories of atmospheric diffusion were proposed in the first half of the twentieth century.

The atmospheric dispersion modeling refers to the mathematical description of contaminant transport in the atmosphere is used to describe the combination of diffusion and advection that occurs within the air the earth’s surface. The concentration of a contaminant released into the air may therefore be described by the advection – diffusion equation by Stockie JM [3].

The advection – diffusion equation has been widely applied in operational atmospheric dispersion model to predict the mean concentration of contaminants in the planetary boundary layer (PBL) which is obtain the dispersion from a continuous point source by Tiziano T et al. [4].

For nearly thirty years it has been known that vertical concentration profiles from field and laboratory experiments of near-surface point sources releases exhibit non-Gaussian distribution [5-7]. In this work diffusion equation is solved in two diffusivity which depend on the vertical height in unstable case. The statistical technique is used in dimensions to obtain the concentration by using separation of variables under the variation of eddy. The diffusion equation of pollutants in air can be written in the form by Arya [8].

Mathematical Model

Equation (1)

where c(x,y.z) is the concentration in the three dimensions x, y and z directions respectively, Ky and Kz are the crosswind and vertical turbulent eddy diffusivity coefficients of the PBL and u is the mean wind oriented in the x direction .

Equation (1) is subjected to the following boundary condition.

Equation (i) Equation (ii) Equation (iii)

Q is the emission rate, hs are the stack height, h is the height of PBL and δ is the Dirac delta function.

By integration with respect to y from -∞ to∞, then one gets:

Equation (2)

Suppose that:

Equation (3)

Since,

Equation (4)

By substituting from equations (3) and (4) into equation (2), one can get:

Equation (5)

Bearing in mind the dependence of the Kz coefficient, h is the height of PBL is discretized in N sub- intervals in such a manner that inside each interval Kz assume average value. Then the value of the average value is:

Equation

The solution of equation (5) is reduced to the solution of “N” problems of the type

Equation (6)

Cy(x, z) is called cross- wind integrated concentration of nth subinterval.

Let the solution of equation (6) using separation variables is in the form.

Equation

Then equation (6) becomes:

Equation (7)

Divided equation (7) on X(x) Z (z) one gets:

Equation (8)

Where α is constant. The solution of the first term of equation (8) can be written as:

Equation (9)

By integration from 0 to x, one gets

Equation (10)

Then equation (10) becomes:

Equation (11)

The second term of equation (8) can be written as:

Equation (12)

Then the solution of equation (12) is written in the form:

Equation (13)

Then the general solution becomes in the form:

Equation (14)

Applying the first boundary condition (i) one gets:

Equation (15) Equation (16)

Substituting by z = 0 then one can get:

C1 = 0

The general solution can be written as:

Equation (17)

Using the boundary condition (ii) one gets:

Equation (18) Equation (19) Equation (20) Equation (21) Equation (22)

Substituting from equation (22) in equation (17) then one gets:

Equation (23)

Using the boundary condition (iii). The equation (23) written as:

Equation (24)

Multiplying equation (24) by Equation then one gets: (25) Equation (26) Equation (27) Equation (28)

Substituting by equation (28) in equation (23) one obtains:

Equation (29)

Then the concentration at n = 0, we have that:

image (30)

At n = 1 one can get:

Equation (31)

For simplicity the crosswind integrating concentration in the form:

Equation (32)

Taking EquationWhere k0 is the von- Karman constant (k0~0.4), Z is the vertical height, hs is the stack height at 115m and w* is the convection velocity scale (Table 1).

