Journal of Scientific Research and Studies Vol. 1(2), pp. 9-16, July, 2014 Copyright © 2014 Author(s) retain the copyright of this article http://www.modernrespub.org/jsrs/index.htm

Full Length Research Paper

Analysis and simulation of 3-D advection diffusion reaction model for pollutant dispersion

Feda A. Zahor1*, Wilson M. Charles2 and Silas Mirau1

1Nelson Mandela African Institution of Science and Technology (NM-AIST), Arusha-Tanzania.

2Department of Mathematics, University of Dar es Salam, Dar es Salaam, Tanzania.

*Corresponding author. E-mail: zahorf@nm-aist.ac.tz

Accepted 28 June, 2014

This paper focuses on modeling vertical movements of a plume of pollutants. A three dimensional advection diffusion model for modeling the prediction of the dispersion of pollutants in the atmosphere was studied. The concentration of contaminants was released from ground level sources and behavior of the dispersion of the plume of pollutants was modeled. The solution was approximately numerical by using finite difference method and incorporating Crank-Nicolson scheme in MATLAB. The effect of embedded parameters was studied and their results are presented graphically in this paper. Key words: 3-D, advection-diffusion-reaction equation, pollutants, finite difference, Crank-Nicolson.

INTRODUCTION A problem of great importance in environmental science is to understand how chemical or biological contaminants are transported through water or air. Mathematical models are the most important tool to understand and predict the behavior of pollutants dispersion as well as to plan for counter measures to improve and protect our environment (Charles et al., 2009). There are several sources of pollution. For instance, in water, the largest source is oil spills. The largest and most recent oil spill in history, larger than both Alaska's Exxon Valdez spill and Ixtoc blowout, is that of BP's Macondo well blowout in the Gulf of Mexico, where 4.4 million barrels of oil were discharged in the ocean (Griggs, 2011). According to images taken from satellite, the spill impacted 180,000 km

2 of ocean surface, which is equivalent to the size of

Oklahoma (Norse and Amos, 2010). Apart from damaging the ocean floor, the spill caused a devastating effect on marine living organisms, especially the endangered Louisiana pancake batfish, whose habitat is entirely encompassed inside the area covered by the spill (Campagna et al., 2011). Oil spills due to blowout, pipeline leakage, ship or plants accidents are not the only source of contaminants in coastal zones. Much smaller accidents also harm the ecosystem at the equivalent degree. Water is also polluted by industrial chemical

wastes discharges and byproducts, discharge of untreated or weakly-treated sewage, surface water flows containing pesticides and so on (Spivakovskaya et al., 2005).

In the case of air, contaminants may constitute almost any natural or artificial constituents of matter having airborne capability (Spellman and Whiting, 2004). They can be in the form solid, liquid droplets, gaseous or combinations of these forms (Ali et al., 2013; Habingabwa et al., 2012). Principal contaminants produced by human activities include fine solid particles, carbon-monoxide (CO) produced as a result of burning fossil fuels, sulphur oxides (SO) produced from various industrial processes, etc (Dominici et al., 2006). These contaminants are the primary cause of environmental health problems. World Health Organization statistics revealed that 2.4 million people die every year from causes directly associated to air pollution (WHO, 2009). In recent decades, economic and industrial growth in Chinese cities caused varieties of urban air pollution problems. One decade later, as a result of high level of air pollution, most of southern cities started to experience the effects of severe acid rain pollution. Due to increased and growing use of fossil fuels, nitrous oxides (NO), carbon monoxide (CO) and photochemical smog; a

MMMMRRRRPPPP

10 J. Sci. Res. Stud. typical results of burning fossil fuels, have brought China a massive air quality deterioration (He et al., 2002).

