Turbulence and dispersion flow of radioisotopes in the atmospheric Boundary layer
By Sawsan Ibrahiem Mohamed El Said B.sc (physics) Ain Shams University D.M. (Meteorology) Cairo University M.Sc. (Meteorology) Cairo University
Thesis
Submitted as requirements for degree of Ph.D. in Meteorology
TO THE Meteorological and Astronomy Department, Faculty of Science, Cairo University Under Supervision of
Prof. Dr. M. M. Abd El-Wahab Prof. Dr. Khaled. S.M. Essa
Professor of Meteorological and Professor o f Mathematics and Astronomy Faculty of Science Theoretical Physic Nuclear Cairo University Research Center Atomic Energy Authority
(2013)
Title of the Ph.D. Thesis
Turbulence and dispersion flow of radioisotopes
in the atmospheric Boundary layer
Name of candidate:
Sawsan Ibrahiem Mohamed El Said
Submitted to the Cairo University for
Degree of Ph.D. in Meteorology
Faculty of Science, Meteorological and Astronomy Department,
Cairo University
(2013)
Title of the Ph.D. Thesis Turbulence and dispersion flow of radioisotopes in the atmospheric Boundary layer
Name of candidate :
Sawsan Ibrahiem Mohamed El Said Submitted to the Cairo University for Degree of Ph.D. in Meteorology Faculty of Science, Meteorological and Astronomy Department . Supervision Committee: Prof. Dr.M. M. Abad El-Wahab Professor of Meteorological, Faculty of Science, Cairo University Prof. Dr. Khaled. S.M. Essa Professor of Mathematics and Theoretical Physic Nuclear Research Center Atomic Energy Authority Assistant Prof. Dr. Mohamed Embay Assistant Professor of Mathematics and Theoretical Physic Nuclear Research Center Atomic Energy Authority Head of the Department of Meteorological and Astronomy , Faculty of Science, Cairo University
Prof. Dr.M.YASFE
Approval sheet Name of candidate :
Sawsan Ibrahiem Mohamed El Said
Title of the Ph.D. Thesis
Turbulence and dispersion flow of radioisotopes in the atmospheric Boundary layer
Submitted to the
Faculty of Science, Cairo University
Approved by : Prof. Dr.M. M. Abad El-Wahab Professor of Meteorological, Faculty of Science Cairo University Prof. Dr. Khaled. S.M. Essa Professor of Mathematics and Theoretical Physic Nuclear Research Center Atomic Energy Authority Assistant Prof. Dr. Mohamed Embay Assistant Professor of Mathematics and Theoretical Physic Nuclear Research Center Atomic Energy Authority
Summary
There is an increase in the study of atmospheric pollution and harmful impact on environment, in this work attention was forward to atmospheric diffusion equation to evaluate the concentration pollution with different methods under different stability conditions.
The material in the present thesis is organized in six chapters in the following way:
Chapter (1), it describe as. In section 1.1, General Introduction, In section 1.2,
Turbulence, In section 1.3 , Turbulence of the atmosphere. In section 1.4,
Atmospheric stability. In section 1.5, Atmospheric pollution . In section 1.6,
Behavior of effluent released to the atmosphere . In section 1.7, Source Types .
In section1.8, Atmospheric Dispersion Theories (Modeling). In section1.9.Comparison between Some Models. In section1.10 , The Planetary
Boundary Layer.
Chapter (2), it describe as: In section 2.1 , Introduction. In section 2.2 ,
Analytical Method. In section 2.3 , Numerical Method. In section 2.4 ,
Statistical method.
In chapter (3), it describe as : In section 3.1, Introduction. In section 3.2,
Analytical solution. In section 3.3, statically methods. Chapter (4), it contain following: In section 4.1, Introduction. In section 4.2,
Proposed model structure. In section 4.3, the effective height. In section 4.4,
Mathematical technique In section 4. 5, Case study. In section 4.6, Verification .
Chapter (5), one can find as: In section 5.1, Introduction. In section 5.2,
Gaussian distributions. In section 5.3, Dispersion parameters schemes. In section 5.4, Result and discussion. In section 5.5 Statistical methods.
Chapter (6), it can be arranged in the following: In section 6.1, Introduction .
In section 6.2, Model formulation. In section 6.3, Results and Discussion. In section 6.4, Statistical method.
Some Major Finding of this theses were summarized in published articles
(41, 42, 43)
ACKNOWLEDGEMENT
All gratitude is due God almighty that has ever guided and helped me to bring- forth to light this thesis.
I would like to thank my parents and my family for the continuous encouragement and help.
My deep appreciation and sincere gratitude are resent to my supervisor, Prof. Dr. Mohamed Magdy Abdel-Wahab, faculty of science Cairo university, for suggestion the interested point of research, supervision and following the following the progress of the work with keep interest, and for his facilities provided to me during the preparation of this thesis, invaluable advice and guidance, for his help and encouragement moral support.
I would like to acknowledge my gratitude and sincere thank fullness to my supervisor, Prof. Dr.Khaled Sadek Mohamed Essa for his help and encouragement moral support.
I would like to acknowledge my gratitude to Dr. Mokhtar
Embaby Assistance professor in Atomic Energy Authority for helpful in this work.
ا ه ة آ ا م م ا وا ء و ا ر د ا
ا اب وا ا ا ا ا ا ف ا ي د ن
أ اه ا ر س ء د م أر د ا ه ة أر د ا ه ة
ر ل در ا آ را
א א ذ א د و א ذ א د و د א ذ א د و د د د א و د د دق د א ذ א د א و א ذ د א وא ط א ذ א وא ط א מ א وא د א و מ א وא ط א מ א وא ط א א وמ א ز א و א وو א ط א ذ ز א و א وو
2013
ﻤﻠﺨﺹ ﺍﻝﺭﺴﺎﻝﺔ
ﻝﻘﺩ ﺫﺍﺩ ﺍﻻﻫﺘﻤﺎﻡ ﺒﺩﺭﺍﺴﺔ ﺍﻨﺘﺸﺎﺭ ﺍﻝﻤﻠﻭﺜﺎﺕ ﺍﻝﺠﻭﻴﺔ ﻭ ﺃﺜﺎﺭﻫﺎ ﺍﻝﻀﺎﺭﺓ ﻋﻠﻰ ﺍﻝﺒﻴﺌﺔ ﻤﻤﺎ ﺩﻓﻌﻨﺎ ﺇﻝﻰ ﺤﺴﺎﺏ ﺘﺭﻜﻴﺯ ﻫﺫﻩ ﺍﻝﻤﻠﻭﺜﺎﺕ ﺒﻁﺭﻕ ﻤﺨﺘﻠﻔﺔ . .
ﻓﻲ ﺍﻝﻔﺼل ﺍﻷﻭل : :
ﺘﻡ ﺩﺭﺍﺴﺔ ﻤﺭﺠﻌﻴﺔ ﻋﻠﻰ ﺍﻨﺘﺸﺎﺭ ﺍﻝﺠﻭﻯ ﻭ ﻤﺘﻐﻴﺭﺍﺕ ﺍﻝﻁﺒﻘﺔ ﺍﻻﻀﻁﺭﺍﺒﻴﺔ ﺍﻝﻤﺨﺘﻠﻔﺔ ﻤﺜل ﺴﺭﻋﺔ ﺍﻝﺭﻴﺎ ﺡ ﻭﻤﻌﺎﻤل ﺍﻝﺘﺸﺘﺕ. ﻭ ﻤﻘﺩﻤﺔ ﻋﻠﻰ ﺍﻷﻀﻁﺭ ﺍ ﺏ ﺍﻝﺠﻭﻯ ﻭﻤﻜﻭﻨﺎﺘﻪ ﻭﺍﻝﻌﻭﺍﻤل ﺍﻝﺘﻰ ﻴﻌﺘﻤﺩ ﻋﻠﻴﻬﺎ ﻭﺩﺭﺍﺴﺔ ﻋﺎﻤﺔ ﻋﻠﻰ ﺍﻷﻨﺘﺸﺎﺭ ﺍﻝﺠﻭﻯ ﻭﺤﺎﻻﺘﻪ ﺍﻝﻤﺨﺘﻠﻔﺔ . ﻭﻤﻘﺩﻤﻪ ﻋﺎﻤﺔ ﻋﻠﻰ ﺍﻝﺘﻠﻭﺙ ﺍﻝﺠﻭﻯ ﻭﺃﻨﻭﺍﻉ ﻤﺤﺘﻠﻔﺔ ﻤﻥ ﺍﻝﻤﺼﺎﺩﺭ ﻭﺍﻝﻨﻤﺎﺫﺝ ﻝﻤﻌﺭﻓﺔ ﺘﺭﻜﻴﺯ ﺍﻝﻤﻠﻭﺜﺎﺕ . ﻭﻗﺩ ﺘﻡ ﺘﻌﺭﻴﻑ ﺃﻨﻭﺍﻉ ﺍﻝﻤﺨﺘﻠﻔﺔ ﺍﻝﻁﺒﻘﺔ ﺍﻝﺤﺩﻴﺔ ﻝﻠﻐﻼﻑ ﺍﻝﺠﻭﻯ.
ﻓﻲ ﺍﻝﻔﺼل ﺍﻝﺜﺎﻨﻲ : :
ﺘﻡ ﺤل ﻤﻌﺎﺩﻝﺔ ﺍﻷﻨﺘﺸﺎﺭ ﺍﻝﺠﻭﻯ ﻓﻰ ﺃﺘﺠﺎﻫﻴﻥ ﺃﺤﺩﻫﻤﺎ ﻓﻰ ﺃﺘﺠﺎﻩ ﺍﻝﺭﻴﺎﺡ ﻭﻵﺨﺭ ﻓﻰ ﺃﺘﺠﺎﻩ ﺭﺃﺴﻰ ﻭﺫﺍﻝﻙ ﻓﻰ ﺤﺎﻝﺔ ﺍﻝﺜﺒﺎﺕ ﺒﻁﺭﻴﻘﺘﻴﻥ ﻤﺨﺘﻠﻔﺘﻴﻥ ﺃﺤﺩﻫﻤﺎ ﺒﻁﺭﻴﻘﺔ ﺘﺤﻭﻴﻼﺕ ﻻﺒﻼﺱ ﺍﻝﻌﻜﺴﻴﺔ ﻭ ﺍﻷﺨﺭﻯ ﺒﻁﺭﻴﻘﺔ ﻋﺩﺩﻴﺔ ﻤﻊ ﺃﺨﺫ ﻓﻰ ﺍﻷﻋﺘﺒﺎﺭ ﺸﻜﻠﻴﻥ ﻤﺨﺘﻠﻔﻴﻥ ﻝﻤﻌﺎﻤل ﺍﻻﻨﺘﺸﺎﺭ ﺍﻝﺩﻭﺍﻤﻰ ﺒﺤﻴﺙ ﻴﻜﻭﻥ ﻫﺫﺍ ﺍﻝﻤﻌﺎﻤل ﺩﺍﻝﺔ ﻓﻲ ﺍﻝﻤﺴﺎﻓﺔ ﺍﻻﻓﻘﻴﺔ ﻭﻴﻜﻭﻥ ﻫﺫﺍ ﺍﻝﺸﻜﻠﻴﻥ ﺍﺤﺩﻫﻤﺎ ﻓﻰ ﺍﻝﺤﺎﻝﺔ ﺍﻝﻌﺎﺩﻴﺔ ﻝﺠﻭ ﻭﺍﻷﺨﺭ ﻓﻲ ﺍﻝﺤﺎﻝﺔ ﺍﻝﻐﻴﺭ ﻤﺴﺘﻘﺭﺓ ﻝﺠﻭ ﻭﻻﻴﺠﺎﺩ ﺘﺭﻜﻴﺯ ﺍﻝﻤﻠﻭﺜﺎﺕ ﺒﺎﻝﻨﺴﺒﺔ ﻝﻤﻌﺩل ﺍﻷﻨﺒﻌﺎﺙ ﻭﻤﻘﺎﺭﻨﺘﻬﺎ ﺒﺎﻝﺤﺴﺎﻴﺎﺕ ﺍﻝ ﻤﻘﺎﺴﺔ ﻭﻗﺩ ﻭﺠﺩﻨﺎ ﺘﻭﺍﻓﻕ ﺒﻴﻥ ﺤﺴﺎﺒﺎﺘﻨﺎ ﻭ ﺍﻝﺤﺴﺎﺒﺎﺕ ﺍﻝﻤﻘﺎﺴﺔ ﻭﺘﻡ ﻋﻤل ﻁﺭﻕ ﺃﺤﺼﺎﺌﻴﺔ ﻝﻠﻭﺼﻭل ﺍﻝﻰ ﺃﺤﺴﻥ ﻨﻤﻭﺫﺝ ﻤﻊ ﺍﻝﺤﺴﺎﺒﺎﺕ ﺍﻝﻤﻘﺎﺴﺔ .
ﻓﻲ ﺍﻝﻔﺼل ﺍﻝﺜﺎﻝﺙ : :
ﺘﻡ ﺤل ﻤﻌﺎﺩﻝﺔ ﺍﻻﻨﺘﺸﺎﺭ ﺍﻝﺠﻭﻯ ﻓﻰ ﺍﺘﺠﺎﻫﻴﻥ ( x, z ) ﺫﻭ ﻝﻙ ﻓﻲ ﺤﺎﻝﺔ ﺍﻝﺜﺒﺎﺕ ﺒﻁﺭﻴﻘﺔ ﺘﺤﻭﻴﻼﺕ ﻻﺒﻼﺱ ﺍﻝﻌﻜﺴﻴﺔ ﻤﻊ ﺃﺨﺫ ﻓﻰ ﺍﻻﻋﺘﺒﺎﺭ ﺍﻥ ﻜﻼ ﻤﻥ ﺴﺭﻋﺔ ﺍﻝﺭﻴﺎﺡ ﻭ ﺍﻻﻨﺘﺸﺎﺭ ﺍﻝﺩﻭﺍﻤﻰ ﻴﻌﺘﻤﺩﺍﻥ ﻋﻠﻰ ﺃﻨﺠﺎﻩ ﺍﻝﺭﺍﺴﻲ ﻭﺫﻝﻙ ﻋﻨﺩﻤﺎ ﻴﻜﻭﻥ ﺍﻝﺠﻭ ﻤﺘﻌﺎﺩل ﻭﻏﻴﺭ ﻤﺴﺘﻘﺭ ﻻﻴﺠﺎﺩ ﺃﻋﻠﻰ ﺘﺭﻜﻴﺯ ﺴﻁﺢ ﺍﻷﺭﺽ ﻭﻤﻘﺎﺭﻨﺘﻬﺎ ﺒﺎﻝﺤﺴﺎﺒﺎﺕ ﺍﻝﻤﻘﺎﺴﺔ ﻭﻗﺩ ﻭﺠﺩﻨﺎ ﺘﻭﺍﻓﻕ ﺒﻴﻥ ﺤﺴﺎﺒﺎﺘﻨﺎ ﻭ ﺍﻝﺤﺴﺎﺒﺎﺕ ﺍﻝﻤﻘﺎﺴﺔ ﻭﺘﻡ ﻋﻤل ﻁﺭﻕ ﺃﺨﺼﺎﺌﻴﺔ ﻝﻠﻭﺼﻭل ﺇﻝﻰ ﺃﺤﺴﻥ ﻨﻤﻭﺫﺝ ﻤﻊ ﺍﻝﺤﺴﺎﺒﺎﺕ ﺍﻝﻤﻘﺎﺴﺔ . .
ﻓﻲ ﺍﻝﻔﺼل ﺍﻝﺭﺍﺒﻊ : ﺘﻡ ﻋﻤل ﻋﻤل ﻨﻤﻭﺫﺝ ﺒﺴﻴﻁ ﻝﺤﺴﺎﺏ ﺘﺭﻜﻴﺯ ﺍﻝﻤﻠﻭﺜﺎﺕ ﻭﺫﻝﻙ ﺒﺎﻋﺘﺒﺎﺭ ﺍﻥ ﺴﺭﻋﺔ ﺍﻝﺭﻴﺎﺡ ﺘﻌﺘﻤﺩ ﻋﻠﻰ ﻜﻼ ﻤﻥ ﺍﻝﻘﻭﺓ ﺍﻻﺴﻴﺔ ﻭﺍﻝﻠﻭﻏﺎﺭﺘﻤﻴﺔ ﻤﻌﺎ ﻭﺘﻡ ﻤﻘﺎﺭﻨﺔ ﺍﻝﺘﺭﻜﻴﺯﺍﺕ ﺍﻝﺘﻰ ﺘﻡ ﺍﻝﻭﺼﻭل ﺍﻝﻴﻬﺎ ﻤﻊ ﺍﻝﺤﺴﺎﺒﺎﺕ ﻗﺩ ﺤﺴﺒﺕ ﻤﻥ ﻗﺒل ﻓﻭﺠﺩﻨﺎ ﺘﻁﺎﺒﻕ ﻫﺫﺍ ﺍﻝﻨﻤﻭﺯﺝ ﻭ ﺍﻝ ﻨﻤﺎﺫﺝ ﺍﻝﺴﺎﺒﻘﺔ ﻓﻰ ﺤﺎﻻﺕ ﺍﻝﺠﻭ ﺍﻝﻤﺨﺘﻠﻔﺔ . .
ﻓﻲ ﺍﻝﻔﺼل ﺍﻝﺨﺎﻤﺱ:
ﺘﻡ ﺤﺴﺎﺏ ﺘﺭﻜﻴﺯﺍﺕ ﺍﻝﻨﻅﺎﺌﺭ ﺍﻝﻤﺸﻌﺔ ﻝﻌﻨﺎﺼﺭ ( I131 , I 135 ,I 133 to and Cs 137 ) ﻭﺫﻝﻙ ﺒﺎﺴﺘﺨﺩﺍﻡ ﺍﺸﻜﺎل ﻤﺨﺘﻠﻔﺔ ﻝﻤﻌﺎﻤﻼﺕ ﺍﻝﺘﺸﺘﺕ (Power law, standard method, Irwin method and Briggs method) ﻭﺘﻡ ﺍﻝﺘﻌﻭﻴﺽ ﺒﻬﻤﺎ ﻓﻰ ﻤﻌﺎﺩﻝﺔ ﺠﺎﻭﺱ ﻭﺘﻡ ﺍﻀﺎﻓﺔ ﻤﻌﺎﻤل ﺍﻻﻀﻤﺤﻼل ﻝﻠﻌﻨﺎﺼﺭ ﺍﻝﻤﺸﻌﺔ ﻭﺘﻡ ﺤﺴﺎﺏ ﺍﻝﻨﺸﺎﻁ ﺍﻻﺸﻌﺎﻋﻰ ﻭﻤﻘﺎﺭﻨﺘﻬﻤﺎ ﻤﻊ ﺘﺭﻜﻴﺯﺍﺕ ﻤﻘﺎﺴﺔ ﻓﻭﺠﺩﻨﺎ ﻤﻁﺎﺒﻘﺔ . ﻭﺘﻡ ﻋﻤل ﻨﻤﻭﺯﺝ ﺍﺤﺼﺎﺌﻰ ﻻﻴﺠﺎﺩ ﺍﺤﺴﻥ ﺘﺭﻜﻴﺯ.
ﻓﻲ ﺍﻝﻔﺼل ﺍﻝﺴﺎﺩﺱ :
ﺘﻡ ﺍﺸﺘﻘﺎﻕ ﺼﻭﺭﺓ ﺠﺩﻴﺩﺓ ﻝﻤﻌﺎﻤﻠﻲ ﺍﻝﺘﺸﺘﺕ ﺍﻝﻤﺴﺘﻌﺭﺽ ﻭﺍﻝﺭﺍﺴﻲ ﻭ ﺘﻡ ﺍﺴﺘﺨﺩﻤﻬﺎ ﻓﻲ ﻤﻌ ﺎﺩﻝﺔ ﺠﺎﻭﺱ ﻋﻠﻰ ﺴﻁﺢ ﺍﻷﺭﺽ ﻭﻋﻠﻰ ﺨﻁ ﺍﻝﻤﺭﻜﺯ ﻭﺍﺴﺘﺨﺩﻤﻨﺎ ﺒﻴﺎﻨﺎﺕ ﻤﻘﺎﺴﺔ ﻤﻥ ﻗﺒل ﻝﻠﻤﻘﺎﺭﻨﺔ ﺒﻴﻥ ﺍﻝﻨﻤﻭﺫﺝ ﺍﻝﺫﻱ ﺘﻡ ﺍﺸﺘﻘﺎﻗﻪ ﻭﻭﺠﺩﻨﺎ ﺘﻁﺎﺒﻕ ﺒﻴﻥ ﺍﻝﺒﻴﺎﻨﺎﺕ ﺍﻝﻤﻘﺎﺴﺔ ﻭﺍﻝﻤﺤﺴﻭﺒﺔ . .
