The effect of added pollutant along a river
on the pollutant concentration described by
one –dimensional advection diffusion
equation.
Mourad Fadl Alla Dimian
Department of Mathematics , Faculty of Science, Ain Shams University Cairo, Egypt
mouradfadl@yahoo.com
Ali Suleiman Wadi
Department of Mathematics , Faculty of Science, Ain Shams University Cairo, Egypt
ali.wadi4@gmail.com
Fayez Nasif Ibrahim
Department of Mathematics , Faculty of Science, Ain Shams University Cairo, Egypt
fayeznasif@yahoo.co.uk Abstract:
Analytical solutions are obtained, by using Laplace transformation method , for one dimensional advection –diffusion equation with variable coefficients in a longitudinal finite initially pollutant concentration free domain . Two cases for the boundary conditions, are studied . The first is the case of uniform continuous input condition and the second is the case of input condition of increasing nature .By writing the equations in the dimensionless form , the five physical parameters controlling the pollutant concentration is reduced to only two dimensionless parameters the dimensionless added pollutant concentration R1and the dimensionless dispersion
2
R .It is found that some physical parameters in the dimensional form have the same effect on the concentration of the pollutant, while other physical parameters have opposite effect. It is shown that the dimensionless concentration pollutant increases, as the dimensionless added pollutant increases along the river. But the concentration decreases , as the dimensionless dispersion increases. The details are demonstrated in graphs. Keywords: Effect ; pollutant ; river ; concentration ; advection ; diffusion.
1- Introduction:
Pollution can be classified as air pollution, soil pollution , surface pollution and ground water pollution. It’s source may be natural or anthropogenic .One type of source of these pollution is a point source ,[Yadav et al. (2011)]. On the other hand there is increasing concern about water quality worldwide , with increased pollution having a serious impact on the environment . Mathematical models have been used extensively to predict water quality , and to provide reliable tools for water quality management in affected areas.
Reference [Jaiswal et al. (2010)] obtained analytical solutions of the one dimensional advective-dispersive equation for solute transport in homogeneous porous media along non – uniform flow. Velocity of the flow is considered exponential function of space variable. The objective of this study is to obtain the effect of the added pollutant concentration R2 and the dispersion R1 on pollutant concentration along the river .
2- Formulation of the problem:
The pollutant concentration from a source along the flow field through a medium of air or water is described by a partial differential equation of parabolic type derived on the principle of conservation of mass and Fick’s Laws of diffusion and is known as the advection- diffusion equation[Yadav et al. (2011) ]. The partial differential equation of parabolic type describing one- dimensional advection diffusion equation with variable coefficients can be written as[Kumar et al. (2009)].
C C
= (D(x, t) - u(x, t)C ) +γ(x, t) (1)
t x x
where C(x, t)(kg km-3) is the pollutant concentration at position x(km) along the longitudinal direction of the
river at time t(years) , D(x, t)(km2 / year)is the pollutant dispersion , if it is independent of position and time , is called dispersion coefficient ,u(km/year) is the medium’s flow velocity , and γ(x, t)(kg km-3 / year) is the added pollutant rate along the river. To study the temporally pollutant dispersion of a uniform input concentration of continuous nature in an initially pollutant free finite domain ,we consider:
γ0
D(x, t) = D f(mt), u(x, t) = u0 0andγ(x, t) = (2)
f(mt)
where m is a resistive whose dimension is inverse of that the time variable t . f(m t) is chosen such that for m = 0 or t = 0 ,f(m t) = 1.Thus f(m t) is an expression in the non – dimensional variable (m t) [Yadav (2011)]. The constants D0 , u0 andγ0in equation (2) may be referred to as initial dispersion coefficient , a uniform velocity and added pollutant rate along the river respectively[Kumar and Jaiswal (2011) ]. Equations (1) and (2) with initial and boundary conditions give:
2 γ
C C C 0
= D0 f(mt) 2 - u0 + (3)
t x x f(m t)
C(x, t) = 0 , 0 x L , t = 0 (4)
C(x, t) = C0 , x = 0 , t 0 (5)
C(x, t)
= 0 , x = L , t 0 , (6)
x
Where the input condition (5) is assumed at the origin and a second type or flux type homogeneous conditions is assumed at the other end x = L ,of the domain. C0 is a reference concentration.