To prevent possible damage as a result of discharge of contaminants, it is extremely important to predict their movements and spreading behavior. While in real life pollution transport phenomena occur in three dimensional domain, most applications suggested so far have put emphasis on one and two dimensions advection-diffusion equations (Habingabwa et al., 2012; Ali et al., 2013; Makungu et al., 2012; Dehghan, 2005; Heitmann and Peurifoy, 2007). Charles et al. (2009) used a stochastic particle model in simulating the advection and diffusion of pollutants in shallow waters. They improved the model and deployment of contaminants, by introducing a colored noise which takes into account the short-term correlated turbulent flow of the particles. Moreover, Dang and Ehrhardt (2006) and Dang et al. (2007) considered the problems of optimal location of industrial enterprises and optimization of emissions from enterprises for ensuring sanitary environment criteria. They also studied the problem of determination of the coefficients of diffusion and the coefficient of transformation of aerosols. Thornburg et al. (2010) developed a probabilistic model based on aerosol physics and fluid dynamics to predict the breathing zone concentration of a particulate contaminant emitted from a surface during activities of variable intensity. The model predicted the particle emission rate, tracked particle transport to the breathing zone, and calculated the breathing zone concentration for two scenarios; one scenario used an Eulerian model based on a Gaussian concentration distribution, to quantify aerosol exposure in the trailing wake of a moving object, the other scenario modeled exposure in a quiescent environment. Hamdi (2007) considered the problem of determining pollution sources in a river by using boundary measurements. He used a two dimensional advection-diffusion reaction equation in the stationary case, and established identifiability and a local Lipschitz stability.

In this work we apply the 3-D advection diffusion reaction equation to model the dispersion of pollutant in air.

MATHEMATICAL MODEL

Consider ( , , , ) to be the concentration of pollutants which is a function of both space and time; the advection-diffusion-reaction equation in one dimension. In three dimensional domains, as in Marchuk (2011) and Berlyand (1982), the equation is given by:

−

+ −

ν φ +

+ +

− + = (1)

Where , , , !, , , , !, , , , ! are the

components of the wind velocity in , and direction respectively; is the falling velocity of the pollutants by

gravity; , , , ! the power of the source; , , , ! is the reaction coefficient of pollutants and , , , !, " , , ! are respectively horizontal and vertical diffusion coefficients (Dang and Ehrhardt, 2006). In this work, we consider the following initial and boundary condition for Equation 1 , , , 0! = $ , , !, (2) % &, ', (, )! = % *, &, (, )! = % *, ', &, )! = % +, ', (, )! =% *, ,, (, )! = % *, ', -, )! = &, (3) Where ., / and 0 are far edges of the domain Ω which is given by Ω = 10, .2 × 10, /2 × 10, 02. NUMERICAL PROCEDURES The finite difference method is one of the techniques for obtaining numerical solutions to Equation 1. We discretize the space domain using central finite difference approximation and applied the Crank-Nicholson scheme for the time discretization. We have refined the mesh size for not only increasing the resolution but also the accuracy of the solution (Ali et al., 2013). The Crank-Nicholson scheme was used as it is unconditionally stable and is second-order accurate (Thornburg et al., 2010; Makungu et al., 2012; Yang et al., 2006). It is worth pointing out that the complicated analytical solution for advection-diffusion equation may exit under specific conditions (Berlyand, 1982; Dang and Ehrhardt, 2006). The space discretization for Equation 1 results in the following scheme: 56,7,85 = − 69:,7,8;6<:,7,8=>? − @6,79:,8;6,7<:,8=>A B − − 6,7,89:;6,7,8<:=>C

+ @69:,7,8;=6,7,8D6<:,7,8>? + 6,79:,8;=6,7,8D6,7<:,8>A B

+ 6,7,89:;=6,7,8D6,7,8<:>C − E,F,G +E,F,G (4)

Where ℎ, ℎ , IJK ℎ are the mesh sizes in , IJK direction respectively and L, MIJK N are the respective discretization indices. With a proper numbering of nodal point of the discretization domain Ω, the scheme represented by the Equation 4 can be written in matrix form as 5Φ5 = O.Φ + P, (5)

Where Φ is the nodes concentration vectors given by:

Φ = φQ,Q,Q,φ=,Q,Q, … ,φS;Q,T,U,φS,T,UV (6)

I, J and K are maximum of internal node discretization in , IJK direction respectively. The total internal nodes n in the domain is given as J = W × X × Y. O is hepta-

diagonal matrix of order J. We denote O±[ to be the

diagonal matrices of order n where the diagonal is at position p above or below the main diagonal respectively. The diagonal elements are defined as follows: O± \×]!^ = _

