electrokinetic remediation of contaminated soil

Nonlinear Analysis, Themy, Methods & Applications, Vol. 30, No. 6, pp. 3391-339X. 1997 Proc. 2nd World Congress of Nonlinear An&sts

0 1997 Elsevier Science Ltd

PII: SO362-546X(!TV)OOO21-7

Printed in Great Britain. All rights reserved 0362-546X/97 $17.00 + 0.00

ELECTROKINETIC REMEDIATION OF CONTAMINATED SOIL

BIYUE LIUt*and ROGER LUTS t Department of Mathematics, University of Wyoming, Laramie, Wyoming

$ Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, hLIA 01609

1. INTRODUCTION

Many phenomena in chemistry, biology, chemical and electrical engineering involve the electro-diffusion of ions and their interactions during the transport process. The fundamental principles governing these phenomena are often the same, and the equations derived from these principles are well accepted and important. See for example the books by Newman [lo], Babskii et.al. [4], and Rubinstein [la]. A recent application of the above principle is to model the removal of toxic ions from contaminated soil by an electric field. The method in many cases works better than the traditional method of flushing. Testing of this method have been successfully carried out in the laboratories [8] and [2], and large scale pilot test has also been conducted in the Netherlands [9].

A very detailed and complete model has recently been proposed by Acar in [3] although similar models have been proposed earlier by other people [B], [ll]. A car et. al. considered multicomponent species transport under coupled hydraulic, electric, and chemical potential differences. In this paper, we ignore hydraulic pressure and assume that fluid flow is slow. Then the principal mechanisms for the transport process are diffusion, electro-migration, and electro-osmosis. The total mass flux N, of species i is given by the sum of the above three effects. In general, diffusion is modeled by -d,Vu, where u, is the molar concentration of species i and d, is the diffusion coefficient. Electra-migration is modeled by -Riziu,V$ where 4 is the electric potential, zi is the charge, and R, is the ionic mobility. It is customary to assume that ionic mobility is proportional to the diffusion coefficient (Nernst-Einstein relation), that is, 0, = pd, where /I depends on the Faraday’s constant, the universal gas constant, and the absolute temperature. Electra-osmosis is due to the drag interaction between the bulk of the liquid in the pore and a thin layer of charged fluid next to the pore wall [13]. In the paper [3], it is modeled by --IceuiV~ where Ic, is the coefficient of electro-osmotic permeability. I f the soil pH is low and the contaminants are charged, then very little electro-osmosis transport occurs [l].

2.THEMODEL

In this paper, we shall consider the above model in two-space dimensions with no electro-osmosis force or convection. However, as pointed out in [3], in applying mass balance, one should take into account consumption and production (&) of the species. Doing so, the transport equation is

&i/at = -div Ni + &, i = 1, . . ..m. (2.l)

where Ni = -&Vui - pzizliVui and m denotes the number of species in the system. The electric

*Present address: Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worces- ter, MA 01609

3392 Second World Congress of Nonlinear Analysts

potential 4 is not known in (2.1). It is assumed to satisfy the electrostatic Poisson’s equation (Gauss Law):

EAT = -,&t~i i=l

(2.2)

where e > 0 is the molar charge and E > 0 is the permittivity of the solvent. This equation is sometimes replaced by the electro-neutrality condition

$ -Wi = 0. (2.3)

Equations (2.1) and (2.2) under zero-flux boundary condition have been studied by Choi and Lui in [6] and [7].

We now briefly explain the term R, in equation (2.1). A more detailed explanation is given in [5]. For simplicity, we shall assume that the toxic ion is cadmium (Cd++) which reacts with some oxide (SOH) in the soil according to the reaction:

SOH + Cd2+ + SOCd+ + H+ . (2.4)

In the above reaction, SOCd+ represents precipitated complexes on the mineral surface. Assuming mass action kinetics, the reaction rate is r-1 = kl[SOH][Cd++] - k2[SOCd+][H+] where kl, k2 > 0 and [B] stands for the molar concentration of B. There is also dissociation-association of water given by

H20 +H++OH-. (2.5)

Assuming mass action kinetic, the reaction rate r, = Kf[H20] - K,[H+] [OH-] where Kf and K,

are positive constants.

