Numerical Simulation for a Three-Dimensional Air Pollution Measurement Model in a Heavy Traffic Area under the Bangkok Sky Train Platform

and Applied Analysis 3 Figure 1: The street tunnel configuration. The air pollutant concentration can be described by a threedimensional advection-diffusion equation as follows: ∂C ∂t + V ⋅ 󳶋C = 󳶋 ⋅ (K ⊗ 󳶋C) + R (x, y, z, t) , (1) where C = C(x, y, z, t) is the air pollutant concentration at point (x, y, z) in Cartesian coordinates and at time t (kg/m3). The vector V is the wind velocity field (m/sec); K is the eddy-diffusivity or dispersion tensor (m/sec). 󳶋 = (∂/∂x)󳨀 →i + (∂/∂y)󳨀 →j + (∂/∂z)󳨀 →k , ⊗ is matrix multiplication, and R(x, y, z, t) describes sources or sinks of air pollutants (sec). If the wind velocity and diffusion coefficient of pollutant are constant, the governing equation becomes ∂C ∂t + u∂C ∂x + V∂C ∂y + w∂C ∂z = kx ∂2C ∂x2 + ky ∂ 2C ∂y2 + kz ∂ 2C ∂z2 + R (x, y, z, t) , (2) where u, V, and w are the constant wind velocity (m/sec) in x, y, and z-directions, respectively, and kx, ky, and kz are the constant diffusion coefficient (m/sec) in x, y, and zdirections, respectively. By the assumption, we assumed that the wind inflow is along the horizontal direction and the dispersion is horizontally isotropic. Consequently, the three-dimensional advection-diffusion equation in (2) can be written as ∂C ∂t + u∂C ∂x + V∂C ∂y = kh ∂ 2C ∂x2 + kh ∂ 2C ∂y2 + kV ∂ 2C ∂z2 + R (x, y, z, t) , (3) where kh is a constant dispersion coefficient in the horizontal direction (m/sec) and kV is a constant dispersion coefficient in the z-direction (vertical) (m/sec) with the appropriate initial and boundary conditions. We consider the components of the tunnel in Figure 4 and themodel of the problem is divided into three zones as shown in Figure 5. The potential air pollutant concentration can be described by C(x, y, z, 0) = f(x, y, z), for all (x, y, z) ∈ Ω. The boundary conditions are as follows: Entrance gate: C(0, y, z, t) = cN 1 , 0 < y < W, 0 < z < H. Margin of entrance gate: (∂C/∂x)(0, y, z, t) = cN 2 , y = 0, W, z = 0,H. Exit gate: (∂C/∂x)(L, y, z, t) = cX. Both side walls: (∂C/∂y)(x, 0, z, t) = cW 1 , (∂C/∂y)(x,W, z, t) = cW 2 . Ground: (∂C/∂z)(x, y, 0, t) = cF. Platform ceiling: (∂C/∂z)(x, y,H, t) = cT, A < y < B. Ceiling parallel gaps: (∂C/∂z)(x, y,H, t) = cG 1 , 0 ≤ y ≤ A, (∂C/∂z)(x, y,H, t) = cG 2 , B ≤ y ≤ W, where cN 1 is the inflow air pollutant concentration at the entrance gate. cN 2 , cX, cW 1 , cW 2 , cF, cT, cG 1 , cG 2 are the average rate of change of air pollutant concentration at the margin of entrance gate, exit gate, both side walls, ground, platform ceiling, and both ceiling parallel gaps, respectively. A is the distance from the right wall to the right-ended platformceiling; see in Figure 5.B is also the distance from the right wall to the left-ended platform ceiling; see in Figure 5. 3. Numerical Techniques The finite difference method is used to approximate the solutions to the governing equation.The domainΩ is divided by xi = iΔx, i = 0, 1, 2, . . . ,M; yj = jΔy, j = 0, 1, 2, . . . , N; zk = kΔz, k = 0, 1, 2, . . . , P; tn = nΔt, n = 0, 1, 2, . . . , Q over three spaces and time coordinate axes, respectively. The approximated air pollutant concentration at point (iΔx, jΔy, kΔz, nΔt) is denoted by Cn i,j,k = C(iΔx, jΔy, kΔz, nΔt) at the grid point (i, j, k, n). The constant spatial and temporal grid spacing are Δx = L/M, Δy = W/N, Δz = H/P, Δt = T/Q, respectively. In this research, an explicit forward time central space (FTCS) method is employed. Consequently, the finite difference equation to (3) becomes Cn+1 i,j,k − Cn i,j,k Δt + u( Cn i+1,j,k − Cn i−1,j,k 2Δx ) + V(C i,j+1,k − Cn i,j−1,k 2Δy ) = Dh (C n i+1,j,k − 2Cn i,j,k + Cn i−1,j,k (Δx)2 ) + Dh(C n i,j+1,k − 2Cn i,j,k + Cn i,j−1,k (Δy)2 ) + DV (C n i,j,k+1 − 2Cn i,j,k + Cn i,j,k−1 (Δz)2 ) + (Δt) Rn i,j,k. (4) 4 Abstract and Applied Analysis Wind outflow Pollution outflow Wind inflow Pollution inflow x L


