Analytical Solution of the Advection Diffusion Equation Taking Power Law and Linearity of Eddy Diffusivity Using Hankel Transform Hanaa M. Taha, Khaled S. M. Essa Environmental Quality Management, 2025 An analytical solution to the advection‐diffusion equation using the Hankel transform and variable separation technique is presented in this work. Taking into account the height “h” of the planetary boundary layer, it is divided into N sub‐intervals, and it was taken to be a variable throughout the work. Two mathematical models are formulated; the first one assumes power‐law profiles for wind speed and eddy diffusivity, while the second assumes constant wind speed and linear eddy diffusivity. The first model is applied to predict the concentrations for both the prairie grass experiment of sulfur dioxide in the United States and the unstable iodine‐135, then the results are compared with the observed data. Additionally, the models are applied to predict concentrations of iodine‐131 under neutral atmospheric conditions, and the comparison between the values of predicted concentrations was included. Also, the concentrations for iodine‐131 under stable atmospheric conditions were predicted, and the results were compared with the observed experimental data. It was found that the first predicted model is within a factor of two and well in agreement with observed data in unstable and stable conditions. The proposed models are within a factor of two with observed data, and the first predicted model is in better agreement with observed data in neutral conditions than the second predicted model. The proposed models achieved 98%, 84%, 70%, and 100% accuracy from observed data of prairie grass in unstable condition, iodine‐135 in unstable condition, iodine‐131 in neutral condition, and iodine‐131 in stable condition, respectively.
THREE-DIMENSIONAL ANALYTICAL MODEL FOR ESTIMATING CONCENTRATION IN A CAPPING INVERSION AND ITS HERMITIZED Khaled S. M. Essa, Sawsan E. M. El SAIED, A. M. MOSALLEM Revue Roumaine De Chimie, 2025 Three-dimensional advection-diffusion equation is estimated in a capping inversion layer where the turbulent eddy diffusivity is as function of power law of vertical height and downwind distance from the source. A theoretical solution of the resulting diffusion equation with the relevant boundary conditions has been estimated using the turbulent eddy diffusivity using the method of Eigen-function expansion. Also, Gaussian plume model, the advection-diffusion equation and its hermiticity equation in a capping inversion layer are calculated. The predicted model, Gaussian and the hermitized of the advection-diffusion equation have been evaluated comparing with observed data from Egyptian Atomic Energy Authority experiment in moderately unstable conditions. The proposed models are discussed qualitatively as well as quantitatively.
Analytical and Numerical Solutions of Concentration With Deposition Under Unstable Condition Khaled S. M. Essa, Sawsan E. M. El Said, Hanaa. M. Taha, Ahmed S. Shalaby Environmental Quality Management, 2025 In order to study the analytical and numerical solutions with Adomain decomposition of the pollutant's concentration from point source, wind speed and vertical turbulent eddy diffusivity are taken into account as functions of the vertical height above the surface layer power law. The top of the boundary layer of height has an elevated inversion layer that limits the concentration, and there is dry deposition on the ground surface. The pollutant's degradation distance is also estimated. The outcomes of the analytical and numerical solutions were compared with Iodine‐135 measured data under unstable conditions at the Egyptian Atomic Energy Authority. One finds that there is a better agreement between the predicted and observed concentrations than between the numerical concentrations. This study compares the analytical and numerical solutions of the advection‐diffusion equation and data of Iodine‐135 concentrations observed under unstable conditions.
COMPARISON BETWEEN ANALYTICAL SOLUTION OF ADVECTION DIFFUSION EQUATION AND GAUSSIAN PLUME MODEL IN THREE DIMENSIONS THROUGH DIFFERENT STABILITIES Khaled S. M. Essa, H. Taha Revue Roumaine De Chimie, 2025 In this research, the concentration of contaminants was obtained by solving the three-dimensional advection-diffusion equation using the separation of variables technique. Different shapes were adopted for the eddy diffusivity and the logarithmic wind speed under all stability conditions. The analytical results of the advection-diffusion equation were calculated through different stabilities. The calculated results for neutral and stable conditions were compared with already existing observed data for Iodine-131 at the Egyptian Atomic Energy, Nuclear Research Center. Also, the calculated results for unstable conditions were compared with already existing observed data for Iodine-135. Comparisons of the analytical results of the advection-diffusion equation, previous work and the Gaussian plume model, in all stability conditions, are included. Using the statistical methodology, it was discovered that there a strong correlation with the observed concentrations in both neutral and unstable settings compared to stable conditions.
