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International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol. 3, Issue 3, Aug 2013, 23-36 Š TJPRC Pvt. Ltd.

NUMERICAL SIMULATION OF ONE -DIMENSIONAL SOLUTE TRANSPORT EQUATION IN AN ADSORBING MEDIUM BY USING DIFFERENTIAL QUADRATURE METHOD SONA RAJ & VIKAS H. PRADHAN Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat, India

ABSTRACT In the present paper, one dimensional solute transport equation has been numerically discussed in an adsorbing medium by using Differential Quadrature Method (DQM) with minimal computational effort. Recently the solute transport equation has been numerically discussed without adsorption. Since the importance of adsorption plays a major role in the transport process, the effects of various equilibrium adsorption isotherms are considered. Linear, Freundlich and Langmuir adsorptions are considered and the numerical results are compared with the analytical solution for linear adsorption. It is also observed that the effect of adsorption is more in Langmuir adsorption.

KEYWORDS: Polynomial Based Differential Quadrature Method, Shu’s Approach, Solute Transport, Dispersion, Adsorption, Porous Media

INTRODUCTION The theoretical and experimental study of the movement of reactive solutes through an absorbing medium has wide role in science and technical fields. Movement of reactive solutes in soil is controlled by the processes such as convection by moving water, hydrodynamic dispersion, and adsorption or exchange of the solutes by the soil matrix. The mathematical simulation of the transport of a reactive solute through soil requires solution of differential equation describing convective-dispersive transport and the equation describing the interaction between the solute and the soil matrix. In chemical engineering, the theory of solute transport has been used successfully to develop chromatography into a powerful tool for chemical separation and analysis. The movement of chemicals through the soil is an important area in the study of soil fertility in agriculture, especially in pest control, salinity control, irrigation, drainage etc. The solute transport has considerable attention in various disciplines such as petroleum engineering, sanitary engineering nuclear waste management groundwater hydrology, soil physics and environmental monitoring. The rate at which a chemical constituent moves through soil is determined by several transport mechanisms such as convection, advection, diffusion, dispersion, adsorption, production and decay. Among these adsorption plays an important role in transport of solutes through any porous medium. The phenomenon in which the mass of a solute accumulates on the solid surface at the solidliquid interface is known as Adsorption. In another words, it is the mass transfer process between the contaminants dissolved in groundwater i.e. aqueous phase and the contaminants adsorbed on the porous medium i.e. solid phase. When the solutes dissolved in the soil matrix, the transport of the solute results in slower relative to water velocity and it results the reduction in dissolved concentration, in other words retardation occurs. It is a reversible reaction i.e. at a given solute concentration, some portion of the contaminant is partitioning out of the solution onto the soil matrix, and some portion is desorbing and re-entering the solution. When the concentration of solute changes, then the relative amount of solute which is adsorbing and desorbing


24

Sona Raj & Vikas H. Pradhan

also will change. Ogata and Banks (1961) and Bear (1972) had presented a theoretical description about adsorption in transport process. Adsorption does not permanently remove solute mass from soil water; it merely retards migration (Wiedemeier et al. 1999). Recently Sona Raj and Vikas H. Pradhan (2013) had given a numerical study on onedimensional solute transport equation without considering adsorption.

MATHEMATICAL DESCRIPTION Mathematical formulation for one-dimensional solute transport subject to adsorption in soil column has been discussed extensively in the literature of van Genuchten and Alves (1982), Bolt (1979), Bosmaand van der Zee (1992). In the present paper linear adsorptive transport of solute through a homogeneous soil column is considered. The governing equation for transport of solute in soil column is given by [17],

C  q 2C C   D 2 v t  t x x

(1)

equation (1) represents one dimensional solute transport equation in a porous medium during saturated flow with adsorption, where C is solute concentration ( gm velocity q is the Darcy velocity ( cm coefficient ( cm

2

cm 3 ), t is time (s), x is the soil depth (cm), v 

q

is the mean flow

s 1 ) and  is the volumetric water content ( cm3cm3 ), D is the dispersion

s 1 ), which is the sum of the effective diffusion and mechanical dispersion[18], q is the adsorbed

concentration ( gm

gm 1 ),  is the bulk density ( gm cm 3 ) .

