IJRIT International Journal of Research in Information Technology, Volume 2, Issue 1, January 2014, Pg:131-143

International Journal of Research in Information Technology (IJRIT) www.ijrit.com

ISSN 2001-5569

A NUMERICAL STUDY FOR THE ESTIMATION OF WATER POLLUTION M. M. RAHAMAN1, L. S. ANDALLAH2 & M. BEGUM3 1

Assistant Professor, Department of Mathematics, Patuakhali Science And Technology University Dumki, Patuakhali, Bangladesh Email: [email protected] 2

Professor, Department of Mathematics, Jahangirnagar University Savar, Dhaka, Bangladesh Email:[email protected]

3

Assistant Professor, Department of Mathematics, Jahangirnagar University Savar, Dhaka, Bangladesh. Email: [email protected] Abstract

This paper considers the advection diffusion equation as an initial boundary value problem (IBVP) for the estimation of water pollution. By implementing a finite difference scheme for the IBVP, we estimate and analyze the extent of water pollution at different times and different points in a one dimensional spatial domain. We present a comparison study of the results for varying parameters and verify the qualitative behavior of the solution.

Keywords: Advection Diffusion Equation, Finite difference scheme, Initial boundary value problem (IBVP), Parameter, Water Pollution.

1. Introduction The mathematical model describing the transport and diffusion processes is the one dimensional advection-diffusion equation (ADE). Mathematical modeling of heat transport, pollutants, and suspended matter in water and soil involves the numerical solution of an Advection diffusion equation. Many researchers are involved for solving the model equation (ADE) by using the finite difference scheme. The Finite Difference Method (FDM) is a powerful tool to solve fluid mechanics and heat transfer problems. Agusto and Bamigbola (2007) studied on the Numerical treatment of the mathematical model for water pollution. This study was examined by various mathematical models involving water pollutant. The authors used the implicit centered difference scheme in space and a forward difference method in time for the evaluation of the generalized transport equation. Changjun and Shuwen (2010) made a numerical simulation on river water pollution using Grey Differential Model. They reported that the truncation error of finite differential method in solving the model was corrected. The authors claimed that the obtained results from the gray model are excellent and reasonable. Kumar et al (2009) presented an analytical solution of one dimensional advection diffusion equation with variable coefficients in a finite domain using Laplace transformation technique. The authors introduced new independent space and time variables in this process. In this study the analytical solution was compared with the numerical solution in case the dispersion is proportional to M. M. RAHAMAN, IJRIT

131

the same linearly interpolated velocity. Badia et al (2005) studied on the Identification of a point source in a linear advection-dispersion-reaction equation: application to a pollution source problem. Park et al (2008) performed an analytical solution of the advection diffusion equation for a ground level finite area source using superposition method. Thongmoon and Mckibbin (2006) compared some Numerical Methods for the Advection-Diffusion Equation. The authors reported that the finite difference methods (FTCS, Crank Nicolson) give better point-wise solutions than the spline methods. Yuste et al (2005) described an explicit finite difference method and a new Von Numann-type stability analysis for fractional diffusion equations. Saeedpanah et al (2007) used the combination of Meshless Local PetrovGalerkin Scheme with Radial Basis Function and the Fuzzy Parameters to Simulate of Water Pollution in Coastal Areas. Romao et al (2009) presented the finite difference methods to investigate error in the numerical solution of 3D convection diffusion equation. With the above discussion in view, our goal is to examine mathematical models and ensuing numerical methods for the estimation of the pollutants at different times and different points of water bodies. In section 2, presents a short discussion on the derivation of a water pollution model treated as ADE. We describe Mathematical Models for the estimation of the pollutants based on ([1], [3], [4], [5], [11], [13]. Based on the study of the general finite difference method for the second order linear partial differential equation ([6],[7],[10],[12],[15]), we develop a explicit finite difference scheme for our water pollution model treated as ADE as an IBVP with two sided boundary conditions in section 3. The problem of computer simulation techniques of the pollution model has become an important area in the field of numerical solution method. In section 3, we also establish the stability condition of the numerical scheme. In section 4, we present an algorithm for the numerical solution and we develop a computer programming code for the implementation of the numerical scheme and perform numerical simulations in order to verify the behavior for various parameters. In section 5, we implement the numerical scheme to estimate and analyze the extent of water pollution at different times and different points through advection diffusion equation as water pollution model. We perform a comparison study of different parameter of water pollution models. Finally the conclusions of the paper are given in the last section.

