Medical, Pharma, Engineering, Science, Technology and Business

**Essa KSM ^{1*}, Mina AN^{2}, Hamdy HS^{2} and Khalifa AA^{2}**

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

^{2}Faculty 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.

**Visit for more related articles at** International Journal of Accounting Research

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.

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

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].

(1)

where c(x,y.z) is the concentration in the three dimensions x, y and z directions respectively, K_{y} and K_{z} 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.

(i) (ii) (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:

(2)

Suppose that:

(3)

Since,

(4)

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

(5)

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

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

(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.

Then equation (6) becomes:

(7)

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

(8)

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

(9)

By integration from 0 to x, one gets

(10)

Then equation (10) becomes:

(11)

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

(12)

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

(13)

Then the general solution becomes in the form:

(14)

Applying the first boundary condition (i) one gets:

(15) (16)

Substituting by z = 0 then one can get:

C_{1} = 0

The general solution can be written as:

(17)

Using the boundary condition (ii) one gets:

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

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

(23)

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

(24)

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

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

(29)

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

(30)

At n = 1 one can get:

(31)

For simplicity the crosswind integrating concentration in the form:

(32)

Taking Where 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) | C_{y}/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.

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

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.

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

where σp and σo are the standard deviations of C_{p} and C_{o} 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 C_{p} and C_{o} 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).

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).

- Demuth C (1978) A contribution to the analytical steady solution of the diffusion equation. Atoms Environ 12: 1255-1978
- Essa KSM (2014) Studying the effect of vertical eddy diffusivity on the solution of diffusion equation. Physical science international Journal 4: 355-365.
- Stockie JM (2011) The mathematics of atmospheric dispersion modeling. Society for industrial and applied mathematics 5: 349-372.
- Tiziano T, Moreira DM, Vilhena MT, Costa CP (2010) Comparison between non- Gaussian puff model and a model based on a time-dependent solution of advection equation. Journal of Environment protection 1: 172-178.
- Elliot WP (1961) The vertical diffusion of gas from continuous source. Int j air water pollut 4: 33-46.
- Malhorta RC, Cermak JE (1964) Mass diffusion in neutral and unstably stratified boundary–layer flouss. J Heat mass trans 7: 169-186.
- Marrouf AA, Mohamed AS, Ismail G, Essa KSM (2013) An analytical solution of two dimensional atmospheric diffusion equation in a finite boundary layer. International Journal of Advanced Research Volume 1: 356-365.
- Arya SP (1995) Modeling and parameterization of near –source diffusion in weak wind. J Appl Met 34: 1112-1122.
- Hanna SR(1989) Confidence limit for air quality models as estimated by bootstrap and Jackknife resembling methods. Atom Environ 23: 1385-1395.

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