>C ∓ a;ab=>C (7)

O±\^ = c>A ∓ _

=>A (8)

O±Q^ = c>? ∓ d

=>? (9)

O±e^ = − @=c>? + =c

>A + =_>C + B (10

So, A is defined in terms of diagonal matrices as: O = O±e + O±Q + O±\ + O±\f] (11)

The off diagonals of sparse matrix A also contain some zero entries which occur periodically. Matrix A can also be expressed as a block diagonal matrix given by:

O = g h h= 0hQ h h=0 hQ h i , h = g j j= 0jQ j j=0 jQ j i (12)

Where B1, B2, C1 and C2 are diagonal matrices of order

IXJ with elements O; \×]!^, OD \×]!^, O;\^ and OD\^ respectively.

O is a block of zeros and C is a tridiagonal matrix of order

I, with elements O±\^ and O±e^. It is interesting to note that,

the block matrix B represented a matrix obtained when two dimensional domain is considered and sub –block C represent the matrix obtained from the one dimensional domain. The vector F is defined as: P = Q,Q,Q, =,Q,Q, … , \;Q,],k , \,],kl (13)

Applying Crank- Nicolson scheme to descretize the time derivative of Equation 5 results into Φ

m9:;Φmn = Q

= ΦoDQ +Φo + Q= FoDQ + Fo! (14)

Which simplify to

Wq − n= OΦoDQ = Wq + n

= OΦo + rm9:Drm.n= (15)

Where m is a time discretization index, and s being step.

Feda et al. 11 It is worth noting that, this scheme is stable and possess good approximation properties, but it may lead to solutions with negative values that are meaningless. To get rid from the problem, all initial and boundary conditions should be nonnegative, then the solution of the corresponding problem is nonnegative, too (Dang et al., 2007). Moreover, the system represented in Equation 15 can have a unique solution if the hepta-diagonal matrix A is positive definite (Saad, 2003). RESULTS AND DISCUSSION In this section, the numerical solutions of the Equation 15 subject to the zero Dirichlet boundary condition and initial condition in the environment where there were no pollutants are computed and illustrated graphically using MATLAB. The influences of all embedded parameters to the dispersion of pollutants were discussed. The parameters used in the model are drift velocity , , !, horizontal and vertical diffusion coefficients and t respectively and chemical transformation coefficient . We assume that all parameters are constant with the respect to space and time and the initial value function g , , ! = 0. We shall use s as unit of time throughout. Unless otherwise stated, the parameter values used in this work are, ℎ = ℎ , ℎ = 0.02, = 0.1, " = 0.05, = 2, s = 0.01ℎ, , , ! = 1.5,1.5,1.5! and = 0.02. The size

of the domain was taken to be Ω = 10,12 × 10,12 × 10,12. The pollutant source issue 80 kilo units of pollutant concentration at selected positions each time. The effect of diffusion coefficients, velocity, and chemical transformation coefficient on pollutants dispersion, are discussed and results are presented. Vertical dispersion Figures 1-3 are the graphical solution of the transient advection diffusion reaction model. They illustrate the evolution of vertical dispersion behavior of contaminants between the intervals of 0.1 h, when the horizontal drift is still. The point-like pollutant source is placed at the center of the plane z = 0 (at the coordinates , , ! = 0.5, 0.5, 0!!, covering a volume of about 0.06

3 cubic

units. The source remained active for 0.1 h before it went off. It suffices to plot vertical dispersion using the plane as the results are symmetric to = 0.5 plane. It can be clearly seen that, the polluted zone expand and the concentration increases near the source when the source is active, and continue to expand with decreased concentration and rose up when the source is stopped (see the color bars). Figures 4-6 represent the approximate solution as given in Figures 1-3 except that, here in this case, the source is active throughout.

Figures 7-8 compare the dispersion of cloud of contaminant at two different vertical velocities (w= 1 and

12 J. Sci. Res. Stud.

Figure 1. Polluted zone after 0.02h.

Figure 2. Polluted zone after 0.2h.