2. Model equation and its derivation In this study we consider the governing equation as water pollution model treated as ADE.

∂c ∂c ∂ 2c +u =D 2 ∂t ∂x ∂x

c

is the concentration at the point x at the time

t

, D is the diffusive constant in the

is the fluid velocity t is the time. With appropriate initial and boundary condition with I.C B.C

c(t 0 , x ) = c 0 ( x); a ≤ x ≤ b c(t , a ) = c a (t ); t 0 ≤ t ≤ T c(t , b) = cb (t )

2.1. Derivation of equation

(1)

x

direction, u

The derivation of the advection diffusion equation relies on the principle of superposition: advection and diffusion can be added together if they are linearly independent. Diffusion is a random process due to molecular motion. Due to diffusion, each molecule in time t will move.

Figure: 1. Schematic of a control volume with cross flow

Either one step to the left or one step to the right (i.e. ± x). Due to advection, each molecule will also move uδt in the cross-flow direction. These processes are clearly additive and independent; the presence of the cross flow does not bias the probability that the molecule will take a diffusive step to the right or the left; it just adds something to that step. The net movement of the molecule is uδt ± δx , and thus, the total flux in the x-direction

J x (above shown in graph), including the advection transport

and a Fickian diffusion term, must be

J x = uc + q x = uc − D Where,

∂c ∂x

uc the correct form of the advection term.

We now use this flux law and the conservation of mass to derive the advection diffusion equation. Consider a cross flow velocity, u

= (u, v, w)

as shown in Figure-1. From the conservation of mass,

the net flux through the control volume is

∂M = ∑ m& in − ∑ m& out ∂t and for the x -direction, we have

(2)

δm& x = (uc − D

∂c ∂c ) δyδz − (uc − D ) δyδz ∂x 1 ∂x 2

we use linear Taylor series expansion to combine the two flux terms, giving

uc 1 − uc 2 = uc 1 − (uc 1 + =−

−D

∂c ∂c +D ∂x 1 ∂x

∂ (uc) δx) ∂x 1

∂ (uc ) δx ∂x = −D

2

=D

∂c ∂c ∂ ∂c + (D + ( D ) δx) ∂x 1 ∂x 1 ∂x ∂x 1

∂ 2c δx ∂x 2

Thus, for the x -direction

δm& x = −

∂ (cu ) ∂ 2c δxδyδz + D 2 δxδyδz ∂x ∂x

The y and z -directions are similar, but with v and w for the velocity components, giving

δm& y = −

∂ (cv) ∂ 2c δyδxδz + D 2 δyδxδz ∂y ∂y

δm& z = −

∂ (cw) ∂ 2c δzδxδy + D 2 δzδxδy ∂z ∂z

substituting these results into (2) and recalling that

M = cδxδyδz We obtain

∂c + ∇.(cu ) = D∇ 2 c ∂t or in Einsteinian notation

∂c ∂cu i ∂ 2c + =D 2 ∂t ∂xi ∂xi This is the desired advection diffusion equation (ADE).

(3)

In the one-dimensional case, direction or

u = (u,0,0)

and there are no concentration gradients in the

y-

z -direction, leaving us with ∂c ∂uc ∂ 2c + =D 2 ∂t ∂x ∂x

Since

u

is constant, then

∂c ∂c ∂ 2c +u =D 2 ∂t ∂x ∂x

(4)

(Advection diffusion equation)

3. Numerical solution of the model equation We consider our specific one dimensional water pollution model problem as an initial and boundary value problem.