Sorption is the physical or chemical process in which one substance becomes attached to another, is estimated experimentally by measuring the solute concentration sorbed on a particular sediment, soil, or rock type. Adsorption [1, 7, 8, 9, 12] is a specific case of sorption, which is the physical adherence of molecules onto the surface of another phase. This process creates a film of the adsorbate on the surface of the adsorbent. In this work, solute is the adsorbate, the material being adsorbed and the solid material is the adsorbent, the material being used as the adsorbing phase. If the process of sorption is much faster than the fluid velocity, then the solute may be considered locally in a condition of constant equilibrium with the sorbed phase [19]. Moreover equilibrium sorption isotherms are depend on several factors, such as surface charge of the sorbing phase, ionic strength, competing counter-ions and their concentrations, the concentration of the sorbed phase, and solution pH. Adsorption is usually described through isotherms, which express the functional relationship between quantity of an adsorbed component and its quantity in the adsorbent, at a constant-temperature adsorption process. There are three types of equilibrium adsorption isotherms Linear Isotherm The simplest expression of equilibrium adsorption is the linear isotherm which gives a linear relationship between the amount of the solute adsorbed by the soil matrix and the concentration of the solute in the soil solution is related as [18]

q  kC where k

(2) is the linear adsorption constant (cm

3

gm 1 ) , also known as partition coefficient or distribution

coefficient. Generally linear adsorption is discussed for low concentrations [18]. Linear adsorption constant k can be


Numerical Simulation of One -Dimensional Solute Transport Equation in an Adsorbing Medium by using Differential Quadrature Method

25

determined from the ratio,

k

M [ soil ] M [ solution ]

M [ soil ] = the amount of solute adsorbed by the soil matrix and q is the mass of solute adsorbed per unit mass of adsorbate.

M [ solution ] = concentration of solute in soil solution. Freundlich Isotherm The relationship between the magnitude of adsorption and concentration of solute can be expressed mathematically by an empirical equation known as Freundlich adsorption isotherm is expressed as [18 ],

q  kF C n where

(3)

k F is the Freundlich adsorption constant, n is chemical specific coefficient. In the special case that n=1,

Freundlich isotherm is the same as the linear isotherm. The number of adsorption sites is large relative to the number of solute molecules then also Freundlich isotherm is applicable. The equation (3) is valid only up to a certain pressure and invalid at higher pressure. The constants

k F and n are not temperature independent, they vary with temperature. When the

concentration of the adsorbate is very high Freundlich adsorption isotherm fails. Freundlich isotherms had no upper bound accounted for the adsorbed concentration; their use is restricted within the concentration limits of experimental data. Langmuir Isotherm Mathematical derivation of Langmuir absorption isotherm was presented by Langmuir for gases were modified to apply to the adsorption of reactive solutes by soil [18]. Langmuir isotherm was developed to limit the adsorbed concentration to the maximum amount of solute that can be adsorbed onto the solid phase. The adsorbed concentration approaches a constant value because there are limited numbers of adsorption sites in the soil matrix. The Langmuir nonlinear equilibrium isotherm is described mathematically with equation as:

q

QmQC 1  QC

where

(4)

Qm is the maximum adsorbed capacity on solid surface ( kgkg 1 ), Q is the Langmuir adsorption constant

(m3 kg 1 ) . If the value of QC  1 , then equation (4) reduces to linear form. When

and

 are constants in time and space and the flow of water is steady in a homogeneous soil profile

equation (1) is reduces to

R

C  2C C  D 2 v t x x

(5)

where R is a dimensionless quantity known as the retardation factor or coefficient of retardation. When there is no interaction between the solute and soil, the adsorption

constant will be zero and R reduces to one. When the solute


26

Sona Raj & Vikas H. Pradhan

transports through any immobile liquid region the value of R will be less than one, it means that only a fraction of liquid phase is participates in the transport process. For adsorbing solute, with a constant adsorption constant ‘k’ and the bulk density of solid ‘  ’ scales the solute movement. Thus, the transport velocity is R times lower and arrival time R times longer compared to a non-adsorbed solute [17]. When the adsorption is linear, Freundlich and Langmuir equilibrium isotherms R is expressed respectively as [4, 18],

R  1

 kL 

(6)

R  1

 nkF C n 1 

(7)

R  1

 QmQ  (1  QC ) 2

(8)

The initial and boundary conditions for the semi finite system containing solute are expressed as follows. In the present paper, to solve the transport equation (5) the initial conditions is specified as [14],

