An analytical model for point source pollutants in an urban area with mesoscale and removal mechanisms
R Latha^{1}, K Lakshminarayanachari^{2}, C Bhaskar^{2}
^{1} Department of Mathematics, Jyothy Institute of Technology, CIIRC, Bengaluru; Department of Mathematics, Visvesvaraya Technological University, Belagavi; Department of Mathematics, Sai Vidya Institute of Technology, Bengaluru, Karnataka, India ^{2} Department of Mathematics, Visvesvaraya Technological University, Belagavi; Department of Mathematics, Sai Vidya Institute of Technology, Bengaluru, Karnataka, India
Date of Submission  30Nov2022 
Date of Acceptance  15Mar2023 
Date of Web Publication  31Aug2023 
Correspondence Address: Dr. K Lakshminarayanachari Department of Mathematics, Sai Vidya Institute of Technology, Bengaluru  560 064, Karnataka; Visvesvaraya Technological University, Belagavi, Karnataka India
Source of Support: None, Conflict of Interest: None  Check 
DOI: 10.4103/ijehe.ijehe_59_22
Aim: Urbanization, rapid industrialization, and related anthropogenic activities are the main reasons for air pollutant emissions and poor air quality. Continued deterioration of air quality in urban areas requires effective measures to control air pollution. The purpose of developing a model is for a better understanding the physical, chemical, and dynamic properties of air pollution and meteorology. Materials and Methods: In this work, an analytical approach has been used to analyze the diffusion of air pollutants emitted from point sources is presented. This article reports the method of solving threedimensional atmospheric diffusion equation using the Fourier transform technique, variable separable method, and series solution to obtain analytical solution. Results: The present work focuses on the effect of dry deposition and wet deposition on the concentration of primary impurities. The concentration of primary impurities decreases when the removal rate of dry and wet deposition increases with respect to height and distance. Conclusion: The focus of this research article is to learn the effects of removal mechanisms such as dry and wet deposition of primary pollutants. It is found that the contaminant concentrations are close to point sources and gradually decrease toward nonsource areas due to diffusion and removal mechanisms in an urban area.
Keywords: Advection–diffusion equation, atmospheric impurities, dry deposition, urban heat island, wet deposition
How to cite this article: Latha R, Lakshminarayanachari K, Bhaskar C. An analytical model for point source pollutants in an urban area with mesoscale and removal mechanisms. Int J Env Health Eng 2023;12:21 
How to cite this URL: Latha R, Lakshminarayanachari K, Bhaskar C. An analytical model for point source pollutants in an urban area with mesoscale and removal mechanisms. Int J Env Health Eng [serial online] 2023 [cited 2023 Sep 24];12:21. Available from: https://www.ijehe.org/text.asp?2023/12/1/21/384961 
Introduction   
Air pollution is considered to be the most important environmental risk factor for health. The smoky air that surrounds cities often contains dangerously high levels of particulate matter are called PM2.5. These pollutants are associated with lung and heart disease and are known to affect cognitive function and the immune system. Air pollution also harms our natural environment. It reduces the oxygen supply to our oceans, makes plant growth difficult, and contributes to climate change. Other harmful environmental impacts include soil and waterway depletion, endangered freshwater sources, and reduced crop yields. Hence, it is of great importance to study the dispersion of pollutants.
Wind speed measures the air pollutants mixed in relation to wind speed, and their direction leads to the widespread diffusion of pollutants. To develop a precise model or method to illustrate the manner, in which air pollution is transported and dispersed for a given locale, information about the pollutant source is needed. This information generally includes surrounding geographic features, quantity and types of pollutants emitted, effluent gas conditions, stack height, and influential meteorological factors.
