Numerical Simulation of an Air Pollution Model on Industrial Areas by Considering the Influence of Multiple Point Sources

A numerical simulation on a two-dimensional atmospheric diffusion equation of an air pollution measurementmodel is proposed. e considered area is separated into two parts that are an industrial zone and an urban zone. In this research, the air pollution measurement by releasing the pollutant from multiple point sources above an industrial zone to the other area is simulated. e governing partial differential equation of air pollutant concentration is approximated by using a finite difference technique. e approximate solutions of the air pollutant concentration on both areas are compared. e air pollutant concentration levels influenced by multiple point sources are also analyzed.


Introduction
A rapid growth of industrial sector can explain that air pollution affects the health of human being who lives around industrial areas.
e air pollution has become a major problem of human life and environment.e purpose of this research was to study the air pollution assessment problem in two adjacent zones: industrial and urban zones by using the atmospheric diffusion model.In [ ], the simulation of two-dimensional advection-diffusion model with a point source was presented.e numerical solutions were solved by using the finite difference techniques.In [ ], the researchers used the mathematical model to simulate the dispersion of sulfur dioxide concentration with the wind and diffusion parameters regarding the reference atmospheric stability.
In [ ], the mass transport model consisted of the stream function, vorticity, and convection-diffusion equation.e smoke dispersion which released into the atmosphere from one and two-point sources was considered with obstacle domain.e approximated solutions were solved by using the finite element techniques.In [ ], the researchers studied the smoke dispersion model in a two-dimensional space by considering two and three point sources with a two obstacles in the domain.In [ ], the two-dimensional advection-diffusion equation with mesoscale wind, eddy diffusivity profiles, and removal mechanisms was introduced.
en, the primary pollutant released into the atmosphere from an area source, which was also considered.In [ ], the researchers studied the two-dimensional advection-diffusion equation of primary and secondary pollutants.
e area source with removal mechanisms and the point source considering on the boundary were proposed.e solutions of air pollution in [ , ] were estimated by using the Crank-Nicolson implicit methods.In [ ], the air-quality model in the three-dimensional with variations of the atmospheric stability classes and wind velocities from multiple sources was analyzed.
e fractional step methods were used in order to predict the air pollutant concentration in [ , , ].In [ ], the atmospheric diffusion model was used to describe the dispersion of air pollutant concentration near an industrial zone.e problem was considered by controlling the air pollution emission under a point source.From the numerical experiments, it was indicated that the air pollution control was necessary for air-quality management.In [ ], the researchers studied the dispersion behavior of air pollution in the tunnel under a Bangkok sky train platform by using the simulation of a three-dimensional air-quality model.is model was considered varied cases on the wind inflow with obstacles.In [ ], the three-dimensional advection-diffusion equation was considered to approximate the concentration of air pollutant in a heavy traffic area under the Bangkok sky train station.e numerical simulations were studied for three cases that were the average of source or sink emissions, the moving of source or sink emissions, and the mix of source and sink emissions.e explicit finite difference scheme was used to solve the air pollutant concentration in [ -].
e source that is smokestack of industrial factory or power plant discharges the air pollution into the system.e genesis of air pollution is the cause of problems.In this research, the simple finite difference methods are used for solving the atmospheric diffusion equation.e diffusion model generally uses Gaussian plume idea, which is the well-known atmospheric diffusion equation.It represents the behavior of air pollution in industrial areas.e dispersion of pollutant concentration from multiple point sources is described by the following three-dimensional advection-diffusion equation:

The Atmospheric Diffusion Equation
where  = (, , , ) is the concentration of air pollutant at (, , ) and time  (/ 3 ), , V, and  are the wind velocity component (/) in -, -, -directions, respectively,   ,   , and   are the diffusion coefficient ( 2 /) in -, -, -directions respectively, and  is the sink rate of air pollutants ( −1 ).e assumptions of ( ) are defined that the concentrations of air pollutant are emitted from continued point sources.e advection and diffusion in -direction are laterally averaged.By the assumption, we can also eliminate all terms in direction.erefore, the governing equation can be written as and  max is a stationary time.us the nondimensional equation of air pollution is as follows: where  = max{  ,   } and  =  max  max / when  < 0 that means the air pollutant concentrations are absorbed from the atmosphere by the chemical reaction.

Numerical Methods
We use the finite difference methods for calculating the nondimensional form of the atmospheric diffusion equation.
In ( ), we get the concentration of  at each time  +1 from   when Δ is a time increment.e solution of concentration at (, , ) is denoted by (  ,   ,   ) =   , .e domain is divided by the grid spacing in -direction and -direction which are Δ and Δ, respectively, where   = Δ and   = Δ.e approximate solutions are obtained by using the following methods.

Forward Time Central Space Scheme.
e first method, we use the forward difference in transient term that is en, the centered difference for the advection and diffusion in -direction and -direction is applied as follows: International Journal of Differential Equations respectively.We substitute ( )-( ) into ( ).It will be us, the forward time central space (FTCS) scheme of the nondimensional mathematical model is where 2 .e stability of the forward time central space scheme can be investigated by using von Neumann stability analysis.We can obtain that the stability condition is 0 ⩽ 2  +   +   ⩽ 1.

Backward Time Central Space Scheme.
e second method, we use the backward difference in transient term that is en, the centered difference for the advection and diffusion in -direction and -direction is utilized as follows: respectively.We substitute ( )-( ) into ( ).It obtains that ( ) erefore, the backward time central space (BTCS) scheme of this research becomes ( ) e stability of the implicit backward time central space scheme can be investigated by using von Neumann stability analysis.We can obtain that the method is an unconditionally stable method.

