Mathematical model for hydraulically aided electrokinetic remediation of aquifer and removal of nonanionic copper

Mathematical model for hydraulically aided electrokinetic remediation of aquifer and removal of nonanionic copper

Engineering Geology 77 (2005) 305 – 315 www.elsevier.com/locate/enggeo Mathematical model for hydraulically aided electrokinetic remediation of aquif...

237KB Sizes 0 Downloads 40 Views

Engineering Geology 77 (2005) 305 – 315 www.elsevier.com/locate/enggeo

Mathematical model for hydraulically aided electrokinetic remediation of aquifer and removal of nonanionic copper Sadataka Shibaa,*, Yushi Hirataa, Tadachika Senob,1 a

Department of Chemical Science and Engineering, Osaka University, Toyonaka, Osaka 560-8531, Japan b Department of Systems Engineering, Shizuoka University, Hamamatsu, Shizuoka 432-8561, Japan Received 1 June 2003; accepted 1 July 2004 Available online 15 September 2004

Abstract One of the most cost-effective in situ technologies for soil and groundwater (i.e., aquifer) remediation is electrokinetic remediation. In electrokinetic remediation, electromigration due to electric field is combined with hydromigration due to hydraulic flow by purge water to remove pollutants from aquifers through the pore water. This study aims at investigating theoretically the role of electromigration (as active movement) of pollutants and the role of hydromigration (as passive movement) of pollutants in electrokinetic remediation, and making it clear that the control variables for electrokinetic remediation are the applied voltage and the hydraulic flow rate. These aims are pursued by construction of a mathematical model based on physico-chemical considerations and by model simulations of the electrokinetic remediation applied to the virtual aquifer polluted by heavy metals of copper sulfate. According to numerical simulations with the model: (1) heavy metal (nonanionic copper) is removed from the upstream anode region and accumulated in the downstream cathode region; (2) to carry away the heavy metal outside the aquifer (global removal), hydromigration by purge water flow is essential; and (3) electromigration contributes mainly to the redistribution of heavy metals within the aquifer (local removal and local accumulation). D 2004 Elsevier B.V. All rights reserved. Keywords: Applied voltage; Control variable; Electromigration; Heavy metal; Hydromigration; Purge flow

1. Introduction

* Corresponding author. Fax: +81 6 6850 6277. E-mail addresses: [email protected] (S. Shiba)8 [email protected] (T. Seno). 1 Fax: +81 53 478 1206. 0013-7952/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2004.07.022

Shortage of water resources arises everywhere in the whole world, from increasing need for not only municipal use but also industrial use. This forces us to exploit more groundwater, which has served for a long time as the major good-quality water resource. However, in many aquifers, groundwater

306

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

has been seriously contaminated by the migration of various hazardous organic and inorganic chemicals from the disposal of municipal and industrial wastes. Continuous increase in water demand makes it urgent to clean up the contaminated aquifers immediately, in addition to the development of new water resources. Many new techniques have been presented for the remediation of groundwater and soil. One of the most cost-effective in situ technologies is electrokinetic remediation, which consists of electromigration and hydromigration. This method utilizes the electromigration of charged pollutants in an electric field and hydromigration by purge water in soil pore. However, understanding of pollutant removal by electromigration in pore water is not only experimentally but also theoretically quite insufficient. This is because the technique is relatively new and innovative for decontamination, although in civil engineering, electroosmotic flow generated by the electrokinetic method has been utilized for practical soil dewatering (Wan and Mitchell, 1976). This paper is devoted, firstly, to developing a mathematical model for electrokinetic remediation based on physico-chemical mass transport theory and, secondly, to simulating nonsteady characteristics of the transport of heavy metals by electromigration and hydraulic flow in pore waters of aquifers. However, only a few studies (Alshawabkeh and Acar, 1992; Eykholt and Daniel, 1994; Rødsand et al., 1995; Segall and Bruell, 1992; Shapiro et al., 1989; Taha et al., 1997) can be consulted to construct such a mathematical model for aquifer remediation, taking the electromigration of heavy metals into consideration. The electrokinetic remediation of aquifers utilizing electromigration consists of: (1) installing electrodes (anode and cathode) into the contaminated zone, which should be saturated with groundwater or purge water; (2) injection of purge water into the anode well and taking out contaminated water from the cathode well, which brings about hydraulic flow in pore water; and (3) applying a fixed low-voltage direct current between the electrodes to start electromigration of charged species, as shown in Fig. 1. Heavy metals are typically ionized into cations (e.g., Cu2+, Pb2+, Cd2+, and so on) in groundwater. In electric fields, an electrokinetic driving force acts on