3 – Analytical solutions:
3 -1 Temporally dependent dispersion along uniform flow: 3-1-1 Uniform continuous input condition:
The medium through which the pollutant dispersion occurs is supposed to be finite domain along the longitudinal direction. To solve advection dispersion equation (3) with the initial and boundary conditions (4) , (5) and (6) analytically using Laplace transform technique , conveniently we introduce a new independent variables X and T [Kumar and Jaiswal (2009),Crank (1975)].
dx dX 1
X = or = (7)
f(m t) dx f(mt) t dt
T = (8)
0 f(mt)
2
C C C
= D0 - u0 + γ0 (9)
2
T X X
C(X, T) = 0 , 0 X X0 (10)
L T = 0, X0 =
f(mt)
C(X, T) = C0 ,X = 0, T 0 (11)
C(X, T)
= 0, X = X0, T 0 (12)
X
It is generally more convenient to work with models written in dimensionless variables. By employ the following definitions:
L
* * *
C = C0 C , X = LX , T = T u0
u0 * L * * *
m = m , t = t , f(mt) = f(m t ) (13)
L u0
Where (* ) denotes dimensionless quantity , hence equations (9 - 12) take the form:
* 2 * *
C C C
= R1 - + R2 (14)
* * *
T X X
where :
D0 Lγ0 R =1 , R2 = ,
Lu0 u C0 0
1
* * * * * *
C (X , T ) = 0 , 0 X X0 = , T =0 (15)
* * f(m t )
* * * * *
C (X , T ) = 1 , X = 0 , T 0 , (16)
* * *
C (X , T ) * * *
= 0 , X = X0 , T 0 (17)
* X
Hence by writing the equations in the dimensionless form , the five physical parameters :
(D0 , u0,γ0 , C0 , L)controlling the pollutant concentration is reduced to only two dimensionless parameters ,the dimensionless added pollutant concentration R1 and the dimensionless dispersion R2 .Now the initial and
boundary value problem (equations (14 - 17 ) ) in the(X ,T )* * domain becomes similar to that of [Cleary and Adrian (1973)] , in (x, t) , quoted as problem B7 [Van Genuchten and Alves (1982)] , hence or otherwise using Laplace transformation technique , the desired analytical solution may be written as follows:
* * * * * * *
C (X ,T ) = A(X ,T ) + B(X ,T ) (18)
* *
* * * * * *
(2X - X )
1 X - T X X + T 1 T
* * 0
A(X , T ) = erfc( ) + exp[ ] erfc( ) + [2 + + ]
* *
2 2 R T R1 2 R T 2 R1 R1
1 1
* * * * * *
X0 2X0- X + T T X0 1 * * * 2
exp[ ] erfc( ) - exp[ - (2X0- X + T ) ]
* *
R1 2 R T πR1 R1 4R T1
1
And
* * * * * * * * *
(X - T ) X - T X + T X X + T
* * *
B(X ,T ) = R {T2 + erfc( ) - ( ) exp[ ]erfc( ) +
* *
2 2 R T 2 R1 2 R T
1 1
1 *
*
X
T 2 * * * 0 1 * * * 2
( ) [ (2X0 - X ) + T + 2R ] exp[1 - (2X0 X + T ) ] -*
4πR1 R1 4R T1
* * * * * *
(2X - X - R ) 1 X 2X - X + T
* 0 1 * * * 2 0 0
[T + + (2X0 - X + T ) ] exp[ ]erfc( )
-*
2 4R1 R1 2 R T
1
* * * * *
X - X 2X - X - T
R1 0 0
exp[ ]erfc( )}
*
2 R1 2 R T
1
Where
* * *
x 1 [exp(m t ) - 1]
* * * * * * *
X = , X0 = , T = , f(m t ) = exp(-m t ).