Let ~1 be the concentration of water, IQ be the concentration of hydrogen ion, ~3 be the concentration of hydroxyl ion, ~4 be the concentration of cadmium ion, and 0s be the concentration of some other inert ion, say chloride. These are the only ions in the aqueous phase. Let us be the concentration of SOH, and u7 be the concentration of SOCd+. These are the only solid phase ions, which do not diffuse or migrate. Assuming the above, the governing equations for our model are

au,/at au2/at au,/at &Q/at adat au,/at au7/at 7

. i=l

= -divN1 - T,,, = -divN:! +rl tr, T.r -div N3 + T, = -divNd - r1 = -div N5

- -1 l-1

= 0

(2.6)

where 21 = 0: 22 = 1, za = -1, z4 = 2, zs = -1, zs = 0, and ~7 = 1. Including 4, there are eight

equations and eight unknowns.

3. REFINED MODEL

Let c > 0 be a small parameter, say, the ratio of the total initial concentration of cadmium ions to

Second World Congress of Nonlinear Analysts 3393

that of the water molecules. We assume that water is in excess so that its expansion in terms of E is u1 = K~/E + O(1). We also assume that ‘1~; = tiZ + O(E) and the reaction rate r, = Kf ~1 - K, u2u3 is very rapid so that equilibrium is attained instantaneously. Under these assumptions, we obtained the following refined model for the functions iii (tildes have been dropped) [5]:

u2u3 = Kw qua - us)/at = -divN2+divN3+rl

du4/at = -divN4 - r1 &,/at = -divNg au7/at = r1

(3.1)

, ‘U2 - U3 + %I - 215 + ‘IL7 = 0

From (2.6), us(~, t) f 217(x, t) = g(3c) where g is determined by the initial data. We now turn to the boundary conditions.

4. THE BOUNDARY CONDITIONS

We assume that the contaminated region R is two-dimensional. I f there is a circulation system, then there is an annulus surrounding the electrodes as shown in Figure 1. Inside the annulus is the circulating fluid and I’l,rz correspond to the interface between the soil and the annulus. If there is no circulation system, then the annulus should be removed in Figure 1 and rl, rz correspond to the surface of the cathode and anode, respectively.

No boundary condition is needed for the solid ions ~6, ‘117. At the outer boundary of 0, all aqueous ions satisfy zero-flux boundary conditions. For the boundary conditions at rl and r2, if there is no circulation system, then at the surface of the anode, all aqueous ions except hydrogen and hydroxyl ions satisfy zero-flux boundary conditions. At the surface of the cathode, the same is true except for cadmium ions. When the cadmium ions arrive at the cathode, we assume that they are removed by electroplating so that u4 = 0 for z E l?l. Boundary conditions for the hydrogen and hydroxyl ions near the electrodes are governed by electrolysis. Such reactions are modeled by the Butler-Volmer kinetics [lo]. The precise formulation is given by (4.la) and (4.lb) below.

If there is a circulation system, then cadmium ions are washed away by the circulation system and hence zl4 = 0 at the interface. Suppose the circulriting fluid contains hydrochloric acid which is completely ionized. Let the concentrations of the hydrogen, hydroxyl and chloride ions in the circulating fluid be denoted by ug,u$, anduE, respectively. The hydrogen, hydroxyl and chloride ions satisfy Dirichlet type boundary conditions. Finally, we assume that the circulation system is conductive so that 4 is known at the interface. We now summarize the boundary conditions. Let El and E2 be the applied voltage on the electrodes at l?l and r2, respectively and let Ez > El.

Case 1. No Circulation System:

I

P2 - Nd . n = %l(#b, t) - Ed at rl (N:, - N3) t n = -4dEz - 46 t)) at rz

N4. n = 0 at l?2, u4 = 0 at rl (4.1) N5-n = 0 at rl, r2

(N2 - N3) . n = 0, N4. n = 0, N5. n = 0 at l?3

where a(4) = ev(m4) - ev--LY24)r cyl, (~2 > 0 and 92 is a similar function with different (~1, (~2.


