Two Dimensional Advection Diffusion Equations in Unstable Case

Khaled Sadek Mohamed Essa; Sawsan Ibrahim Mohamed El Saied

doi:doi:10.11648/j.engmath.20210501.12

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Two Dimensional Advection Diffusion Equations in Unstable Case

Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied

Published in Engineering Mathematics (Volume 5, Issue 1)

Received: 12 June 2021 Accepted: 29 June 2021 Published: 9 July 2021

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Abstract

The aim of this work is to solve the diffusion equation in two dimensions to obtain normalized crosswind integrated concentrations using the Laplace Transform technique, taking into account that the wind speed is constant but the vertical diffusivity differs from the friction velocity and the Monin -Obukhov length. A comparison of the calculated values and the observed concentrations taken from the northern part of Copenhagen, Denmark and also Inshas, Cairo, Egypt for trace hexafluoride (SF₆) through unstable condition were made. It has compared the current and observed concentration one finds that the current concentration agreement well with the observed data. The results showed an agreement between the measurements and the simulations. The values for NMSE and FB are relatively close to zero, and COR, FA2 is relatively close to one.

Published in	Engineering Mathematics (Volume 5, Issue 1)
DOI	10.11648/j.engmath.20210501.12
Page(s)	7-12
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Laplace Transforms Technique, Wind Speed, Advection-Diffusion Equations, Eddy Diffusivity

References

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[2]	Chatterjee, A. and Singh, M. K. (2018) Two-dimensional advection -dispersion equation with depth-dependent variable source concentration. Pollution, 4 (1): 1-8.
[3]	Demuth, C., (1978). "A contribution to the analytical steady solution of the diffusion equation". Atom. Environ. 12, 1255.
[4]	Hanna Steven R. Gary A. Briggs and Rayford P. hosker. Jr. (1982) Handbook on atmospheric diffusion. Technical Information Center, U. S. Department of Energy.
[5]	Hanna, S. R., (1989) Confidence limit for air quality models as estimated by bootstrap and Jackknife resem¬bling methods. Atom. Environ. 23, 1385-1395.
[6]	Gryning. S. E., and Lyck., (1984) Atmospheric dispersion from elevated sources in an urban area: Comparison between tracer experiments and model calculations. J. Climate Appl. Meteor., 23, pp. 651-660.
[7]	Gryning, S. E., Holtslag, A. A. M., Irwin, J. S., Sivertsen, B. (1987) Applied dispersion modeling based on meteorological scaling parameters. Atmos. Environ. 21 (1), 79-89.
[8]	John M. (2011) The mathematical of atmospheric dispersion modeling. Society for Industrial Applied Mathematics. 53. 349-372.
[9]	Khaled S. M Essa and S. E. M. Elsaid (2015) Solving the advection diffusion equation in three dimensions in neutral case. Pyrex Journal of Ecology and the Natural Environment Vol 1 (1) pp. 001-006.
[10]	Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied, Ayman Marrouf (2018) Analytical Solution of Time Dependent Diffusion Equation in Stable Case. American Journal of Environmental Science and Engineering; 2 (2): 32-36.
[11]	Murray R. Spiegel (1992) Advanced mathematics for engineers and scientists. McGill Publishing House.
[12]	Demuth C (1978) A contribution to the analytical steady solution of the diffusion equation. Atoms Environ 12: 1255-1978 2.
[13]	Essa KSM (2014) Studying the effect of vertical eddy diffusivity on the solution of diffusion equation. Physical science international Journal 4: 355-365.
[14]	Stockie JM (2011) The mathematics of atmospheric dispersion modeling. Society for industrial and applied mathematics 5: 349-372.
[15]	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.
[16]	Elliot WP (1961) The vertical diffusion of gas from continuous source. Int j air water pollut 4: 33-46.
[17]	Malhorta RC, Cermak JE (1964) Mass diffusion in neutral and unstably stratified boundary–layer flouss. J Heat mass trans 7: 169-186.
[18]	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.
[19]	Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied (2019) Mathematical Solution of Two Dimensional Advection-Diffusion Equations. Journal of Chemical, Environmental and Biological Engineering Volume 3, Issue 1, Pages: 8-12.
[20]	Essa KSM, Mina AN, Hamdy HS and Khalifa AA (2016) Theoretical Solution of the Diffusion Equation in Unstable Case. Int J Account Res, 4: 1.
[21]	Khaled S. M. Essa, Ahmed S. Shalaby, Mahmoud A. E. Ibrahim, And Ahmed M. Mosallem1 2020) Analytical Solutions of the Advection–Diffusion Equation with Variable Vertical Eddy Diffusivity and Wind Speed Using Hankel Transform. Pure Appl. Geophys. 177, 4545–4557.