Contents Page ACKNNTOWLEDGEME Abstract ABBREVIATIONS Chapter I Review study on atmospheric diffusion and dispersion parameters of atmospheric boundary layer 1.1 General Overview 1 1.2 Turbulence 4 1.2.1 Description of turbulence 5 1.2.2 Turbulent flux 5 1.2.3 Parameterization of turbulence 9 1.3 Turbulence of the atmosphere 11 1.3.1 Types of Turbulences. 11 1.3.2 Turbulence diffusion 12 1.4 Atmospheric stability 12 1. 5 Atmospheric pollution. 16 1 6 Behavior of effluent released to the atmosphere 17 1.7 Source Types 18 1.8 Atmospheric Dispersion Theories (Modeling). 19 1.8.1 Eulerian models 20 1.8.2 Lagrange models 23 1.8.3 Gaussian plume and puff models 23 1.8.4 Ground Dispersion Formula 24
1.8.5 Plume rise 26 1.9 Comparison between Some Models 28 1.9.1 Simplified methods 28 1.9.2 Pasquill Gifford type methods 28 1.9.3 Temperature lapse rate method 29 1.9.4 Wind fluctuation method 29 1.9.5. The effect of friction 31 1.10 The Planetary Boundary Layer 31 1.10 .1 Definition the Planetary Boundary Layer 31 1.10 .2 .The Depth the Planetary Boundary Layer 32 1.10 .3 .The types of the Planetary Boundary layer 32 1.10 .3.1 Definition the surface layer 32 1.10 .3.2 Definition the Ekman layer 32 1 1.11. Model evaluation methodologies 34 1.12. The objective of this study 34
Page Chapter II Analytical and Numerical Solutions of Crosswind Integrated Concentration by using Different eddy diffusivities methods 2.1 Introduction 37 2.2 Analytical Method 39 2.3 Numerical Method 42 2.4 Statistical method 43 Chapter III Maximum integrated ground level concentration under two stability classes 3.1 Introduction 56 3.2 Analytical solution 57 3.2 a In neutral case 57 3.2 .b In unstable case 62 3.3 Statically method 70 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions 4 1 Introduction 71 4.2 Proposed model and its components 72 4 3 The effective height 73 4.4 Mathematical technique 73 4. 5 Case study 79 4.6 Verification 85 Chapter V Calculating Isotopes concentrations using different schemes of dispersion parameters 5.1 Introduction 88 5.2 Gaussian distributions 89 5.3 Dispersion parameters schemes 91 3 a Power –law method 91 5.3.b Standard method 92 5.3.c Briggs method 92 5.3.d Irwin method 93 5.4 Result and discussion 94 5.5 Statistical method 112 Chapter VI Derivation the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model 6.1 Introduction 116 6.2 Model formulation 117 6.3 Results and Discussion 123 6.3 Statistical method 126 Major Conclusion 128 130 Reference 131 141 List of Figures
Page
Figure (1.1). Instantaneous smokestack plume 6 Figure (1.2) .Time series of C and w measured at fixed point M. C and w 7 are the time – averaged values Fig. (1.3) Behavior of effluent release to the atmosphere 17 Fig. (2.1) Comparison between analytical and observed normalized 50 crosswind integrated concentration. Fig. (2.2) Comparison between analytical and observed normalized 51 crosswind integrated concentration and down distance Fig. (2.3) Comparison between numerical and observed normalized 51 crosswind integrated concentration. Fig. (2.4) Comparison between numerical cross wind integrated 52 concentration and down distance. Fig. (2.5) Comparison between analytical, numerical and observed 52 normalized crosswind integrated concentration. Fig. (2.6) Comparison between analytical, numerical and observed 54 normalized crosswind Integrated concentration and down distance. Fig. (2.7) Ratio of predicted and observed normalized crosswind 52 concentrations via observed normalized crosswind concentrations for all models Fig. (3.1) show that comparison between the observed, predicated and 68 maximum ground level concentrations. Fig. (3.2) show comparisons between down distance per height and the 68 observed, predicated and maximum ground level concentrations Fig. (3.3) show comparison between maximum downwind distances the 69 observed, predicated and maximum ground level concentrations. Fig. (4.1) shows the coordinate system direction of the mean wind. 73 Fig. (4.2). The variation of the concentration of Iodine (I 138 ) with distance 76 from the reactor. Fig. (4.3). Variation of the concentration at the plume axis over emission rate 82 with effective height in neutral, stable and unstable Figure (4.4) Show the relation between the effective height and 82 Co /Q using our proposed model and power, logarithmic laws in neutral case. Fig. (4.5) Show the relation between the effective height and Co /Q using our 84 proposed model and power, logarithmic laws in stable case Figure (4.6) Show the relation between the effective height and 84 Co /Q using our proposed model and power, logarithmic laws in unstable case. Fig. (5.1) Comparison between predicated, observed and distance for 131 I 97 98 from 1 to 13 experiments in Briggs, Power law, standard and Irwin methods respectively.
Page
99 100 Fig. (5.2): Comparison between predicated, observed and distance for 135 I in case power law, standard, Briggs and Irwin methods from 1 to 13 experiments respectively Fig. (5.3): Comparison between predicated, observed and distance for 137 Cs 106 107 in case power law, standard, Irwin and Briggs methods from 1 to 13 experiments respectively. Fig. (5.4): Comparison between predicated, observed and distance for 133I in 108 109 case power law, Standard, Briggs and Irwin methods from 1 to 13 experiments respectively. Fig. (6.1).Observed and predicated ground crosswind integrated centerline concentration, normalized with emission C y(x, 0, 0)/Q: scatter diagram for the solution of equation (6.18) using equations (6.14), (6.16), (6.19), (6.20) 124 and (6.21),(6.22) Fig.(6.2).observed and predicated ground crosswind integrated concentration, 124 normalized with emissions(x, 0)/Q: scatter diagram for the solution of equation (6.17) using equations (6.16), (6.19) and (6.21)
List of table Page
Table (1.1): Estimation of Pasquill stability classes. 13 Table (2.1) Estimates of the power (p) in urban areas for six Stability Classes 47 based on information by Irwin (1979b). Table (2.2) Values of wind speed at 10 m and 115 m and downwind distance 48 through unstable and neutral stabilities at northern part of Copenhagen. Table (2.3) Comparison between Observed and different analytical, numerical 49 normalized crosswind integrated concentrations Cy/Q (10 4 sm 3). Table (2.4) Ratio of predicted and observed normalized crosswind concentrations 50 for all models Table (2.5) Comparison between our different models according to standard 55 statistical performance measure Table (3.1) Comparison between the observed integrated crosswind ground level 57 concentration and predicated ground level concentration under different stability, height and wind speed. Table (3.2) .Comparison between our two models according to standard statistical 70 performance measure Table (4.1) Wind speed, the plume rise, effective height and the concentration at 80 the Axis of the plume at the reactor release over emission rate in different stability condition. Table (4.2), Observed wind speed, the plume rise, effective height and our 81 proposed model, using power and using logarithmic Laws of the concentration at the axis of the plume at the reactor release over emission rate in neutral classes. Table (4.3), Observed wind speed, the plume rise, effective height and our 83 proposed model ,using power law and Using logarithmic Law of the concentration at the axis of the plume at the reactor release over emission rate in stable classes L=55m and β=5. Table (4.4), Observed wind speed, the plume rise, effective height and our 85 proposed model ,using power law and Using logarithmic Law of the concentration at the axis of the plume at the reactor release over emission rate in unstable classes L= 2.5m and β=15 Table (5.1). Estimated of the power (p) in urban areas for six stability classes 91 based on information by Irwin (1979). Table (5.2), values of the dispersion parameters for the Pasquill stability classes. 92 Table (5.3), values of the dispersion parameters for the Pasquill stability classes. 92 Table (5.4).Formulas produced by Briggs (1973) for σy (x) and σ z (x) 93 Table (5.5).The values of the standard deviation of the wind direction in 93 horizontal and vertical directions for different stability classes. Table (5.6).Source strength ‘Q’ (Bq) and decay distance ‘λ ‘for the studied fission 94 radionuclide and Meteorological data (downwind distance ‘x’, wind speed ‘U’, stability classes and different effective heights). Table (5.7) .Comparison between predicated and observed concentrations 95 Table (5.8). Comparison between predicated and observed concentrations for 131 I Page (standard method) from 1 to 13 experiments. 95 Table (5.9). Comparison between predicated and observed concentrations for I 131 97 (power law) from 1 to 13 experiments. Table (5.10). Comparison between predicated and observed concentrations for 97 I131 from 1 to 13 experiments in Irwin method. 101 Table (5.11). Comparison between predicated and observed concentrations for I135 in case standard method from 1 to 13 experiments. Table (5.12). Comparison between predicated and observed concentrations for 101 I135 in case Briggs method from 1to 13 experiments. Table (5.13). Comparison between predicated and observed concentrations for 102 I135 case power law from 1 to 13 experiments Table (5.14). Comparison between predicated and observed concentrations. 102 Table (5.15) .Comparison between predicated and observed concentrations for 103 Cs 137 in case power law from 1 to 13 experiments Table (5.16). Comparison between predicated and observed concentrations for 104 Cs 137 in case standard method from 1 to 13 experiments. Table (5.17). Comparison between predicated and observed concentrations for 104 Cs 137 in case Irwin method from 1 to 13 experiments. Table (5.18), Comparison between predicated and observed concentrations for 105 Cs 137 in case Briggs method from 1 to 13 experiments. Table (5.19). Comparison between predicated and observed 110 Concentrations for I 133 in case power law from 1 to 13 experiments. Table (5.20) .Comparison between predicated and observed concentrations for 110 I133 in case standard method from 1 to 13experiments. Table (5.21). Comparison between predicated and observed concentrations for 111 I133 in case Briggs method fro 1 to 13 experiments. Table (5.22). Comparison between predicated and observed concentrations and 111 distance for I 133 in case Irwin method from 1 to 13 experiments. Table (5.23).Average Predicated and observed concentration 112 for I 131 under different methods Table (5.24) .Shows average predicated and observed concentrations for I 135 112 under different methods. Table (5.25), shows average predicated and observed concentrations for Cs 137 113 under different methods. Table (5.26). Show average predicated and observed concentrations for I 133 from 113 1 to 13 experiments. Table (5.27) Comparison between our four method for I 131 , I 135 , according to 114 standard statistical performance measure Table (5.28) Comparison between our four method for I 133 and Cs 137 according to 114 standard statistical performance measure Table (6.1). Observed and Model ground – level centerline concentration C(x, 0, 0)/ 125 at different distances, wind speed and effective height from the source. Table (6.2). Observed and Model ground –level concentration 126 Cy(x, 0)/Q at different distances, wind speed and effective height from the source. Table (6.3): Comparison between different Models ground – level centerline 126 concentration C(x, 0, 0)/Q and observed concentrations. Table (6.4): Comparison between different Models ground – level concentration 127 C(x, 0,)/Q and observed concentrations.
CHAPTER I Review study on atmospheric diffusion and dispersion parameters of boundary layer
1.1 General Overview
The atmosphere is an important pathway for the transport of either airborne contaminant or radioactive releases from a nuclear power plants to the environment and thereby the man.
It is necessary therefore to have adequate information about this pathway in order to estimate the dispersion of radioactive to the population in the region and thus be able to assess the radiological impact on man. The present guide describes the meteorological phenomena and mechanisms involved in the dispersion of the released effluents in the atmosphere, discusses methods which may be to calculate the concentration and deposition in the region specifies the data needed for input to the models.
As a matter of fact, each industry has its own waste. Generally most of
these waste are hazardous to man and his environment. One of the main
and new industries is the nuclear power industry. Due to the increasing
importance of nuclear power industries and the release of different
effluents to air then through different pathways such as water and soil,
later, they will find their way to man. So it became important to study the
behavior of the radio contaminants in the environment. So to achieve an
improved understanding of the fate of contaminant radionuclide it is necessary to characterize the processes occurring in the receiving environment. This is clearly illustrated by the atmospheric dispersion models. Atmospheric dispersion modeling is essential to predicting the path and danger from an atmospheric plume of hazardous materials.
Atmospheric dispersion models are computer simulation programs, which combine information about the source of a release and observations of wind and weather conditions with theories of atmospheric behavior to predict the spread and travel of contaminants. The most widely used model is the Gaussian plume model. It is still the basic workhorse for dispersion calculations due to its great advantages of being computationally simple and for many practical applications or preliminary analysis. In the simple case of flat, homogenous rough terrain with time independent flow and source conditions and with sources located outside the area of influence of roughness elements, Gaussian plume models (GPMs) may yield acceptable estimates of concentration values. (Giovanni, L.; and Paolo, M.; (1998)).
The heart of any atmospheric diffusion model is to estimate the concentration of a pollutant at a particular receptor point by calculating from some basic information about the source of the pollutants, surrounding buildings and the meteorological conditions. To determine the dispersion conditions of the lower atmosphere, the wind velocity and wind direction must be known, as well as stability parameters, which include the temperature gradient (temperature lapse rate), solar radiation, cloud amount and fluctuation of wind direction (σθ). Such information could be obtained by an intensive field measurement program of several weeks in all distributed in the various seasons. One of the most important parameters in plume dispersion modeling is the plume growth, more commonly referred to as dispersion coefficients (σ) these plume width parameters grow with diffusion time and their rate of growth depends on meteorological variables. Various parameterizations exist for the vertical and horizontal plume dispersion (σy and σz standard deviation of distribution of concentration in the lateral and vertical directions respectively).Radioactive air concentrations can be calculated by using these various dispersion parameters. By comparing these results with that observed, the best method can be determined. The variation of the horizontal wind speed and direction with height is important in evaluating diffusion from stacks. The wind speed must be zero at the earth's surface and then rise to the gradient value at a few hundred meters. The Friction force causes the air to be turned slightly toward the low pressure with an angle of 15 o by day over smooth surfaces, changing to as much as 50 o by night over rough terrain.
Above the surface the wind turns slowly in a clockwise direction, so that this angle is reduced with height until at a few hundred meters the observed wind is equal to the gradient wind and runs parallel to the isobars. The variation of wind direction with altitude is known as the Ekman Spiral
(Anli, K.; and Maithili, S.; (1996)). So the atmosphere near the surface of the earth can be divided into three layers: the free atmosphere, the Ekman layer and the surface layer. The Ekman layer and the surface layer constitute the planetary boundary layer. A good first approximation for wind speed at some upper elevation is given by:
p U z=U 1*(Z\Z1) (1.1 ) where: U z: is the wind speed at some upper elevation (z).
U1: is the wind speed at reference height (z 1).
P: is a varying exponent (from 12 50), which increases with stability and is affected by surface roughness.
1.2 Turbulence
So far we have discussed the role of buoyancy in driving vertical motions in the atmosphere, but we have yet to quantify the rates of vertical transport. Buoyancy in an unstable atmosphere accelerates both upward and downward motions; there is no preferred direction of motion. One observes considerable irregularity in the vertical flow field a characteristic known as turbulence. In this section we describe the turbulent nature of atmospheric flow, obtain general expressions for calculating turbulent transport rates and infer characteristic times for vertical transport in the atmosphere.
1.2.1 Description of turbulence
There are two limiting regimes for fluid flow: laminar and turbulent.
Laminar flow is smooth and steady; turbulent flow is irregular and fluctuating. One finds empirically (and can justify to some extent theoretically) that whether a flow is laminar or turbulent depends on its dimensionless Reynolds number Re:
Re=UL/ ϑ (1.2)
Where U is the mean speed of the flow, L is a characteristic length defining the scale of the flow and ϑ is the kinematics viscosity of the fluid
( ϑ = 1.3x10 5 m 2 s 1 for air at 273K and 1atm). The transition from laminar to turbulent flow takes place at Reynolds numbers in the range
1000 10,000. Flows in the atmosphere are generally turbulent because the relevant values of U and L are large. This turbulence is evident when one observes the dispersion of a combustion plume emanation from a cigarette, a barbecue or a smokestack.
1.2.2 Turbulent flux
Consider a smokestack discharging a pollutant X (Figure 1 1). We wish to determine the vertical flux F (molecules cm 2 s 1) of X at some point M downwind of the stack. The number of molecules of X crossing a horizontal surface area dA centered on M during time dt is equal to the
number nXwdtdA of molecules in the volume element wdtdA, where w is the vertical wind velocity measured at point M and n X is the number concentration of X. The flux F at point M is obtained by normalizing to unit area and unit time:
Figure (1.1). Instantaneous smokestack plume
nw dt dA F=x = nwnCw = dt dA x a x (1.3)
Where n a is the number air density of air and C X is the mixing ratio of x.
We will drop the subscript X in what follows. From equation (1.3), we can determine the vertical flux F of pollutant X at point M by continuous measurement of C and w. Due to the turbulent nature of the flow, both C and w show large fluctuations with time, as illustrated schematically in
Figure (1 1) (Daniel J.Jacob,1999). Since C and w are fluctuating quantities, so is F. We are not really interested in the instantaneous value of
F, which is effectively random, but in the mean value F = n aCw over a useful interval of time t (typically 1 hour). Let C and w represent the mean values of C and w over t.
Figure (1.2) .Time series of C and w measured at fixed point M. C and w are the time – averaged values.
We decompose C (t) and w (t) as the sums of mean and fluctuating components:
Ct() = Ct()() + Ct' (1.4 ) wt() = wt()() + wt'
Where C and w are the fluctuating components; by definition, C = 0 and w = 0. Replacing (1.4) into (1.3) yields:
′ ′ ′′ F= na ( Cw +++ Cw Cw Cw ) = na () Cw+ Cw′ + Cw ′ + Cw ′′ = (1.5) na () Cw+ Cw′ ′ The first term on the right hand side F A = n aCw is the mean adjective flux driven by the mean vertical wind with second term, F T= n aCw is turbulent flux driven by the covariance between C and w. The mean wind w is generally very small relative to w because atmospheric turbulence applies equally to upward and downward motions, as discussed above. In the troposphere, F T usually dominates over F A in determining rates of vertical transport. One can apply the same distinction between mean adjective flux and turbulent flux to horizontal motions. Mean winds in the horizontal direction are ~1000 times faster than in the vertical direction, and are more organized; so that F A usually dominates over F T as long as t is not too large (say less than a day). You should appreciate that the distinction between mean adjective flux and turbulent flux depends on the choice of
t; the larger t, the greater the relative importance of the turbulent flux.
To understand the physical meaning of the turbulent flux considers our point M located above the centerline of the smokestack plume. Air parcels moving upward through M contain higher pollutant concentrations than air parcels moving downward; therefore, even with zero mean vertical motion of air there is a net upward flux of pollutants. An analogy can be drawn to a train commuting back and forth between the suburbs and the city during the morning rush hour. The train is full traveling from the suburbs to the city and empty traveling from the city to the suburbs. Even though there is no net motion of the train when averaged over a number of trips (the train is just moving back and forth) there is a net flow of commuters from the suburbs to the city. The use of collocated high frequency measurements of
C and w to obtain the vertical flux of a species as described above is called the eddy correlation technique. It is so called because it involves determination of the covariance, or correlation, between the “eddy”
(fluctuating) components of C and w.
1.2.3 Parameterization of turbulence
So far, our discussion of atmospheric turbulence has been strictly empirical. In fact, no satisfactory theory exists to describe the characteristics of turbulent flow in a fundamental manner. In atmospheric chemistry models one must resort to empirical parameterizations to estimate turbulent fluxes. We present here the simplest and most widely used of these parameterizations. Let us consider the smokestack plume described previously. The instantaneous plume shows large fluctuations but a time averaged photograph would show a smoother structure (In this time averaged, smoothed plume there is a well defined plume centerline, and a decrease of pollutant mixing ratios on both sides of this centerline that can be approximated as Gaussian. We draw a parallel to the Gaussian spreading in molecular diffusion, which is a consequence of the linear relationship between the diffusion flux and the gradient of the species mixing ratio (Fick’s Law): ∂C F= − n D (1.6) a ∂ z
Here F is the molecular diffusion flux and D (cm 2 s 1) is the molecular diffusion coefficient. Fick’s law is the postulate on which the theory of molecular diffusion is built. Molecular diffusion is far too slow to contribute significantly to atmospheric transport but the dispersion process resulting from turbulent air motions resembles that from molecular diffusion. We define therefore by analogy an empirical turbulent diffusion coefficient k z as:
∂C F= − n k (1.7) a z ∂ z
Where F the turbulent flux and C is now is the time averaged mixing ratio.
Equation (1.7) defines the turbulent diffusion Parameterization of turbulence. Because k z is an empirical quantity, it needs to be defined experimentally by concurrent measurements of F and ∂C/∂z. The resulting value would be of little interest if it did not have some generality. In practice, one finds that k z does not depend much on the nature of the diffusing species and can be expressed with some reliability in the lower troposphere as a function of:
(a) The wind speed and surface roughness (which determine the mechanical turbulence arising from the collision of the flow with obstacles). (b) The heating of the surface (which determines the buoyant turbulence)
(c) The altitude (which determines the size of the turbulent eddies).
2 5 2 1 Order of magnitude values for k z are 10 10 cm s in a stable
atmosphere,10 4 10 6 cm 2 s 1 in a near neutral atmosphere and 10 5 10 7cm 2 s 1
in an unstable atmosphere.
1.3 Turbulence of the atmosphere
Turbulence flow represented by eddy motion (an eddy motion is very important in diluting concentrations of pollutants).
1.3.1 Types of Turbulences
Atmospheric turbulence or eddies may be classified as mechanical and thermal turbulence:
Mechanical turbulence is caused by air motion over the rough surface of the earth, either natural or man made (shearing stress). This type of turbulence prevails on windy nights with neutral advection.
Thermal turbulence caused by solar heating (convection) during the day this turbulence prevails on sunny days with light winds. When the surface is smooth the airflow is smooth and the velocity profile becomes very steep near the ground. For rougher surfaces more mechanical turbulence generated and the velocity profile becomes less steep and reaches deeper into the atmosphere. The most useful quantitative description of turbulence is the statistical standard deviation of the wind fluctuations taken over a period of approximately one hour. 1.3.2 Turbulence diffusion
Eddy diffusion is the transport of pollutants from regions of high concentrations to region of low concentrations depending on the fluctuations of the wind and the strength of the average wind. Wind distribution, temperature gradient, and stability variables must be known to determine the effluent trajectory. The more rapid the decrease of temperature with height, the more unstable the air and the greater the vertical motions. During the daytime, when incoming solar radiation is greatest, the air tends to be unstable, and the lighter the wind, the greater the turbulence. Then vertical mixing brings pollutants downward from the free atmosphere to the surface layer, and these can be deposited to the ground surface at night, although eddy diffusion serves as a process for carrying pollutants upward from the ground, but this is not always the case. Eddy sizes increase with altitude and so does the rate of spreading of a plume of contaminants. Very large eddies often appearing with thermal turbulence during a sunny afternoon result in large motions of the plume but little dispersion.
1.4 Atmospheric stability
Atmospheric stability is its tendency to resist or enhance vertical motion.
Intensity of turbulence and atmospheric diffusion are strongly dependent upon atmospheric stability. It is an important parameter in models used to estimate downwind concentration of radionuclide relaqeased by nuclear facilities, Pasquill, (1961) assumed that the concentration distribution is Gaussian both in the horizontal as well as in the vertical. He introduced the concept of expressing atmospheric stability in terms of stability categories ranging from A, most unstable, to F, most stable. Later a more stable category, G, was added. Determination of stability classes categories:
Table (1.1): Estimation of Pasquill stability classes.
Surface wind speed at 10 m Solar Radiation Night time cloud cover
(m/sec) Strong Moderate fraction
Slight > 4/8 < 3/8
<2 A A B B
2 3 A B B C E F
3 5 B B C C D E
5 6 C C D D D D
>6 C D D D D
A Stability category can be determined according to observed wind speed, cloud cover and insulation conditions as shown in table (1.1).
Where:
A: extremely unstable B: moderately unstable
C: slightly stable D: neutral
E: slightly stable F: moderately stable
B Stability with respect to vertical motions is more precisely referred as
‘hydrostatic stability’. Atmosphere is either:
1 Unstable: Buoyancy forces enhance vertical motion,
2 Stable: Buoyancy forces oppose vertical motion and 3 Neutral: Buoyancy forces vanish.
The lapse rate in the lower portion of the atmosphere has a great influence on the vertical motion of air. So, stability depends on the temperature lapse rate.
When a rising parcel of air is warmer than the surrounding air, it is accelerated upward and the atmosphere is said to be unstable. The vertical acceleration “a” is given by: a = g. (T a T e)/T a (1.8 )
Where: g is the acceleration of gravity.
Ta is the temperature of the rising or descending air and.
Te is the temperature of the environmental air (Kelvin).
Under certain atmospheric conditions, when a volume of air is lifted to a higher altitude, it arrives there colder and denser than the surrounding air. In this case, there is a downward buoyancy force and the volume of displaced air is forced back toward the level of origin. In such circumstances, the atmosphere is stable. If this volume of arising air having the same density as the surrounding air, there will be no net buoyancy force. The volume will be remaining at the new level, and the atmosphere is said to have neutral stability.
Stability criteria for dry and moist air:
Stability criteria for dry air may thus be summarized with respect to height as:
1 ∂θ / ∂z > 0 (γ < Γ, stable) 2 ∂θ / ∂z = 0 (γ = Γ, neutral)
3 ∂θ / ∂z < 0 (γ > Γ, unstable)
Where: ∂θ is the potential temperature.
∂z is the small distance, in which the air will move.
γ is the ambient (or actual lapse rate) and
Γ is the dry adiabatic lapse rate (= 0.98 oC/10m).
