Analytical Solution of Time Dependent Diffusion Equation in Stable Case

Khaled Sadek Mohamed Essa; Sawsan Ibrahim Mohamed El Saied; Ayman Marrouf

doi:doi:10.11648/j.ajese.20180202.12

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Analytical Solution of Time Dependent Diffusion Equation in Stable Case

Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied, Ayman Marrouf

Published in American Journal of Environmental Science and Engineering (Volume 2, Issue 2)

Received: 27 May 2018 Accepted: 8 June 2018 Published: 9 August 2018

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Abstract

The normalized integrated concentration of pollutant has been obtained after solving temporaly diffusion equation using the method of separation variable considering the eddy diffusivities which measuring at night or at any time in high inversion layer in the stable condition. The dataset is observed from the “Project prairie Grass” (Barad 1958) which is measured using wind speed at 1.5m and downwind distance during the experiment at 50, 200 and 800 m in stable case for runs from 1 to 10. Comparison between the estimated and observed normalized integrated concentration at a different downwind distance for all runs at t = 30 minutes is calculated.

Published in	American Journal of Environmental Science and Engineering (Volume 2, Issue 2)
DOI	10.11648/j.ajese.20180202.12
Page(s)	32-36
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), 2018. Published by Science Publishing Group

Keywords

Project Prairie Grass, Laplace Transform, Normalized Concentration, Diffusion Equation, Stable Condition

References

[1]	Akula Venkatram (2004): On estimating emissions through horizontal fluxes. Atmospheric Environment38, 1337-1344.
[2]	Barad M. L. (Ed) (1958). Project prairie Grass, A field program in diffusion, vol. 1. Geophysics Research Paper no. 59, Geophysics Research Directorate, Air Force Cambridge Research Center.
[3]	Doran J. C., and Horst T. W. (1985). ”An evaluation of Gaussian plume-depletion models with dual-tracer field measurement”. Atmos. Environ., 19, 939-951.
[4]	Eckman, R. M., (1994). “Ro-examination of empirically derived formulas for horizontal diffusion from surface sources”. Atmospheric Environment: 28, 265-272.
[5]	Hanna Steven R. Gary A. Briggs and Rayford P. hosker Jr. (1982). “Handbook on Atmospheric Diffusion”. Technical Information Center, U. S. Department of Energy.
[6]	Hanna, S. R., 1989, “confidence limit for air quality models as estimated by bootstrap and Jacknife resembling methods”, Atom. Environ. 23, 1385-1395.
[7]	Hazewinkel, Michiel, ed. (2001), “Delta-function”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4.
[8]	John E. Till, Helen A. Grogan (2008). “Radiological risk assessment and environmental analysis”.
[9]	Khaled S. M. Essa, Sawsan E. M. Elsaid and Fawzia Mubarak (2015).” Time dependent Advection-Diffusion Equation in Two Dimensions” Journal of Atmosphere, vol. 1, issue1, pages 8-16.
[10]	Michael R. K. Thambynayagam, R. K. M (2011). The Diffusion Handbook: Applied Solutions for Engineers. McGraw-Hill.
[11]	Palazzi, E., De Faveri, M., Fumarola, G., Ferraiolla, G., (1982), Diffusion from a steady source of short duration, Atmospheric Environment 16, 2785–2790.
[12]	Panofsky, H. A., H. Tennekes, D. H. Lenschow, and J. C. Wyngaard, (1977).”The characteristics of turbulent velocity components in the surface layer under convective conditions, boundary layer meteorology. 11:355-361.

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APA Style

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. https://doi.org/10.11648/j.ajese.20180202.12

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ACS Style

Khaled Sadek Mohamed Essa; Sawsan Ibrahim Mohamed El Saied; Ayman Marrouf. Analytical Solution of Time Dependent Diffusion Equation in Stable Case. Am. J. Environ. Sci. Eng. 2018, 2(2), 32-36. doi: 10.11648/j.ajese.20180202.12

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AMA Style

Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied, Ayman Marrouf. Analytical Solution of Time Dependent Diffusion Equation in Stable Case. Am J Environ Sci Eng. 2018;2(2):32-36. doi: 10.11648/j.ajese.20180202.12

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@article{10.11648/j.ajese.20180202.12,
  author = {Khaled Sadek Mohamed Essa and Sawsan Ibrahim Mohamed El Saied and Ayman Marrouf},
  title = {Analytical Solution of Time Dependent Diffusion Equation in Stable Case},
  journal = {American Journal of Environmental Science and Engineering},
  volume = {2},
  number = {2},
  pages = {32-36},
  doi = {10.11648/j.ajese.20180202.12},
  url = {https://doi.org/10.11648/j.ajese.20180202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajese.20180202.12},
  abstract = {The normalized integrated concentration of pollutant has been obtained after solving temporaly diffusion equation using the method of separation variable considering the eddy diffusivities which measuring at night or at any time in high inversion layer in the stable condition. The dataset is observed from the “Project prairie Grass” (Barad 1958) which is measured using wind speed at 1.5m and downwind distance during the experiment at 50, 200 and 800 m in stable case for runs from 1 to 10. Comparison between the estimated and observed normalized integrated concentration at a different downwind distance for all runs at t = 30 minutes is calculated.},
 year = {2018}
}