* * * * *
f(m t ) f(m t ) m
3-1-2 Input condition of increasing nature:
In the case when the velocity of the flow u0is small at the river , the source of input concentration may increase with time t due to the accumulation of pollutants at the origin x = 0. This type of situation may be described by a mixed type or third type condition written as follows:
C
-D(x, t) + u(x, t)C = u C0 0 at x = 0, t 0 (19)
x
Substituting equations (2) , (7) , and (8) into (19) we get:
C
-D0 + u C0 = u C0 0 at X = 0, T 0 (20)
X
By using equation (13) , equation (20)in dimensionless form takes the form:
*
C * * *
-R1 + C = 1 at X = 0, T 0 (21)
* X
Now the initial and boundary value problem composed of advection – diffusion equation (14) ,initial
* * * * * * *
C (X , T ) = A (X , T )1 + B (X , T )1 (22)
where:
1
* * * * * 2 * * * * *
1 X - T T (X - T ) 1 X T X X + T
* * 2
A (X , T )1 = erfc( ) + ( ) exp[- ] - (1 + + )exp( )erfc( ) *
* *
2 2 R T πR1 4R T1 2 R1 R1 R1 2 R T
1 1
1 *
* *
X
4T 1 * * * 0 1 * * * 2 1 * * 3T
2
+ ( ) .[1 + (2X0 - X + T )]exp[ - (2X0 - X + T ) ] - [2X0 - X + + *
πR1 4R1 R1 4R T1 R1 2
1 * * * 2
(2X0 - X + T ) ]exp[ 4R1
* * * *
X0 2X0 - X + T
]erfc[ ]
*
R1 2 R T
1
and:
1
* * 2
* * * (X -T )
1 X -T T
* * * * * 2 * *
B (X , T )1 = R {T +2 (X - T + R )erfc[1 ] - ( ) [X + T + 2R ] .exp[-1 ] *
2 2 R T1 * 4πR1 4R T1
* * * * *
* * 2 * *
* R (X + T ) * X + T R X -X 2X -X -T
T 1 X 1 0 0
+ [ - + ]exp[ ]erfc[ ] - exp( )erfc[ ]
2 2 4R1 R1 2 R T* 2 R1 2 R T*
1 1
* * * * * *
(2X - X ) X 2X - X + T
R1 0 1 * * * * * * 1 * * * 3 0 0
+ [1- + (2X - X + T ).(2X - X + 3T )0 0 (2X - X + T ) ].exp[0 ]erfc[ ]
2 3
2 2R1 2R 6R R1 2 R T*
1
1 1
1 *
*
* R T X
1 * * 7T 1 * * * 2 1 2 0 1 * * * 2
-[-1+ (2X - X +0 ) + (2X - X + T ) ](0 ) exp[ - (2X - X + T ) ]}0
2 *
2R1 3 6R π R1 4R T
1 1
where:
* * *
x 1 (exp[m t ] - 1)
* * * * * * *
X = , X0 = , T = , f(m t ) = exp[-m t ].
* * * * *
f(m t ) f(m t ) m
From the dimensionless numbers : R =1 D0 Lu0
, and R2 = Lγ0 u C0 0
we notice that:
(i) The effect of D0 on the concentration of the pollutant is opposite to the effect of Lu0.
(ii) The effect of Lγ0 is opposite to the effect ofu C0 0 .
(iii) L (Length of the river ) has the same effect asγ0 .
(iv) u0has the same effect as C0. 4-Conclusion and Discussions:
The dimensionless concentration values are evaluated from the analytical solution discussed in section
(3) in a finite domain 0x 1 (i. e L = 1 km is chosen) , m = 0.1(year)-1
-3 2
C0 = 1(kg km ), u0 = 0.11(km / year), D0 = 0.21(km / year).
2
R where R2
= 0 , 3 ,6 ,9 ; t = 0.4 (years) and R1 = 1.9091 from the figure it is clear that:
The effect of R2 is small near the upstream x = 0.and the effect of R2 onC* is dominant at downstream x = 1 , the value of C*corresponding to R2 = 9 ten times more than the value of C* corresponding to R2 = 0,near the downstream x = 1.As expected at any position of the river x = constant as R2 increases (either
Lγ0increases or u C0 0decreases) C*increases .For any constant value of R2 as x increases *
C decreases due to the dispersion of the pollutant in the medium .