Stability criteria for moist air are:
1 γ < Γs absolutely stable
2 γ = Γs saturated neutral
3 Γs < γ < Γ conditionally unstable
4 γ = Γ dry neutral
5 γ > Γ absolutely unstable.
Where Γs is the adiabatic lapse rate for stable condition.
Another way for determining the stability class’s categories is the fluctuation of wind direction. By using the standard deviation of wind direction fluctuations in the lateral (horizontal) ( σy) and vertical ( σz) directions. σy and σz differ according to height of release, building wake effects and surface roughness. For example, σy values for stable classes are greater (for flat desert area) than those for non desert areas, while the σz values are lower). σy and σz provide the best estimates of the true stability, but ( ∂T/ ∂z) method is very commonly used because it is easier and less expensive, The most widely used ( σy and σz) correlation based on the
Pasquill stability classes have been those developed by (Gifford (1961)).
Analytical expressions for σy and σz as functions of x can be represented by the power low expressions (Which is more useful for downwind distances less than 10 km),
p σy = ax (1.9 )
q σz = bx (1.10)
Where "x" in meters, and a, b and q depend on the stability class and the
averaging time.
1. 5 Atmospheric pollution
Atmospheric pollution is the presence of substances in the ambient atmosphere, resulting from the activity of man or from natural processes, causing adverse effects to man and the environment. Or it is the presence of substances or energy in the atmosphere in such quantities and of such duration liable to cause harm to human, plant, or structures, or changes in the weather and climate, or interference with the comfortable enjoyment of life or property or other human activities. From produced radionuclide, the most significant radionuclide is 14 C, 90 Sr, 137 Cs and Pu isotopes. Following rapid upward convective movement in the atmosphere, these radionuclide were deposited on the surface of the earth as fallout, which comprises three components, namely: local (12%), troposphere (10%) and stratospheric (78%). Troposphere fallout was dominantly dispersed in the latitude of the test, whereas stratospheric fallout was globally dispersed. The dispersion of air borne substances in the atmosphere depends on two different transport mechanisms: For one, air pollutants are carried along with existing wind flows (advection) (mean airflow that transports the pollutants downwind), for another they are distributed by diffusion movements due to atmospheric turbulence (turbulent velocity fluctuations that disperse the pollutants in all directions.
1-6- Behavior of effluent released to the atmosphere
Fig. (1.3). Behavior of effluent release to the atmosphere
When the effluent enters the atmosphere with a certain velocity and temperature, which is generally different from those of the ambient atmosphere, it has a vertical component due to the effects of vertical velocity and difference of temperature until they are dissipated. This upward rise of the effluents is called plume rise ( h) figure (1.3). The movement of the effluent with the wind with the air, this process is called atmospheric diffusion. The combination of transport and diffusion is called atmospheric dispersion. The effluent, while undergoing plume rise transport and diffusion, may also be subject to processes such as:
Radioactive decay and build up of daughter products.
Wet deposition:
Rainout / snow out, vapor or aerosol is scavenged by water droplets or snowflakes in cloud and falls out as precipitation.
Washout, vapor or aerosol is scavenged below the rain cloud by falling precipitation and Fogging vapor or aerosol is scavenged by the water droplets present in the fog.
Dry deposition: a Sedimentation of aerosols or gravitational settling (for particulate diameter is too large. b Impaction of aerosols and absorption of vapors and gases on obstacles in the path of the wind. c Formation of aerosols and coalescence of aerosols.
Re suspension of materials deposited on surfaces.
1.7 Source Types
In atmospheric dispersion calculations, the different kind’s sources are
usually classified in terms of spatial configuration and duration of release.
There are two types, Instantaneous sources classified according to release time as: puff (a release time of a few seconds up to a few minutes for a
travel time of a few hours) and short period (a release time of up to a few
hours).According to spatial configuration there are: point sources, line
sources, area sources and volume sources. The primary difference between
a long line source and an isolated point source is the elimination of the
lateral spreading of the effluent as a factor in reducing concentration. Line
sources and area sources are not generally encountered in the analysis of
radioactive releases from nuclear power plants, because the air
concentration and deposition rate calculations become complicated. But
they can be treated by subdivision into a number of effective point sources,
which were analytically integrated in the x and y directions. The area
sources are regarded as a series of line sources confined in a circle, it is
assumed to be homogenous and circular. Dispersion during the daytime
depends on the location of the source. It occurs under stable conditions
when mechanical turbulence is suppressed. The vertical component is
suppressed more than the horizontal with the result that plume width is
greater than its thickness. It is most likely at night when the earth’s surface
is cooled by outgoing radiation. Such that the observer has the same
position relative to plume’s centerline.
1.8 Atmospheric Dispersion Theories (Modeling)
Atmospheric dispersion modeling is the use of numerical methods to compute the time history of air pollutant concentration. This means that the central purpose of dispersion modeling is to describe the relationship between pollutant emission, transmission, and ambient air concentrations of one or several air pollutants as a function of space and time in a mathematically exact way. Diffusion models have been grouped according to the complexity of the numerical modeling required, by source configuration, by the distance of material transport and by their relation to particular regulatory requirements. Such methods can take into account the space time variations of the wind and stability field terrain complexity as well as the various factors that affect the concentration. Theoretically there are three approaches: a The gradient transfer (K theory approach). b The statistical theory approach. c The dimensionless analysis approach (similarity theories).
A fourth class of diffusion theory called second or more generally higher order closure theory.
There are two basic, broad category models:
1.8.1- Eulerian models
Which solve the advection diffusion equation on a fixed grid. It is. The
common way of treating heat and mass transfer phenomena Eulerian
methods are typically applied when complex emission sceneries are
considered. Requiring solutions at all grid points, i.e. the Eulerian method
is based on carrying out a material balance over an infinitesimal region fixed in space. By performing a mass balance on a small control volume, a simplified diffusion equation, which describes a continuous cloud of material dispersing in a turbulent flow, can be written as (Robert
Macdonald .(2003)): dC dC d dC d dC d dC +U = Kx + Ky + Kz +S (1.11) td d x dx d x dy d y d z d z where:
C is the average concentration of diffusing point (x, y, z) (kg/m 3).
U is mean wind velocity along the x axis (m/s).
Kx, K y and K z are the eddy diffusivities coefficients along x, y and z axes respectively (m 2/s). x is along –winds coordinate measured in wind direction from the source
(m). y is cross wind coordinate direction (m). z is vertical coordinate measured from the ground (m).
S is source/ sinks term (kg/m 3 s).
A term –by term interpretation of equation (1.11) is as follows: dC dC +U dt dx is time rate of change and advection of the cloud by the mean wind.
d dC d dC K K y z dy dy and dz dz represent turbulent diffusion of material relative to the center of the pollutant cloud. (The cloud will expand over time due to these terms).
S source term which represents net production (or destruction) of pollutant due to sources (or removal).
Equation (1.11) is grossly simplified, due to the following assumptions:
1) The pollutant concentrations do not affect the flow field (passive dispersion).
2) Molecular diffusion and longitudinal (along wind) diffusion are neglected.
3) The flow is incompressible.
The wind velocities and concentrations can be decomposed into a mean and fluctuating component with the average value of the fluctuating
(stochastic) component equal to zero. Mean values are based on time averages of 10 60 minutes.
4) The turbulent fluxes are linearly to the gradients of the mean concentrations as in equation (1.11).
5) The mean lateral and vertical wind velocities v and w are zero, so we have also restricted our analysis to steady wind flow over an idealized flat terrain.
1.8.2- Lagrange models
In which the advection and diffusion components are calculated independently. Concentration changes are described relative to the moving fluid. The variations of the ensemble average concentration and other dispersion quantities are then derived from the calculated trajectory distribution. This approach has simplicity, flexibility, and ability to incorporate temporal and spatial variations in turbulence properties, in complex terrain, where other techniques such as similarity or gradient transfer theory are inappropriate (although the gradient transfer theory is relatively easy to apply and it is physically more accurate) and there are some models which either Lagrange, Eulerian or mixed of them; such as:
1.8.3 Gaussian plume and puff models
This is Eulerian or Lagrange model. It is the most widely used model in flat homogenous rough terrain with time independent flow and source conditions. It is a simple, straighter forward and makes use of readily available meteorological data. It is usually considered adequate for the short range, up to tens of km from the source. It is also used to describe the dispersion of airborne matter in cases where dry deposition had to be taken into account.The mean value of air concentration"χ"of a radionuclide (or any other airborne material) is a function of position and time χ = χ(x, y, z, t).
The object of an atmospheric transport and diffusion model is to specify χ and its frequency distribution at any point in space. The Gaussian Plume formula for concentration from a continuous point source of strength Q without interference from the ground at a mean wind speed U measured at plume level is:
2 2 2 2 χ (x, y, z)= [Q/ {2πσ yσz U} ] exp [ 1/2 {(y /σy ) + (z /σz )] (1.12) where:
χ is the mean concentration of the effluent at a point (x, y, z) (g.m 3 or
Ci.m 3).
Q is the “source strength” i.e. the rate at which the effluents are released sometimes called the emission rate (Ci s 1), (also, refer to total release (Ci))
U is the mean wind speed (m s 1). x, y, z refer to a co ordinate system at the center of the moving cloud with the x, y, z in downwind, crosswind and vertical directions respectively and σi (i = x, y, z) are the plume dispersion coefficients or plume growth (standard deviation of distribution of concentration) in the x, y, and z directions (m), Equation (1.11) assumes a straight line plume in the direction of the wind and does not take into account wind direction changes in space and time. The Gaussian formula is only an approximation. In practice departures from this approximation may be encountered especially in the vertical direction (z) in strong wind shear and large distances.
1.8.4-Ground Dispersion Formula
The dispersion formula for continuous release from a point source at effective height H is derived from the basic equation (1.12) by assuming perfect reflection at the ground surface. Taking the origin on the ground vertically below the source (e.g. at the base of the stack) and adding the effect of the mirror image source gives:
2 2 2 2 χ (x,y,z) =[Q/(2πσ yσz U)] exp ( y /2σy )* {exp [ (Z H) /2σz ] +exp [
2 2 (Z+H) /2σz ]} (1.13)
Where the second of the exponential terms within the curly brackets in the term due to reflection (Z H) is the above ground real source and (Z+H) is the imaginary source below the ground). The mean concentration for a ground level source with no plume is given by setting H=o in equation (1.13).
2 2 2 2 χ(x,y,z)=[Q/(πσ yσzU)] exp. ( 1/2[y /σy +Z /σz ]) (1.14)
The right hand side of equation (1.13) contains a multiplicative factor of 2 which is a conventional device to account for the assumed reflection of the plume by the ground plane. The mean ground level concentration for an elevated release is given by setting Z=o in equation (1.14):
2 2 2 2 χ (x,y,0) = [Q/(πσ yσzU)]exp ( y /2σy + H /2σz ) (1.15)
For elevated sources, the average plume centerline, defined by the mean maximum concentration, descended within a short distance from the source until it reached the ground. The position of this maximum concentration can be explained by the probability distribution of the vertical velocity. Ground level concentration (GLC) being sensitive to the ratio of plume height to σ z
GLC is the maximum in the neighborhood of the stack. The worst case of
GLC (twice a concentration) may be occurring when the base of the inversion layer coincides with the effective stack height.
Also a “worst” wind speed exists at which the ground level concentration is a maximum at any downwind location. This maximum is different from the largest ground level concentration that is achieved as a function of downwind distance for any given wind speed. This is due to that wind speed u affects the GLC through two terms having opposite effects: Firstly: the
GLC is inversely proportional to wind speed, such that the lighter the wind the larger the GLC. Secondly : the plume rise h is inversely proportional to wind speed, such that the lighter the wind, the higher the plume rises and the smaller the ground level concentration. To investigate the maximum GLC we must define plume rise and the location χm
1.8.5-Plume rise
Most chimney plumes, especially those resulting from combustion processes, are made up of gases whose composition is very similar to air.
The density difference comes about due to the higher than ambient temperature in the plume. This higher temperature and the vertical velocity of the exhaust gases cause the plume to be buoyant and induce the phenomenon called plume rise. If the exhaust gas density is lighter than the density of the surrounding air, the plume has a positive buoyancy and vice versa. A plume of hot gases passage through 4 stages:
Initial or Jet phase, in which the vertical momentum is converted to horizontal within a distance of about 2 5 stack diameters above the stack.
Thermal phase, in which mixing occurred due to self generated turbulence.
The most dominant effect is the total excess of heat.
Breakup phase, in which atmospheric turbulence dominated the mixing and the plume breakup into distinct parcels.
Diffuse phase, in which plume comes together again with more diffusion,
The effective stack height (H) is taken to be the sum of the actual stack height h s and the plume rise h [figure (1.3)] defined as the height at which the plume becomes passive and subsequently follow the ambient air motion,
H = h s + h (1.16)
Based on the initial source conditions, plume can be categorized as buoyant, forced, and jet plume. Plume rise formula can be expressed as:
b a h=E x / U (1.17)
Where a, b and E depending on the stability class.
To investigate the properties of the maximum GLC with respect to distance from the source and wind speed, we begin with the Gaussian plume equation evaluated along the plume centerline (y=0) at the ground (z=0) and at x = χm (the distance of maximum concentration) then: 2 2 χ (x,0,0)=[Q/(πσ yσzU)] exp ( H /2σz ) (1.17 )
We want to calculate the location χm of the maximum GLC for any given wind speed. By differentiating equation (1.16) with respect to χ and setting the resulting equation = 0 we find that:
2 3 2 (1/σy*dσy/dx) – (1/σz*dσz/dx) +(h /σz *dσz/dx) – (h/σz *dh/dx)=0 (1.18)
By using the expressions equations (1.16), (1.17), (1.18) and (1.15) for H, σy and σz as a function of χ, we obtain the following implicit equation for χm,
2 χm ={ 2Q/eπUH }*(σz/σy) (1.19 )
½ This maximum value occurs at the distance where σz = H/ (2)
1.9- Comparison between Some Models
1.9.1- Simplified methods
It is particularly useful when there is no site specific meteorological information for the site and comparisons with other variables such as population distribution have to be made, or when one is only dealing with small releases. The disadvantage of this method is that the reliability of dispersion estimates from such models is not good but may be used in preliminary site survey.
1.9.2- Pasquill-Gifford type methods
Here, Six stability classes from A to F are used as following: 1 Very Unstable (A) 2 Moderately Unstable (B)
3 Slightly Unstable (C) 4 Neutral (D)
5 Slightly Stable (E) 6 Moderately Stable (F).
The σy and σz are obtained from graphs as functions of the downwind distance for each stability class. Here the advantage is that the isolation data from a nearby weather station may be considered to be acceptable for the site. But the disadvantage is that the method seems to give less precision in the classification of a stable situation than an unstable one.
1.9.3- Temperature lapse rate method
It uses the bulk vertical temperature gradient between two levels in the atmosphere to characterize both the horizontal and vertical turbulence. The relationship is generally applicable in smooth and even terrain, it may require some modification if the climatic zone is different. The advantage of this method is that the parameter T/ Z is reasonably simple to measure even in very low wind speed
1.9.4- Wind fluctuation method
Here, σy and σz can be obtained from the quantity σθ (τ x/Uβ) and the quantity σΦ (τ x/Uβ), which are the standard deviation of wind direction fluctuations in the lateral and vertical directions. The evaluation of σθ and σΦ should be done at the plume height level, which for elevated releases may be above 100 m. The advantage of this method is that, it is a direct indication of dispersion and that the change in stability conditions can be continuously seen on a strip chart recorder. While the disadvantage is that, it is difficult to design butane sensitive enough to give good indication at low wind speeds and yet rugged enough to withstand harsh field conditions. The bushier in is sensitive to precipitation, birds, etc., which tend to cause inaccuracies in the value. The concentrations in the atmosphere are controlled by four types of processes:
• Emissions. Chemical species are emitted to the atmosphere by a variety
of sources. Some of these sources, such as fossil fuel combustion, originate
from human activity and are called anthropogenic. Others, such as
photosynthesis of oxygen, originate from natural functions of biological
organisms and are called biogenic. Still others, such as volcanoes, originate
from non biogenic natural processes.
• Chemistry. Reactions in the atmosphere can lead to the formation and
removal of species.
• Transport. Winds transport atmospheric species away from their point of
origin.
• Deposition. All material in the atmosphere is eventually deposited back to
the Earth’s surface. Escape from the atmosphere to outer space is
negligible because of the Earth’s gravitational pull. Deposition takes two
forms: “dry deposition” involving direct reaction or absorption at the Earth’s surface, and “wet deposition” involving scavenging by
precipitation. A general mathematical approach to describe how the above
processes determine the atmospheric concentrations of species will be
given in the form of the continuity equation. Because of the complexity and
variability of the processes involved, the continuity equation cannot be
solved exactly, these two models represent respectively the simplest
applications of the Eulerian and Lagrangian approaches to obtain
approximate solutions of the continuity equation.
1.9.5. The effect of friction
Near the surface of the earth, an additional horizontal force exerted on
the atmosphere is the friction force. As air travels near the surface m it
loses momentum to obstacles such as trees m, buildings, or ocean waves.
1.10 The Planetary Boundary Layer
1.10 .1 Definition the Planetary Boundary Layer
The neglect of the frictional forces due to molecular viscosity and heating
Due to molecular diffusion can be justified on the basis of scale analysis.
Near the ground strong wind shears and surface heating continually lead to the development of turbulent eddies. These eddies are very effective mixing agents which serve to transfer heat and water vapor away from the earth's surface, and momentum towards the earth's surface at a rate many orders of magnitude faster than the mixing rate for molecular diffusion. This turbulent transport has appreciable influence on the motion throughout a layer; this layer is called (The Planetary Boundary Layer)
1.10 .2 .The Depth the Planetary Boundary Layer
Depth may range from about 30m in conditions of large static stability to
more than 3km in highly convective conditions. For average multitude
conditions the planetary boundary layer extends through the lowest
kilometer of the atmosphere and it conditions about 10% of mass of the
atmosphere. For a statically stable atmosphere, the turbulent mixing in the
boundary layer is generated primarily by dynamical instability due to the
strong vertical shear of the wind near the ground.
1.10 .3 .The types of the Planetary Boundary layer
The turbulent mixing is mechanically driven –not thermally driven. In
case the planetary boundary layers divide into two sub layers (the surface
and Ekman layers).
1.10 .3.1 Definition the surface layer
The surface layer is confined to the lowest few meters of the atmosphere
is a layer in which the velocity profile is adjusted so that the horizontal
frictional stress is nearly independent of height.
1.10 .3.2 Definition the Ekman layer
The Ekman layer which extends from the top of the surface layer to a height of about 1km, is a layer which there is a three way balance between the carioles force, pressure gradient force and viscous stress .The Ekman layer does provide a qualitatively valid description of those dynamical properties of the boundary layers flow which are important for the analysis of the multitude synoptic scale motions. This study duo to knowledge of the structure and amplitude of the turbulent eddies for the vertical momentum transport.
Power low
The wind gradient was describing wind profile at the lower from 0.5 to
13m height:
− B ∂ u v ∗ z l = (a) ∂ z N z 0 z 0
θ5 −θ 2.0 Where B is a function of Richardson's number ( B = f ) and u is u 2 the mean wind speed in the consider layer. Integrating the equation (a), we get that:
l − B v z u = ∗ − 1 N l − B z () 0 Rossby (MohammedA.El Shahawy, 2000) found that the mixing length can
be expressed not only through z but also roughness parameters z 0 must be taken into consideration l = N(z − z0 ).For the special case when ground is planted or irrigated by introduced the height of visitation l = N(z − d ) Laukhtman found that the non dimensional coefficients of turbulence can be z expressed terms of z = and stratratrifications n l
1− ξ k n = Az n where A is a constant, ξ is a function of stratification. The coefficient of turbulence 1 − ξ k = ANvlz 1.11. Model evaluation methodologies The following standard statistical performance measures that characterize
the agreement between prediction (Cp=Cpred/Q) and observations
(Co=Cobs/Q):
Where σ p and σ o are the standard deviations of C p and C o
respectively. Here the over bars indicate the average over all
measurements (N m). A perfect model would have the following
idealized performance: NMSE = FB = 0 and COR = FAC2 = 1.0
1.12. The objective of this study 1 Advection diffusion equation is solved in two dimensional space (x, z) using Laplace transform and Adomian decomposition methods to obtain the normalized crosswind integrated concentration employing analytical and numerical forms respectively. Two forms models of the eddy diffusivities as well as the wind speed at the released point were used in the solution. Two calculated models were compared with observed data measured at Copenhagen in Denmark by using statistical technique. 2 The advection diffusion equation (ADE) is solved in two directions ways to obtain the crosswind integrated ground level concentration in neutral and unstable conditions. Laplace transform technique was used considering the wind speed and eddy diffusivity depends on the vertical height. The maximum ground level concentration is estimated. Comparison between observed data from Copenhagen (Denmark).
3 A model is suggested for the diffusion of material from a point source in
an urban atmosphere is incorporated. The plume is assumed to have a well
defined edge at which the concentration falls to zero. The vertical wind
shear is estimated using combination between logarithmic and power laws
under different stability conditions. The problem of diffusion and advection
of conservative material as it travels downwind is calculated. The
concentrations estimated from this model were compared favorably with
the field calculated of other investigators such as power and logarithmic
law models. Also we calculate the ground level concentration of the Iodine
(I 138 ) which agrees with the observed concentration value after adjusting its
source strength.
4 We use Gaussian plume formula in order to estimate concentration from
a continuous point source of strength Q with interference from the ground at mean wind speed U and taking the dilution factors. Using four methods such as power law, standard, Briggs and Irwin methods to calculate the dispersion parameters σ y and σ z and comparison between predicated and
131 133 135 137 observed concentrations at different distances for I, I, I and Cs, respectively.
5 To estimate the schemes of dispersion parameters in the lateral direction
(σ y) and the vertical direction (σ z) in unstable stability by using wind speed in power law and comparing between our work and (algebraic and integral formulations ) with observed data of SF 6 are taken from Copenhagen in
Denmark. CHAPTER II Analytical and Numerical Solutions of Crosswind Integrated Concentration by using Different eddy diffusivities methods
2.1-Introduction
Analytical solution for the Eulerian and Lagrangian particle models are usually obtained just for stationary conditions and by assuming strong assumptions on the wind speed profiles and turbulent parameters. In the analytical solutions of the diffusion – advection equation, authors assumed constant wind velocity along the whole Planetary Boundary Layer (PBL) or following a power law wind velocity (Van Ulden, 1978; Pasquill and Smith,
1983; Seinfeld, 1986.Tirabassi et al., 1986; Sharan et al., 1996). (Lin and
Hildemann, 1997). (Essa et al. 2007). (Adomian, 1994). (Adomian, 1988).
Wazwaz (2001). ADM, El Sayed and Abdel – Aziz (2003). El gamel (2007).
(John, 2011).
The advection and diffusion of emitted pollutants from area sources are one of the very important problems because of bearing its direct effect on calculating dispersion of containment of urban area.
Air dispersion model based on its analytical solution had several advantages over numerical models because all of the parameters are explicitly expressed in mathematical form. Where the Mathematical techniques can properly predict dispersion and transport of atmospheric pollutants are an essential element in the development of warning and control strategies , proper forecast of atmospheric boundary –layer height and its vertical mean wind speed which provide a basis for predictions of air concentrations under meteorological conditions that vary horizontally and vertically.