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TY  - JOUR
T1  - Analytical Solution of Time Dependent Diffusion Equation in Stable Case
AU  - Khaled Sadek Mohamed Essa
AU  - Sawsan Ibrahim Mohamed El Saied
AU  - Ayman Marrouf
Y1  - 2018/08/09
PY  - 2018
N1  - https://doi.org/10.11648/j.ajese.20180202.12
DO  - 10.11648/j.ajese.20180202.12
T2  - American Journal of Environmental Science and Engineering
JF  - American Journal of Environmental Science and Engineering
JO  - American Journal of Environmental Science and Engineering
SP  - 32
EP  - 36
PB  - Science Publishing Group
SN  - 2578-7993
UR  - https://doi.org/10.11648/j.ajese.20180202.12
AB  - The normalized integrated concentration of pollutant has been obtained after solving temporaly diffusion equation using the method of separation variable considering the eddy diffusivities which measuring at night or at any time in high inversion layer in the stable condition. The dataset is observed from the “Project prairie Grass” (Barad 1958) which is measured using wind speed at 1.5m and downwind distance during the experiment at 50, 200 and 800 m in stable case for runs from 1 to 10. Comparison between the estimated and observed normalized integrated concentration at a different downwind distance for all runs at t = 30 minutes is calculated.
VL  - 2
IS  - 2
ER  -

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Author Information

Khaled Sadek Mohamed Essa

Department of Mathematics and Theoretical Physics, NRC, Atomic Energy Authority, Cairo, Egypt
Sawsan Ibrahim Mohamed El Saied

Department of Mathematics and Theoretical Physics, NRC, Atomic Energy Authority, Cairo, Egypt
Ayman Marrouf

Department of Mathematics and Theoretical Physics, NRC, Atomic Energy Authority, Cairo, Egypt

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APA Style

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. https://doi.org/10.11648/j.ajese.20180202.12

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ACS Style

Khaled Sadek Mohamed Essa; Sawsan Ibrahim Mohamed El Saied; Ayman Marrouf. Analytical Solution of Time Dependent Diffusion Equation in Stable Case. Am. J. Environ. Sci. Eng. 2018, 2(2), 32-36. doi: 10.11648/j.ajese.20180202.12

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AMA Style

Khaled Sadek Mohamed Essa, Sawsan Ibrahim Mohamed El Saied, Ayman Marrouf. Analytical Solution of Time Dependent Diffusion Equation in Stable Case. Am J Environ Sci Eng. 2018;2(2):32-36. doi: 10.11648/j.ajese.20180202.12

Copy | Download

@article{10.11648/j.ajese.20180202.12,
  author = {Khaled Sadek Mohamed Essa and Sawsan Ibrahim Mohamed El Saied and Ayman Marrouf},
  title = {Analytical Solution of Time Dependent Diffusion Equation in Stable Case},
  journal = {American Journal of Environmental Science and Engineering},
  volume = {2},
  number = {2},
  pages = {32-36},
  doi = {10.11648/j.ajese.20180202.12},
  url = {https://doi.org/10.11648/j.ajese.20180202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajese.20180202.12},
  abstract = {The normalized integrated concentration of pollutant has been obtained after solving temporaly diffusion equation using the method of separation variable considering the eddy diffusivities which measuring at night or at any time in high inversion layer in the stable condition. The dataset is observed from the “Project prairie Grass” (Barad 1958) which is measured using wind speed at 1.5m and downwind distance during the experiment at 50, 200 and 800 m in stable case for runs from 1 to 10. Comparison between the estimated and observed normalized integrated concentration at a different downwind distance for all runs at t = 30 minutes is calculated.},
 year = {2018}
}

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TY  - JOUR
T1  - Analytical Solution of Time Dependent Diffusion Equation in Stable Case
AU  - Khaled Sadek Mohamed Essa
AU  - Sawsan Ibrahim Mohamed El Saied
AU  - Ayman Marrouf
Y1  - 2018/08/09
PY  - 2018
N1  - https://doi.org/10.11648/j.ajese.20180202.12
DO  - 10.11648/j.ajese.20180202.12
T2  - American Journal of Environmental Science and Engineering
JF  - American Journal of Environmental Science and Engineering
JO  - American Journal of Environmental Science and Engineering
SP  - 32
EP  - 36
PB  - Science Publishing Group
SN  - 2578-7993
UR  - https://doi.org/10.11648/j.ajese.20180202.12
AB  - The normalized integrated concentration of pollutant has been obtained after solving temporaly diffusion equation using the method of separation variable considering the eddy diffusivities which measuring at night or at any time in high inversion layer in the stable condition. The dataset is observed from the “Project prairie Grass” (Barad 1958) which is measured using wind speed at 1.5m and downwind distance during the experiment at 50, 200 and 800 m in stable case for runs from 1 to 10. Comparison between the estimated and observed normalized integrated concentration at a different downwind distance for all runs at t = 30 minutes is calculated.
VL  - 2
IS  - 2
ER  -

Copy | Download