Figure(2) ,shows the variation of C*with R1 whereR1 = ( 1 , 2 , 2.5 , 3), R2 = 1.9091 and t = 0.4 (years) , from the figure it is clear that :
The effect of R1 is big near the downstream . As expected at any position of the river x = constant as R1
increases (eitherD0increases orLu0 decreases) C*decreases .
Figure(3) ,shows the variation of C*with time t where the input values are : R1 = 1.9091 , R2 = 3 and t = 0.1 ,0.4 , 0.7 and 1. From the figure it is clear that :
As the time t increases C*increases at any cross-section x = constant .Near the downstream x = 1 , the increase of C*with the time t is dominant. This is due to the small value ofu0 and the accumulation of the pollutant with the time t. In the special case when R2= 0, our results coincide with the results obtained by [Kumar et al. (2009)].
Figures (4) ,(5) and (6) represent temporal dependent of the dimensionless pollutant concentration *
C on the added pollutant concentration R2 ,dispersion R1 and time t along a uniform flow having input of increasing nature through a homogeneous medium , described by equation (22) . Figure (4) ,shows the
variation of C*with R2 where the input values are R1 = 1 .9091 , t = 0.4 (years)and R2 = 1 ,3 ,6 ,9 . From the figure it is clear that :as R2 increases the value of C*at x = 0 , increases this is due to the term
* C -R1 *
X
at equation (21) . As R2 increases at any section x = constant ,we notice that *
C increases. For any
fixed value of R2 as x increases ,we notice that C* decreases and reaches its minimum value at x =
x
0 ,say .Figure (5) ,represents the variation ofC* with R1 where R1 = (1.9 , 2.4 ,2.9 ,3.4 ) ,t = 0.4(years) and
2
R = 0.36 .From the figure it is clear that. Near the origin x = 0 , the value of * C
* X
is negative. Hence from
equation (21) it is clear that both the terms
* C -R1 *
X
and *
C are positive, consequently C*must be less than 1.
Also as R1 decreases, the value of C*near x = 0 must increase. Near the upstream 0.3 x 1 , an opposite
effect is noticed. This is due to the fact that the effect of the term
* C -R1 *
X
is small and as R1 increases ,
u
0decreases and consequently C* must increase at this region .
As the time t increasesC* increases at x = 0.The increase of C*with time t is dominant along the river .This is
due to the effect of the term
* C -R1 *
X
in equation (21), and the accumulation of the pollutant with time t. for the special case when R2 = 0 ,our results coincide with the results obtained by [Kumar et al. (2009)].
0
R2 0 , 3 , 6 , 9
0.2 0.4 0.6 0.8 1.0
0.2 0.4 0.6 0.8 1.0
x
* C
2 1
*
Figure(1) : Shows C for diffrerent values of R where R and t are constant, along
uniform flow of uniform input described by solution (equation 18).
R1 3 , 2.5 , 2 , 1
0 0.2 0.4 0.6 0.8 1.0
0.2 0.4 0.6 0.8 1.0
x
* C
1 2
*
Figure(2) : Shows C for diffrerent values of R where R and t are constant, along
R2 1 ,3 ,6 ,9
0
0.2 0.4 0.6 0.8 1.0
0.1 0.2 0.3 0.4 0.5
x
*C
2 1
*
Figure(4) : Shows C for diffrerent values of R where R and t are constant, along
uniform flow of input increasing nature described by solution (equation 22).
t 0.1 , 0.4 , 0.7 , 1
0
0.2 0.4 0.6 0.8 1.0
0.2 0.4 0.6 0.8 1.0
x
*C
2 1
*
Figure(3) : Shows C for diffrerent values of t where R and R are constant, along
0
R1 3.4 , 2.9 , 2.4 , 1.9
0.2 0.4 0.6 0.8 1.0
0.05 0.10 0.15
x
*C
1 2
*
Figure(5) : Shows C for diffrerent values of R where R and t are constant, along
uniform flow of input increasing nature described by solution (equation 22).
t 0.1 , 0.4 , 0.7 , 1
0
0.2 0.4 0.6 0.8 1.0
0.05 0.10 0.15 0.20 0.25 0.30 0.35
x
*C
2 1
*
Figure(6) : Shows C for diffrerent values of t where R and R are constant, along
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