Analytical solutions are useful in examining the accuracy and performance of the numerical models through studies of the analytical solution that allows valuable insight to be gained regarding the behavior of a system [Essa et al.
(2007)]. The Adomian decomposition method (ADM) has been applied to wide class of stochastic and deterministic problems in many interesting mathematical and physical areas (Adomian, 1994). Adomian gave a review of the decomposition method in (Adomian, 1988). Bellomo and Monaco (1985) have used ADM in solving random nonlinear differential equations, Wazwaz (2001) found the numerical solution of sixth order boundary value problem by ADM,
El Sayed and Abdel – Aziz (2003) compared between Adomian decomposition method and wavelet – Galerkin method for solving integral differential equations. El Gamel (2007) compared between the Sine –Galerkin and the modified decomposition methods for two – point boundary –value problems.
In this chapter, advection diffusion equation is solved in two dimensional space (x, z) using Laplace transform and Adomian decomposition methods to obtain the normalized crosswind integrated concentration employing analytical and numerical forms respectively. Two forms models of the eddy diffusivities as well as the wind speed at the released point were used in the solution. Two calculated models were compared with observed data measured at Copenhagen in Denmark by using statistical technique.
2.2- Analytical Method
Referred to equation (1.11). For steady state, taking dc/dt=0 and the diffusion in the x axis direction are assumed to be zero compared with the adjective in the same directions, hence:
∂ C ∂ ∂ C ∂ ∂ C U = K y + K z (2.1 ) ∂ x ∂ y ∂ y ∂ z ∂ z we must assume that k y =k z =k(x)
Integrating equation ( 2.1) with respect to y,we obtain the normalized crosswind integrated concentration c y (x,z) of contaminant at a point (x,z) of the atmospheric advection–diffusion equation is on the form
(Essa et al. 2006):
2 ∂ C y (x, z) ∂C y (x, z) k =u (2.2 ) ∂ z ∂ x
Equation (3.2) is subjected to the following boundary condition:
1 It is assumed that the pollutants are absorbed at the ground surface i.e.
∂cy ( x, z ) k= − vcxzg y (), atz = 0 ∂z (i) where v g is the deposition velocity (m/s). 2 The flux at the top of the mixing layer can be given by:
∂c( x, z ) K ∂z = 0 at z=h (ii)
3 The mass continuity is written in the form:
u c y (x,z) =Q δ(z h) at x=0 (iii)
Where δ is Dirac delta function, Q is the source strength and h is mixing height.
4 The concentration of the pollutant tends to zero at large distance of the source, i.e.
Cy(x, z) =0 at z=∞ (iv)
Applying the Laplace transform on equation (2.2) to have:
~ ∂ 2 C (s, z) u s ~ u y − C ()s, z =− C (),0 z ∂ z 2 k y k y (2.3)
Substituting from equation (iii) in equation (2.3), we obtain that:
∂2C% ( s , z ) u s Q y −Csz% (), =−δ () zh − ∂z2 ky k (2.4)
where C˘ y(s,z) = L p{ c y(x,z) ; x→s} , and L p is the operator of the Laplace transform :
∂cy ( x, z ) L = scsz()%y(), − coz y () , ∂ x Equation (2.4) is nonhomagneus differential equations. The general solution of this equation consists of two parts solutions, one part from solution is homogeneous solution and a second part is special solution. To solve the homogeneous solution from equation (2.4).Let: (Q / k) δ ( z h ) =0 to have(Shamus, 1980):
2 (2.5) ∂ C% y ( s , z ) u s 2 −C% y () s, z = 0 ∂z k
The solution of equation (2.5) su su z − z (2.6) k k C%y () sz. = ce1 + ce 2
From the boundary condition (iv), we find c 1=0, and Using the boundary condition (iii) after taking Laplace transform, we get:
Q c=δ () z − h 2 k s (2.7)
Substituting from equation (2.7) in equation (2.6), we obtain that:
s u − z (2.8) Q k Csz%y (). = δ ()z− h e k s
To solve the special solution from equation (2.4), we get:
2 ∂ u s Q − Csz% y (), =−δ () zh −⇒ ∂ z2 k k Q −δ ()z − h k C% () s, z = y ∂ 2 u s 2 − ∂ z k
∂ R( z ) Where: D = y( z ) = ∂z Q D− m mz− mz Then, the general solution to this form is given by: e(∫ e Rzdz( ) )
The special solution is given that:
s u 1 − h (2.9) Csz% ,= 1 − e k y () h suk Summation the equations (2.8) and (2.9) we have the general of solution equation (2.5) on the form: su su −z − h Q k 1 k (2.10) Csz%y (), = δ () zh−e + 1− e k s h suk
Taking the inverse Laplace transform for the equation (2.10), we get the crosswind integrated concentration is on the from:
h2 u h2 u − − Cy (x , z ) h u 4kx 1 1 4 kx = e+ − e (2.11 ) Q 2 π k3 x 3 hπ xuk h π xuk
2.3 Numerical Method
2 ∂ C y u ∂ C y (x , z ) ∂ C y (x , z ) 2 = = A (2.12) ∂ z k ∂ x ∂ x
where: A=u / k
Equation (2.12) can be solved using Adomian decompositions method as follows: Lczzy( xz,) = ALc xy ( xz , ) ∂2 ∂ whereL=, L = (2.13) z z∂z2 x ∂ x
1 Multiplying both sides of this equation by L zz
−1 Cy( xz,) = c0 + ALLC zzxy ( xz , ) z z (2.14 ) L−1= c + ALLC − 1 () xz, dzdz zz∫0 ∫ 0 ()0 zzxy assuming that:
Co=M(x) + z N(x) (2.15)
Where M and N are unknown function which will be determined from boundary condition using equation (2.15) to get the general solution in the from:
z z ∂c c= An dzdz n +1 ∫0 ∫ 0 (2.16) ∂x
Put n=0
2 2 z z∂c0 z z ∂M ∂ N ∂Mz ∂ Nz cA1 = ∂∂= zzA+z =+ A A (2.17) ∫0 ∫ 0 ∫0 ∫ 0 ∂x ∂x ∂ x ∂x2! ∂ x 3!
Assuming the solution has the form:
∞ Wn= ∑ c n n =0
d M z 2 d N z 2 W1 =c0 + c1 = M ()()x + z N x = A + A (2.18) d x !2 d x !3 By differentiating the equation (2.18) with respect to z and multiplying by k z, we obtain:
2 ∂W1 ∂ M z ∂ N K z = K z N()x + AZ K z + A kz (2.19 ) ∂ Z ∂ x !2 ∂ x
Using the boundary condition (i) at z=0, we have that:
∂ W k 1 = k N ( x) = − v M ( x) z ∂ z z g − v k (2.20) ∴ N (x) = g M (x) ⇒ M ( x) = − z N ( x) k z v g
Using the boundary condition (ii) at z=h, we obtain that:
dM h 2 d N N ()x + A h + A = 0 d x !2 d x h 2 d N A h d N N ( x) + A + K + N ( x) k u = 0 o * (2.21) !2 d x − v g d x d N k A h u − v 2 v = 0 * g g dx N ( x) v h 2 A v − 2 A h k g g
Integrating equation (2 .21) from 0 to x, we obtain that:
2 h A k 0 − 2 v g h A ( h A v − 2 k ) g x N ( x ) = N 0 ( x ) e (2.22) Using the boundary condition (iii), we get that:
Q Nx0 () =δ () zh − u
Substituting N 0(x) in equation (2.22), to have:
2 (A h k 0 u * − v g )x Q A h ()h v − 2 k N ()x = δ ()z − h e g u (2.23 ) substituting equations (2.20) and (2.23) into equation (2.15), we obtain that:
−kx() kx() c= − zNx =− + zNx =−( BzNx + )() 0 () () (2.24) vg v g
Where : B= k /v g
d N 2(Ahk0 u * − v g ) = N (2.25) d x Ahhv− 2 k ()g
d M k 2(Ahk0 u * − v g ) =− N (2.26) dx v Ahhv−2 k g ()g
Substituting equations (2.25) and (2.26) into equation (2.19), we obtain that:
z 3 k z 2 c 1 = ( A D ) ( − ) N (2.27) 3 ! v g 2 ! where
(2hAk0 − 2 vg ) x D = hA hAv− 2 k ()g
Similarity, we get z 5 k z 4 c = A D 2 − N 2 () 5 ! v g 4 ! z 7 k z 6 c = A D 3 − N 3 () (2.29) 7 ! v g 6 ! z 9 k z 8 c = A D 3 − N 4 () 9 ! v g 8 !
The general solution:
i d k 2 Ah −v dk d x g 2uAh −v x n g Cy ()x,z vg Ah()vh −2x d x = e g Q u vh −k ∑ A kh vh −2x ()g i=1 ()g (2.30) −KZ2i Z 2i+1 + vg ()2i ! ()2i + !1
We can obtain the wind speed at source height 115m as follows:
p U115 = U 10 (z/10) (2.31 )
Where:
U115 is the wind speed at 115m.
U10 is the wind speed at 10m height. z is the physical height .
p is a parameter estimated by Irwin (1979), which is related to stability classes, is given in Table (2.1).
Table (2.1) Estimates of the power (p) in urban areas for six Stability Classes based on information by Irwin (1979b).
Stability Very Moderately Slightly Neutral Slightly Moderately Classes Unstable Unstable Unstable (D) Stable (E) Stable (F) (A) (B) (C) Urban p 0.19 0.21 0.32 0.30 0.36 0.46
Table (2.1), shows the values of the power (P in urban areas for six stability classes. In the present model we used two methods for the calculation of the eddy diffusivity depends on the downwind distance (x) .The first method taking k in the from 'k 1(x) =0.04ux ' and the second method are referenced to (Arya,
2 1995) where k takes in the form:' k2(x) =0.16(σ w)x' .
σw is the standard deviation vertical velocity.
The used data set was observed from the atmospheric diffusion experiments conducted at the northern part of Copenhagen, Denmark, under unstable conditions (Gryning and Lyck, 1984; Gryning et al., 1987). The tracer sulfur hexafluoride (SF 6) was released from a tower at a height of 115m without buoyancy. The values of different parameters such as stability, wind speed at
10m (U10 ), wind speed at 115m (U115 ), and downwind distance during the experiment are represented in Table 2.2.
Table (2.2) shows the values of wind speed at 10 m and 115 m and downwind distance through unstable and neutral stabilities at northern part of Copenhagen.
Table (2.3) shows the Comparison between Observed and different analytical, numerical normalized crosswind integrated concentrations. Table (2.4) shows the ratio of predicted and observed normalized crosswind concentrations for all models. We compared between the final results obtained using four models. One look for which is the most optimum model to be used .Figures (2.1) and (2.2), show the variation between the observed and predicted normalized crosswind concentrations using analytical and numerical methods of the trace sulfur
Hexafluoride (Sf 6) respectively. We find that both models can be considered agree with observed data expect for some points.
Table (2.2) Values of wind speed at 10 m and 115 m and downwind distance through unstable and neutral stabilities at northern part of Copenhagen.
Distance U U Stability 10 115 (x) (m) Run no. (m/s) (m/s)
1 Very unstable (A) 2.1 3.34 1900 1 Very unstable (A) 2.1 3.34 3700 2 Slightly unstable (C) 4.9 10.71 2100 2 Slightly unstable (C) 4.9 10.71 4200 3 Moderately unstable (B) 2.4 4.01 1900 3 Moderately unstable (B) 2.4 4.01 3700 3 Moderately unstable (B) 2.4 4.01 5400 5 Slightly unstable (C) 3.1 4.93 2100 5 Slightly unstable (C) 3.1 4.93 4200 5 Slightly unstable (C) 3.1 4.93 6100 6 Slightly unstable (C) 7.2 11.45 2000 6 Slightly unstable (C) 7.2 11.45 4200 6 Slightly unstable (C) 7.2 11.45 5900 7 Moderately unstable (B) 4.1 6.85 2000 7 Moderately unstable (B) 4.1 6.85 4100 7 Moderately unstable (B) 4.1 6.85 5300 8 Neutral (D) 4.2 8.74 1900 8 Neutral (D) 4.2 8.74 3600 8 Neutral (D) 4.2 8.74 5300 9 Slightly unstable (C) 5.1 11.14 2100 9 Slightly unstable (C) 5.1 11.14 4200 9 Slightly unstable (C) 5.1 11.14 6000
Table (2.3) Comparison between Observed and different analytical, numerical normalized crosswind integrated concentrations Cy/Q (10 4 s/m 3).
Down 4 3 Run Cy/Q *10 (s/m ) Stability distance no. Analytical Analytical Numerical Numerical (m) Observed model 1 model 2 model 1 model 2 1 Very unstable (A) 1900 4.48 8.95 3.59 2.08 6.48 1 Very unstable (A) 3700 3.37 4.64 4.93 3.79 2.31 2 Slightly unstable (C) 2100 1.29 6.28 7.36 4.03 5.38 2 Slightly unstable (C) 4200 1.02 3.14 2.04 1.27 2.95 3 Moderately unstable (B) 1900 5.08 10.92 1.05 1.32 8.2 3 Moderately unstable (B) 3700 3.17 6.30 8.94 3.40 6.22 3 Moderately unstable (B) 5400 1.80 8.30 1.20 6.25 4.3 5 Slightly unstable (C) 2100 4.64 9.47 1.18 3.55 6.72 5 Slightly unstable (C) 4200 1.80 9.01 1.69 8.75 5.84 5 Slightly unstable (C) 6100 0.91 12.19 3.76 1.53 4.97 6 Slightly unstable (C) 2000 1.56 5.30 2.02 2.83 3.96 6 Slightly unstable (C) 4200 0.98 2.53 1.44 7.24 2.22 6 Slightly unstable (C) 5900 0.60 1.98 5.31 1.18 1.83 7 Moderately unstable (B) 2000 2.12 8.11 1.81 2.63 6.7 7 Moderately unstable (B) 4100 1.64 3.96 1.46 6.09 3.25 7 Moderately unstable (B) 5300 1.33 3.06 1.01 8.62 2.23 8 Neutral (D) 1900 2.83 10.31 5.14 7.11 4.16 8 Neutral (D) 3600 1.30 5.45 9.14 1.50 2.02 8 Neutral (D) 5300 0.66 4.37 4.32 2.42 1.52 9 Slightly unstable (C) 2100 1.22 6.86 5.97 3.50 4.58 9 Slightly unstable (C) 4200 0.94 3.43 1.05 7.70 3.11 9 Slightly unstable (C) 6000 0.72 2.40 1.60 1.18 2.59
Table (2.4) Ratio of predicted and observed normalized crosswind concentrations for all models
4 3 Cy/Q *10 (s/m ) Analytical Analytical1/ 2/ Numerical 1/ Numerical 2 Observed Observed observed / observed Observed 0.691 1.381 0.554 0.321 6.48 1.461 2.010 2.134 1.641 2.31 0.240 1.167 1.368 0.749 5.38 0.345 1.064 0.692 0.431 2.95 0.619 1.332 0.128 0.161 8.2 0.510 1.014 1.437 0.547 6.22 0.418 1.931 0.279 1.453 4.3 0.690 1.409 0.176 0.528 6.72 0.309 1.543 0.289 1.498 5.84 0.184 2.452 0.757 0.308 4.97 0.393 1.339 0.510 0.715 3.96 0.443 1.138 0.649 3.261 2.22 0.328 1.083 2.902 0.645 1.83 0.316 1.210 0.270 0.393 6.7 0.505 1.217 0.449 1.874 3.25 0.596 1.372 0.453 3.865 2.23 0.681 2.479 1.236 1.709 4.16 0.646 2.696 4.525 0.743 2.02 0.436 2.878 2.842 1.592 1.52 0.267 1.497 1.303 0.764 4.58 0.301 1.102 0.338 2.476 3.11 0.279 0.926 0.618 0.456 2.59
Fig. (2.1) Comparison between analytical and observed normalized crosswind integrated concentration .
Fig. (2.2) Comparison between analytical and observed normalized crosswind integrated concentration and down distance
Fig. (2.3) Comparison between numerical and observed normalized crosswind integrated concentration .
Fig. (2.4) Comparison between numerical cross wind integrated concentration and downwind distance.
Fig. (2.5) Comparison between analytical, numerical and observed normalized crosswind integrated concentrations.
Fig. (2.6) Comparison between analytical, numerical and observed normalized crosswind Integrated concentration and downwind distance.
Fig. (2.3). Show that the variation of analytical and observed normalized crosswind concentration data with down distance. Fig. (2.4), show that the variation of numerical and observed normalized crosswind concentrations data downwind distance .One find that analytical model(2.1) and (2.2)and numerical model (2.1) have points agree with the observed data , while the others points are over predicated. Fig. (2.5). Shows that the variations of the analytical and numerical normalized crosswind integrated concentrations with the observed normalized crosswind integrated concentrations. Fig. (2.6).Shows that the normalized crosswind integrated concentrations of analytical, numerical and observed data with downwind distances.
Fig. (2.7) Ratio of predicted and observed normalized crosswind concentrations via observed normalized crosswind concentrations for all models
Fig. (2.7).shows that analytical models (1) and (2) have most points near observed data, while others over predicated. In the other hand numerical model
(1) has most data inside a factor of 2 (FAC2).also numerical model (2) has most points are over predicted.
2.4 -Statistical method
Now, the statistical method is presented and comparison among analytical, statically and observed results will be offered (Hanna1989). Referred to section
1.11.
Table (2.5) Comparison between our different models according to standard statistical performance measure Models NMSE FB COR FAC2 Analytical model 1 0.79 0.71 0.71 0.48 Analytical model 2 0.30 0.40 0.78 1.56 Numerical model 1 0.66 0.04 0.11 1.19 Numerical model 2 0.79 0.19 0.08 1.09
From the statistical method, we find that the four models are factors of 2 with observed data. Regarding to NMSE, the analytical models (1), (2) and numerical model (1) are better than numerical model (2) .the analytical model
(2) and numerical model (1) are also the best regarding to FB. The correlation of analytical model(1) equals (0.71) and analytical model (2) equals (0.78) which are stronger to the observed data than the correlation of numerical model
1 which equals ( 0.11).
CHAPTER III Maximum integrated ground level concentration under two stability classes
CHAPTER III Maximum integrated ground level concentration under different two stability classes
3.1-Introduction
The analytical solution of the atmospheric diffusion equation containing different shapes depending on Gaussian and non Gaussian solutions. An analytical solution with power law for the wind speed and eddy diffusivity with the realistic assumption was studied by (Demuth, 1978). The solution has been implemented in the KAPPA G model (Tirabassi et al. 1986). (Lin and Hildemann, 1997) extended the solution of (Demuth, 1978) under boundary conditions suitable for dry deposition at the ground. The mathematics of atmospheric dispersion modeling is studied by (John, 2011).
Estimating of crosswind integrated Gaussian and non Gaussian concentration through different dispersion schemes was studied by Essa and
Fouad (2011). Analytical solution of diffusion equation in two dimensions using the two forms of eddy diffusivities was estimated by Essa et al. (2011).
In this chapter the advection diffusion equation (ADE) is solved in two directions ways to obtain the crosswind integrated ground level concentration in neutral and unstable conditions. Laplace transform technique was used considering the wind speed and eddy diffusivity depends on the vertical height. The maximum ground level concentration is
56 CHAPTER III Maximum integrated ground level concentration under two stability classes estimated. Comparison between observed data from Copenhagen (Denmark) and predicted concentration data was presented.
3.2-Analytical solution
Referred to equation (1.11).For steady state, taking dc/dt=0 and the
diffusion in the x axis direction are assumed to be zero compared with the
adjective in the same directions, hence Arya (1995):
∂ C ∂ d C ∂ d C U = K y + K z (3.1) ∂ x ∂ y d y ∂ z d z
Integrating the equation (3.1) with respect to y, we obtain the normalized crosswind integrated concentration C y(x,z) of contaminant at a point (x,z) of the atmospheric advection–diffusion equation is written in the form (Essa et al. 2006) :
2 ∂Cy ∂ C y ∂ C y ∂ K z u() z= K z 2 + (3.2) ∂z∂ z ∂ z ∂ z
3.2. a In neutral case
Where:
k(z) =k v w * z (3.3 ) kv is the von Karman constant which is set to 0.4 and w * is the convective scaling parameter.
Substituting from equation (3.3) in equation (3.2), we get that:
∂C k w z ∂2 C k w ∂C y = v * y + v * y 2 (3.4) ∂ x u()z ∂ z u()z ∂ z
57 CHAPTER III Maximum integrated ground level concentration under two stability classes
Applying the Laplace transform on equation (3.4) respect to the direction x to have:
∂C Lp = sCszC% y()(), − y 0, z (3.5) ∂x
Where L p is the operator of the Laplace transform.
Substituting from (3.5) in equation (3.4), we get that:
2 ∂Csz%y( ,) 1 ∂ Csz % y ( , ) us u 2 + −Csz%y(), =− Cz y () 0, (3.6) ∂z z∂ z kwzv* kwz v *
Equation (3.6) is subjected to the following boundary condition
1 The flux at the top of the mixing layer can be given by:
∂Cy kz =0 at z= 0, h (i) ∂z
2 The mass continuity is written in the form:
u (z) C y(x, z) =Q δ (z hs) at x=0 (ii)
Where δ is Dirac delta function, Q is the source strength and h s are stack height.
58 CHAPTER III Maximum integrated ground level concentration under two stability classes
3 The concentration of the pollutant tends to zero at large distance of the source, i.e.
Cy(x, z) =0 at x, z→ ∞ (iii)
Substituted from (ii) in equation (3.6) we obtain that:
2 ∂Csz%y( ,) 1 ∂ Csz % y ( , ) us Q 2 + −Csz%y(), =−δ() zh − s (3.7) ∂z z∂ z kwzv* kwz v * integrated the equation (3.7) respect to z, we obtain that:
∂C% y ( s, z ) uszln( ) Q −C% y () s, z = − (3.8) ∂z kwv* kwh v * s
Equation (2.8) is nonhomagneus differential equations. The general solution of this equation consists of two parts solutions, one part from solution is homogeneous solution and a second part is special solution.
To solve the homogeneous solution from equation (2.8).