Case 2. With Circulation System containing HCE:

u2 = 218, u3 = I& uq =,o, us = us at I’i and I’2

4 = El at II, 4= E2 at I2

(Nz-Nf).n = 0, Nd.n=O, Ns.n=O at r3

h Cathode Anode

(4.2)

Figure 1. Cross section of contaminated site R with a circulation system.

5. STEADY-STATE SOLUTIONS

It was shown in [5] that for the case with no circulation system, the steady-state solutions are given by 214 = 0,217 = 0, $J is a solution of the differential-integral equation

(d2uz + d3u3)A+ + V&u2 + d3u3) ‘04 + (d2Au2 - d3Au3) = 0.

with boundary conditions given by (4.la) and (4.lb) where

(5.1)

N2 - N3 = -V(dzuz - d3u3) - (d2u2 + d3u3)V&

Also, ug = Ce$ where C = sn ug(x, 0) dx/ Jo eb(5) dx, and

(5.2)

i

‘u2 = (Gee + JC2e2+ + 4K,,,)/2

(5.3)

u3 = (-Ce+ + JC2e2# + 4K,)/2

If we substitute these relations into (5.1), then we obtain an integral-differential equation for 4.

For the case with circulation system, one possible steady-state solutions are ‘112 = u;, 2~3 = u$, 214 = 0, ug = UE in R and 4 satisfies the equation Ad = 0 in Q, 04. n = 0 at I’s, 4 = El at Ii, and 0 = E2 at l?2.


In the paper [5], we have computed the solutions of the time-dependent and steady-state equations for the one-dimensional case. For the rest of this paper, the refined model (3.1) is solved numerically in the twedimensional case. For the circulation system with boundary condition (4.2), we obtain numerical solutions for both the time-dependent and steady-state cases. For the no circulation system, we obtain a numerical solution for the steady-state case.

6. NUMERICAL SOLUTIONS

In our numerical algorithm, for time-dependent problems, we use iteration to solve the fully nonlinear system of equations at every time step t, = nAt in the following way:

For iterations Ic = O,...,l, starting from $” = u:-~,$‘@ = 4n-1, the implicit Euler method is used to solve the equation

n,k u7,t = -(hq wk-l + kzu;~k-‘)u;~k + klg(z)Ut’k-’ .

Next, uGTk, ujl’” are obtained by solving the algebraic system:

n,k n,k ‘112 u3 = K,

n,k n>k = u2 -‘113

qk-1 + 2uy,k-’ _ $k

Then u:‘~, uz’” are obtained by solving the following equations with the corresponding boundary conditions:

-&.k - u;-‘) = dsAU;‘k _ dspv$,“>“-l . Vu;yk - &pA@‘,k-lu;‘k .

Finally, we obtain &‘pk by solving

(i pdiU”‘k)A$n’k + (5 pdiVul’k) V4n’k = - & zidiAuFk i=2 i=2

4 with the imposed boundary conditions. After the last iteration, we take ui r4ny’ as the numerical solution $,dn at t,, and move to the next time step.

Numerical solutions of the model with a circulation system is computed with data d2 = 2.0,ds = l.O,dd = 0.5,ds = 1.0,~ = l.O,kl = & = 1.0, Kw = l.O,g(z) = l.O,El = 0.0, and Ez = 10.0 on a 50 x 50 mesh over the unit square with At = 2/502. For the initial conditions, u7 = 0.05, ‘1~4 is shown in Figure 1 a), us shapes like a flat hill (not shown), and the graph of uz is given in Figure 2 a), which is a flat hill with a cave. Figures 1 b)-d) g ive the results of our computations for 214 (cadmium ion) at different time steps t, = 2OA1, lOOAt, 300At. It is clear from the figures that as t increases, the solution converges to the steady-state ‘114 = 0. (Note the change in scale on the z-axis.) Figures 2 b)-d) give the numerical solution u2 (hydrogen ion) at time steps t, = At, 20At, 300At. From the figures, the solution converges rapidly to a constant steady-state solution as time increases. The non- physical wiggle along the boundary I’2 is caused by the numerical solver. Here we used ELLPACK to


solve the linear elliptic equation at each iteration. We used uniform mesh due to the non-flexibility of ELLPACK. We expect that these wiggles will disappear if the mesh is refined around the hole. Figures 3 a) and b) plot the electrical potential 4 at time steps t, - - At, 20At. From the figures, the solution

also converges to the steady-state solution (Figure 3 c) as time increases.