Introduction
Nowadays if we are talking about pollution, surely one of the pollution sources that we face and have a big effect on society is "air pollution."Air pollution does not only affect one society but also the problem for human life and environment that everyone all over the world should realize.Air pollution is harmful to human health because it releases pollutants and dirty air which caused asthma, lungs, and cancer.Moreover, it is a major factor which affects environmental resources as well as human-made structures and facilities and contributes to climate change.
Sources of air pollution can be classified into two types which are natural sources and artificial sources.Natural sources of pollution come from natural phenomena such as volcanic eruptions, forest fires, biological decay, pollen grains, marshes, and radioactive materials.On the other hand, artificial sources are those created by human beings such as thermal power plants, vehicular emissions, fossil fuel burning, and agricultural activities.Air pollution can occur in many forms but in general it occurs in the form gas and particulate contaminants which are in our atmosphere.Gaseous pollutants include carbon monoxide (CO), nitrogen oxides (NOx), sulfur dioxide (SO2), ozone (O3), and various gaseous.
Some people might wonder if indoor or outdoor air is more polluted.According to studies of scientists, indoor air pollution is often more harmful than outdoor air pollution, especially because we spend most of the time per day indoor inside our home or office.The air inside our homes and offices can sometimes be much more polluted compared to outdoor air and thus presents a major health threat.In their latest study, the British scientists measured air quality inside and outside three residential buildings with different types of energy use.What they found was that the levels of one of the most common air pollutants, nitrogen dioxide (NO2), in kitchens in the city center apartments with gas cookers were as much as three times higher than the levels measured outdoors and well above clean air quality standards.
In [1], also in October 2016, more than 140 countries reached an agreement to reduce the use of these chemicals which are used in air conditioners and refrigerators and to find greener alternatives over time.David Doniger, director of NRDC's Climate and Clean Air program, wrote, "NRDC estimates that the agreed HFC phase-down will avoid the equivalent of more than 80 billion tons of CO2 over the next 35 years."Moreover, Walke said, "make good choices about transportation.When you can walk, ride a bike, or take public transportation.For driving, choose cars that get better miles per gallon of gas or choose an electric car."The sources of smog and soot are similar.Walke said, "both come from cars and trucks, factories, power plants, incinerators, engines."So, a wise decision is encouraged to make our world green.
In 1961, [2] studied the pollution of the air (smoke, polycyclic hydrocarbons, carbon monoxide, and lead) by motor vehicles in two London road tunnels.It was found that the concentration of air pollution in the tunnels does not appear to be high but the effect of traffic on the concentration of smoke, polycyclic hydrocarbons, carbon monoxide, and lead in the air of city streets deserves continued study.In 2002, [3] studied average air pollutant concentration during weekdays and found it to be higher than during the weekend.The test result showed that the average air pollutant concentrations for the three urban sites are noticeably higher than the suburban site.Our analysis revealed that an obvious way to reduce the build-up of pollutant concentration on Bangkok streets would be to speed up the flow of traffic and prevent long periods of idling in congested streets.In 2004, [4] studied the stability conditions for several different numerical techniques which were developed and compared for solving the three-dimensional advection-diffusion equation with constant coefficient.The results of a numerical experiment were presented, and the accuracy and central processor time needed were discussed and compared.In 2006, [5] studied the numerical methods for solving the advectiondiffusion equation.It was solved by using cubic splines to estimate first and second derivatives and also by solving the same problem using two standard finite difference schemes (the FTCS and Crank-Nicolson methods).The numerical results were compared with analytical solutions.It was found that, for the examples studied, the finite difference methods yielded better pointwise solutions than the spline methods.In 2016, [6] studied the three-dimensional air quality model.The considered domain was divided into two zones: a factory zone and a residential zone.The modifications of the atmospheric stability classes and wind velocities from multiple point sources were also analyzed by using the threedimensional fractional step method.In 2017, [7] studied a three-dimensional advection-diffusion equation by using the explicit forward difference method.The wind inflows are considered in two cases: there is wind inflow only in direction and there are wind inflow in -direction and direction.Moreover, we added the obstacles along the middle into the tunnel.The results of the model are satisfactory.
Currently in Bangkok, Thailand, air pollution from car exhaust on the street, which contains particulates, especially from old cars or diesel cars, is harmful to people's health.Scientists are concerned that the particulates carrying toxic chemicals, such as nitrous oxide and carbon monoxide, when deeply in haled, can be harmful to people's health.Bangkok Transit System (BTS) provides an effective route of urban transport for Bangkok people because BTS facilitates speed and convenience for transportation.The major source of air pollution under Bangkok sky train platform comes from vehicle exhaust, mobile source, and others sources including smoke from restaurants, construction, and building demolition.Therefore, it is also causing some of the environmental impacts, especially the air pollutant impact to the vicinity area around its platform with high traffic and large amount of people.
These days, an increasing in population caused heavy traffic and air pollution on the road.Air pollution around the platform in an area under BTS platform has increased dramatically.So, if we know the value of the concentration of pollution that is likely to occur from the existing pollution accumulation or may be from sources of emissions, such as from car smoke, we might be able to control the concentration of air pollution in that area not to exceed the standard.As already mentioned, we recognized the importance of air pollution.Therefore, the purpose of this research is to approximate the concentration of air pollutant in the area under the Bangkok sky train platform with moving air pollutant sources by heavy traffic of vehicles in different time by using the finite difference technique.It can help control the pollutant from the traffic and crowded people in this area.It will be beneficial to human and environment.However, this area should be implemented into the wind inflow directions near the tunnel because it affects the concentration of air pollutant.Then the wind inflow directions are an important factor of the model.So, we distinguish two cases: there is wind inflow only in -direction and there is wind inflow in and -directions.