Let: −Q =0 in equation (3.8) and the solution is obtaining that: kv w* h s
s uln z C% s, z z y () kv w * = c1 e (3.9) Q
Substituted from (ii) after using Laplace transform in equation we obtain that:
c1 = (1/ u s ) δ (z hs) (3.10)
Substituted from equation (3.10) in equation (3.9):
59 CHAPTER III Maximum integrated ground level concentration under two stability classes
suln z suln h z s h Csz%y (),δ ()z− h Csz % y () , 1 s =s ekwv * ⇒ = e kwv * Q us Qus (3.11)
To solve the special solution from equation (3.8), we get:
∂ uszln( ) Q − C% y () s, z = − ∂ z kwv * kwhv* s
C% y () s, z 1 ⇒ = − Q u sln( z ) kwv* h s D − kv w *
∂ R( z ) Where: D = y( z ) = ∂z Q D− m mz− mz Then, the general solution to this form is given by: e (∫ e R( z ) dz )
The special solution is given that:
s u 1 − h Csz% ,= 1 − e k y () h suk (3.12) From equation (3.11) and equation (3.12) we have the general of solution equation (3.8) on the form:
suln h s uh()ln hz− ln z C% () s, z 1hs 1 − s y =ekwv * + e kwv ∗ (3.13) Q us ushhs ln
Taking Laplace inverse transform on equation (3.13), we get that:
C( x, z ) 1 1 y = + (3.14) Q u hln () h uhln h− z ln z u x − s s ()()() ()() uhs ()ln () h x − kv w ∗ k w v ∗
Where:
60 CHAPTER III Maximum integrated ground level concentration under two stability classes
−1 −− 11 − 1 1 L()()() AB= L AL B, L = 1 s 1 1 ,L−1 () exp−() as = andL− 1 () exp () as = xa+ xa −
L 1 is the operator of the Laplace inverse transform by (Shamus, 1980).
To estimate the integrated ground level concentration put z=0 in equation
(3.14), to have:
Cy ( x ,0) 1 1 = + (3.15) Q u hln () h u hln h u x − s s ()()() uhs ()ln () h x − kv w ∗ k w v ∗
Differentiating equation (3.15) with respect to x and equating the result to zero, we get the maximum downwind distance on the form:
u ( h s ln (h ) )(h − h s ) ln( h ) x max . = (3.16) k v w * ()1 + h s ln (h )
Substituting from equation (3.16) in equation (3.15), we get the maximum integrated ground level concentration:
61 CHAPTER III Maximum integrated ground level concentration under two stability classes
C (x , 0) 1 max. = Q 1 2 uhh()()()− ln( hh )()() ln h uh ln () h u s s− ss 1 2 kv w ∗ kw1+ () h() ln () h v∗ s 1 + 1 (3.17) 2 uhh()()()− ln( hh )()() ln h u() h()ln () h u h()ln () h s s − s 1 2 kv w ∗ kw1+ () h() ln () h v∗ s
3.2. b In unstable case
Where: k (z) =k v w * z (1 z /h) (3.18)
Where h is the height of the ABL.
Substituting from equation (3.18) in equation (3.2):
z 2z kv w* z 1− 2 kv w* 1− ∂C h ∂ C y h ∂ C y = + (3.19) ∂ x u ()z ∂ z 2 u ()z ∂ z
Applying the Laplace transform on equation (3.19) with respect to x to have that:
∂C Lp = sCszC% y()(), − y 0, z (3.20) ∂x
Where L p is the operator of the Laplace transform
62 CHAPTER III Maximum integrated ground level concentration under two stability classes
Substituting from (3.20) in equation (3.19), to have:
2z 2 1− ∂Csz%y(),h ∂ Csz % y () , u s u + −Csz%y(), =− Cz y () 0, (3.21) ∂z 2 z2∂z z 2 z 2 z− kwzv* − kwz v * − h h h
Substituted from (ii) in equation (3.21) to obtain that:
2z 2 1− ∂Csz%, ∂ Csz % , y() h y () us Qδ() z− h s + −C%y () s, z =− ∂zz22∂z z 2 z 2 (3.22) z− kwzv* − kwz v * − h h h
Integrated equation (3.22) respect to z, we have:
z− h u s ln ∂C% s, z y () z Q (3.23) +C% y () s, z = − ∂z kv w * h s kv w* h s 1 − h
Equation (3.23) is nonhomagneus differential equations. The general solution of this equation consists of two parts solutions, one part from solution is homogeneous solution and a second part is special solution.
To solve the homogeneous solution from equation (3.23), Let:
−Q =0, in equation (3.23) and the solution is obtaining that: hs kv w* h s 1− h
z − h s u ln z − z kv w * C% () s, z y = c e Q 2 (3.24)
63 CHAPTER III Maximum integrated ground level concentration under two stability classes
Substituted from (ii) after Laplace transform in equation (3.24) we obtain that:
1 c=δ () z − h 2 u s s (3.25)
Substituted from equation (3.25) in equation (3.24) :
h− h s u ln s h − s z k w v * C% () s, z 1 y = e Q u s (3.26)
To obtain the special solution from equation (3.23), Let :
z− h u s ln ∂C%y () s, z z Q +C%y () s, z = − (3.27) ∂z kv w * hs kv w* h s 1− h
Then, the solution of equation (3.27) is given by:
z − h s u ln z − z kv w * C% () s, z 1 y = e (3.28) Q hs kv w∗ h s − 1 h
Summation the equations (2.28) and (2.26) we have the general solution equation (2.23) on the form:
64 CHAPTER III Maximum integrated ground level concentration under two stability classes
h− h s z − h su ln su ln hs − z z k w − z v * kv w * C% () s, z 1 1 y =e + e (3.30) Q us hs kv w∗ h s −1 h
Taking Laplace inverse on equation (3.30), we get that:
C( x, z ) 1 1 y = + (3.31) Q h− h −z h u ln s u ln h z hs s u x − kwhv∗ s −1 x + k w h kv w ∗ v ∗ where:
−1 −− 11 − 1 1 L()()() AB= L AL B, L = 1 s 1 1 ,L−1 () exp−() as = andL− 1 () exp () as = xa+ xa −
, L 1 is the operator of the Laplace inverse transform by (Shamus, 1980).
To estimate the integrated ground level concentration put z=0 in equation
(3.31), we get that:
C( x ,0) 1 1 y = + (3.32) Q h hs − h s u ln kwhv∗ s −1 x h h u x − s kv w ∗
Differentiating the equation (3.32) with respect to x and render the result to zero, we get the maximum downwind distance:
65 CHAPTER III Maximum integrated ground level concentration under two stability classes
h− h u ln s h s x max . = (3.33) h kw kwh1 −s − u v∗ v ∗ s h
Substituting from equation (3.33) in equation (3.31), we get the maximum integrated ground level concentration:
C( x ,0 ) 1 y = + Q hh− hh − ulns u ln s hs h s u − h kv w ∗ kw kwh1−s − u v∗ v ∗ s h 1 (3.34) h− h u ln s h hs s kv w∗ h s −1 h hs kwv∗ kwh v ∗ s 1− − u h
The used data set was observed from the atmospheric diffusion experiments conducted at the northern part of Copenhagen, Denmark, under unstable conditions (Gryning and Lyck, 1984; Gryning et al., 1987). Table (3.1), show that comparison between the observed ground level concentration and estimated ground level concentration, maximum ground level concentration, maximum down distance per height.
66 CHAPTER III Maximum integrated ground level concentration under two stability classes
Table (3.1) Comparison between the observed integrated crosswind ground level concentration and predicated ground level concentration under different stability, height and wind speed.
4 3 Cyglc.. (x,0)/Q*10^ (s/m ) distance Run U (x) No. Stability h(m) (m/s) (m) w* observed predicated Maximum xmax. x/h Very 1 unstable(A) 1980 3.34 1900 1.8 6.48 5.01 2.15 1.19 0.96 Very 1 unstable(A) 1980 3.34 3700 1.8 2.31 2.62 1.30 0.71 1.87 Slightly 2 unstable (C) 1920 7.79 2100 1.8 5.38 4.36 2.51 3.22 1.09 Slightly 2 unstable (C) 1920 7.79 4200 1.8 2.95 2.26 1.41 1.79 2.19 Moderately 3 unstable(B) 1120 3.82 1900 1.3 8.2 5.01 5.89 3.32 1.70 Moderately 3 unstable (B) 1120 3.82 3700 1.3 6.22 2.61 3.67 1.97 3.30 Moderately 3 unstable (B) 1120 3.82 5400 1.3 4.3 1.80 2.30 1.21 4.82 Slightly 5 unstable (C) 820 4.93 2100 0.7 6.72 4.50 10.74 9.92 2.56 Slightly 5 unstable (C) 820 4.93 4200 0.7 5.84 2.27 7.12 5.42 5.12 Slightly 5 unstable (C) 820 4.93 6100 0.7 4.97 1.57 4.42 2.97 7.44 Slightly 6 unstable (C) 1300 11.45 2000 2 3.96 4.35 2.60 4.27 1.54 Slightly 6 unstable (C) 1300 11.45 4200 2 2.22 2.21 1.42 2.30 3.23 Slightly 6 unstable (C) 1300 11.45 5900 2 1.83 1.60 0.85 1.37 4.54 Moderately 7 unstable (B) 1850 6.52 2000 2.2 6.7 4.57 1.58 1.65 1.08 Moderately 7 unstable (B) 1850 6.52 4100 2.2 3.25 2.32 0.88 0.91 2.22 Moderately 7 unstable (B) 1850 6.52 5300 2.2 2.23 1.81 0.62 0.64 2.86 8 Neutral(D) 810 6.68 1900 2.2 4.16 4.89 0.02 1.89 2.35 8 Neutral(D) 810 6.68 3600 2.2 2.02 2.68 0.05 1.17 4.44 8 Neutral(D) 810 6.68 5300 2.2 1.52 1.85 0.07 0.73 6.54 Slightly 9 unstable (C) 2090 8.11 2100 1.9 4.58 4.34 2.15 2.93 1.00 Slightly 9 unstable (C) 2090 8.11 4200 1.9 3.11 2.26 1.21 1.63 2.01 Slightly 9 unstable (C) 2090 8.11 6000 1.9 2.29 1.60 0.70 0.94 2.87
67 CHAPTER III Maximum integrated ground level concentration under two stability classes
Fig. (3.1) show that comparison between the observed, predicated and maximum ground level concentrations.
Fig. (3.2) show comparisons between downwind distance per height and the observed, predicated and maximum ground level concentrations
68 CHAPTER III Maximum integrated ground level concentration under two stability classes
Fig. (3.3) show comparison between maximum downwind distances and the observed, predicated and maximum ground level concentrations.
Fig. (3.1) Shows comparison between the observed and predicated normalized maximum ground level concentrations. We find that the predicated model is better than maximum ground level concentrations with observed data. Fig. (3.2) Shows comparison between downwind distance per height via the observed and predicated normalized maximum ground level concentrations. We find that the observed and predicated normalized ground level concentrations are better than maximum ground level concentrations.
Fig, (3.3) Show comparison between maximum downwind distances via the observed, predicated normalized maximum ground level concentrations. We
69 CHAPTER III Maximum integrated ground level concentration under two stability classes find that the observed and predicated normalized ground level concentrations are better than maximum ground level concentration.
3.3 Statistical method
Now, the statistical method is presented and a comparison among analytical, statically and observed results will be offered (Hanna
1989).Referred to section 1.11.
Table (3.2) .Comparison between our two models according to standard statistical performance measure
Models NMSE FB COR FAC2 Predicated model 0.26 0.32 0.67 0.80 Maximum model 0.62 0.52 0.70 0.51
From the statistical method, one can find that the two models are factors of 2 with observed data. Regarding to NMSE, the maximum model is better than predicated model. .the maximum model and predicated model are also the best regarding to FB. The correlation of predicated model equals (0.67) and maximum model equals (0.70).
70 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
4-1-Introduction
Most models for urban air pollution are based on Gaussian plume diffusion or Sutton’s equations or the K theory that require a digital computer. There are some simpler models such as Miller and Holzworth
(1967), Hanna (1982) and Anon (1989). In these models the assumptions used, for example uniform wind, uniform mixing in the mixing layer, ground terrain, etc. are not very realistic and, therefore, these assumptions restrict the use of the model for some special cases which don’t normally occur in real life. The diffusion from a point source using logarithmic law of the wind speed is studied by Essa and maha (2007), and the diffusion from a point source using power law of the wind speed is studied by Essa and Refaat
(2010).
In this chapter, a model is suggested for the diffusion of material from a point source in an urban atmosphere is incorporated. The plume is assumed to have a well defined edge at which the concentration falls to zero. The vertical wind shear is estimated using combination between logarithmic and power laws under different stability conditions. The problem of diffusion and advection of conservative material as it travels downwind is calculated.
71 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions The concentrations estimated from this model were compared favorably with the field calculated of other investigators such as power and logarithmic law models. Also we calculate the ground level concentration of the Iodine (I 138 ) which agrees with the observed concentration value after adjusting its source strength.
4.2-Proposed model structure
A point source normal to mean wind direction with height “h” was situated at the ground level with emission rate" Q" employing diffusion and advection of air pollutant. The problem is two –dimensional in nature because homogeneity in the lateral direction. Fig. (4.1) describes the coordinate system direction of the mean wind. The effective height denoted by H =h s + h, where "h s "is the stack height and " h" is the plume height which increases as the plume travel downwind. The mean wind speed is denoted by u(z). The analysis that follows assumes steady state conditions; that is, the variables of interest; for example the mean wind speed, and stability of the atmosphere, don’t change in the time interval of interest. The ground surface is treated as a complete reflector of matter; that is, no removal occurs.
72 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
Fig. (4.1) shows the coordinate system direction of the mean wind.
4-3 The effective height
Defining the plume height " h" of diffusing matter as the distance from the stack height" h s" to the point at which concentration has fallen to one tenth of the surface value. The plume height has been calculated adopting the following equation (IAEA Safety Guide (1983)).
h = 3 ( wuD / ) 1
Where "w" is the exit velocity of the pollutants, and "D 1" is the internal stack diameter. The effective stack height H equals:
H =h + h=h + (3 w/u)D s s 1
4.4 -Mathematical technique
The concentration of a pollutant down wind of a source, relative to its concentration at the point of emission is determined largely by two processes. Advection is the movement of the pollution carried by the wind; it
73 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions averaged over the time taken for it to blow from the source to any given receptor. Stronger wind causes more rapid motion at the point of emission; the highest concentration is found source to any given receptor. Stronger wind causes more rapid motion at the point of emission; the highest concentration is found when it is calm. Turbulence is the instantaneous random motion of the air, which causes the pollution to spread out perpendicular to the wind direction. Turbulence is initiated in the atmosphere by the drag of the movement of the lower layers of air over a rough surface, a process that is closely related to deposition.
Under steady state condition, the ambient concentration C(x, z) of contaminant at a point (x, z) of the atmospheric advection– diffusion equation under several hypotheses is written on the form (Essa etal. 2006)
∂C ∂ ∂ C uz() = Kz() ∂x ∂ z ∂ z (4.1)
Where u is the mean wind velocity and k is turbulent diffusion coefficient.
Diffusion in the x and y directions is neglected.
A very simple approach, namely the principle of conservation of mass with a steady state can be written as (Essa et al. 2006):
H Q= ∫ U()() zCzdz (4.2) 0
And the concentration in the form of power series is written as (Essa etal. 2006)
74 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
2 Cz() z z =+1 α + α + ...... 1( ) 2 ( ) (4.3) Cz0 () H H
where:
C is averaged concentration (Bq/m 3).
Q is actual emission rate of the point source (Bq)
α 1, α 2 are constant
U (z ) is average plume velocity (m /s). z is plume height in meters (m).
3 C0 (z) is the concentration at the edge of the plume (Bq/m ).
H is the effective height of the plume (m).
The number of terms chosen in the above series will depend upon desired goodness of fit to the observed data as shown in Fig. (4.2). It was found that the series in Eq. (4.5) gives a fairly good fit to observed data even if only the first two terms are retained; that is
C( z ) z =1 + α (4.4 ) C0 () z H
This is an equation of straight line. The value α will depend upon the concentration desired at the edge of the plume. If the edge of the plume is defined as having r percent of the concentration, (Essa. etal. (2006)):
α= 1+ 0.01 r and if r = 0 (4.4.1)
C/C o=1 (z/H) (4.4.2)
75 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
Fig. (4.2), the variation of the concentration of Iodine (I 138 ) with distance from the reactor. a- In neutral case Suppose that the wind speed is in the form as follows because the combustion between the power law and logarithms law:
z+ z ln 0 z U() z= U 0 (4.4.3) 1 z+ z ln 1 0 z 0 where:
U( z ) is the average wind speed at height z
76 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
z o is roughness height in urban area (m) (0.5 3m).
U1 is the observed wind speed at height 10m.z1 is the height of plume at
10m.
Similarly one can generate the (C o/Q) term for each stability condition as:
Substituting of equations (4.4.3), (4.4.2) into equation (4.2) and integrating we obtain the concentration at the edge over the emission rate in the form:
−1 zz+ HZ + ln1 0 2HZHZ2++ 4 2 2 ln0 −− 3 HZH 2 2 ()0 0 0 C0 z 0 Z 0 = Q U 4 H (4.5)
b - In stable case
Suppose that the wind speed is in the form as follows because the combustion
between the power law and logarithms law, taking L >0 (L=55m) and β=5 as
follows:
zz+ 0 zz 1 − 0 ln + β z L U z= U 0 (4.4.4) () zz10+ zz 10 + ln + β z0 L
U( z ) is the average wind speed at height z zo is roughness height in urban area (m).
U1 is the observed wind speed at height 10m.z1 is the height of plume at
10m.
77 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions Substituting of equations (4.4.4), (4.4.2) into equation (4.2) and integrating we obtain the concentration at the edge over the emission rate in the form:
zz+0 zz 1 − 0 ln +β C z L 0 =0 Q U 1 (4.6) H+ z 66H2++ 132 zHz 66 2 ln0 −−−+ 66 zHHzHH 992 6 2 2 3 ()00 0 0 z 0 132 H c- In unstable case
Suppose that the form of the wind speed is in the form as follows because the combustion between the power law and logarithms law, taking L < 0
(L= 2.5m) and β=15 as follows:
zz+0 zz 1 − 0 ln + β z L U z= U 0 (4.4.5) () zz10+ zz 10 + ln + β z0 L
Substituting of equations (4.4.5), (4.4.2) into equation (4.2) and integrating we obtain the concentration at the edge over the emission rate in the form:
zz+0 zz 1 − 0 ln +β C z L 0 =0 Q U 1 (4.7) H+ z 2.0H2++ 4.0 zHz 2.0 2 ln0 −−+− 2.0 zHH 3.02 12.0 zHH 2 4.0 3 ()00 0 0 z0 25 H
78 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions 4. 5-Case study
The derived equations were employed at first research rector at Inshas (Egypt). A continuous Ventilation system is provided with the reactor to the areas where radioactive gases, volatile materials and suspended particles can exist due to either leakage or airborne radioactivity. The total ventilation rate which could be emitted from the reactor stacks of 43 m height, 1 m internal diameter, and exist velocity 4 m/s is 39965 m3/hr, (Report 53 of Reactor Physics Department). As α is taken 1.
Table (4.1), shows the calculated values of U, h, H and Co/Q in different stability conditions. The last three columns in all tables are given after 48 hours that are the usual continuous operation time of the reactor. We notice that the values of Co/Q in neutral condition are almost the same values in stable conditions but the values in unstable condition are the smallest . Fig. (4.3), shows the relation ship between the effective height and Co/Q in different stability conditions. From this figure that the values of Co/Q in neutral condition are almost the same values as in stable conditions
.Fig. (4.4) and Table (4.2). Show the association between Effective height and the values of Co/Q. If Comparing proposed model, power and logarithmic laws, it can be found that there is agreement between proposed model and power law in neutral case but in the case of the logarithmic law the curve take the same shaped and the values are larger.
79 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
Table (4.1) Wind speed, the plume rise, effective height and the concentration at the axis of the plume at the reactor release over emission rate in different stability condition.
3 3 3 3 h(m) H (m) Co/Q*10 Co/Q*10 Co/Q*10 (sec/m ) U (sec/ m 3) (sec/m 3) in unstable (m/s) in neutral in stable 5.27 2.28 45.28 7.72 7.66 5.58 5.31 2.26 45.26 7.66 8.75 6. 34 5.34 2.25 45.25 7.62 8.39 6. 09 6.37 1.88 44.88 6.47 6.97 5. 09 5.17 2.32 45.32 7.86 8.98 6. 50 4.45 2.70 45.70 9.02 9.27 6. 70 5.1 2.35 45.35 7.96 9.11 6. 59 4.81 2.49 45.49 8.40 9.55 6. 89 4.81 2.26 45.26 8.40 9.20 6. 65 4.86 2.47 45.47 8.32 8.77 6. 36 5.36 2.24 45.24 7.60 8.40 6. 09 5.19 2.31 45.31 7.83 8.59 6. 23 4.41 2.22 45.22 9.09 8.88 6. 43 5.54 2.17 45.17 7.37 8.42 6. 11 5.2 2.31 45.31 7.81 9.50 6. 86 5.61 2.14 45.14 7.28 8.28 6. 01 5.79 2.07 45.07 7.07 7.86 5. 72 6.27 1.91 44.91 6.56 7.85 5. 71 5.93 2.02 45.02 6.92 8.35 6. 06 6.01 2.00 45.00 6.83 8.19 5. 95 5.41 2.22 45.22 7.53 8.50 6. 17 5.75 2.09 45.09 7.12 8.65 6. 27 5.26 2.28 45.28 7.73 9.05 6. 55
80 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
Table (4.2), Observed wind speed, the plume rise, effective height against proposed model, using power and using logarithmic Laws of the concentration at the axis of the plume at the reactor release over emission rate in neutral classes.
Using h(m) H(m) proposed using logarithmic Model power Law Law U Co/Q *10^3 Co/Q*10^3 Co/Q*10^3 (m/s) (sec/m 3) (sec/m 3) (sec/m 3) 5.27 2.28 45.28 7.72 8.17 33.62 5.31 2.26 45.26 7.66 8.11 33.64 5.34 2.25 45.25 7.62 8.07 32.67 6.37 1.88 44.88 6.47 8.83 28.10 5.17 2.32 45.32 7.86 8.32 34.62 4.45 2.70 45.70 9.02 9.57 39.09 5.1 2.35 45.35 7.96 8.43 34.59 4.81 2.49 45.49 8.40 8.9 36.73 4.81 2.26 45.26 8.40 8.13 33.64 4.86 2.47 45.47 8.32 8.8 35.57 5.36 2.24 45.24 7.60 8.04 32.67 5.19 2.31 45.31 7.83 8.29 33.58 4.41 2.22 45.22 9.09 7.97 32.70 5.54 2.17 45.17 7.37 7.79 31.82 5.2 2.31 45.31 7.81 7.27 33.59 5.61 2.14 45.14 7.28 7.7 31.84 5.79 2.07 45.07 7.07 7.48 31.03 6.27 1.91 44.91 6.56 6.94 28.79 5.93 2.02 45.02 6.92 7.31 30.24 6.01 2.00 45.00 6.83 7.22 29.47 5.41 2.22 45.22 7.53 7.97 32.70 5.75 2.09 45.09 7.12 7.53 31.01 5.26 2.28 45.28 7.73 8.19 33.62
81 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
Fig. (4.3). Variation of the concentration at the plume axis over emission rate with effective height in neutral, stable and unstable
Figure (4.4) Show the relation between the effective height and Co /Q using our proposed model and power, logarithmic laws in neutral case.
82 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
Figure (4.5) and Table (4.3) Show the relation ship between Effective height and
Co/Q for proposed model, power low, and logarithmic law. It can be found that there is agreement between proposed model and power law in stable case, but for the unstable condition the values are little bit larger than other two models.
Table (4.3). Observed wind speed, the plume rise, effective height against proposed model ,using power law and Using logarithmic Law of the concentration at the axis of the plume at the reactor release over emission rate in stable classes L=55m and β=5.