The advection–diffusion equation is analytically solved for the steady state by making appropriate assumptions about the wind speed profile and the eddy diffusion coefficient. Griffiths^{[1]} predicted knowledge of large winds is insufficient to predict urban air pollution. Sharan^{[2]} formulated a mathematical model of air pollution propagation in weak winds, taking into account all directional propagation and advection along average winds. The analytical solution of the advection–diffusion equation is the basis for the explanation and understanding of the dispersion phenomenon. Pasquill and Smith^{[3]} determined, it is expressed in a mathematically closed form using all parameters, making it easy to investigate the effect of individual parameters on pollutant concentrations. Dilley and Yen^{[4]} investigated the effects of mesoscale wind on the pollutant distribution of infinite sources. Arora et al.^{[5]} presented an analytical approach for investigating the delayed removal process of air pollutants from line sources using variable wind profiles and diffusion coefficients. Ermak.^{[6]} obtained the analytical solution of the atmospheric dispersion equation, assuming that the eddy diffusion coefficient is constant as a function of the downdraft distance and the average wind speed. A generalized mathematical model explaining the crosswindintegrated concentration is presented for the spread of pollutants released from continuous sources of the atmospheric boundary layer with surface deposition given by Pramod and Maithili.^{[7]}
The solution for airpollutant enrichment is obtained through an integrated and diverse embedding process that takes into account chemical reaction processes by Naresh and Nath.^{[8]} The effects of mesoscale wind and chemical reactions about pollutants in the presence and absence of mesoscale wind were investigated by Joseph and Agarwal.^{[9]} The solution of the twodimensional convection–diffusion equation using the Laplace transform method is formulated by Buske.^{[10]} They also discussed solving the threedimensional (3D) convection–diffusion equation for wind speed profiles. Wortmann et al.^{[11]} proposed a solution to the convection–diffusion equation by adopting a generalized integral transform method. Pal and Khan^{[12]} reported a timedependent model of air pollutants from point sources and concluded that pollutant concentrations decrease as pollutants escape from the top of the inversion layer. The analytical approach was developed by Khan et al.^{[13]} to analyze the removal mechanism of nonreactive air pollutants for various complex air conditions. Narayanachari KL et.al.,^{[14]} used a numerical model to investigate the causes of mesoscale airflow in the distribution of air pollutants. Lakshminarayanachari et al.^{[15]} have developed a twodimensional numerical model for studying the diffusion of primary pollutants in residential areas with chemical reactions and dry deposition. Many researchers have reported the occurrence of health problems due to air pollution. Pope^{[16]} found that air pollution causes chronic bronchitis, leading to increased mortality. However, the work reported so far does not integrate the effects of mesoscale airflow on point sources.
The study by Lakshminarayanachari et al.^{[17]} elucidates that the mesoscale wind prevents the diffusion of pollutants that would increase the concentration of pollution in the atmosphere and also studied that urban heat island (UHI) increases the concentration of pollutants and helps to move vertically upward, resulting in pollution becoming more serious. Verma et al.^{[18]} have presented analytical model of the diffusion of air impurities with variable wind velocity and constant removal rate. The solutions of advection–diffusion equation were obtained by analytical approach, with the assumption that eddy diffusivities to be linear functions of downwind distance with physically relevant boundary conditions. They reported this under the assumption that pollutants would be transported horizontally by large winds and the mesoscale winds generated would transport them both vertically and horizontally. However, these works did not compact with the consequences of mesoscale air current for the point source.
This paper sought to develop an analytical method for studying the spread of air pollutants emitted from elevated reference points integrated into the mesoscale airflow generated by the UHI. This task uses the convection–diffusion equation to analyze the diffusion of air pollutants in the presence of large winds, eddy diffusion, and thermal diffusion. In this article, we study the effect of removal mechanisms such as dry deposition and wet deposition on the concentration of pollutants along vertical height and horizontal distance.
Materials and Methods   
Mathematical model formulation
Based on the gradient transport theory, the diffusion equation for describing the dispersion of atmospheric pollutants in the atmosphere is given by Equation 1:
Where C represents the concentration pollutants in the atmosphere. u, v, and w are the wind speed and k_{x}, k_{y} and k_{z} are diffusivities in x, y, and z direction. α represents the removal rate of atmospheric pollutants naturally.