∂c ∂c ∂ 2c +u =D 2 ∂t ∂x ∂x With I.C B.C

(5)

c(t 0 , x ) = c 0 ( x); a ≤ x ≤ b c(t , a ) = c a (t ); t 0 ≤ t ≤ T c(t , b) = cb (t )

Like many other numerical approaches, our approach begins with a discretization of the domain of the independent variables x and t. In Mathematics, the finite difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives. A finite difference method proceeds by replacing the derivatives in the differential equation by the finite difference approximations. This gives a large algebraic system of equations to be solved in place of the differential equation, something that is easily solved on a computer.

3.1 explicit centered difference scheme Consider the Advection diffusion equation

∂c ∂c ∂ 2c +u =D 2 ∂t ∂x ∂x

(6)

In order to develop the scheme, we discretize the x − t plane by choosing a mesh width h ≡ ∆x space and a time step k ≡ ∆t . The finite difference methods we will develop produce approximations

c in ∈ R n to the solution c( xi , t n ) at the discrete points by

xi = ih

i = 0,1,2,3.....

,

t n = nk

,

n = 0,1,2,3.....

n

n

Let the solution c ( x i , t n ) be denoted by C i and its approximate value by c i . Simple approximations to the first derivative in the time direction can be obtained from

∂c Cin +1 − Cin ≈ + o(∆t ) ∂t ∆t Centered difference discretization in spatial derivative:

Discretization of

∂c Cin+1 − Cin−1 ≈ + o(∆x 2 ) ∂x 2∆x

∂ 2c is obtain from second order centered difference in space. ∂x 2 ∂ 2 c C in−1 − 2Cin + Cin+1 ≈ + o(∆x 2 ) ∂x 2 ∆x 2

We obtain the difference methods

cin +1 − cin cn − cn c n − 2cin + cin+1 + u i +1 i −1 = D i −1 ∆t 2∆x ∆x 2 n +1

=> ci

n +1

=> ci

=(

D ∆ t u∆ t n D ∆t D ∆ t u∆ t n + )c i −1 + (1 − 2 2 )cin + ( 2 − )c i +1 2 2 ∆x 2 ∆x ∆x ∆x ∆x

γ

γ

= (λ + )cin−1 + (1 − 2λ )cin + (λ − )cin+1 2 2

=> ci = (λ + n

γ n −1 γ )ci −1 + (1 − 2λ )cin −1 + (λ − )cin+−11 2 2

Where γ = u

∆t ∆t , λ =D ∆x ∆x 2

Lemma: 3.2. Stability of the explicit centered difference scheme (8) is given by the conditions

0≤u

∆t ≤ 1, ∆x

0≤D

∆t 1 ≤ 2 2 ∆x

Proof: The explicit centered difference scheme for (6) is given

(7)

(8)

(9)

(10)

cin +1 = ( n +1

=> ci

Where

D∆t u∆t n D∆t D∆t u∆t n + )ci −1 + (1 − 2 2 )cin + ( 2 − )ci +1 2 2 ∆x 2 ∆x ∆x ∆x ∆x

γ

γ

= (λ + )cin−1 + (1 − 2λ )cin + (λ − )cin+1 2 2 γ =u

∆t , ∆x

λ=D

(11)

∆t ∆x 2

The equation (11) implies that if

γ

0 ≤ λ + ≤1 2 0 ≤ 1− 2λ ≤ 1

γ

0 ≤ λ − ≤1 2 Then the new solution is a convex combination of the two previous solutions. That is the solution at new time-step

( n + 1)

at a spatial node

i

is an average of the solutions at the previous time-step at

the spatial-nodes i − 1 , i and i + 1 . This means that the extreme value of the new solution is the average of the extreme values of the previous two solutions at the three consecutive nodes. In our model the characteristics speed

Then we have

γ =

u

must be positive in the positive

x − direction.

u ∆t ≥0 ∆x

We can conclude that the explicit centered difference scheme (11) is stable for

Where

0≤γ =u

∆t ≤1 ∆x

and

0≤λ =D

∆t 1 ≤ 2 2 ∆x

4. Algorithm for the numerical solution of model To find the numerical solution of the model, we have to accumulate some variables which are offered in the following algorithm. Input:

nx and nt tf

the number of spatial and temporal mesh points respectively.