Using h H logarithmic Using (m) (m) law proposed Power aw 3 3 Co/Q*10 U Model Co/Q*10 2 (m/s) Co/Q*10 3 (sec/m 2) (sec/m ) (sec/m 3) 4.43 2.28 45.28 7.66 8.7 17.99 3.81 2.26 45.26 8.75 9.93 20.97 4 2.25 45.25 8.39 9.5 19.64 4.92 1.88 44.88 6.97 7.87 16.62 3.7 2.32 45.32 8.98 10.19 20.90 3.57 2.70 45.70 9.27 10.52 21.61 3.64 2.35 45.35 9.11 10.34 21.66 3.45 2.49 45.49 9.55 10.85 22.38 3.6 2.26 45.26 9.20 10.45 21.63 3.8 2.47 45.47 8.77 9.95 20.97 3.99 2.24 45.24 8.40 9.52 19.63 3.89 2.31 45.31 8.59 9.75 20.27 3.75 2.22 45.22 8.88 10.07 20.94 3.98 2.17 45.17 8.42 9.54 19.63 3.47 2.31 45.31 9.50 10.79 22.39 4.06 2.14 45.14 8.28 9.37 19.67 4.3 2.07 45.07 7.86 8.9 18.51 4.31 1.91 44.91 7.85 8.88 18.52 4.02 2.02 45.02 8.35 9.46 19.65 4.11 2.00 45.00 8.19 9.27 19.04 3.94 2.22 45.22 8.50 9.63 20.30 3.86 2.09 45.09 8.65 9.81 20.26 3.67 2.28 45.28 9.05 10.27 21.68
83 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
Fig. (4.5) Show the relation ship between the effective height and C o /Q using proposed model and power, logarithmic laws in stable case
Figure (4.6) Show the relation ship between the effective height and C o /Q using proposed model and power, logarithmic laws in unstable case
84 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
Table (4.4), Observed wind speed, the plume rise, effective height and proposed model ,using power law and Using logarithmic Law of the concentration at the axis of the plume at the reactor release over emission rate in unstable classes L= 2.5m and β=15
proposed Using Using model power Law logarithmic law h(m) H(m) Co/Q*10 6 Co/Q*10 6 Co/Q*10 6 U(m/s) (sec/m 3) (sec/m 3) (sec/m 3)
4.43 2.28 45.28 55.75 9.68 73.97 3.81 2.26 45.26 63.43 11.13 76.15 4 2.25 45.25 60.86 10.65 75.80 4.92 1.88 44.88 50.87 8.78 72.22 3.7 2.32 45.32 65.01 11.44 76.60 3.57 2.70 45.70 66.99 11.82 77.64 3.64 2.35 45.35 65.91 11.61 77.13 3.45 2.49 45.49 68.93 12.19 78.07 3.6 2.26 45.26 66.53 11.73 77.07 3.8 2.47 45.47 63.57 11.16 76.74 3.99 2.24 45.24 60.99 10.67 75.79 3.89 2.31 45.31 62.32 10.1 76.26 3.75 2.22 45.22 64.28 11.3 76.67 3.98 2.17 45.17 61.12 10.69 75.78 3.47 2.31 45.31 68.60 12.13 77.48 4.06 2.14 45.14 60.09 10.24 75.29 4.3 2.07 45.07 57.20 9.96 74.40 4.31 1.91 44.91 57.08 9.94 74.41 4.02 2.02 45.02 60.60 10.6 75.82 4.11 2.00 45.00 59.46 10.38 75.34 3.94 2.22 45.22 61.65 10.79 75.73 3.86 2.09 45.09 62.73 11.0 76.22 3.67 2.28 45.28 65.46 11.52 77.17
Figure (4.6) and Table (4.4) Show the relation ship between Effective height and
Co/Q for proposed model, power law and logarithmic law. It was found that there is agreement between proposed model and logarithmic law in unstable case, while The values of C o/Q are smaller for the other two models.
85 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
4.6-Verification
The dosage ˝D˝ is the time integral of the concentration during the passage of a point source puff is given by:
t D= ∫0 Cdt (4.8)
Where C is the concentration in g/m 3 or Bq/m 3.
For an instantaneous point source and assuming non absorbing boundary, the mass balance condition is: z t Q=∫0 ∫ 0 UzCzdtdz( ) ( ) = Constant (4.9)
Using Eq. (6) we get the emission rate in the form:
z Q=∫ 0 U( zDdz) = Constant (4.10) where:
Q=35 Bq, where the wind speed (u) is 2.8 m/s and the lapse rate ( T/ Z) is o.36(0C/100m).
This is the case of stable, using Eq. (4.8); one can get the concentration at the axis of the
3 plume (C 0) equals 1.210 Bq/m . Then the concentration at the ground modifies to Q is the total material discharge per unit second of the source g/s or Bq/s.
U is the wind speed m/s.
D is the dosage of material in g. s. m 3 or Bq. s. m 3.
86 Chapter IV A Simple Model for Pollutant Diffusion Emitted from a Point Source under different Atmospheric Conditions
For a point source locate at height H 1=27m (height of the source of the Research
Reactor from the ground. For Iodine (I 138 ), the height of the plume (H) is 31.29m, the total material discharge per unit second (Q) is
C(ground)= 1.210 (1 (H 1/H))= 1.210 (1 (27/31.29))=0.1659 Bq/m
The observed concentration at a distance x=300 m (with H=31.29m) was only 0.16
Bq/m 3.The calculated emission rate is:
Q= (Cobs .Q obs. )/C cal. = ((0.16) (35))/ (0.1659) =33.75 Bq
Using the calculated value of Q, the calculated concentration at the ground is:
C (ground) =0.1739 Bq/m3.
87 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
CHAPTER V
Isotopes concentrations using different schemes of dispersion parameters
5.1-Introduction
A simple model for atmospheric dispersion in short range is the
Gaussian plume model (Curtiss and Rabl.1996). One of the most important parameters in plume dispersion modeling is the plume growth, more commonly referred to as dispersion coefficients (σ) (Yadav and
Sharan, 1996). These plume width parameters depends on meteorological variables (Panofsky and Dutton, 1984).
Various parameterizations exist for the vertical and lateral plume dispersion, the dispersion parameters are expressed as functions of downwind distance and wind speed is parameterized as a power law function of the vertical height above the ground (Irwin, 1979).
Since the Gaussian plume model is expressed in terms of the
dispersion parameters σ y and σ z the appropriate selection of lateral and
vertical dispersion parameters is much targeted. We select the four
different methods namely, power law , Briggs, Irwin and standard
method for calculating σ y and σ z to select the most accurate one (Essa.
and Fouad, (2011)).
88 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Atmospheric dispersion scientists and modelers seek to characterize air pollution spread in terms of important parameters representing the actual state of the atmospheric turbulence. not too long ago this was impossible and scientists as late as the 1960 and 1970 instead invented methods for dispersion parametersation based on synoptic classification schemes (time of day, cloud cover and mean wind speed) (Torben Mikkelsen,2003).
In this work, we use Gaussian plume formula in order to estimate concentration from a continuous point source of strength Q with interference from the ground at mean wind speed U and taking the dilution factors.
5.2- Gaussian distributions
The Gaussian plume formula for concentration from a continuous
point source of strength Q with interference from the ground at a mean
wind speed U and taking the dilution factors is as follows (Essa, et al.)
(2005):
−(( 2/π) *()Vd / U ) x dx χxyzQ, ,= / 2 πσσ + CAU / *exp − λ xU / * exp () (){}y z w () ∫ 2 0 H σ exp z 2 (5.1 ) 2σz *exp−y22 /2σ * exp −− zH2 /2 σ 2 +−+ exp zH 2 /2 σ 2 ()y{ ()() z z } where:
χ is the mean concentration of the effluent at a point (x, y,z), (Bq m 3).
89 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Q is the source strength (Bq).
U is the mean wind speed (m s 1). x,y,z refer to a downwind, crosswind and vertical coordinate system at the center of the moving cloud.
σi (i=x,y,z) are the plume dispersion coefficients in the x,y and z directions respectively (m) (Neiuwstadt and van Dop, 1984; Zannetti,
1990, Shleien, 1992; Faw and Shultis, 1993).
Exp ( x λ /U) is the radioactive decay for the specified nuclide.
A is the cross sectional area of the building normal to the wind.
Cw is the ‘shape factor’ that represents the fraction of ‘A’ over which the plume is dispersed; C w =0.5 is a conservative value which is commonly used.
Vd is deposition velocity (m/s).
H is the effective stack height {h s (stack height) + h (plume rise)} (m).
Wet deposition was neglected, as the annual precipitation was very small. Dry deposition was taken into consideration during calculations by assuming the deposition velocity, V d =0.01m/s for Iodine and 0.035 for
− 2/π *()V / U (( ) d ) x dx Cesium, where exp ∫ 2 is due to dry deposition. 0 H σ z exp 2 2σ z
The wind speed is taking in the form (Irwin, 1979):
p U=U 10 (h s /10)
90 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Where:
U is the wind speed, at any vertical height (m/s).
U10 is the wind speed at 10 m height. hs is the stack height (m). p is a parameter estimated by Irwin (1979), which is related to stability classes as shown in table (5.1):
Table (5.1). Estimated of the power (p) in urban areas for six stability classes based on information by Irwin (1979).
Stability A B C D E F classes P o.15 0.15 0.20 0.25 0.40 0.60
5.3-Dispersion parameters schemes
We select the four different methods namely, power law , Briggs, Irwin and standard method for calculating σ y and σ z to select the most accurate one(Essa and Fouad, 2011), as follows.
5.3-a Power –law method
In this method, σ y and σ z can be calculated from the following
formula:
σ = c x m y (5.2)
n σz =d x (5.3)
Where c, d, m, n values (Panofsky and Dutton, 1984) differ according to stability classes, as follows:
91 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.2), values of the dispersion parameters for the Pasquill stability classes.
Stability σy (m) σz (m) C M D n A B 1.46 0.71 0.01 1.54 C 1.52 0.69 0.04 1.17 D 1.36 0.67 0.09 0.95 E F 0.79 0.70 0.40 0.67
5.3-b Standard method
In this method, σ y and σ z can be analytically expressed, based on
(Pasquill, 1974.Giffort, 1972) curves, using the following forms:
r x σ y = p (5.4 ) x 1+ a
s x σ z = q (5.5 ) x 1+ a
Where r, s, p and q are constants depending on the atmospheric stability.
These values are given in the following table:
Table (5.3), values of the dispersion parameters for the Pasquill stability classes.
Stability A B C D E F Classes R (m/km) 250 202 134 78.7 65.6 37 S (m/km) 102 96.2 72.2 47.5 33.5 2 A (km) 0.927 0.370 0.283 0.707 1.07 1.17 P 0.189 0.162 0.134 0.135 0.137 0.134 Q 1.918 0.101 0.102 0.465 0.624 0.70
5.3-c Briggs method
In this method, σy and σ z can be calculated from the following table according to (Briggs (1973)).
92 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.4). Formulas produced by Briggs (1973) for σy (x) and σ z (x)
Stability σ y (x) classes σz (x) A 0..32x (1+0.0004x) 1/2 0.24x (1+0.001x )1/2
B 0.32x (1+0.0004x) 1/2 0.24x (1+0.001x)1/2 C 0..32x (1+0.0004x) 1/2 0.20x D 0.16x (1+0.0004x) 1/2 0.14x (1+0.0003x) 1/2 E 0.11x (1+0.0004x) 1/2 0.08x (1+0.00015x) 1/2 F 0.11x (1+0.0004x) 1/2 0.08x(1+0.00015x) 1//2
5.3-d Irwin method
In this method, σ y and σ z are calculated using the following formula:
σ θ x σ y ()x = x (5.6) 1+ 0.9 1000 U
σ z (x ) = σφ x (5.7)
Where σ θ and σ φ are the standard deviation of the wind direction in the
horizontal and vertical directions, respectively. Specification of σ θ and σ φ
can be found in Gifford, (1976) and Hanna et al., (1982). Based on the
Pasquill stability classes from A to F.
Table (5.5). The values of the standard deviation of the wind direction in horizontal and vertical directions for different stability classes.
Stability classes σθ (deg) σφ (deg) A 25 10 B 20 8 C 15 6.5 D 10 5.5 E 5 2.5 F 2.5 1
93 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.6), shows the source strength and decay constant (AEA, 2006)
131 133 135 137 for I, I, I and Cs through thirteen experiments, downwind
distance, wind speed, stability and the effective height for two stacks.
Table (5.6).Source strength ‘Q’ (Bq) and decay distance ‘λ ‘for the studied fission radionuclide and Meteorological data (downwind distance ‘x’, wind speed ‘U’, stability classes and different effective heights).
Exp. Q (I 131 ) Q (I 133 ) Q (I 135 ) Q (Cs 137 ) Downwind U Stabilit H distance ‘x’ (m/s) y (m) (m) classes 1 28114286 2811429 1028571 0.555429 92 4 A 49 2 28700000 2870000 1050000 0.567 96 4 A 48 3 1171429 117142.9 42857.14 0.023143 97 6 B 45 4 12885714 1288571 471428.6 0.254571 98 4 C 46 5 13471429 1347143 492857.1 0.266143 99 4 A 45 6 140557143 1405714 514285.7 0.277714 100 4 D 45 7 27528571 2752857 1007143 0.543857 115 4 E 47 8 28524286 2752857 1043571 0.563529 132 4 C 46 9 28260714 2826071 1033929 0.558321 134 4 A 47 10 2928571.4 292857.1 107142.9 0.057857 165 3 D 28 11 4100000 410000 150000 0.081 184 2 B 28.3 12 1171428.6 117142.9 42857.14 0.023143 200 3 A 30.8 13 2342857.1 234285.7 85714.29 0.046286 300 3 A 30.6 λ 9.95*10 7 9.25*10 5 7.3*10 10 3.8*10 4
5.4-Result and discussion
Using table (5.6) isotopes concentrations can be calculated at different
shapes of dispersions parameters (power law, standard, Briggs and Irwin),
so during thirteen experiments to know the best isotopes concentration
and comparing with observed concentrations.
Tables (5.7), (5.8), (5.9) and (5.10) shows the variations from isotopes
concentrations for 131 I at different shapes of dispersions parameters
(Briggs, Standard, power law, and Irwin methods), so during thirteen
experiments respectively.
94 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.7) .Comparison between predicated and observed concentrations for 131 I (Briggs method) from 1 to 13 experiments.
Predicated concentration (Bq / m 3) Observed 1 2 3 4 5 6 7 8 9 10 11 12 13 0.025 0.03 0.01 0.03 0.03 0.03 0.03 0.03 0.04 0.02 0.05 0.01 0.02 0.04 0.037 0.30 0.05 0.06 0.11 0.11 0.15 0.05 0.03 0.02 0.02 0.03 0.08 0.02 0.09 0.17 0.30 0.26 0.28 0.30 0.08 0.04 0.26 0.06 0.06 0.08 0.22 0.04 0.20 0.26 0.10 0.04 0.10 0.11 0.01 0.22 0.10 0.23 0.19 0.27 0.08 0.02 0.27 0.01 0.09 0.17 0.18 0.19 0.10 0.09 0.07 0.40 0.36 0.05 0.14 0.29 0.19 0.16 0.20 0.09 0.19 0.01 0.08 0.11 0.19 0.22 0.18 0.37 0.37 0.37 0.45 0.38 0.49 0.24 0.25 0.03 0.69 0.27 0.06 0.54 0.04 0.63 0.63 0.63 0.12 0.36 0.37 0.02 0.17 0.17 0.22 0.35 0.37 0.36 0.03 0.17 0.17 0.17 0.03 0.35 0.40 0.30 0.11 0.12 0.18 0.30 0.39 0.24 0.21 0.30 0.08 0.17 0.42 0.10 0.10 0.04 0.05 0.05 0.50 0.10 0.10 0.10 0.01 0.30 0.30 0.30 0.42 0.64 0.70 0.11 0.21 0.13 0.22 0.59 0.68 0.66 0.24 0.52 0.52 0.52 0.67 0.39 0.50 0.22 0.47 0.26 0.64 0.28 0.47 0.42 0.48 0.49 0.49 0.58 0.67 0.47 0.63 0.42 0.53 0.49 0.54 0.40 0.70 0.47 0.11 0.11 0.16 0.16
Table (5.8). Comparison between predicated and observed concentrations for 131 I (standard method) from 1 to 13 experiments. Predicated concentration (Bq /m 3) Observed 1 2 3 4 5 6 7 8 9 10 11 12 13 0.025 0.01 0.01 0.01 0.02 0.01 0.02 0.02 0.03 0.02 0.03 0.03 0.03 0.02 0.037 0.04 0.01 0.01 0.08 0.09 0.08 0.02 0.02 0.02 0.02 0.03 0.03 0.02 0.091 0.07 0.10 0.04 0.04 0.09 0.04 0.08 0.08 0.08 0.08 0.02 0.10 0.05 0.197 0.14 0.08 0.03 0.04 0.04 0.04 0.08 0.09 0.09 0.37 0.01 0.04 0.07 0.272 0.05 0.24 0.10 0.07 0.07 0.07 0.01 0.05 0.05 0.02 0.02 0.61 0.01 0.188 0.07 0.22 0.14 0.06 0.07 0.06 0.01 0.01 0.01 0.01 0.02 0.06 0.01 0.447 0.06 0.17 0.38 0.17 0.22 0.17 0.50 0.59 0.56 0.65 0.72 0.01 0.12 0.123 0.03 0.05 0.02 0.02 0.02 0.02 0.38 0.04 0.39 0.05 0.06 0.02 0.03 0.032 0.04 0.05 0.03 0.02 0.04 0.02 0.04 0.04 0.04 0.06 0.03 0.02 0.03 0.42 0.03 0.62 0.03 0.02 0.19 0.02 0.39 0.04 0.04 0.04 0.58 0.02 0.03 0.42 0.04 0.01 0.04 0.16 0.21 0.12 0.25 0.26 0.25 0.26 0.68 0.05 0.10 0.67 0.54 0.69 0.29 0.22 0.30 0.22 0.47 0.49 0.48 0.50 0.70 0.20 0.40 0.67 0.24 0.01 0.24 0.43 0.67 0.63 0.18 0.18 0.73 0.19 0.65 0.57 0.51
95 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.9). Comparison between predicated and observed concentrations for 131 I (power law) from 1 to 13 experiments.
Predicated concentration (Bq /m 3)
Observed 1 2 3 4 5 6 7 8 9 10 11 12 13 0.025 0.04 0.13 0.04 0.02 0.02 0.02 0.04 0.05 0.04 0.05 0.06 0.02 0.04 0.037 0.03 0.05 0.03 0.01 0.01 0.01 0.03 0.03 0.03 0.03 0.28 0.01 0.02 0.091 0.07 0.01 0.07 0.03 0.03 0.03 0.07 0.07 0.07 0.07 0.70 0.03 0.06 0.197 0.01 0.05 0.01 0.06 0.06 0.71 0.01 0.01 0.01 0.01 0.02 0.19 0.01 0.272 0.18 0.03 0.18 0.08 0.08 0.08 0.17 0.18 0.18 0.02 0.16 0.07 0.01 0.188 0.49 0.13 0.49 0.13 0.15 0.04 0.06 0.21 0.10 0.05 0.21 0.21 0.24 0.447 0.26 0.43 0.36 0.65 0.41 0.65 0.70 0.18 0.31 0.29 0.24 0.32 0.63 0.123 0.04 0.09 0.05 0.06 0.05 0.02 0.35 0.61 0.06 0.13 0.44 0.05 0.31 0.032 0.03 0.01 0.01 0.04 0.03 0.04 0.05 0.04 0.09 0.01 0.01 0.04 0.08 0.42 0.02 0.05 0.72 0.33 0.35 0.03 0.71 0.73 0.72 0.08 0.21 0.03 0.06 0.42 0.30 0.14 0.50 0.60 0.47 0.79 0.33 0.53 0.30 0.59 0.40 0.35 0.71 0.67 0.21 0.08 0.21 0.22 0.52 0.32 0.42 0.50 0.39 0.46 0.25 0.38 0.68 0.67 0.33 0.04 0.33 0.36 0.01 0.64 0.68 0.19 0.40 0.26 0.24 0.31 0.61
Table (5.10). Comparison between predicated and observed concentrations for 131 I from 1 to 13 experiments in Irwin method. Predicated concentration (Bq/m 3)
Observed concentration (Bq/m 3) 1 2 3 4 5 6 7 8 9 10 11 12 13 0.025 0.03 0.02 0.03 0.03 0.03 0.03 0.06 0.03 0.01 0.01 0.01 0.01 0.03 0.037 0.06 0.03 0.02 0.03 0.03 0.03 0.06 0.06 0.06 0.06 0.01 0.01 0.02 0.091 0.08 0.09 0.03 0.04 0.04 0.04 0.08 0.08 0.08 0.08 0.01 0.01 0.03 0.197 0.21 0.06 0.17 0.10 0.10 0.10 0.20 0.21 0.21 0.21 0.02 0.03 0.01 0.272 0.05 0.23 0.02 0.02 0.02 0.02 0.05 0.05 0.05 0.05 0.01 0.01 0.21 0.188 0.49 0.07 0.37 0.37 0.37 0.22 0.48 0.49 0.49 0.49 0.05 0.07 0.02 0.447 0.53 0.53 0.63 0.63 0.63 0.16 0.48 0.57 0.54 0.54 0.26 0.37 0.11 0.123 0.10 0.09 0.17 0.17 0.17 0.04 0.09 0.10 0.10 0.10 0.01 0.01 0.40 0.032 0.02 0.01 0.10 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.24 0.34 0.10 0.42 0.10 0.54 0.30 0.30 0.30 0.05 0.10 0.10 0.10 0.10 0.01 0.01 0.04 0.42 0.02 0.10 0.52 0.52 0.52 0.01 0.52 0.02 0.02 0.02 0.33 0.27 0.77 0.67 0.01 0.49 0.99 0.79 0.88 0.07 0.14 0.15 0.46 0.46 0.51 0.21 0.60 0.67 0.26 0.23 0.11 0.16 0.11 0.12 0.25 0.03 0.58 0.58 0.27 0.38 0.11
96 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
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97 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Fig. (5.1) Comparison between predicated, observed and distance for 131 I from 1 to 13 experiments in Briggs, Power law, standard and Irwin methods respectively.
