Analytical Solution of the Advection Diffusion Equation Taking Power Law and Linearity of Eddy Diffusivity Using Hankel Transform

ABSTRACT 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.