STUDY THE EFFECT OF WIND VARIANCE, EULERIAN AND THE LAGRANGIAN INTEGRAL TIME SCALES ON THE DISPERSION PARAMETERS Khaled S. M. Essa, H. Taha Revue Roumaine De Chimie, 2025 The simplest meaningful statistical measures of dispersion, we can compute the mean-square parameters displacement which must be increasing functions of time. This is in contrast to the variances of Lagrangian velocity fluctuations which must be independent of time in a stationary and homogeneous field of turbulence and equal to the corresponding Eulerian velocity variances. For small diffusion time, the mean-square particle displacement increases in proportion to square of diffusion time “t2”. For large diffusion time, the mean-square diffusion eventually becomes proportional to Lagrangian integral time scale “TiL” and diffusion time “t”. Considering these dispersion parameters in Gaussian diffusion model in three dimensions. These Gaussian and analytical concentrations results are compared with experimental data of Iodine-135 which collected in a convective boundary layer from Inshas which located at Egyptian Atomic Energy Authority.
URBAN DIFFUSION AND AIR QUALITY MODELS Khaled S. M. Essa, Ahmed Mosallem Revue Roumaine De Chimie, 2025 In this research, the Atmospheric Turbulence and Diffusion Laboratory (ATDL) is another sample of Gaussian-based urban diffusion models. Also, the simplest box model without chemical transformation can be driven to get average concentration using the lateral and vertical standard deviations values of Brookhaven National Laboratory through different stabilities. The two models (Gaussian and Box models) are applied to obtain the concentration in three dimensions to compare with experimental data of Iodine-135(I135). Also, using the two models to obtain the normalized crosswind concentrations to compare with data from Copenhagen-Denmark for Sulphur-Hexi-Fluoride (SF6).
COMPARISON BETWEEN GAUSSIAN AND ADVECTION DIFFUSION EQUATION OF EULERIAN AND LAGRANGIAN INTEGRAL TIME SCALES IN STABLE CONDITION , Khaled S. M. ESSA, Hanna M. TAHA, and Revue Roumaine De Chimie, 2024 The displacement mean-square parameters (X^2 ) ̅(t),(Y^2 ) ̅(t)and (Z^2 ) ̅(t)need to be increasing functions of time. In a stable and homogeneous field of turbulence, the variances of Lagrangian velocity fluctuations must be equal to the corresponding Eulerian velocity variances and independent of time. The mean-square particle displacement rises proportionately to the square of the diffusion time "t2" for small diffusion times. The mean-square diffusion finally becomes proportional to the diffusion time (t) and the Lagrangian integral time scale (TiL) for long diffusion times. The three-dimensional advection-diffusion equation and Gaussian model are computed using these dispersion values. The results of these concentrations are contrasted with Iodine-131 experimental data that was gathered at the Egyptian Atomic Energy Authority under steady conditions.
EVALUATION GAUSSIAN CONCENTRATION USING DISPERSION PARAMETERS IN PUFF PLUME MODEL Khaled S. M. ESSA, , Sawsan I. M. ELSAID, and Revue Roumaine De Chimie, 2024 The proposed concentrations model was compared to observed data from air diffusion studies for hexafluoride (SF6) conducted in the northern area of Copenhagen, Denmark, and earlier work for the Gaussian puff model. The predicted concentrations and earlier studies were found to be within a factor of two of each other, indicating that the predicted data is compatible with the observed concentrations data. Under unstable conditions, the statistical statistics show a reasonable agreement between the expected and observed concentrations at the Copenhagen Experimental. In contrast to the previous work's results, which had an NMSE of 0.3 and an FB of 0.21, we discovered that the data obtained using the statistical method had an NMSE of 0.2 and an FB of 0.25. The projected COR is 0.95, compared to 0.97 in a previous computation. The observed concentration data was 88% of the expected value, compared to 78% in previous research. The expected data are consistent with the observed concentration measurements. When compared to previous work, the Copenhagen experiment was conducted under unstable conditions, and statistical data show good agreement between predicted and observed concentrations.
Analytical concentration of pollutants with deposition using wind speed as power and logarithmic law KHALED S. M. ESSA, SOAD M. ETMAN, MAHA S. EL-OTAIFY, M. EMBABY Mausam, 2024 The mathematical formulation of the concentration of the diffusing particles in air was derived by solving analytically the advection-diffusion equation taking into consideration: (1) the vertical variation of wind speed and eddy diffusivity with height above ground. (2) the vertical diffusion is limited by an elevated impenetrable inversion layer located at the top of the atmospheric boundary layer (ABL) of height h. (3) the dry deposition of the diffusing particles at the ground surface which was included through the boundary conditions. A power law profile is used to describe the vertical variation of eddy diffusivity with height, while the sum of power law profile and logarithmic law is used to describe the vertical variation of wind speed with height above ground surface. The decay distance of a pollutant along the wind direction was derived. The present solution was evaluated against the dataset from Hanford diffusion experiment in stable conditions. The results are discussed and presented in illustrative figures.
Analytical solution of diffusion equation in two dimensions using two forms of eddy diffusivities Romanian Reports of Physics, 2011
Estimating of crosswind integrated Gaussian and non gaussian concentration by using different dispersion schemes Australian Journal of Basic and Applied Sciences, 2011
Using dispersion modeling for ground level concentration Mausam, 2011