98 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
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99 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Fig. (5.2): Comparison between predicated, observed and distance for 135 I in case Power law, standard, Briggs and Irwin methods from 1 to 13 experiments respectively
100 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.11). Comparison between predicated and observed concentrations for 135 I in case standard method from 1 to 13 experiments. Observed Predicated concentration (Bq/m 3) concentration (Bq/m 3) 1 2 3 4 5 6 7 8 9 10 11 12 13 0.03 0.03 0.12 0.04 0.08 0.04 0.08 0.01 0.08 0.09 0.01 0.03 0.07 0.05 0.041 0.06 0.1 0.03 0.07 0.03 0.06 0.01 0.07 0.07 0.05 0.03 0.05 0.04 0.037 0.01 0.04 0.13 0.03 0.01 0.03 0.29 0.03 0.03 0.01 0.01 0.24 0.04 0.05 0.03 0.29 0.01 0.31 0.16 0.31 0.32 0.31 0.32 0.03 0.13 0.26 0.06 0.04 0.03 0.82 0.02 0.05 0.03 0.03 0.05 0.05 0.06 0.03 0.02 0.04 0.07 0.145 0.12 0.19 0.12 0.27 0.27 0.22 0.54 0.37 0.06 0.06 0.22 0.24 0.16 0.14 0.11 0.24 0.31 0.33 0.2 0.23 0.22 0.3 0.05 0.05 0.12 0.2 0.16 0.092 0.07 0.17 0.05 0.17 0.17 0.14 0.01 0.17 0.15 0.82 0.06 0.12 0.15 0.032 0.02 0.29 0.07 0.01 0.07 0.01 0.05 0.01 0.01 0.02 0.04 0.01 0.04 0.24 0.23 0.33 0.17 0.15 0.17 0.14 0.15 0.15 0.15 0.17 0.21 0.12 0.27 0.24 0.22 0.33 0.14 0.29 0.26 0.19 0.36 0.23 0.1 0.32 0.38 0.37 0.25 0.34 0.14 0.26 0.22 0.18 0.29 0.17 0.18 0.18 0.19 0.17 0.24 0.27 0.18 0.34 0.31 0.32 0.3 0.17 0.33 0.25 0.31 0.27 0.19 0.15 0.28 0.26 0.25
Table (5.12). Comparison between predicated and observed concentrations for 135 Iin case Briggs method from 1to 13 experiments. Predicated concentration (Bq/m 3) Observed 2 3 4 5 6 7 8 9 10 11 12 13 0.03 0.03 0.08 0.09 0.03 0.11 0.09 0.01 0.04 0.01 0.01 0.07 0.14 0.041 0.03 0.05 0.01 0.07 0.03 0.02 0.02 0.04 0.04 0.04 0.03 0.03 0.037 0.03 0.01 0.04 0.05 0.05 0.03 0.04 0.02 0.02 0.01 0.02 0.04 0.05 0.05 0.16 0.02 0.02 0.02 0.04 0.04 0.04 0.04 0.39 0.13 0.03 0.04 0.01 0.03 0.05 0.02 0.02 0.02 0.02 0.01 0.02 0.04 0.03 0.02 0.145 0.06 0.40 0.37 0.39 0.04 0.08 0.28 0.08 0.05 0.05 0.15 0.31 0.14 0.08 0.28 0.40 0.47 0.47 0.10 0.11 0.08 0.19 0.19 0.28 0.17 0.092 0.01 0.07 0.03 0.29 0.23 0.05 0.06 0.10 0.03 0.08 0.06 0.18 0.032 0.09 0.01 0.02 0.02 0.01 0.04 0.18 0.03 0.04 0.04 0.01 0.13 0.24 0.37 0.28 0.10 0.12 0.18 0.21 0.20 0.31 0.26 0.27 0.43 0.25 0.24 0.10 0.16 0.30 0.18 0.42 0.16 0.31 0.31 0.16 0.16 0.23 0.45 0.34 0.20 0.29 0.15 0.39 0.39 0.31 0.16 0.35 0.22 0.22 0.37 0.35 0.34 0.29 0.16 0.14 0.21 0.15 0.36 0.33 0.32 0.45 0.39 0.31 0.13
101 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.13). Comparison between predicated and observed concentrations for 135 I case power law from 1 to 13 experiments
Observed 3 concentration Predicated concentration (Bq/m ) (Bq/m 3) 1 2 3 4 5 6 7 8 9 10 11 12 13 0.03 0.01 0.04 0.01 0.03 0.01 0.04 0.01 0.01 0.01 0.03 0.03 0.03 0.01 0.041 0.04 0.02 0.00 0.02 0.04 0.02 0.04 0.04 0.04 0.04 0.09 0.02 0.03 0.037 0.09 0.05 0.00 0.04 0.09 0.04 0.09 0.09 0.09 0.01 0.02 0.04 0.01 0.05 0.02 0.01 0.03 0.01 0.02 0.01 0.02 0.02 0.02 0.02 0.03 0.03 0.02 0.04 0.02 0.01 0.03 0.02 0.02 0.10 0.02 0.02 0.03 0.02 0.03 0.01 0.02 0.145 0.12 0.09 0.37 0.06 0.27 0.17 0.07 0.27 0.12 0.17 0.17 0.27 0.16 0.14 0.13 0.17 0.13 0.33 0.24 0.10 0.23 0.13 0.23 0.15 0.24 0.16 0.24 0.092 0.07 0.05 0.17 0.02 0.07 0.09 0.17 0.02 0.11 0.07 0.07 0.11 0.06 0.032 0.02 0.01 0.02 0.01 0.02 0.01 0.02 0.02 0.02 0.02 0.03 0.07 0.01 0.24 0.30 0.16 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.24 0.20 0.13 0.12 0.22 0.32 0.22 0.52 0.52 0.22 0.22 0.12 0.22 0.32 0.34 0.28 0.22 0.19 0.39 0.49 0.29 0.19 0.38 0.29 0.39 0.29 0.29 0.18 0.34 0.26 0.29 0.11 0.36 0.16 0.44 0.11 0.16 0.37 0.37 0.24 0.14 0.11
Table (5.14). Comparison between predicated and observed concentrations for 135 I in case Irwin method from 1 to 13 experiments
Observed Predicated concentration (Bq/m 3) concentration
(Bq/m 3) 1 2 3 4 5 6 7 8 9 10 11 12 13 0.03 0.02 0.05 0.03 0.01 0.02 0.09 0.02 0.02 0.02 0.02 0.02 0.1 0.03 0.041 0.02 0.03 0.03 0.1 0.02 0.03 0.02 0.02 0.02 0.02 0.02 0.09 0.1 0.037 0.03 0.04 0.04 0.01 0.03 0.06 0.03 0.03 0.03 0.03 0.03 0.01 0.04 0.05 0.08 0.11 0.03 0.04 0.08 0.01 0.08 0.08 0.08 0.08 0.08 0.03 0.05 0.04 0.02 0.05 0.01 0.08 0.02 0.08 0.02 0.02 0.02 0.02 0.02 0.08 0.06 0.145 0.02 0.19 0.01 0.08 0.02 0.08 0.02 0.07 0.02 0.02 0.02 0.08 0.09 0.14 0.1 0.24 0.01 0.05 0.1 0.03 0.1 0.13 0.1 0.01 0.01 0.04 0.19 0.092 0.04 0.07 0.05 0.02 0.04 0.02 0.13 0.17 0.16 0.04 0.04 0.15 0.19 0.032 0.09 0.03 0.01 0.04 0.09 0.17 0.08 0.09 0.09 0.09 0.09 0.06 0.05 0.24 0.04 0.09 0.05 0.02 0.04 0.06 0.04 0.04 0.04 0.04 0.04 0.02 0.05 0.24 0.12 0.14 0.20 0.31 0.17 0.1 0.26 0.22 0.17 0.27 0.17 0.28 0.27 0.34 0.35 0.25 0.33 0.45 0.35 0.2 0.24 0.19 0.38 0.28 0.28 0.22 0.37 0.34 0.25 0.33 0.25 0.35 0.2 0.24 0.19 0.18 0.28 0.28 0.22 0.37 0.32
When comparing between the final results obtained using the four
methods. One look for which is the all experiments to be used for figures
102 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
(5.1) and (5.2) this show the variation between the observed and
predicted normalized crosswind concentrations using four methods of 131 I
and 135 I respectively. It was found that both models can agree with
observed data expect for some points.
Tables (5.11), (5.12), (5.13) and (5.14) shows the variations from isotopes
concentrations for 135 I at different shapes of dispersions parameters
(standard, Briggs, power law and Irwin methods), so during thirteen
experiments respectively.
Table (5.15) .Comparison between predicated and observed concentrations for 137 Cs in case power law from 1 to 13 experiments Predicated concentration (Bq/m 3)
Observed 1 2 3 4 5 6 7 8 9 10 11 12 13 0.002 0.004 0.002 0.022 0.005 0.002 0.002 0.001 0.044 0.044 0.002 0.038 0.002 0.004 0.004 0.042 0.030 0.472 0.016 0.017 0.002 0.035 0.036 0.036 0.037 0.015 0.015 0.042 0.005 0.028 0.024 0.026 0.025 0.022 0.035 0.002 0.028 0.002 0.002 0.023 0.048 0.028 0.007 0.001 0.068 0.260 0.051 0.056 0.037 0.016 0.018 0.017 0.002 0.044 0.010 0.001 0.009 0.083 0.472 0.200 0.013 0.138 0.001 0.028 0.029 0.029 0.030 0.020 0.012 0.083 0.007 0.008 0.140 0.022 0.078 0.036 0.062 0.278 0.092 0.028 0.028 0.013 0.118 0.008 0.007 0.017 0.060 0.065 0.017 0.009 0.005 0.124 0.206 0.127 0.031 0.008 0.026 0.017 0.019 0.009 0.017 0.047 0.045 0.036 0.056 0.001 0.019 0.001 0.028 0.011 0.032 0.009 0.006 0.009 0.153 0.317 0.003 0.037 0.004 0.007 0.008 0.003 0.008 0.011 0.032 0.009 0.002 0.002 0.002 0.005 0.002 0.003 0.001 0.008 0.006 0.008 0.001 0.004 0.003 0.002 0.004 0.013 0.005 0.092 0.155 0.023 0.002 0.005 0.043 0.005 0.005 0.001 0.002 0.013 0.008 0.039 0.008 0.005 0.341 0.010 0.006 0.092 0.006 0.009 0.005 0.037 0.004 0.039 0.009 0.014 0.008 0.047 0.113 0.017 0.008 0.003 0.009 0.035 0.004 0.001 0.001 0.014
103 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.16). Comparison between predicated and observed concentrations for 137 Cs in case standard method from 1 to 13 experiments.
Predicated concentration (Bq/m 3)
Observed 1 2 3 4 5 6 7 8 9 10 11 12 13 0.002 0.040 0.020 0.022 0.020 0.021 0.002 0.043 0.044 0.044 0.046 0.038 0.002 0.040 0.004 0.042 0.030 0.472 0.016 0.017 0.002 0.035 0.036 0.036 0.037 0.015 0.015 0.042 0.005 0.028 0.024 0.026 0.025 0.022 0.035 0.002 0.028 0.002 0.002 0.023 0.048 0.028 0.007 0.001 0.068 0.260 0.051 0.056 0.037 0.016 0.018 0.017 0.002 0.044 0.010 0.001 0.009 0.083 0.472 0.200 0.013 0.138 0.001 0.028 0.029 0.029 0.030 0.020 0.012 0.083 0.007 0.008 0.140 0.022 0.078 0.036 0.062 0.278 0.092 0.028 0.028 0.013 0.118 0.008 0.007 0.017 0.060 0.065 0.017 0.009 0.005 0.124 0.206 0.127 0.031 0.008 0.026 0.017 0.019 0.009 0.017 0.047 0.045 0.036 0.056 0.001 0.019 0.001 0.028 0.011 0.032 0.009 0.006 0.009 0.153 0.317 0.003 0.037 0.004 0.007 0.008 0.003 0.008 0.011 0.032 0.009 0.002 0.002 0.002 0.005 0.002 0.003 0.001 0.008 0.006 0.008 0.001 0.004 0.003 0.002 0.004 0.013 0.005 0.092 0.155 0.023 0.002 0.005 0.043 0.005 0.005 0.001 0.002 0.013 0.008 0.039 0.008 0.005 0.341 0.010 0.006 0.092 0.006 0.009 0.005 0.037 0.004 0.039 0.009 0.014 0.008 0.047 0.113 0.017 0.008 0.003 0.009 0.035 0.004 0.001 0.001 0.014
Table (5.17). Comparison between predicated and observed concentrations for 137 Csin case Irwin method from 1 to 13 experiments.
Predicated concentration 3 Observed (Bq/m ) concentration (Bq/m 3) 1 2 3 4 5 6 7 8 9 10 11 12 13 0.002 0.013 0.02 0.05 0.02 0.006 0.064 0.059 0.043 0.002 0.009 0.005 0.007 0.011 0.004 0.011 0.11 0.05 0.06 0.005 0.056 0.002 0.041 0.001 0.002 0.069 0.008 0.009 0.005 0.016 0.26 0.07 0.02 0.007 0.078 0.031 0.087 0.002 0.009 0.001 0.002 0.013 0.007 0.041 0.00 0.02 0.01 0.020 0.207 0.061 0.002 0.002 0.001 0.005 0.003 0.035 0.009 0.010 0.60 0.04 0.06 0.005 0.049 0.021 0.006 0.004 0.013 0.008 0.005 0.003 0.007 0.097 0.08 0.04 0.03 0.047 0.486 0.012 0.007 0.005 0.012 0.007 0.048 0.006 0.007 0.053 0.01 0.02 0.02 0.253 0.026 0.003 0.005 0.003 0.002 0.009 0.007 0.004 0.019 0.019 0.02 0.08 0.07 0.009 0.095 0.006 0.004 0.001 0.005 0.005 0.010 0.010 0.006 0.005 0.00 0.02 0.10 0.002 0.023 0.006 0.008 0.005 0.002 0.001 0.009 0.008 0.002 0.019 0.10 0.08 0.06 0.009 0.097 0.003 0.001 0.003 0.014 0.001 0.004 0.009 0.004 0.004 0.03 0.01 0.01 0.002 0.018 0.003 0.008 0.005 0.004 0.002 0.008 0.006 0.008 0.030 0.02 0.08 0.01 0.001 0.002 0.001 0.002 0.005 0.008 0.008 0.004 0.009 0.009 0.074 0.01 0.04 0.01 0.200 0.087 0.087 0.004 0.005 0.007 0.006 0.004 0.008
104 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.18), Comparison between predicated and observed concentrations for 137 Cs in case Briggs method from 1 to 13 experiments. Predicated concentration (Bq/m 3) Observed 1 2 3 4 5 6 7 8 9 10 11 12 13 0.002 0.001 0.001 0.022 0.009 0.004 0.006 0.002 0.001 0.002 0.002 0.007 0.002 0.003 0.004 0.039 0.039 0.002 0.002 0.004 0.004 0.006 0.002 0.006 0.006 0.001 0.002 0.005 0.005 0.105 0.105 0.010 0.005 0.007 0.006 0.005 0.004 0.007 0.003 0.006 0.010 0.009 0.007 0.003 0.003 0.001 0.003 0.008 0.004 0.009 0.004 0.002 0.008 0.005 0.001 0.003 0.009 0.002 0.002 0.003 0.004 0.003 0.003 0.008 0.003 0.005 0.004 0.004 0.003 0.009 0.007 0.003 0.002 0.001 0.006 0.007 0.008 0.006 0.007 0.004 0.002 0.009 0.001 0.005 0.007 0.003 0.003 0.006 0.003 0.002 0.004 0.003 0.008 0.001 0.008 0.003 0.009 0.002 0.019 0.010 0.008 0.001 0.004 0.002 0.005 0.006 0.003 0.005 0.005 0.006 0.003 0.003 0.006 0.007 0.007 0.005 0.001 0.001 0.001 0.002 0.002 0.005 0.005 0.002 0.005 0.007 0.002 0.074 0.007 0.003 0.001 0.001 0.001 0.002 0.002 0.004 0.004 0.009 0.003 0.002 0.004 0.004 0.005 0.014 0.003 0.004 0.004 0.007 0.008 0.004 0.004 0.005 0.004 0.003 0.008 0.008 0.008 0.002 0.002 0.002 0.002 0.005 0.005 0.003 0.003 0.009 0.008 0.003 0.009 0.050 0.005 0.005 0.002 0.002 0.002 0.004 0.005 0.003 0.006 0.006 0.008 0.004
Tables (5.15), (5.16), (5.17) and (5.18) shows the variations from isotopes
concentrations for 137 Cs at different shapes of dispersions parameters
(power law, standard, Irwin and Briggs methods), during thirteen
experiments respectively.
105 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
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106 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Fig. (5.3): Comparison between predicated, observed and distance for 137 Cs in case power law, standard, Irwin and Briggs methods from 1 to 13 experiments respectively.
107 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
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108 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Fig. (5.4): Comparison between predicated, observed and distance for 133 I in case power law, Standard, Briggs and Irwin methods from 1 to 13 experiments respectively.
109 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Overall the final results obtained using four methods. One may select which
are the all experiments to be used. For figures (5.3) and (5.4), it shows the
variation between the observed and predicted normalized crosswind
concentrations using four methods of 133 I and 137 Cs respectively. Therefore
one can find that both models may be considered with acceptable agreement
with observed data expect for few points.
Table (5.19). Comparison between predicated and observed concentrations for 133 I in case power law from 1 to 13 experiments.
Predicated concentration (Bq/m 3) observed 1 2 3 4 5 6 7 8 9 10 11 12 13 0.02 0.074 0.108 0.145 0.205 0.214 0.205 0.237 0.053 0.145 0.247 0.065 0.065 0.014 0.031 0.036 0.053 0.027 0.125 0.130 0.125 0.067 0.277 0.274 0.284 0.398 0.140 0.076 0.055 0.089 0.133 0.070 0.003 0.003 0.003 0.007 0.007 0.071 0.073 0.127 0.103 0.054 0.063 0.218 0.311 0.124 0.118 0.250 0.118 0.122 0.126 0.125 0.144 0.181 0.174 0.064 0.15 0.202 0.304 0.177 0.008 0.008 0.008 0.017 0.018 0.018 0.185 0.287 0.259 0.296 0.15 0.366 0.166 0.366 0.366 0.366 0.366 0.366 0.366 0.366 0.366 0.361 0.366 0.366 0.275 0.633 0.005 0.633 0.633 0.633 0.633 0.633 0.633 0.633 0.633 0.633 0.633 0.332 0.393 0.167 0.066 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.167 0.168 0.167 0.167 0.028 0.016 0.023 0.097 0.045 0.047 0.045 0.096 0.099 0.098 0.010 0.042 0.014 0.043 0.2 0.304 0.506 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.304 0.344 0.304 0.304 0.2 0.520 0.742 0.520 0.520 0.520 0.520 0.520 0.520 0.520 0.520 0.520 0.520 0.520 0.36 0.888 0.718 0.988 0.988 0.688 0.588 0.988 0.879 0.789 0.988 0.889 0.879 0.988 0.36 0.564 0.487 0.106 0.364 0.664 0.844 0.106 0.564 0.864 0.964 0.564 0.664 0.106
133 Table (5.20) .Comparison between predicated and observed concentrations for I in case standard method from 1 to 13experiments .
Predicated concentration (Bq/m 3) Observed 1 2 3 4 5 6 7 8 9 10 11 12 13 0.02 0.02 0.0316 0.0929 0.0102 0.0107 0.0102 0.3274 0.2263 0.2242 0.2324 0.0325 0.0929 0.0372 0.031 0.358 0.027 0.0751 0.0083 0.0086 0.0083 0.1191 0.1829 0.1812 0.1878 0.0263 0.0751 0.2272 0.055 0.01 0.001 0.3297 0.0363 0.3787 0.3627 0.4559 0.0287 0.4559 0.3066 0.1154 0.3297 0.587 0.063 0.03 0.008 0.355 0.3905 0.4075 0.3905 0.8563 0.0864 0.8563 0.0847 0.1242 0.355 0.001 0.15 0.2315 0.0241 0.0613 0.0067 0.007 0.0067 0.047 0.1492 0.1478 0.1531 0.2144 0.0613 0.1478 0.15 0.055 0.0933 0.0006 0.0006 0.0007 0.0006 0.01 0.0014 0.0014 0.0001 0.2062 0.589 0.0001 0.275 0.2 0.3802 0.1062 0.1168 0.1219 0.0012 0.5617 0.5856 0.5617 0.6546 0.3716 0.1062 0.0007 0.393 0.32 0.0047 0.1622 0.1784 0.186 0.1784 0.9121 0.3949 0.9121 0.054 0.5676 0.1622 0.1307 0.028 0.0702 0.0078 0.0162 0.0018 0.0019 0.0018 0.0001 0.0396 0.0392 0.0406 0.0569 0.0162 0.0081 0.2 0.21 0.0635 0.0002 0.0002 0.0002 0.0002 0.3971 0.0004 0.3971 0.1148 0.5761 0.1646 0.6009 0.2 0.24 0.0009 0.1051 0.0116 0.12 0.1156 0.5351 0.0587 0.5351 0.627 0.0368 0.1051 0.3709 0.36 0.33 0.07 0.0002 0.0002 0.0002 0.0002 0.0482 0.0005 0.0482 0.4999 0.6998 0.2 0.04 0.36 0.87 0.0006 0.7575 0.8332 0.8667 0.8332 0.7392 0.8444 0.7392 0.0937 0.0265 0.7575 0.0002
110 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.21). Comparison between predicated and observed concentrations for 133 I in case Briggs method For 1 to 13 experiments. Observed Predicated concentration (Bq/m 3) 1 2 3 4 5 6 7 8 9 10 11 12 13 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.031 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.055 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.063 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.275 0.393 0.393 0.393 0.393 0.393 0.393 0.393 0.393 0.393 0.393 0.393 0.393 0.393 0.393 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.028 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95
Table (5.22). Comparison between predicated and observed concentrations and distance for 133 I in case Irwin method from 1 to 13 experiments. Observed Predicated concentration (Bq/m 3) 1 2 3 4 5 6 7 8 9 10 11 12 13 0.02 0.01 0.05 0.03 0.04 0.03 0.07 0.01 0.06 0.04 0.01 0.03 0.05 0.04 0.031 0.02 0.01 0.05 0.01 0.03 0.01 0.05 0.02 0.04 0.03 0.01 0.02 0.04 0.055 0.06 0.03 0.02 0.04 0.05 0.02 0.06 0.04 0.02 0.01 0.05 0.07 0.04 0.063 0.03 0.05 0.02 0.02 0.02 0.03 0.04 0.04 0.04 0.04 0.02 0.13 0.03 0.15 0.24 0.15 0.33 0.18 0.15 0.13 0.20 0.16 0.08 0.37 0.14 0.29 0.22 0.15 0.20 0.06 0.40 0.37 0.39 0.04 0.08 0.83 0.08 0.11 0.05 0.25 0.31 0.275 0.40 0.24 0.18 0.40 0.47 0.47 0.10 0.11 0.08 0.19 0.19 0.58 0.72 0.393 0.01 0.01 0.72 0.93 0.29 0.29 0.95 0.56 0.40 0.80 0.28 0.39 0.18 0.028 0.02 0.09 0.01 0.11 0.03 0.04 0.14 0.18 0.77 0.54 0.54 0.56 0.13 0.2 0.04 0.37 0.08 0.10 0.02 0.02 0.15 0.11 0.09 0.57 0.57 0.43 0.85 0.2 0.10 0.10 0.06 0.71 0.74 0.74 0.52 0.67 0.31 0.16 0.16 0.23 0.45 0.36 0.20 0.20 0.17 0.55 0.39 0.39 0.31 0.43 0.93 0.22 0.22 0.37 0.35 0.36 0.71 0.79 0.56 0.14 0.91 0.91 0.61 0.63 0.83 0.89 0.89 0.31 0.13
Tables (5.19), (5.20), (5.21) and (5.22) show the variations in isotopes
concentrations for 133 I at different shapes of dispersions parameters
(power law, standard, Briggs and Irwin methods) during thirteen
experiments respectively.