Governing Equation
A street tunnel is a place for foot or vehicular road traffic, where the street is flanked by buildings on both sides, including the top area that is also closed.The street tunnel configuration is shown in Figure 1.An overhead part of the street is the sky train platform and both sides of the street are composed of sections of building.In this research, the simulation of configuration of street tunnel is divided into two cases.
Case 1. Assume that the wind is flowing only in -direction.The considered street tunnel is illustrated in Figure 2(a).The wind direction field is shown in Figure 2(b).
Case 2. Assume that the wind is flowing in and directions.The considered street tunnel is illustrated in Figure 3(a).The wind direction field is shown in Figure 3(b).
The considered domain is restricted by Ω = {(,, ); 0 ≤  ≤ , 0 ≤  ≤ , 0 ≤  ≤ }, where  is the platform width (m),  is the platform length (m), and  is the platform height (m) over the street tunnel.The air pollutant concentration can be described by a threedimensional advection-diffusion equation as follows: where  = (, , , ) is the air pollutant concentration at point (, , ) in Cartesian coordinates and at time  (kg/m 3 ).The vector  is the wind velocity field (m/sec);  is the eddy-diffusivity or dispersion tensor (m 2 /sec). = , ⊗ is matrix multiplication, and (, , , ) describes sources or sinks of air pollutants (sec −1 ).
If the wind velocity and diffusion coefficient of pollutant are constant, the governing equation becomes where , V, and  are the constant wind velocity (m/sec) in , , and -directions, respectively, and   ,   , and   are the constant diffusion coefficient (m 2 /sec) in , , and directions, respectively.By the assumption, we assumed that the wind inflow is along the horizontal direction and the dispersion is horizontally isotropic.Consequently, the three-dimensional advection-diffusion equation in (2) can be written as where  ℎ is a constant dispersion coefficient in the horizontal direction (m 2 /sec) and  V is a constant dispersion coefficient in the -direction (vertical) (m 2 /sec) with the appropriate initial and boundary conditions.We consider the components of the tunnel in Figure 4 and the model of the problem is divided into three zones as shown in Figure 5.The potential air pollutant concentration can be described by (, , , 0) = (, , ), for all (, , ) ∈ Ω.The boundary conditions are as follows: Entrance gate: (0, , , ) =   1 , 0 <  < , 0 <  < .
Margin of entrance gate: (/)(0, , , ) =   2 ,  = 0, ,  = 0, .where   1 is the inflow air pollutant concentration at the entrance gate.  2 ,   ,   1 ,   2 ,   ,   ,   1 ,   2 are the average rate of change of air pollutant concentration at the margin of entrance gate, exit gate, both side walls, ground, platform ceiling, and both ceiling parallel gaps, respectively. is the distance from the right wall to the right-ended platform ceiling; see in Figure 5.  is also the distance from the right wall to the left-ended platform ceiling; see in Figure 5.
In this research, an explicit forward time central space (FTCS) method is employed.Consequently, the finite difference equation to (3)   Rearrangement of (4) gives in which   = Δ/Δ,   = VΔ/Δ,   =  ℎ Δ/(Δ) 2 ,   =  ℎ Δ/(Δ) 2 , and The stability condition of the proposed finite difference scheme, which can be investigated by using the von Neumann method [4,8], is stable if both are satisfied.The finite difference scheme for the left end and the right end of the fictitious points is as follows:

Numerical Experiments
In this section, there are three simulations of released air pollutant phenomena demonstrated by using the finite difference in (5).In all simulations, the air is flowing along the -direction from the entrance to the exit gates.There are two parallel gaps along the celling; see Figures 1 and 4.There is no potential ambient air pollution.There are two buildings that were bracing the areas as well; see Figures 1 and 4. All of building walls are nonabsorbing air pollution materials.Since there is no potential air pollution, the initial condition is assumed by (, , ) = 0.
Simulation A (source or sink emissions are averaged).In this example, we consider two cases.In the first case,  is the constant of source ( > 0), which are 0.001, 0.004, and 0.007 sec −1 .In the second case,  is the constant of sink ( < 0), which are −0.001,−0.004, and −0.007 sec −1 .The results of Simulation A are shown in  Simulation B (source or sink emissions are moving).In this example, we consider two cases.In the first case,  is the function of source and sink ( > 0,  < 0), that is, 0.001sin(), 0.003sin(), and 0.005sin() sec −1 .In the second case,  is the function of source ( > 0), which are 0.001|sin()|, 0.003|sin()|, and 0.005|sin()| sec −1 .The results of Simulation B are shown in Figures 12-17 and 24.

Discussion
The air pollutant concentrations are calculated by using a finite difference technique.Whether sources or sinks, it affected the air pollutant concentrations.The comparison of sources or sinks for Simulations A, B, and C are shown in Table 1.Figures 6-7 and 9-10 show the air pollutant concentration levels after passing 30 seconds in contour plot and surface plot between  = 0.007 (source) and  = −0.007(sink), respectively.Figures 8 and 11 compare the air pollutant concentration levels where  is the constant in first case and second case of Simulation A, respectively.From the results, if we take more source rate into our system, we can see that the concentration of air pollutant levels has increased (see Figure 8).Therefore, the concentration varied with the sources.Furthermore, the sink can lower the concentration of air pollutant levels (see Figure 11).Furthermore, Figures 12-13 and 15-16 show the air pollutant concentration levels after passing 30 seconds in contour plot and surface plot between  = 0.001sin() (source-sink) and  = 0.003|sin()| (source), respectively.Figures 14 and 17 compare the air pollutant concentration levels where  is the constant in the first case and second case of Simulation B, respectively.As a result,  is a function of both source and sink and the concentration of air pollutant has increased and decreased (see Figure 14).That is, it is increased when  is the source.On the other hand, if  is sink, the concentration of air pollutant has decreased.Moreover, Figures 18-20 show the air pollutant concentration levels after passing 30 seconds in surface plot where  is the constant in first, second, and third cases of Simulation C, respectively.Figure 21

Conclusion
The released vehicles air pollutant can be assumed by source functions.The source functions are defined by many methods such as averaged collected data methods or numerical interpolations.The simulations show that the air pollution problems arise by external and internal vehicles that released

Table 1 :
Comparison of sources or sinks for Simulations A, B, and C. We can see that, under the platform area, air pollutant level is higher than the outside level due to air flow obstacle.