In the current problem, the point source is kept at a height of h_{s} meters from ground level. Here, it assumed that pollutants are discharged from a point source at a constant rate and the pollutants move in a horizontal direction parallel to the largescale airflow. The point source is being fixed at the origin with coordinates x=0, y=0. The diffusion of pollutants is determined at a downwind length l = 5 km.
To develop this mathematical model, the following assumption has been made:
 The release of atmospheric pollutants point source is steady rate
 The condition for a steady state is considered to be,
 The direction of x coordinate is aligned with the average wind speed (u = U and v = 0)
 The horizontal advection of the airflow dominates the parallel diffusion.
Considering the above assumptions, the Equation 1 becomes as Equation 2.
Where x, y, and z are the coordinate axes, U (x) is the speed of the airflow in the x direction which found to vary with vertical distance from the ground level. It can be written in the following form Dilley and Yen.^{[4]}
Where U_{0} represents the average speed of the wind.
W (z) = U_{0}(az) indicates air current velocity along the direction of z.
Usually, K_{y}>K_{z} in the atmospheric conditions.
The boundary constraints in Equation 2 are:
 Impurities are discharged from the elevated point source with the concentration of Q positioned at a coordinate (0,0,h_{s})
Where, h_{s}stack height, δDirac delta function, and Hmixing height
 Concentration of impurities becomes zero as it tends to be away from the point source in y direction, i.e.,
C (x, y, z) =0 when y → ± ∞ (4)
 The impurities are reflected at height h from the ground surface i.e.,
Where v_{d} is the dry deposition.
 There is some diffusion flux at the vertical height H from the ground surface
Where w_{s} is the wet deposition.
Recent studies show that largescale wind speeds are considered constant. That is, u = U_{0} is presumed that the horizontal mesoscale wind fluctuates in the vertical direction equal to u. The vertical mesoscale air flow w (z) can be calculated by integrating the continuity equation.
w_{e}= aU_{0}z (iii)
Where a is the proportionality constant.
U (x) = u + u_{e}= U_{0} (1ax) (iv)
Method of solution
Then, by employing the following dimensionless parameter procedure, the partial differential equation (2), which describes the diffusion of air pollutants and boundary constraints, is reduced to dimensionless to arrive at the solution:
Where is the reference diffusivity and U_{0} is the reference wind velocity. On dropping asterisks (*), the Equation 2 and the boundary conditions given by Equations 3–6 may be made in the dimensionless form as found below
The Equation 7 is solved by employing Fourier transform technique using boundary conditions given by Equation 8 to Equation 11.
Taking Fourier transform of the Equation 7 with respect to “y” to get
Where is the Fourier Transform of c with respect to y and p
By considering the Fourier transform, the boundary conditions become
In this method of separation of variables using the Equation 12 and the following trial solution:
Where Z (z) is a function of z only and X(x) is a function of only x.
By substituting the Equation 17 in Equation 12, the ordinary differential equations are obtained
The term λ^{2} represents the separation constant
The solution of Equations 18 and 19 are of the form
Where a_{0}, a_{1}, and C_{1} are arbitrary constants and
After substituting the values of X (x) and Z (z) from the Equations 20 and 21, the following equation is obtained
The term C_{1} is considered to be 1 without loss of generality. After using the boundary conditions,
Again using the boundary conditions, and as well as applying
and
the solution can be written as
After taking the inverse Fourier transform of the Equation 25, it can be written as
Results   
The results of this research work are presented in [Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6] to analyze the diffusion of air pollutants in the civic area.  Figure 1: Concentration versus distance of primary pollutants for various values of dry deposition
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 Figure 2: Concentration versus crosswind distance of primary pollutants for various values of wet deposition
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 Figure 3: Concentration versus height of primary pollutants for different values of dry deposition
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 Figure 4: Concentration versus height of primary pollutants for various values of wet deposition
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 Figure 5: Variation of Concentration with distance for different values of crosswind distances when vd=0.0001
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 Figure 6: Variation of cConcentration with vertical height for different distances when vd=0.0004
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[Figure 1] depicts the variation of pollutant concentration with downwind distance for various values of dry deposition. It can be observed from the graph that, as distance (x) increases, the concentration of pollutant decreases. It can also be noticed that the concentration is more near the point source at origin then it is gradually decreasing after the point source and reaching zero.