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Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied. (2021). Two Dimensional Advection Diffusion Equations in Unstable Case. Engineering Mathematics, 5(1), 7-12. https://doi.org/10.11648/j.engmath.20210501.12

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ACS Style

Khaled Sadek Mohamed Essa; Sawsan Ibrahim Mohamed El Saied. Two Dimensional Advection Diffusion Equations in Unstable Case. Eng. Math. 2021, 5(1), 7-12. doi: 10.11648/j.engmath.20210501.12

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AMA Style

Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied. Two Dimensional Advection Diffusion Equations in Unstable Case. Eng Math. 2021;5(1):7-12. doi: 10.11648/j.engmath.20210501.12

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@article{10.11648/j.engmath.20210501.12,
  author = {Khaled Sadek Mohamed Essa and Sawsan Ibrahim Mohamed El Saied},
  title = {Two Dimensional Advection Diffusion Equations in Unstable Case},
  journal = {Engineering Mathematics},
  volume = {5},
  number = {1},
  pages = {7-12},
  doi = {10.11648/j.engmath.20210501.12},
  url = {https://doi.org/10.11648/j.engmath.20210501.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20210501.12},
  abstract = {The aim of this work is to solve the diffusion equation in two dimensions to obtain normalized crosswind integrated concentrations using the Laplace Transform technique, taking into account that the wind speed is constant but the vertical diffusivity differs from the friction velocity and the Monin -Obukhov length. A comparison of the calculated values and the observed concentrations taken from the northern part of Copenhagen, Denmark and also Inshas, Cairo, Egypt for trace hexafluoride (SF6) through unstable condition were made. It has compared the current and observed concentration one finds that the current concentration agreement well with the observed data. The results showed an agreement between the measurements and the simulations. The values for NMSE and FB are relatively close to zero, and COR, FA2 is relatively close to one.},
 year = {2021}
}

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TY  - JOUR
T1  - Two Dimensional Advection Diffusion Equations in Unstable Case
AU  - Khaled Sadek Mohamed Essa
AU  - Sawsan Ibrahim Mohamed El Saied
Y1  - 2021/07/09
PY  - 2021
N1  - https://doi.org/10.11648/j.engmath.20210501.12
DO  - 10.11648/j.engmath.20210501.12
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 7
EP  - 12
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20210501.12
AB  - The aim of this work is to solve the diffusion equation in two dimensions to obtain normalized crosswind integrated concentrations using the Laplace Transform technique, taking into account that the wind speed is constant but the vertical diffusivity differs from the friction velocity and the Monin -Obukhov length. A comparison of the calculated values and the observed concentrations taken from the northern part of Copenhagen, Denmark and also Inshas, Cairo, Egypt for trace hexafluoride (SF6) through unstable condition were made. It has compared the current and observed concentration one finds that the current concentration agreement well with the observed data. The results showed an agreement between the measurements and the simulations. The values for NMSE and FB are relatively close to zero, and COR, FA2 is relatively close to one.
VL  - 5
IS  - 1
ER  -

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Author Information

Khaled Sadek Mohamed Essa

Department of Mathematics and Theoretical Physics, Nuclear Research Center, Egyptian Atomic Energy Authority, Cairo, Egypt
Sawsan Ibrahim Mohamed El Saied

Department of Mathematics and Theoretical Physics, Nuclear Research Center, Egyptian Atomic Energy Authority, Cairo, Egypt

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APA Style

Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied. (2021). Two Dimensional Advection Diffusion Equations in Unstable Case. Engineering Mathematics, 5(1), 7-12. https://doi.org/10.11648/j.engmath.20210501.12

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ACS Style

Khaled Sadek Mohamed Essa; Sawsan Ibrahim Mohamed El Saied. Two Dimensional Advection Diffusion Equations in Unstable Case. Eng. Math. 2021, 5(1), 7-12. doi: 10.11648/j.engmath.20210501.12

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AMA Style

Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied. Two Dimensional Advection Diffusion Equations in Unstable Case. Eng Math. 2021;5(1):7-12. doi: 10.11648/j.engmath.20210501.12

Copy | Download

@article{10.11648/j.engmath.20210501.12,
  author = {Khaled Sadek Mohamed Essa and Sawsan Ibrahim Mohamed El Saied},
  title = {Two Dimensional Advection Diffusion Equations in Unstable Case},
  journal = {Engineering Mathematics},
  volume = {5},
  number = {1},
  pages = {7-12},
  doi = {10.11648/j.engmath.20210501.12},
  url = {https://doi.org/10.11648/j.engmath.20210501.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20210501.12},
  abstract = {The aim of this work is to solve the diffusion equation in two dimensions to obtain normalized crosswind integrated concentrations using the Laplace Transform technique, taking into account that the wind speed is constant but the vertical diffusivity differs from the friction velocity and the Monin -Obukhov length. A comparison of the calculated values and the observed concentrations taken from the northern part of Copenhagen, Denmark and also Inshas, Cairo, Egypt for trace hexafluoride (SF6) through unstable condition were made. It has compared the current and observed concentration one finds that the current concentration agreement well with the observed data. The results showed an agreement between the measurements and the simulations. The values for NMSE and FB are relatively close to zero, and COR, FA2 is relatively close to one.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Two Dimensional Advection Diffusion Equations in Unstable Case
AU  - Khaled Sadek Mohamed Essa
AU  - Sawsan Ibrahim Mohamed El Saied
Y1  - 2021/07/09
PY  - 2021
N1  - https://doi.org/10.11648/j.engmath.20210501.12
DO  - 10.11648/j.engmath.20210501.12
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 7
EP  - 12
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20210501.12
AB  - The aim of this work is to solve the diffusion equation in two dimensions to obtain normalized crosswind integrated concentrations using the Laplace Transform technique, taking into account that the wind speed is constant but the vertical diffusivity differs from the friction velocity and the Monin -Obukhov length. A comparison of the calculated values and the observed concentrations taken from the northern part of Copenhagen, Denmark and also Inshas, Cairo, Egypt for trace hexafluoride (SF6) through unstable condition were made. It has compared the current and observed concentration one finds that the current concentration agreement well with the observed data. The results showed an agreement between the measurements and the simulations. The values for NMSE and FB are relatively close to zero, and COR, FA2 is relatively close to one.
VL  - 5
IS  - 1
ER  -

Copy | Download