Run no Date PG Stability Kn h     (m) W* U10 (ms-1) Distance (m) Cy/Q( 10-4 sm-2)           
                Observed     Computed  
          w* (m/s)       Separation Gaussian Laplace Present
1 20-9-78 A 0.14375 1980 1.8 3.34 1900 6.48 7.17 5.16 7,7046931 5.997743
1 20-9-78 A 0.14375 1980 1.8 3.34 3700 2.31 5.13 2.52 3,488227 5.997164
2 26-9-78 C 0.14375 1920 1.8 3.82 2100 5.38 3.7 2.29 4,61996 5.405101
2 26-9-78 C 0.14375 1920 1.8 3.82 4200 2.95 2.18 1.18 2,306918 5.404535
3 19-10-78 B 0.103819 1120 1.3 3.82 1900 8.20 9.8 4.51 8,410968 9.106625
3 19-10-78 B 0.103819 1120 1.3 4.93 3700 6.22 7.53 2.65 3,220596 7.05554
3 19-10-78 B 0.103819 1120 1.3 4.93 5400 4.30 7.44 2.58 1,613861 7.054574
5 9-11-78 C 0.055903 820 0.7 4.93 2100 6.72 9.30 3.63 6,580095 5.738531
5 9-11-78 C 0.055903 820 0.7 6.52 4200 5.84 7.87 2.44 2,044103 7.123884
5 9-11-78 C 0.055903 820 0.7 6.52 6100 4.97 7.86 2.41 1,00388 7.122991
6 30-4-78 C 0.159722 1300 2 6.52 2000 3.96 3.57 1.63 3,751729 7.117371
6 30-4-78 C 0.159722 1300 2 6.68 4200 2.22 2.50 0.82 1,703804 4.517276
6 30-4-78 C 0.159722 1300 2 6.68 5900 1.83 2.20 0.68 1,005404 4.516596
7 27-6-78 B 0.175694 1850 2.2 6.68 2000 6.70 5.27 2.51 5,917179 4.515889
7 27-6-78 B 0.175694 1850 2.2 7.79 4100 3.25 3.52 1.17 2,893086 2.749032
7 27-6-78 B 0.175694 1850 2.2 7.79 5300 2.23 3.06 0.79 2,123662 2.748846
8 6-7-78 D 0.175694 810 2.2 8.11 1900 4.16 8.39 4.20 7,124995 2.640299
8 6-7-78 D 0.175694 810 2.2 8.11 3600 3.02 6.21 2.80 3,123895 5.789855
8 6-7-78 D 0.175694 810 2.2 8.11 5300 1.52 5.89 2.18 1,518876 5.788336
9 19-7-78 C 0.151736 2090 1.9 11.45 2100 4.58 3.43 2.20 4,902058 4.100087
9 19-7-78 C 0.151736 2090 1.9 11.45 4200 3.11 2.77 1.13 2,485021 1.659012
9 19-7-78 C 0.151736 2090 1.9 11.45 6000 2.59 2.49 0.81 1,682239 1.65896

Table 1: Comparison between the predicated and observed crosswind- integrated concentration normalized with the emission source rate at different boundary layer height,downwind distance, wind speed, scaling convection velocity and distance for the different runs.

Figure 1 shows that the predicted normalized crosswind integrated concentrations values of the present, separation, Laplace and Gaussian predicted models and the observed via downwind distance.

international-journal-accounting-down-wind

Figure 1: The variation of the three predicted and observed models via down wind distances.

Figure 2 shows that the predicted normalized crosswind integrated concentrations values of the present, separation, Laplace and Gaussian predicted models via the observed.

international-journal-accounting-observed-concentrations

Figure 2: The variation of the three predicted before and present models via observed concentrations.

From the above two figures, we find that there is agreement between the present, Laplace, Gaussian predicted normalized crosswind integrated concentrations with the observed normalized crosswind integrated concentration than predicted concentration using separation technique.

Model Evaluation Statistics

Now, the statistical method is presented and comparison between predicted and observed results will be offered by Hanna [9]. The following standard statistical performance measures that characterize the agreement between prediction Equation Equation Equation Equation Equation

where σp and σo are the standard deviations of Cp and Co respectively. Here the over bars indicate the average over all measurements. A perfect model would have the following idealized performance: NMSE = FB = 0 and COR= FAC2 = 1.0.

Where σp and σo are the standard deviations of Cp and Co respectively. Here the over bars indicate the average over all measurements. A perfect model would have the following idealized performance: NMSE = FB = 0 and COR = 1.0 (Table 2).

Models NMSE FB COR FAC2
Present 0.2 -0.21 0.52 1.42
Laplace model 0.18 0.1 0.64 0.95
Separation model 0.22 -0.19 0.6 1.38
Gaussian model 0.58 0.58 0.8 0.59

Table 2: Comparison between Laplace, separation and Gaussian models according to standard statistical performance measure.

From the statistical method, we find that the four models are inside a factor of two with observed data. Regarding to NMSE and FB, the present, Laplace and separation predicted models are well with observed data than the Gaussian model. The correlation of present, Laplace and separation predicated model equals (0.52, 0.64 and 0.60 respectively) and Gaussian model equals (0.80).

Conclusion

The crosswind integrated concentration of air pollutants is obtained by using present model by separation technique to solve the diffusion equation in two dimensions. Considering that the eddy diffusivity depends on the vertical distance in unstable case. One finds that there is agreement between the present, Laplace and separation predicted normalized crosswind integrated concentrations with the observed normalized crosswind integrated concentrations than the predicted Gaussian model.

From the statistical method, one finds that the predicted models are inside a factor of two with observed data. Regarding to NMSE and FB, the present, Laplace and separation predicted models are well with observed data than the Gaussian model. The correlation of present, Laplace and separation predicated model equals (0.52, 0.64 and 0.60 respectively) and Gaussian model equals (0.80).

References

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