Figure 3. Polluted zone after 0.5h.

w=1.5) while other parameters remain unchanged. It can be observed that the cloud of pollutants disperse much

Figure 4. Polluted zone after 0.02h.

Figure 5. Polluted zone after 0.2h.

Figure 6. Polluted zone after 0.5h.

faster vertically when the vertical velocity is increased. We also examined the effect of coefficient of chemical

0.02ℎ0.02ℎ

Figure 7. Polluted zone after 0.21h.

Figure 8. Polluted zone after 0.34h.

Figure 9. Pollution level at 0.1h.

reaction on the vertical dispersion of cloud of contaminants. Figures 9-12 depict its effect. Two different

Feda et al. 13

Figure 10. Pollution level at 0.25h.

Figure 11. Pollution level at 0.37h.

Figure 12. Pollution level at 0.5h.

values of were used and the results were compared at four different time levels, t = 0.1ℎ, t = 0.25 ℎ, t = 0.37 ℎ

14 J. Sci. Res. Stud.

Figure 13. Polluted zone after 0.02h.

Figure 14. Polluted zone after 0.2h. and 0.5 ℎ respectively. It can be observed that increasing the coefficient of chemical reaction significantly reduces the size of polluted zone and therefore the coefficient act as a pollutant sink parameter. Horizontal dispersion In this section, we discuss the effect of different parameter on the horizontal expansion of polluted zone. Figures 13-15 illustrate the time evolution of horizontal dispersion of pollutant observed at the plane z = 0.5. The

source is placed at the origin covering 0.063 cubic unit

and remained active for only 0.1ℎ. It is observed that the polluted zone expands horizontally while the concentration rises slowly, reaching maximum value at 0.2 ℎ before decreasing again while rising past the plain = 0.5

Figure 15. Polluted zone after 0.5h.

Figure 16. Polluted zone at z=0.04.

Figures 16-21 represent the approximated results when the throughout-active source is placed at the center of z = 0; plane the horizontal drift is still, that is, , ! = 0,0!. We pick a scenario when time t = 0.25 ℎ and examine the level of contamination at some different vertical position, namely, z = 0.04, z = 0.2, z = 0.4, z = 0.6, z = 0.8 and z = 1.

The results show that concentration of contaminants decreases as you rise up vertically. This is because the source remains active-throughout the time the process is being examined. The horizontal velocity also contributes a lot in the dispersion of contaminants. Figures 22-23 compares the results at two different time levels, t = 0.25 ℎ and t = 0.4 ℎ respectively, with horizontal velocities (, ! = 1,1!IJK , ! = 1.5,1.5!). The effect is observed at the plane z = 0.3. The results show that increasing the velocity results in the polluted zone being expanded.

Figure 17. Polluted zone at z=0.2.

Figure 18. Polluted zone at z=0.4.

Figure 19. Polluted zone at z=0.6.

Feda et al. 15

Figure 20. Polluted zone at z=0.8.

Figure 21. Polluted zone at z=1.

Figure 22. Polluted zone after t=0.25h.

16 J. Sci. Res. Stud.

Figure 23. Polluted zone after t=0.4h.

Conclusion This work developed a pollutant concentration model based on three dimensional advection diffusion reaction equations to track the dynamics of pollutants in air/water. The model has three external parameters, namely the pollutant diffusion coefficient, the drift velocity of air and the pollutant chemical activity coefficient. The polluting source is characterized by its location, its polluting power; that is, the amount of released pollutants into per unit time and the period of time during which the source is active. The numerical method used to solve the problem is a positive finite difference scheme based on Crank-Nicolson method.

The resulting algorithm was implemented in MATLAB taking the advantage of sparse matrix technology to reduce the computational time and memory concerns; for instance, a normal identity matrix of order 1000 would require 8Mb of temporary storage while a sparse identity matrix of the same order would require only 24Kb of the temporary storage. Different scenarios were discussed and analyzed; for example, the influence of the model's external parameters on the dynamics of polluted zones is highlighted. The results reveal that light pollutant with a high diffusion coefficient expands quickly in the affected region but also dissipates quickly when the polluting source is stopped. A dense pollutant slowly expands and stays longer in the affected region even after the source is stopped especially when the drift velocity is low.