C ( x, 0)  C0

;

x0

(9)

Equation (2) describes that initially the solute concentration in the soil at the time of inflow the solute begins with zero concentration i.e. the solute concentration in the soil at the time when solute transport begins. The boundary condition at x=0 describes the solute concentration at the inflow end in a uniform flow system, where a well mixed solute enters the system by advection across the boundary is transported away from the boundary by advection and dispersion. Here the boundary condition used at the inflow boundary is [14], C (0, t) = C f

;

t 0

(10)

where C f is the known measured concentration of the solute in the influent water. The use of simpler boundary conditions indicates that the concentration gradient across the boundary equal zero as flow begins, which helps for the estimation of mass of solute in the system at early times. The boundary at x=L describes the solute concentration at the outflow end in a uniform flow system. The out flow boundary of the system being simulated is far enough away from the solute source that the boundary will not affect the concentration of the solute within the domain. This type of medium is described as a semi-finite medium. A flux type homogeneous boundary condition is specified at the far end of semi finite soil column. This boundary condition is used for the displacement process, which represents zero concentration gradient at the lower end of soil column, is expressed as [14],

C  L, t   0 x

:t

0

(11)

where L is the column length. Equation (11), which leads to a continuous distribution at x=L has been discussed by various authors (Wehner and Wilhelm 1956, Pearson 1959, Van Genuchten and Wieranga 1974, Bear 1979). If the system discharges to a large well mixed reservoir region and the influence of additional solute is not significantly alter reservoir concentrations, and then equation (10) is used in the inflow boundary. If the reservoir is small or not well mixed,


Numerical Simulation of One -Dimensional Solute Transport Equation in an Adsorbing Medium by using Differential Quadrature Method

27

the concentration of solute would equal to the solute concentration at discharge end of the system. Thus there is no concentration gradient across the boundary; this is specified in equation (11). The analysis made by Van Genuchten and Alves[12] in 1982 about the difference between predicted concentrations obtained using analytical solutions for finite system having flux type boundary condition in terms of two dimensionless numbers known as the column Peclet number (P) and the number of displaced pore volumes (T) is given as[17],

P

vL D

(12)

T

vt L

(13)

where D is the dispersion coefficient, v is the mean flow velocity and L is the maximum length of the domain. P is the ratio of advective transport rate to the dispersive transport rate. They were experimentally proved that when the value of T is greater than 0.25, the predicted concentration at points near out flow boundary begins to differ significantly and when T approaches to 1 the difference will increase. Also when the value of P increases the solutions for, the magnitude of difference and distance inward from the outflow boundary, will less diverge[11,17].

DIFFERENTIAL QUADRATURE METHOD The Differential Quadrature Method (DQM) is a powerful numerical method for solving the linear and nonlinear partial differential equations. DQM requires less computer time and storage as compared with standard numerical methods such as finite element and finite difference methods. In this method a partial derivative of a function is approximated by a weighted linear sum of the function values at given discrete points. The weighting coefficients depend only on the grid spacing. Using these weighting coefficients any partial differential equation can be easily reduced to a set of algebraic equations. Because of their high accuracy, generality in a variety of problems and straight forward implementation Differential quadrature (DQ) methods have been distinguished. The basic procedure in the DQM is the determination of weighting coefficients. Richard Bellman 1972 introduced the DQM. Bellman used two procedures to obtain the weighting coefficients. In first procedure, he used a simple function as test functions but when the sampling points are relatively large the coefficient matrix become ill conditioned [2]. Second procedure has similarity with the first except coordinates of grid points, which should be chosen as the roots of the Nth order Legendre polynomial. C. Shu had given a remarkable contribution to generalize the idea of the DQ. The differential quadrature method is a numerical technique for solving differential equations where we approximate the derivatives a function at any location by a linear summation of all the functional values at a finite number of grid points; then the equation can be transformed into a set of ordinary differential equations. In the present paper we used Polynomial based Differential Quadrature (PDQ) Method [5] for solving equation (5). This method has the basis of the quadrature method in deriving the derivatives of a function and it follows that the partial derivative of a function with respect to a space variable can be approximated by a weighted linear combination of the function values at some intermediate points in that variable. The selection of locations of the sampling points plays an important role in the accuracy of the solution of the differential equations. Shu suggested that the solution of a partial differential equation can be accurately approximated by a polynomial of high degree. Suppose that the degree of the approximated polynomial is N-1. This approximated polynomial constitutes an N-dimensional linear vector space VN with


