International Journal of Differential Equations

Volume 2019, Article ID 2319831, 10 pages

https://doi.org/10.1155/2019/2319831

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

^{1}Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand^{2}Centre of Excellence in Mathematics, Commission on Higher Education (CHE), Si Ayutthaya Road, Bangkok 10400, Thailand

Correspondence should be addressed to Nopparat Pochai; moc.oohay@htam_pon

Received 8 July 2018; Revised 16 September 2018; Accepted 27 January 2019; Published 11 February 2019

Guest Editor: Manoj Kumar

Copyright © 2019 Pravitra Oyjinda and Nopparat Pochai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A numerical simulation on a two-dimensional atmospheric diffusion equation of an air pollution measurement model is proposed. The 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. The governing partial differential equation of air pollutant concentration is approximated by using a finite difference technique. The approximate solutions of the air pollutant concentration on both areas are compared. The air pollutant concentration levels influenced by multiple point sources are also analyzed.

#### 1. Introduction

A rapid growth of industrial sector can explain that air pollution affects the health of human being who lives around industrial areas. The air pollution has become a major problem of human life and environment. The 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 [1], the simulation of two-dimensional advection-diffusion model with a point source was presented. The numerical solutions were solved by using the finite difference techniques. In [2], 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 [3], the mass transport model consisted of the stream function, vorticity, and convection-diffusion equation. The smoke dispersion which released into the atmosphere from one and two-point sources was considered with obstacle domain. The approximated solutions were solved by using the finite element techniques. In [4], 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 [5], the two-dimensional advection-diffusion equation with mesoscale wind, eddy diffusivity profiles, and removal mechanisms was introduced. Then, the primary pollutant released into the atmosphere from an area source, which was also considered. In [6], the researchers studied the two-dimensional advection-diffusion equation of primary and secondary pollutants. The area source with removal mechanisms and the point source considering on the boundary were proposed. The solutions of air pollution in [5, 6] were estimated by using the Crank-Nicolson implicit methods. In [7], the air-quality model in the three-dimensional with variations of the atmospheric stability classes and wind velocities from multiple sources was analyzed. The fractional step methods were used in order to predict the air pollutant concentration in [2, 4, 7]. In [8], the atmospheric diffusion model was used to describe the dispersion of air pollutant concentration near an industrial zone. The 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 [9], 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. This model was considered varied cases on the wind inflow with obstacles. In [10], 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. The 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. The explicit finite difference scheme was used to solve the air pollutant concentration in [8–10].

The source that is smokestack of industrial factory or power plant discharges the air pollution into the system. The 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.

#### 2. Governing Equation

##### 2.1. The Atmospheric Diffusion Equation

The 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. The dispersion of pollutant concentration from multiple point sources is described by the following three-dimensional advection-diffusion equation:where is the concentration of air pollutant at and time , , and are the wind velocity component in -directions, respectively, , and are the diffusion coefficient in -directions respectively, and is the sink rate of air pollutants .

The assumptions of (1) are defined that the concentrations of air pollutant are emitted from continued point sources. The advection and diffusion in -direction are laterally averaged. By the assumption, we can also eliminate all terms in -direction. Therefore, the governing equation can be written asThe initial condition is assumed under the cold start assumption. That is,for all and . The boundary conditions are assumed thatfor all where is the length of the domain in -direction and is the height of the inversion layer. The concentration at the point sources is assumed to be the constant variables asfor 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 (2). The nondimensional variables are denoted by letting , , , , , , , and when . We define , , , and is a stationary time. Thus the nondimensional equation of air pollution is as follows:where and when that means the air pollutant concentrations are absorbed from the atmosphere by the chemical reaction.

#### 3. Numerical Methods

We use the finite difference methods for calculating the nondimensional form of the atmospheric diffusion equation. In (9), we get the concentration of at each time from when is a time increment. The solution of concentration at is denoted by . The domain is divided by the grid spacing in -direction and -direction which are and , respectively, where and . The approximate solutions are obtained by using the following methods.

##### 3.1. Forward Time Central Space Scheme

The first method, we use the forward difference in transient term that isThen, the centered difference for the advection and diffusion in -direction and -direction is applied as follows:respectively. We substitute (10)-(14) into (9). It will beThus, the forward time central space (FTCS) scheme of the nondimensional mathematical model iswhere , , . The 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 .

##### 3.2. Backward Time Central Space Scheme

The second method, we use the backward difference in transient term that isThen, the centered difference for the advection and diffusion in -direction and -direction is utilized as follows:respectively. We substitute (17)-(21) into (9). It obtains thatTherefore, the backward time central space (BTCS) scheme of this research becomesThe 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.

#### 4. Numerical Experiment

The two-dimensional atmospheric diffusion equation (9) with a dimension will be considered. The uniform wind velocities and constant diffusion coefficients are introduced. We choose that the wind velocities in -direction and -direction are and , respectively. The diffusion coefficients in -direction and -direction are and , respectively. The grid spacing is m. and the time interval is 20 . In this research, we present two cases. The first case considers a point source when the concentration is . The second case considers two-point source when the concentration is and . The air pollutants in (8) are released into our system. These examples are solved by using the forward time central space and the backward time central space schemes in (16) and (23), respectively, with the initial and boundary conditions (3) to (7).

In Figure 1, model of the problem is shown. The physical problem composed of two zones: an industrial zone and an urban zone with the stable wind along the -axis and -axis. The 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. The 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 2.