, the right end point of

(0, T )

( xd ) , the right end point of (0, b) C0 , the initial concentration density, apply as a initial condition

C a , Left hand boundary condition

Cb , Right hand boundary condition

D , Diffusion rate

u , velocity Output:

c ( x, t )

the solution matrix

Initialization: dt =

T −0 , the temporal grid size nt

dx =

b−0 , the spatial grid size nx

gm = u *

dt , the courant number dx

ld = D *

dt ( dx ) 2

Step1. Calculation for concentration profile of explicit centered difference scheme For

n = 1 to nt For

i=2

to

nx

C(n +1,i) = (λ + γ / 2) *C(n, i −1) + (1− 2*λ) *C(n, i) + (λ − γ / 2) *C(n, i +1) end end Step2; Output

c( x, t )

Step3: Figure Presentation Step4: Stop

5. Numerical experiments and results discussion:

In this section we present the numerical simulation results for various parameters. We consider the initial concentration

c(0, x) = 0

and the constant boundary value

c (t ,0) = 1

and c (t ,100)

=0.

We run the program and attain the concentration profiles as presented in figure below.

Figure 4.1: Concentration distribution at different time In fig. 4.1, the curve marked by “solid line” shows the concentration profile for 4 minutes and the curve visible by “dash-dot line” represents the concentration profile for 8 minutes. The curve marked by “solid line (red)” shows the concentration profile for 12 minutes and the curve visible by “dash line” represents the concentration profile for 16 minutes. The curve marked by “dot line” shows the concentration profile for 24 minutes. We have seen that the pollutant concentration is increasing with respect to time.

Figure 4.2: Concentration distribution at different position In fig. 4.2, the curve marked by “solid line (blue)” shows the concentration profile for initial position and the curve visible by “dot line” represents the concentration profile for 6 meter. The curve marked by “dash dot line” shows the concentration profile for 12 meter and the curve visible by “solid line” represents the concentration profile for 40 meter. The curve marked by “dash line” shows the

concentration profile for 100 meter. We have shown that the pollutant concentration is increasing in a fixed position with respect to time.

Figure 4.3: Concentration distribution for varying the velocity and diffusion rate at time t=24 min In fig. 4.3, the curve marked by “dash dot line” shows the concentration profile for the velocity term

u = 0 .01m / s and diffusion rate D = 0.05m 2 / s and the curve visible by “dash line” represents the concentration profile for the velocity term u = 0.05 m / s and diffusion rate D = 0.25m 2 / s . We have seen that when the velocity and diffusion rate is high, the pollution is spreading faster.

Figure 4.4:Concentration distribution for varying the velocity and diffusion rate In fig. 4.4, Concentration distribution for each velocity and diffusion rate at time t=24 min. The curve marked by “dash dot line” shows the concentration profile for the velocity term

u = 0.01m / s and

diffusion rate D = 0.05m / s and the curve visible by “dot line” represents the concentration profile 2

for the velocity term

u = 0.05m / s

= 0.25m 2 / s . The curve marked by velocity term u = 0.1m / s and diffusion

and diffusion rate D

“dash line” shows the concentration profile for the

rate D

= 0.5m 2 / s . We have shown that the contaminant concentration with a higher velocity and

diffusion rate increases at a higher rate than that with a lower velocity and diffusion rate.

Figure 4.5:Concentration distribution for varying velocity rate In fig. 4.5, Concentration distribution for each velocity rate at time t=24 min. The curve marked by “dash dot line” shows the concentration profile for the velocity term

u = 0.01m / s

and diffusion

= 0.5m / s and the curve visible by “solid line” represents the concentration profile for the 2 velocity term u = 0.05m / s and diffusion rate D = 0.5m / s . In fig.-4.5, the profile for varying

rate D

2

the velocity term,We have seen that when the velocity rate is low, the pollution is spreading slower.