28

Sona Raj & Vikas H. Pradhan

the operation of vector addition and scalar multiplication , and can be expressed in different forms and N is the total number of grid points[5]. Considering a one-dimensional problem over a closed interval [0, L]. If N is total number of grid points in the interval with coordinates as,

0  x1 , x2 , x3 ,....xn  L . For the accuracy, choice of sampling points plays a vital role. In

the present problem we choosed uniform grid, that means the grid has same step sizes [3,5]. i.e.

x  xk  xk 1 ; where k = 1,2,3,..........,11

(14)

Bellman et al. (1972) assumed that a function C(x) is sufficiently smmoth over the interval [0, L], then the first (1)

order derivative C ( xi ) at any grid point can be approximated as

Cx(1)  xi    aij . C  x j  , for i  1, 2,....N ; N

j 1

(15)

By using Bellman’s first and second approaches, Shu generalised the weighting coefficients as follows. The off- diagonal terms of the weighting coefficient matrix of the first order derivative are given by[5]

aij 

M 1  xi 

x  x M   x  1

i

j

for i  j;

i, j  1, 2,3....N

(16)

j

where M 1  xi   M 1  x j  

N

 x

 xk  ;

i

j 1, j i

 x N

j 1, j i

j

i, k  1, 2, 3.....N

 xk  ; i, k  1, 2, 3....N

(17)

(18)

and diagonal terms are given by N

aii   aij ; I  1, 2,3,...., N

(19)

j 1

Equations (10) & (13) are two formulations to compute the weighting coefficients for first order derivative. For the discretization of the second order derivative, we can introudce a similar approximation form given by[3] C x  2   xi  

Where

Cx 

2

N

b j 1

ij

.C  x j  , for i=1,2,....N

 xi  is the

(20)

second order derivative of f(x) at xi , bij is the weighting coefficient of the second order

derivative , is expressed as[3],   1 bi j  2aij  aii   , for i  j xi  x j    

bi i  

(21)

N

j 1, j 1

bi j

(22)


29

Numerical Simulation of One -Dimensional Solute Transport Equation in an Adsorbing Medium by using Differential Quadrature Method

bij is calculated from Equation (15) when i  j and bii from Equation (16). In general, for the discretization of mth order derivative [3]

Cx  m   xi    wij  m  . C  x j  , for i, j  1, 2,....N ; m  2,3,...., N  1 N

(23)

j 1

the weighting coefficient matrix of higher order derivative may obtained through the recurrence relationship expressed as [5,6],

wij

(m)

 wij ( m1)  ( m 1)  m  aij wii   ; for i, j  1, 2,....N ; m  2,3,...., N  1  x  x i j  

(24)

and the diagonal terms of the weighting coefficients are,

wii ( m )  

N

j 1,i  j

wij ( m ) ; for i, j  1, 2,....N ; m  2,3,...., N  1 (25)

NUMERICAL SOLUTIONS Applying the Differential Quadrature discretization to the governing equation (5),

R

dC C  R i ; i  1, 2,3,....., N t dt

where

D

v

(26)

Ci  C ( xi ) , i.e. the concentration of solute at xi

N  2C  D bij C j ; i  1, 2,3,...., N  x 2 j 1

(27)

N C  v  aij C j ; j  1, 2,3,......, N x j 1

where C j  C ( x j ) , i.e. the concentration of solute at

(28)

xi

After applying Differential Quadrature Method equation (1) becomes system of ordinary differential equations of the form

R

N N dCi  D  bij C j v  aij C j dt j 1 j 1

, for i  1, 2,....N

(29)

Applying the Differential Quadrature discretization to the initial condition expressed in equation (10) as,

C ( xi , 0)  C0

;

0  xi  L

(30)

Now apply the Differential Quadrature discretization to the boundary condition expressed in equation (11) and (12). The implementation of Dirichlet boundary condition expressed in equation (11) is straight forward for numerical computation, the numerical condition at input boundary is written as


30

Sona Raj & Vikas H. Pradhan

C (0, t ) = C f

;

t 0

(31)

The Neumann boundary condition at the out flow boundary given in Equation (12) can be discretizes as N

 a C =0 j 1

ij

j

for i  1, 2,....N

(32)

The function value at end boundary can be expressed by the unknown interior point function values, namely N 1

CN  where

 aNj C j j 2

(33)

aNN C N  C ( L, t )

The system of ordinary differential equations obtained by PDQ method in equation (30) is solved by using fourth order Runga- Kutta Method [15] for the initial condition and boundary conditions expressed in equations (31), (32) & (33). All programming solutions are obtained by some codes generated in MATLAB.