[Figure 2] reveals the variation of pollutant concentration with downwind distance for various values of wet deposition. It can be observed from the graph that, as distance (x) increases, the concentration of pollutant decreases.
[Figure 3] gives the variation of pollutant concentration profile with height for various values of dry deposition values. From the figures, it is observed that the concentration is decreasing when height is increasing. Furthermore, it is observed that the concentration of impurities is high near the point source at origin, then it is gradually decreasing. Further, it is found that the concentration of pollutants is decreasing when dry deposition velocity increases.
[Figure 4] indicates the variation of pollutant concentration profile with height for various values of wet deposition values. From the figure, it is observed that the concentration is decreasing when height is increasing. Furthermore, the pollutant concentration is high near the point source at origin, then it is gradually decreasing. Further, the concentration of pollutants is decreasing when wet deposition increases.
[Figure 5] and [Figure 6] exhibit the result of the concentration profile against the vertical height for various values of downward distance for different values of dry deposition values. Here, it is observed that, as vertical height increases, the concentration of pollutant is decreasing when v_{d} = 0.0001 and v_{d} = 0.0004. It is also observed that the concentration of pollutant is also decreasing when the downwind distance increases for both values of v_{d} = 0.0001 and v_{d} = 0.0004.
[Figure 6] shows the variation of concentration with vertical height for different distances when vd = 0.0004.
Discussion   
The UHI consequence produces its own mesoscale winds, and accordingly, it prevents the diffusion of the pollutants which will result in enhancing the pollution in an environment. The UHIs add to the development of haze of contaminated pollutants and also help these pollutants to circulate in an upward direction, thus making the problem of pollution more severe. It should be understood that the reason for the conversion of large cities into “UHIs” is attributed to anthropogenic aspects. Hence, combined efforts should be made in the task of UHI and for the creation of cooler and healthier cities.
The developed analytical model for studying the effects of primary pollutant concentrations in the horizontal and vertical directions discharged from an elevated point of reference at the source of the city was examined. Appropriate boundary conditions are given to explain the point source. Contaminants are expected to undergo removal mechanisms such as dry and wet deposits. The problem was solved using the Fourier transform, variable separable method, and series solutions.
An analytical solution for the computation of air concentration discharged from the point source undergoing dry and wet deposition process is presented. The results of this article are presented in [Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6] to analyze the diffusion of air pollutants in the civic area. The concentration of primary pollutants decreases as removal mechanisms dry and wet deposition rate increases. The results of the present work are compared with Verma et al.^{[18]} and Nirmaladevi et al.,^{[19]} and good agreement was found.
The results of the current research findings can be used to increase the level of credibility in complex model predictions and to identify variables such as wind velocity and atmospheric solidity, which should be investigated more closely by complex modeling studies.
Conclusion   
Analysis of the concentration of pollutants released from point sources in the presence of wet and dry deposits in the city is presented using a 3D analytical model. The focus of this research article is to learn the effects of the removal mechanism of dry and wet deposition of primary pollutants. The results were analyzed for pollutants primarily in terms of horizontal distance and vertical height. It is observed that, as the proportion of removal mechanisms increases, the concentration of pollutants with respect to the horizontal distance and vertical height of the urban area decreases. Contaminant concentrations are close to point sources and gradually decrease toward nonsource areas due to removal mechanisms and diffusion.
Acknowledgments
The researchers wish to extend their sincere gratitude and appreciation toward the support from the Management and Principal of Sai Vidya Institute of Technology, Bengaluru. The researchers also wish to extend their sincere gratitude and appreciation toward the support from the Management and Principal of Jyothy Institute of Technology, Bengaluru.
Ethics code
This paper has never previously been published. This work was drafted by all of the authors.
Financial support and sponsorship
Nil.
Conflicts of interest
There are no conflicts of interest.
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5], [Figure 6]