111 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
5.4 Statistical method
Now, the statistical method is presented and comparison among analytical, statically and observed results will be offered (Hanna 1989).
Refer section (1.11).
Table (5.23).Average Predicated and observed concentration 131 for I under different methods. Average predicated concentration (Bq/m 3)
observed Briggs Standar concentratio metho d Power Irwin n (Bq/m 3) d Method Law Method 0.03 0.06 0.06 0.07 0.04 0.04 0.15 0.12 0.10 0.13 0.09 0.17 0.14 0.11 0.05 0.2 0.20 0.19 0.07 0.13 0.27 0.16 0.22 0.07 0.06 0.19 0.26 0.36 0.34 0.31 0.45 0.37 0.25 0.63 1.31 0.12 0.23 0.21 0.16 0.12 0.03 0.27 0.19 0.11 0.07 0.42 0.16 0.42 0.35 0.16 0.42 0.44 0.55 0.51 0.22 0.67 0.52 0.42 0.83 0.67 0.67 0.41 0.54 0.83 0.63
Table (5.24) .Shows average predicated and observed concentrations 135 for I under different methods.
Observed Average predicated concentration (Bq/m 3) concentration Standard Briggs Irwin (Bq/m 3) method method method Power law 0.03 0.06 0.59 0.05 0.06 0.041 0.05 0.68 0.05 0.05 0.037 0.07 0.86 0.08 0.07 0.05 0.19 0.14 0.20 0.19 0.04 0.14 0.70 0.11 0.14 0.145 0.26 0.43 0.28 0.26 0.14 0.23 1.16 0.19 0.23 0.092 0.21 0.65 0.19 0.21 0.032 0.05 0.77 0.03 0.05 0.24 0.24 0.61 0.22 0.24 0.24 0.34 0.37 0.45 0.34 0.34 0.28 0.88 0.27 0.28 0.34 0.34 0.87 0.45 0.34
112 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
Table (5.25), shows average predicated and observed concentrations for 137 Cs under different methods. Average predicated concentrations Observed Irwin Briggs standard concentration power law method method method 0.002 0.0308 0.023769 0.004769 0.075154 0.004 0.036879 0.032615 0.009077 0.109085 0.005 0.038073 0.045846 0.021692 0.115877 0.007 0.037342 0.031308 0.004154 0.1868 0.009 0.046088 0.063385 0.004077 0.206546 0.007 0.050754 0.067462 0.004692 0.113562 0.007 0.034696 0.031923 0.004231 0.116577 0.019 0.041373 0.025692 0.004692 0.112446 0.006 0.050435 0.014538 0.003846 0.050554 0.002 0.0322 0.030769 0.008692 0.222392 0.004 0.031258 0.008462 0.005308 0.124815 0.008 0.042031 0.013846 0.004615 0.2315 0.009 0.040852 0.041692 0.007846 0.040377
Table (5.26). Show average predicated and observed concentrations for 133 I from 1 to 13 experiments.
Observed Average predicated concentration (Bq/m 3) concentration power Standard Briggs (Bq/m 3) law method method Irwin method 0.02 0.002 0.108 0.010 0.012 0.031 0.010 0.114 0.014 0.016 0.055 0.025 0.261 0.031 0.035 0.063 0.000 0.301 0.028 0.033 0.15 0.059 0.097 0.034 0.038 0.15 0.350 0.069 0.324 0.317 0.275 0.585 0.505 0.588 0.581 0.393 0.159 0.296 0.193 0.198 0.028 0.001 0.023 0.004 0.005 0.2 0.323 0.178 0.293 0.291 0.2 0.537 0.202 0.474 0.466 0.36 0.906 0.124 0.807 0.835 0.36 0.528 0.576 0.564 0.532
113 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters
131 135 Table (5.27) Comparison between four method for I, I according to standard statistical performance measure Predicated 131 I 135 I Concentration model NMSE FB COR FAC2 NMSE FB COR FAC2 Briggs method 0.29 0.06 0.78 1.94 3.84 1.33 0.19 10.11 Standard method 0.21 0. 02 0.83 1 0.24 0.33 0.86 1.84 Power law 0.16 0.15 0.92 1.49 0.24 0.33 0.86 1.84 Irwin method 0.86 0.08 0.66 1.29 0.35 0.37 0.83 1.77
133 137 Table (5.28)) Comparison between four method for I and Cs according to standard statistical performance measure
Predicated 133 I 137 Cs Concentration model NMSE FB COR FAC2 NMSE FB COR FAC2
Briggs method 0.86 0.38 0.79 1.14 1.00 0.08 0.20 1.37 Standard method O. 57 0.22 0.54 2.07 21.73 1.79 0.03 25.97 Power law 1.08 0.42 0.75 1.11 4.06 1.41 0.42 7.62 Irwin method 0.86 0.38 0.78 1.16 4.36 1.32 0.12 6.35
Tables (5.23), (5.24), (5.25) and (5.26) shows the average predicated and
131 135 133 137 observed from isotopes concentrations for I , I , I to and Cs by
thirteen experiments at different shapes of dispersions parameters (power
law, standard, Briggs and Irwin methods), respectively.
From the statistical method given in Table (5.27), the predicted
concentrations for 131 I lie inside factor of 2 with observed data.
Regarding to NMSE, it is found that Briggs, Standard and power
methods are better than Irwin method. Regarding to FB and correlation
coefficient of all methods they coincide are better with observed data.
Similarly for 135 I it is within a factor of 2 with observed data except
Briggs method. Regarding to NMSE, FB and COR the predicted
concentrations for I 135 are better with observed data except the Briggs
114 CHAPTER V Isotopes concentrations using different schemes of dispersion parameters method. In Table (5.28): It was noticed that the predicted concentrations for 133 I lie inside factor of 2 with observed data. Regarding to NMSE, the predicted concentrations for 133 I using standard method is close to the observed data than the other methods. Regarding FB, the predicted concentrations for 133 I using all methods are better with the observed data. The correlation in all methods is stronger to the observed data except in the case of standard method. It was also found that the predicted concentrations for 137Cs using Briggs method lie inside factor of
2 with observed data. For NMSE and FB, the predicted concentrations for 137 Cs using Briggs method are closer to the observed data than the other methods. But the correlation of the predicted concentration for 137 Cs using power law in a good agreement to the observed data than other methods.
115 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
6.1 Introduction
The study and employment of operational short–range atmospheric dispersion models for environmental impact assessment have demonstrated to be of large use in the evaluation of ecosystems perturbation in many distinct scales, Meyer JFC.Diniz Gl. (2007).
In operational applications, the classical Gaussian diffusion models are largely employed in assessing the impacts of existing and proposed sources of air contaminants on local and urban air guilty (Arya SE. (1999). Simplicity, associated to the Gaussian analytical model, makes this approach particularly suitable for regulatory usage in mathematical modeling of the air pollution, such models are quite useful in short forecasting. The lateral and vertical dispersion parameters , respectively σ y and σ z ,represent the key turbulent parameterization in this approach they contain the physical ingredients that describe the dispersion process and , consequently, express the spatial extent of the contaminant plume under the effect of the turbulent motion in the PBL.
Abdul Wahab SA. (2006).
In this section, one can estimate the schemes of dispersion parameters in the lateral direction (σ y) and the vertical direction (σ z) in unstable stability by
116 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model using wind speed in power law and comparing between our work and
(algebraic and integral formulations ) with observed data of SF6 are taken from Copenhagen in Denmark.
6.2 Model formulation
The concentration associated from point source with strength Q, is expressed as (Akula Venkatram (2004)):
s Cxyz(, , ) A Bz y 2 =exp − exp − (6.1) Q Z 2 2π σ yU Z 2σ y
C is the average concentration of diffusing point (x, y, z) (kg/m 3).
U is mean wind velocity along the x axis (m/s). x is along –winds coordinate measured in wind direction from the source (m). y is cross wind coordinate direction (m). z is vertical coordinate measured from the ground (m).
σy is the plume dispersion parameter in the lateral directions.
Where the value of the parameter, s, depends on the stability (s=0.75 and A=
1.42 in unstable case (Sven – Erik Gryning and etals, (1983)).
The mean plume height, Z , is defined by
∞ ∫0 zC( x, yz , ) dz Z( x ) = ∞ (6.2) ∫0 C() x, yz , dz
And the mean plume velocity , U , is defined by
117 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
∞ ∫0 U( zC) ( xyzdz, , ) U = ∞ (6.3) ∫0 C() x, yz , dz assuming that the mean wind speed, U (z), can be described by a power law so that:
p z U z= U () r (6.4) Z r
Ur is a reference velocity at height Zr (U r =3.42m/s and Z r =4m, Veranth et al.,
2003). the value of the power, p, between 0.15 and 0.20 in unstable case
Let
s 1 Bz Bz =X ⇒ X s = Z Z 1 1 (6.5) Z Z −1 ∴=z Xs ⇒= dz XdX s B B s
Substitution from equations (6.1) and (6.5) in equation (6.2), we obtain that:
Γ (2 / s ) B = Γ 1 / s () (6.6) ∞ p −1 Γ()p =∫0 xexp( − xdx ) where Г (p) is the gamma function.
Substitution from equations (6.4), (6.1) and (6.5) in equation (6.3), we obtain the mean plume velocity
118 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
p +1 p Γ Z s U() z= U r (6.7) Zr B 1 Γ s
Van Ulden (1978) shows that the mean plume height, Z can be calculated from: d Z K( q Z ) = (6.8) d x U() qZ qZ and
1 s 2 ()1 − s Γ s q= s 1 (6.9) Γ s
Where K (z) is the eddy diffusivity parameterization which is led the K theory assumption.
According to Pleim and Chang (1992), the form of K (z) in an unstable case is:
z Kz() = kwz* 1 − h (6.10) where k is the von Karman constant which is set to 0.4 , w* is the convection scaling parameter and h is the effective height of release above the ground and it is estimated from (Briggs 1969): h=h s+ h where h s is physical stack height (115m).
119 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
h =3(W/U115 ) D where, W, is the exits velocity (4 m/s).
D is the internal stack diameter (1 m). and
p U115 = U 10 (hs /10)
U10 is the wind speed at 10 m height.
Substituting from equations (6.4), (6.10) in equation (6.8) and integrated the equation (6.8), the mean plume height Z can be derived as:
1 p +1 h2 kw Zp x Z = * r 2 p p (6.11) q1 − Ur q p + 1
Substituting from equation (6.11), in equation (6.7), the mean plume velocity
U can be obtained:
p p+1 p +1 2 p Γ hkwZx U s U() z = * r r p p 1 (6.12) 2 p ()Zr B Γ q1− Ur q p +1 s
We estimate the horizontal spread using Eckman’s (1994) hypothesis that:
d σ σ σ y =v = v (6.13) d x U p p +1 p +1 2 p Γ hkwZx U s * r r p p 1 2 p ()Zr B Γ q1− Ur q p +1 s
120 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model where
p p +1 p +1 2 p Γ hkwZx U s U() z = * r r p p 1 2 p ()Zr B Γ q1− Ur q p +1 s where σ v is the standard deviation of the wind speed in the lateral direction,(0.92 m/s) Veranth et al. (2003).
By Integrating the equation (6.13) with respect to x, we obtain the plume dispersion parameter in the lateral direction ( σ y ) as follows:
σ x() Z B p σ ()x = v r y p (6.14) p+1 p +1 2 p Γ h kw Z x s * r U p r 1 2 p Γ q1− Ur q p +1 s
Also estimating the vertical spread using Eckman’s (1994) hypothesis that:
d σ σ σ z =w = w (6.15) d x U p p +1 p +1 Γ hkwZx2 p U * r r s p p 1 2 p ()Zr B Γ q1− Ur q p +1 s where
p p +1 p +1 2 p Γ hkwZx U s U() z = * r r p p 1 2 p ()Zr B Γ q1− Ur q p +1 s
121 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model where σ w is the standard deviation of the wind speed in the vertical direction,(0.31 m/s) Veranth et al. (2003). Integrating equation (6.15) with respect to x, one can obtain the plume dispersion parameter in the vertical direction (σz) as follows:
σ x() Z B p σ ()x = w r z p (6.16) p +1 p +1 2 p Γ h kw Z x s * r U p r 1 2 p Γ q1− Ur q p +1 s
Then Gaussian expressions for the ground crosswind – integrated concentration
and the normalized ground –level concentration along
The plume centerline respectively is given by (Arya .SE. 1999) as follows:
2 Cy ( x , 0 ) 2 1 − h = exp 2 (6.17) Qπ U σ z 2 σ z
2 Cy ( x ,0,0 ) 1 −h = exp 2 (6.18) Qπ U σz σ y 2σ z
From the previous works, the plume dispersion parameters in the vertical and lateral directions (σ z and σy) respectively are given by Lidiane Buligon, etal. (2008) in the form:
σ 2 0 .4 2 ψ 2/3x 2 z = 2 1 / 3 (6.19) z i 1+ () 2.9 ψ x and
122 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
2 σ 0 .5 5 ψ 2/3x 2 y = 2 1 / 3 (6.20) z i 1+ () 2.24 ψ x
Also,the plume dispersion parameters in the vertical and lateral directions (σ z and σy) respectively are given by Pasquill and Smith (1983) as follows:
2 1/3 σ 2 0.29 sin( 0.98 π Ψ x n ′) z = ∞ dn ′ 2 2∫0 2 5/3 (6.21) z i π n′()1+ n ′
2 2 1/3 ′ σ y 0.66 sin( 0.75 π Ψ x n ) = ∞ dn ′ 2 2∫0 2 5/3 (6.22) z i π n′()1 + n ′
6.2 Results and Discussion
The used data set was observed from the atmospheric diffusion experiments conducted at the northern part of Copenhagen, Denmark, under unstable conditions (Gryning and Lyck, 1984; Gryning et al., 1987). The tracer sulfur hexafluoride (SF6) was released from a tower at a height of 115m without buoyancy. There are two Gaussian models. The First is measured at ground surface and the other at the plume centerline. In this work, there are three predicated normalized concentrations (proposed model and two previous models).
123 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
Fig. (6.1).Observed and predicated ground crosswind integrated centerline concentration, normalized emission CY( x, 0, 0)/Q: scatter diagram for the solution of equation (6.18) using equations (6.14), (6.16), (6.19), (6.20) and (6.21), (6.22)
Fig. (6.2) .observed and predicated ground crosswind integrated concentration, normalized with emission C y(x, 0)/Q: scatter diagram for the solution of equation (6.17) using equations (6.16), (6.19) and (6.21)
124 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
Fig. (6.1) and (6.2) show that the observed and predicated scatter diagram of crosswind integrated concentrations of centerline and ground level respectively using Gaussian model with vertical and lateral dispersion parameters given by (Equations (14)and(16), proposed model) and (Equations
(19).(20), algebraic formulation), (Equations (21).(22), integral formulation) respectively. From the two figures one finds that there are some predicated data which are agreement with observed data (one to one correspondence) and others lie inside the factor of two.
Table (6.1). Observed and Model ground – level centerline concentration C(x, 0, 0)/Q at different distances, wind speed and effective height from the source.
C(x,0,0)/Q (s/m 3) C(x,0,0)/Q Distance (s/m 3) Equations Equations Equations Run H U115 (x) Observed (14), (19).(20), (21).(22) no. (m) (m/s) (m) W* (16),(18) (18) ,(18) 1 119 3 1900 1.8 10.5 8.61 5.34 6.37 1 119 3 3700 1.8 2.14 7.42 2.17 2.55 2 117 8 2100 1.8 9.85 5.55 7.67 8.71 2 117 8 4200 1.8 2.83 1.80 2.93 3.48 3 118 4 1900 1.3 16.33 14.56 13.74 15.87 3 118 4 3700 1.3 7.95 5.77 5.95 6.98 3 118 4 5400 1.3 3.76 1.09 3.72 4.32 5 117 5 2100 0.7 15.71 15.49 17.51 19.36 5 117 5 4200 0.7 12.11 10.49 20.94 21.73 5 117 5 6100 0.7 7.24 4.94 11.49 13.14 6 116 11 2000 2 4.75 5.43 7.52 8.69 6 116 11 4200 2 7.44 2.94 8.02 8.91 6 116 11 5900 2 3.37 8.29 3.24 3.8 7 117 7 2000 2.2 1.74 2.74 2.07 2.44 7 117 7 4100 2.2 9.48 9.78 5.55 6.54 7 117 7 5300 2.2 2.62 4.12 2.03 2.41 8 117 7 1900 2.2 1.15 1.74 1.44 1.7 8 117 7 3600 2.2 9.76 3.22 8.43 9.62 8 117 7 5300 2.2 2.64 1.96 4.06 4.69 9 116 8 2100 1.9 0.98 3.34 2.59 2.96 9 116 8 4200 1.9 8.52 1.52 6.86 7.85 9 116 8 6000 1.9 2.66 6.75 2.55 3.04
125 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
Table (6.2). Observed and Model ground – level concentration Cy(x, 0)/Q at different distances, wind speed and effective height from the source
C(x,0)/Q C(x,0)/Q (s/m 3) (s/m 3) Run U115 distance Observed Equations Equations Equations no. h(m) (m/s) (x) (m) W* (16),(17) (19),(17) (21),(17) 1 119 3 1900 1.8 6.48 3.72 6.06 6.58 1 119 3 3700 1.8 2.31 4.61 3.96 4.28 2 117 8 2100 1.8 5.38 4.66 3.64 3.79 2 117 8 4200 1.8 2.95 2.46 2.48 2.68 3 118 4 1900 1.3 8.2 7.92 7.35 7.72 3 118 4 3700 1.3 6.22 1.23 5.22 5.6 3 118 4 5400 1.3 4.3 2.21 4.22 4.52 5 117 5 2100 0.7 6.72 8.24 8.54 8.77 5 117 5 4200 0.7 5.84 2.63 6.04 5.71 5 117 5 6100 0.7 4.97 4.65 5.73 5.96 6 116 11 2000 2 3.96 3.30 4.9 5.19 6 116 11 4200 2 2.22 4.41 3.14 3.18 6 116 11 5900 2 1.83 1.06 2.31 2.47 7 117 7 2000 2.2 6.7 4.63 1.9 2.04 7 117 7 4100 2.2 3.25 1.09 3.69 4.25 7 117 7 5300 2.2 2.23 1.48 2.14 2.73 8 117 7 1900 2.2 4.16 5.16 4.12 2.31 8 117 7 3600 2.2 2.02 2.91 3.12 4.28 8 117 7 5300 2.2 1.52 2.50 2.56 3.31 9 116 8 2100 1.9 4.58 3.04 3.53 2.71 9 116 8 4200 1.9 3.11 2.19 2.34 3.7 9 116 8 6000 1.9 2.59 4.04 1.85 2.54
6.3 Statistical verification
Now, the statistical evaluation is presented and comparison among analytical, statically and observed results will be offered (Hanna 1989).Referred to section 1.11.
Table (6.3): Comparison between different Models ground – level centerline concentration C(x, 0, 0)/Q and observed concentrations.
Predicated models C(x, 0, 0)/Q NMSE FB COR FAC2 Equations (14),(16),(18) 0.28 0.12 0.72 1.22 Equations (19).(20),(18) 0.18 0.02 0.83 1.12 Equations (21).(22),(18) 0.18 0.14 0.86 1.28
126 CHAPTER VI Derivation of the Schemes of Lateral and vertical dispersion parameters: Application in Gaussian plume model
Table (6.4): Comparison between different Models ground – level concentration (x, 0,)/Q and observed concentrations.
Predicated models CY(x, 0)/Q NMSE FB COR FAC2 Equations (16), (17) 0.13 0.07 0.90 1.09 Equations (19), (17) 0.11 0.03 0.72 1.05 Equations (21), (17) 0.14 0.03 0.64 1.15
From the above Table (6.3), it can be found that the predicted concentrations for all models lie inside factor of 2 with observed data.
Regarding to NMSE, we find that two previous works are better than proposed one. Regarding to FB and correlation coefficient of all models are performing well with observed data. In Table (6.4): similarly it is noticed that the predicted concentrations for all models lie inside factor of 2 with observed data. Regarding to NMSE, all the predicted concentrations are better to the observed data.
127 Major Conclusions
1 We have used an analytical and numerical solution of two dimensional atmospheric diffusion equation by Laplace transform and Adomian decomposition methods respectively to calculate normalized crosswind concentrations for continuous emits sulfur hexafluoride (SF 6). In this model the vertical eddy diffusivity depends on the downwind distance and it is
2 .(0.16σw u x= ٢ calculated using two methods (k 1 =0.04 u x and k
Graphically, we can observe that analytical models 2 and numerical model 1 have most points inside a factor of two with the observed data. The others two models are over predicted. From the statistical method, we find that the four models are factors of 2 with observed data .Regarding to NMSE, the analytical model 2 and numerical model 1 are better than the other model.
Also the analytical model 2 and numerical model 1 are the best regarding to
FB. The correlation of analytical model 1 and analytical model 2 equal 0.71 and 0.78 respectively which are stronger to the observed data than the correlation of numerical model 1 which equals 0.11.
2 The advection diffusion equation (ADE) is solved in two directions ways to obtain the crosswind integrated ground level concentration in neutral and unstable conditions. Laplace transform technique was used considering the wind speed and eddy diffusivity depends on the vertical height. The maximum
128 ground level concentration is estimated. Comparison between observed data from Copenhagen (Denmark) and predicted concentration data was presented.
3 A model is suggested for the diffusion of material from a point source in an urban atmosphere is incorporated. The plume is assumed to have a well defined edge at which the concentration falls to zero. The vertical wind shear is estimated using combination between logarithmic and power laws under different stability conditions. The problem of diffusion and advection of conservative material as it travels downwind is calculated. The concentrations estimated from this model were compared favorably with the field calculated of other investigators such as power and logarithmic law models. Also we calculate the ground level concentration of the Iodine (I 138 ) which agrees with the observed concentration value after adjusting its source strength.
4 we use Gaussian plume formula in order to estimate concentration from a continuous point source of strength Q with interference from the ground at mean wind speed U and taking the dilution factors . Using four methods such as power law, standard, Briggs and Irwin methods to calculate the dispersion parameters σy and σ z , an comparison between predicated and observed concentrations at different distances for I 131 ,I 133 ,I 135 and Cs 137 , respectively .
5 We estimated new schemes of dispersion parameters in the lateral direction
(σ y) and the vertical direction (σ z) in unstable stability by using wind speed in
129 power law and calculating Gaussian plume model at ground and at plume centerline. We used observed data of the tracer sulfur hexafluoride (SF 6) which was released from a tower at a height of 115m without buoyancy at the northern part of Copenhagen, Denmark, under unstable conditions. There are two Gaussian models; The First is measured at ground surface and the other at the plume centerline. There are three predicated normalized concentrations
(our model and two previous models). From the two figures one finds that there are some predicated data which are agreement with observed data (one to one) and others lie inside the factor of two. From the statistical, we find that the predicted concentrations for all models lie inside factor of 2 with observed data. Regarding to NMSE, all the predicted concentrations are better to the observed data. Regarding to FB and correlation coefficient of all methods are better with observed data.
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