Figure 4.6: Concentration distribution for varying diffusion rate In fig. 4.6 represents concentration distribution for each diffusion rate at time t=24 min. The curve marked by “dot line” shows the concentration profile for the velocity term

u = 0.01m / s and

diffusion rate D = 0.05m / s and the curve visible by “dash line” represents the concentration profile 2

for the velocity term u = 0.01m / s and diffusion rate D = 0.5m / s . In fig.-4.6 the profile for varying the diffusive term, we have seen that when the diffusion rate is high, the pollution is spreading faster. 2

6. Conclusion In this study we have discussed the numerical solution of advection diffusion equation as water pollution models. We have also given the explicit central difference scheme in space and forward difference method in time for the estimation of the generalized transport equation as advection diffusion equation and have given profiles for different parameter values. We have seen that the pollution is spreading with varying the Diffusion term and Advection term with respect to time and space.

7. References [1].

F.B. Agusto and O.M. Bamigbola, Numerical Treatment of the Mathematical Models for Water Pollution, Research Journal of Applied Sciences 2(5): 548-556, 2007.

[2].

Changjun Zhu and Shuwen Li, Numerical Simulation of River Water Pollution Using Grey Differential Model, Journal of computers, No. 9, September 2010.

[3].

Atul Kumar, Dilip Kumar Jaiswal and Naveen Kumar, Analytical solution of one dimensional Advection diffusion equation with variable coefficients in a finite domain, J.Earth Syst. Sci.118, No.5, October 2009, pp. 539-549.

[4].

A EI Badia, T Ha-Duong and A Hamdi, Identification of a point source in a linear advection Dispersion-reaction equation: application to a pollution source problem, Institute of Physics Publishing, Inverse Problem 21 (2005) p.1-17

[5].

Young-San Park ,Jong-Jin Baik, Analytical solution of the advection diffusion equation for a ground level finite area source, Atmospheric Environment 42 (2008)9063-9069

[6].

M.Thongmoon and R.Mckibbin, A comparison of some numerical methods for the advection Diffusion equation, Inf.Math.Sci.2006, Vol.10, pp49-52.

[7].

S.B. Yuste and L.Acedo, An explicit finite difference method and a new Von Numann-type Stability analysis for fractional diffusion equations, Vol.42, No. 5, pp.1862-1874, 2005, Society for Industrial and Applied Mathematics.

[8].

I Saeedpanah and E. Jabbari, Simulation of Water Pollution in Coastal Areas Using combination of the Meshless Local Petrov- Galerkin Scheme with Radial Basis Functions and the Fuzzy Parameters, Journal of coastal research, Special issue 50, 2007.

[9].

Changiun Zhu,Liping Wu and Sha Li, A Numerical Simulation of Hybrid Finite Analytic Method For Ground Water Pollution, Advanced Materials Research Vol.121-122(2010), pp 48-51.

[10]. L.F. Leon, P.M.Austria, Stability Criterion for Explicit Scheme on the solution of Advection Diffusion Equation, Maxican Institute of Water Technology.

[11].

John A.Trangestein, Numerical Solution of Partial Differential Equation.

[12].

L.S.Andallah, Finite Difference Method-Explicit centered Difference Scheme, lecturer note, Department of Mathematics, Jahangirnagar University 2008.

[13].

Randall J.Le Veque, Numerical methods for conservation law, second edition, 1992, Springer.

[14].

M.K.Jain, S.R.K.Iyengar, R.K.Jain, Computational Methods for Partial Differential Equations, Book published by New Age International (p) Ltd, Reprint: 2007.

[15]. Romao, Silva and Moura, Error analysis in the numerical solution of 3D convection diffusion equation by finite difference methods, Thermal technology, 2009, vol-08, p-12-17. [16]. Halil Karahan, Solution of weighted finite difference techniques with Advection Diffusion Equation using spreadsheets, Department of civil engineering, Pamukkale University, Denizli, Turkey, 2006

a numerical study for the estimation of water pollution

to examine mathematical models and ensuing numerical methods for the estimation of the pollutants at different times ... In section 2, presents a short discussion on the derivation of a water pollution model treated as ADE. We describe ..... ground level finite area source, Atmospheric Environment 42 (2008)9063-9069. [6].

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