RESULTS AND DISCUSSIONS Analytical solution for the one-dimensional solute transport equation having adsorption given in equation (5) in a semi finite soil column having initial and boundary conditions expressed in (10)-(12) was given by Ogata and Banks (1961) as [6],

  Cf  Rx  vt C ( x, t )  erfc  1   2  2  DRt  2  

    exp  vx  erfc  Rx  vt   1   D   2  DRt  2  

    

(34)

where erfc is the complementary error function. In order to verify the accuracy of numerical solutions it is necessary to compare the solutions with existing analytical solutions. For the validity of the obtained numerical solution we have to compare the results obtained by applying DQM with the existing analytical solution given in equation (34). The comparative results of DQM with analytical solution are given in Table I, II, III, IV & V for time t=5sec,10sec, 15 sec, 20 sec, 25 sec and 50 sec respectively. The graphical comparison is shown in Figure.1.

Figure 1: Comparison of DQM and Analytical Solutions for at Different Time Relative Distance (x/L) Relative Concentration (C/ C f )


31

Numerical Simulation of One -Dimensional Solute Transport Equation in an Adsorbing Medium by using Differential Quadrature Method

Figure 2: The Effect of Change of Relative Concentration (C/ C f ) with Respect Time at Different Points of the Depth of Soil Table 1: The Comparative Results of Concentration C(x, t) at Different Relative Distance(x/L) for t Time t=5 sec Relative Distance(x/L) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exact Values at t=5 sec 0.8359 0.6303 0.4225 0.2490 0.1280 0.0570 0.0219 0.0072 0.0021 0.0004

DQM Values at t=5 sec 0.8362 0.6296 0.4223 0.2493 0.1282 0.0568 0.0202 0.0028 0.0023 0.0121

Table 2: The Comparative Results of Concentration C(x, t) at Different Relative Distance(x/L) fort Time t=10 sec Relative Distance(x/L) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exact Values at t=10 sec 0.9237 0.8182 0.6900 0.5509 0.4142 0.2922 0.1927 0.1186 0.0679 0.0362

DQM Values at t=10 sec 0.9238 0.8182 0.6901 0.5508 0.4141 0.2922 0.1934 0.1207 0.0722 0.0412

Table 3: The Comparative Results of Concentration C(x, t) at Different Relative Distance (x/L) Fort Time t=15 sec Relative Distance(x/L) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exact Values at t=15 sec 0.9566 0.8943 0.8136 0.7174 0.6109 0.5010 0.3946 0.2979 0.2151 0.1484

DQM Values at t=15 sec 0.9566 0.8943 0.8136 0.7174 0.6111 0.5013 0.3951 0.2982 0.2145 0.1454


32

Sona Raj & Vikas H. Pradhan

Table 4: The Comparative Results of Concentration C(x, t) at Different Relative Distance (x/L) for Time t=20 sec Relative Distance(x/L) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exact Values at t=20 sec 0.9729 0.9333 0.8802 0.8138 0.7356 0.6486 0.5566 0.464 0.3753 0.2939

DQM Values at t=20 sec 0.9729 0.9333 0.8802 0.8138 0.7357 0.6486 0.5565 0.4636 0.3742 0.2922

Table 5: The Comparative Results of Concentration C(x, t) at Different Relative Distance (x/L) Fort Time t=25 sec Relative Distance(x/L) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exact Values at t=25 sec 0.9822 0.9558 0.9197 0.8731 0.8162 0.7498 0.6758 0.5966 0.5151 0.4344

DQM Values at t=25 sec 0.9822 0.9558 0.9197 0.8731 0.8162 0.7497 0.6756 0.5963 0.5147 0.4342

Table 6: The Comparative Results of Concentration C(x, t) at Different Relative Distance (x/L) Fort Time t=50 sec Relative Distance(x/L) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Exact Values at t=50 sec 0.9968 0.9919 0.9849 0.9753 0.9626 0.9463 0.9259 0.9011 0.8717 0.8375