Figures 1 a)-d) Concentration of cadmium ion 214 at t = 0,2OAt, lOOAt, 300At

1.5

1

:i,

1 - 1 0.5 -

0.5 O- 0.5

0 0 0 0.5 1

0 0.5 1

Figures 2 a)-d) C oncentration of hydrogen ion ~2 at t = 0, At, 20At, 300At


For the no circulation system case, 4 is determined by (5.1). Since ~2, u3 can be expressed in terms of 4 as in (5.3), we can rewrite (5.1) in the following way. Let p = Ce@ and q = p2 + 4K,. Then one can show that

V(d2u2 + d3u3) = (dzuz - d3u3) $p#J

d2Au2 - d3Au3 = (dzuz + d3ua) L A$ + lower order terms . A

Thus, 4 satisfies the equation

(dm + dm)(1 + $$A& + (fi + f2)lV42 = 0, (6.1)

where fl = (dzuz - daua)T(l + T), f2 = (dzuz + daug)s, s = T - r3, r = p/&r with the boundary conditions (4.la,b,e). We solve this problem by an iterative method. At every iteration, we solve a linearized elliptic equation (6.1) with ‘1~2, u3 obtained from the previous iteration. The result of the computation is plotted in Figure 3 d). Here we used the data oi = 1.0,i = 1,2,3,4, El = 0.1, E2 = 1.0 in the boundary conditions (4.1) and the definitions of gl, g2.

1 1

0 0.5 0

1 0.5 1

15 -

1 1

0 0.5 1

Figures 3 a)-b) Electrical potential 4 at time At, 20At. Figure 3 c) Steady-state solution of 4 with circulation system, d) Steady-state solution of r#~ with no circulation system.

REFERENCES

1. Y.B. ACAR and A.N. ALSHAWABKEH, Principles of Electrokinetic remediation, Environ. Sci. Tech- nolo., 27(1993), 2639-2647.

2. Y.B. ACAR and A.N. ALSHAWABKEH, Electrokinetic Remediation, I. Pilot-Scale Tests with Lead- Spiked Kaolinite, J. Geotechnical Eng., March (1996), 173-185.

3. Y.B. ACAR and A.N. ALSHAWABKEH, II. Theoretical Model, J. Geotechnical Eng., March (1996), 186-196.


4. V. G. BABSKII, M. Yu. ZHUKOV and V.I. YUDOVICH, Mathematical Theory of Electrophoresis- Applications to Methods of Fractionation of Biopolymers, Naukova Dumka, Kiev, 1989.

5. Y.S. CHOI and R. LUI, A mathematical model for the electrokinetic remediation of contaminated soil, J. Hazardous Materials, 44 (1995) 61-75.

6. Y.S. CHOI and R. LUI, Global stability of solutions of an electrochemistry model with multiple species, J. Differential Equatios, 116 (1995), 306-317.

7. Y.S. CHOI and R. LUI, Multi-dimensional electrochemistry model, Arch. Rational Mech. Anal., 130 (1995), 315-342.

8. R.A. JACOBS, M.Z. SENGUN, R.E. HICKS and R.F. PROBSTEIN, Model and experiments on soil remediation by electric fields, Journal of Environmental Sciences and Health, ASS(S) (1994).

9. R. LAGEMAN, Electroreclamation: applications in the Netherlands, Environ. Sci. Technol., 27 (1993), 2648-2650.

10. J. NEWMAN, Electrochemical Systems, Prentice-Hall, 1973.

11. R.F. PROBSTEIN and R.E. HICKS, Removal of contaminants from soils by electric fields, Science, 260 (1993), 498-503.

12. I. RUBINSTEIN, Electra-difiusion of ions, SIAM Studies in Applied Math., SIAM, 1990.

13. A.P. SHAPIRO and R.F. PROBSTEIN, Removal of contaminants from saturated clay by electroosmosis, Environ. Sci. Technol., 27 (1993), 283-291.

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