Numerical Experiment
e two-dimensional atmospheric diffusion equation ( ) with a dimension 1, 000 × 1, 000  2 will be considered.e uniform wind velocities and constant diffusion coefficients are introduced.We choose that the wind velocities in direction and -direction are 0.1 and 0.05 /, respectively.e diffusion coefficients in -direction and -direction are 4.5×10 −1 and 4.5×10 −5  2 /, respectively.e grid spacing is Δ = Δ = 25 m. and the time interval is .In this research, we present two cases.e first case considers a point source when the concentration is 0.5 / 3 .e second case considers two-point source when the concentration is 0.25 and 0.25 / 3 .e air pollutants in ( ) are released into our system.ese examples are solved by using the forward time central space and the backward time central space schemes in ( ) and ( ), respectively, with the initial and boundary conditions ( ) to ( ).
In Figure , model of the problem is shown.e physical problem composed of two zones: an industrial zone and an urban zone with the stable wind along the -axis and axis.e point sources are laid along the -axis.We assume that the primary air pollutants are released from a factory smokestack by a single point source and coupled point sources on industrial zone.e emissions of air pollution are influenced on the urban zone by the rate of air pollutant absorption.In the numerical experiment, the considered domain of solutions is shown in Figure .

Discussion
e air pollutant emission from multiple point sources above an industrial zone to the urban area is presented.e finite difference techniques introduced two methods for calculating the air pollutant concentrations.figures, it is apparent that the results of the forward time central space scheme are close to the results of the backward time central space scheme, when there is no sink of pollutant absorption ( = 0).Figures and illustrate that the sink of pollutant absorption ( = 10 −4 ) is added to the base of urban zone.e air pollutant concentration near human living goes down and the two methods also give the close result.In Figures and , the computed approximate solutions which are calculated by using the forward time central space and the backward time central space schemes are compared.We can see that the results of added sink case and without sink case are quite similar.ese graphs also indicate that the forward time central space scheme gives the computed solutions close to the backward time central space scheme.Figures and demonstrate that the air pollutant concentration at the height  = 25 m. and  = 50 m.are solved by using the forward time central space scheme.e added sink case is less concentration than the without sink case.erefore, the sink can lower the overall pollutant levels.gives less computing time than the backward time central space computing time.On the other hand, the forward time central space also has disadvantages that are the limitation of the grid spacing due to the stability condition.

Data Availability
e calculated air pollution measurement data used to support the findings of this study are available from the corresponding author upon request.
Figures and compare the air pollutant concentrations between two cases: a single point source and coupled point sources, respectively.From the both International Journal of Differential Equations figures, it is apparent that the results of the forward time central space scheme are close to the results of the backward time central space scheme, when there is no sink of pollutant absorption ( = 0).Figures and illustrate that the sink of pollutant absorption ( = 10 −4 ) is added to the base of urban zone.e air pollutant concentration near human living goes down and the two methods also give the close result.In Figures and , the computed approximate solutions which are calculated by using the forward time central space and the backward time central space schemes are compared.We can see that the results of added sink case and without sink case are quite similar.ese graphs also indicate that the forward time central space scheme gives the computed solutions close to the backward time central space scheme.Figures and demonstrate that the air pollutant concentration at the height  = 25 m. and  = 50 m.are solved by using the forward time central space scheme.e added sink case is less concentration than the without sink case.erefore, the sink can lower the overall pollutant levels.Figure establishes the various concentrations when we take more sink rate into our system.e comparison of computing

e
simple air pollution measurement models which are released air pollutants by a single point source and coupled point sources are proposed.e traditional finite difference methods such as forward time central space and backward time central space schemes can be used to approximate the air pollutant levels for each points and times.e results of this study show that the air pollutant concentrations of forward International Journal of Differential Equations pollutant concentration levels a er hours passed which are computed by the forward time central space scheme ( = 0): (a) one-point source and (b) two-point source.pollutant concentration levels a er hours passed which are computed by the backward time central space scheme ( = 0): (a) one-point source and (b) two-point source.pollutant concentration levels a er hours passed which are computed by the forward time central space scheme ( = 10 −4 ): (a) one-point source and (b) two-point source.pollutant concentration levels a er hours passed which are computed by the backward time central space scheme ( = 10 −4 ): (a) one-point source and (b) two-point source.timecentral space are close to the air pollutant concentrations of backward time central space.In the case of a coupled point sources problem, the overall concentration levels of air pollution are less than a single point source problem.erefore, the influence of multiple point sources and the variable rate of sink are also considered.It obtains that the higher sink rate does decrease pollutant levels around human living.e both finite difference methods are used to compute the numerical solutions of air pollution by MATLAB.e forward time central space has advantages that the pollutant concentration between the forward time central space and the backward time central space schemes ( = 0) at  = 0 m. and  = 600 m. pollutant concentration between the forward time central space and the backward time central space schemes ( = 10 −4 ) at  = 0 m. and  = 400 m.
pollutant concentration between cases: added sink and without sink (computed by the forward time central space scheme) at  = 25 m. and  = 600 m. pollutant concentration between cases: added sink and without sink (computed by the forward time central space scheme) at  = 50 m.and  = 600 m. pollutant concentration with the variant values of sink rate (computed by the forward time central space scheme) at  = 0 m. and  = 600 m.
where   is the position of the point source  in the -direction and    is the concentration value at the point source of .2.2.The Nondimensional Form Equation.Now, we introduce the dimensionless form of equation ( ). e nondimensional variables are denoted by letting  = / max ,  = /  ,  = /  ,  = / max ,   =   /   max ,   =   /   max ,  = / max , and  =  max / max when  = / max .We define