DQM Values at t=50 sec 0.9968 0.9919 0.9849 0.9753 0.9626 0.9463 0.9259 0.9012 0.8718 0.8375

When adsorption happens the concentration of solute will be less throughout the depth of soil as comparing to the situation when there is only dispersion [13, 14]. Because when adsorption happens, a mass of solute will get adsorbed by the pore matrix. The concentration

C(x, t) values are evaluated by DQM are for relative distance (x/L) where L is chosen

as 20cm [17]. The total number of nodes N, used for obtaining all results is 11. The initial concentration

C0 is zero. The

relative concentration (C/ C f ) is obtained for different time in seconds where C f is 1gcm-3 [17]. The dispersion coefficient ‘D’ is 4cm2 s-1 and pore velocity ‘v’ is 1cms-1 [17] and R=1.65. Figure 1 represents the comparative results of DQM with analytical solution for different time and in Figure 2, the change of relative concentration with respect to time for different relative distances is presented.


33

Numerical Simulation of One -Dimensional Solute Transport Equation in an Adsorbing Medium by using Differential Quadrature Method

Figure 3: The Linear, Freundlich and Langmuir Adsorption Isotherms at t=4 sec In Figure 3, the Linear adsorption concentration, Freundlich adsorption concentration and Langmuir adsorption concentrations are obtained from the equations (2) , (3) and (4), where k = 2, k F = 2, Q = 2, n =0.7,

Qm =3. The graphical

representation of all three equilibrium adsorption isotherms shows that Langmuir adsorption concentration is more than Freundlich and Linear adsorptions. All these parametric values are chosen from the experimental data [16].

Figure 4: Transport with Linear Adsorption Isotherms. The Effect of various Linear Adsorption Constant at t=25sec From Figure 4, it is clear that when linear adsorption constant is increasing the concentration of the solute will decrease as depth of soil column increases.

Figure 5: Transport with Freundlich Adsorption Isotherms. The Effect of various Chemical Specific Coefficient at t=25sec As the value of increases concentration of solute will increase as depth of soil increases, it is representing in Figure 5. Thus the Freundlich adsorption isotherm having greater chemical specific coefficient will have more solute concentration in each points as depth increases.


34

Sona Raj & Vikas H. Pradhan

Figure 6: Transport with Langmuir Adsorption Isotherms. The Effect of various Langmuir Adsorption Constants at t=25sec When the Langmuir adsorption constant

Q is increases the concentration of solute will decreases as soil depth

increases, thus from the graphical representation it is clear that for low values of

Q concentration of solute will be more.

CONCLUSIONS The comparative results of obtained Differential Quadrature solutions with the analytical solutions are found in better accuracy for very less number of nodes. The method is quite easy for implementation than the conventional numerical methods. In this work a combination of the PDQ method in space and fourth order Runga-Kutta scheme in time were used for obtaining the numerical results. The study on the effects of different adsorption isotherms clarifies equilibrium adsorption phenomenon in the solute transport process.

ACKNOWLEDGEMENTS One of the authors (Sona Raj) acknowledges the financial assistance provided by Ministry of Human Resource and Development (MHRD), Government of India, India.

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Bear .J and Verruijt A. (1990) Modeling Groundwater Flow and Pollution (Theory and Applications of Transport in Porous Media), D. Reidel Pub.Co. Dordrecht, Holland.

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Bellman R.E., B.G. Kashef, and J. Casti, (1972). Differential Quadrature: A technique for the rapid solution of nonlinear partial differential equations, J Computational physics, 10, 40-52.

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Bellomo N. (1997). Nonlinear Models and Problems in Applied Sciences from Differential Quadrature to Generalized Collocation Methods, Mathematical Computational. Modelling, 26 (4), 13-34.

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Bolt, G. H. (1979), Soil Chemistry B. Physico - Chemical Models, Elsevier, New York.

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Chang Shu (2000). Differential Quadrature and its Applications in Engineering. Springer, Great Britain, 1-174.

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Charles W. Bert and Moinuddin Malik (1996), Differential Quadrature Method in computational mechanics: A review, J Applied Mechanics Reviews, 49 (1), 1.

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Cussier E.L. (2007). Diffusion- Mass Transfer in fluid systems, Cambridge University, 3,

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Numerical Simulation of One -Dimensional Solute Transport Equation in an Adsorbing Medium by using Differential Quadrature Method

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