The vertical dispersion of pollutants was also discussed and the results show that vertical diffusion and velocity w, plays a major role in the dispersion. The drift velocity moves the polluted zone from one region to another at the rate which is proportional to its magnitude. The eventual chemical reaction between the pollutant and one or more air components acts as a pollution sink and slows down the expansion of polluted zone.

REFERENCES Ali AO, Mtega NA, Wilson MC (2013). The advection-diffusion model for

atmospheric pollutants dispersion. Int. J. Ecol. Econom. Stat. 30(3):56-72.

Berlyand MJ (1982). Moderne probleme der atmospharischen diffusionund der ver-schmutzung der atmosphere akadeirie-verlag. Campagna C, Short FT, Polidoro BA, McManus R, Collette BB, Pilcher.

Charles W, Heemink A, van den Berg E (2009). Coloured noise for dispersion of contaminants in shallow waters. Appl. Math. Model. 33(2):1158-1172.

Dang QA, Ehrhardt M (2006). Adequate numerical solution of air pollution problems by positive difference schemes on unbounded domains. Math. Comput. Model. 44(9):834-856.

Dang QA, Ehrhardt M, Tran G, Le D (2007). On the numerical solution of some problems of environmental pollution. Air pollut. Res. Adv. pp. 171-200.

Dehghan M (2005). On the numerical solution of the one-dimensional convection-diffusion equation. Math. Prob. Eng. 1:61-74.

Dominici F, Peng RD, Bell ML, Pham L, McDermott A, Zeger SL, Samet JM (2006). Fine particulate air pollution and hospital admission for cardio-vascular and respiratory diseases. Jama 295(10): 1127-1134.

Griggs JW (2011). Bp gulf of mexico oil spill. Energy Law J. 32(1):57. Habingabwa M, Ndahayo F, Berntsson F (2012). Air pollution tracking

using pdes. Rwanda J. 27(1): 63-69. Hamdi A (2007). Identification of point sources in two-dimensional

advection-diffusion-reaction equation: application to pollution sources in a river. Stationary case. Inv. Prob. Sci. Eng. 15(8):855-870.

He K, Huo H, Zhang Q (2002). Urban air pollution in china: current status, characteristics, and progress. Ann. Rev. Energy Environ. 27(1):397-431.

Heitmann N, Peurifoy S (2007). Stabilization of the evolutionary convection-diffusion problem: introduction and experiments. In: Proceedings of the SSHE-MA Spring 2007 Conference.

Makungu J, Haario H, Mahera WC (2012). A generalized 1-dimensional particle transport method for convection diffusion reaction model. Afrika Matematika. 23(1):21-39.

Marchuk GI (2011). Mathematical models in environmental problems. Elsevier.

Campagna C, Short FT, Polidoro BA, McManus R, Collette BB, Pilcher NJ, Sadovy de Mitcheson Y, Stuart SN, Carpenter KE (2011). Gulf of Mexico oil blowout increases risks to globally threatened species. BioScience 61(5):393-397.

Norse EA, Amos J (2010). Impacts, perception, and policy implications of the deepwater horizon oil and gas disaster. Envtl. Law Inst., Washington, DC.

Saad Y (2003). Iterative methods for sparse linear systems. Siam. Spellman FR, Whiting NE (2004). Environmental engineer's

mathematics handbook. CRC Press. Spivakovskaya D, Heemink A, Milstein G, Schoenmakers J (2005).

Simulation of the transport of particles in coastal waters using forward and reverse time diffusion. Adv. Water Resour. 28(9):927-938.

Thornburg J, Kominsky J, Brown GG, Frechtel P, Barrett W, Shaul G (2010). A model to predict the breathing zone concentrations of particles emitted from surfaces. J. Environ. Monit. 12(4):973-980.

WHO (2009). Global health risks: mortality and burden of disease attributable to selected major risks. World Health Organization.

Yang Y, Chen R, Yung EK (2006). The unconditionally stable Crank Nicolson FDTD method for three-dimensional Maxwell’s equations. Microw. Opt. Technol. Lett. 48(8):1619-1622.