Fig. 1. Schematic representation of electrokinetic remediation. Chemicals [i.e., cations (heavy metals) and anions] are transported by the hydraulic flow caused by purge water in conjunction with electromigration caused by the applied voltage.

the ionized heavy metals (cations) and causes electromigration in pore water. The force is directed toward the cathode and superposed on the hydraulic flow due to purge water. Therefore, the heavy metals are accelerated and migrate to the cathode well where they are swept out through the drainage of waste water. On the other hand, because the electrokinetic driving force on anions (e.g., CO32, SO42, and so on) is directed toward the anode (i.e., in the direction opposite to the hydraulic flow), the migration of anions is retarded or even driven in opposite to the hydraulic flow. Then, the cationic heavy metals are separated from anions and transported with increasing speed to the downstream well.

2. Mathematical model of electrokinetic remediation The mathematical model is based on mass conservation law and consists of governing equations and its boundary conditions. The model is characterized especially by electromigration of charged species under the electric field. Electromigration can be easily controlled by the applied voltage between the buried electrodes. The model contains three kinds of chemical reaction as: (a) the homogeneous liquid-phase reaction (dissociation equilibrium) in pore water, which appears in the governing equation as the reaction term; (b) the heterogeneous interface reaction between pore water and soil surface (adsorption and desorption), which is considered in the governing equation as the retardation factor; and (c) the electrode reaction at electrode surface (electrolysis), which

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

appears in boundary conditions for the governing equations. 2.1. Governing equation The governing equation developed here is described by the advective diffusion equation with homogeneous and heterogeneous chemical reactions. The work of Shapiro et al. (1989) is a good reference, although they do not consider the injection of purge water, which carries away the chemicals outside the remediation domain through the electrode wells. Superposing the electromigration of the chemical species k on steady uniform hydraulic flow of injected purge water, the equation of mass conservation (diffusion equation), which considers the advection, dispersion, and sorption of chemical species k in aquifers (i.e., saturated homogeneous isotropic media) can be described in 1-D form as follows as:   B B  4 Ck us þ uezk ðnRd k Ck Þ þ Bt Bz ¼ Dk4

B2 Ck þ nRaq k Bz2

ð1Þ

where t = time (s); z = space coordinate parallel to electric field (cm); C k = concentration of chemical species k (mol l1); u s = pore flow velocity due to hydraulic flow (cm s1); n = porosity of soil (); Rd k = retardation factor () defined by Eq. (3), mentioned later; R kaq = molar rate due to liquid-phase chemical reactions (mol l1s 1); and Dk* and * = effective diffusion coefficient (cm2s1) and u ezk effective electromigration velocity (cm s1), respectively, defined by:    n n 4 D4 D ; u ð2Þ k ; uezk ¼ k ezk s2 s2

velocity due to the injected purge flow rate regardless of the microscopic local electroosmotic flow. Chemical reactions in the governing equation are: (a) homogeneous liquid-phase reaction (dissociation equilibrium) in pore water, which generates the molar rate R kaq in pore water, and (b) heterogeneous interface reaction between pore water and soil surface (adsorption and desorption), which generates the molar rate R ksp, although R ksp is not included in Eq. (1) explicitly. Instead of including R ksp in the governing equation explicitly, the retardation factor Rd k is introduced to take account of R ksp satisfactorily (Inoue and Kaufman, 1963) as:   ð1  nÞq Bsk Kd k ; ðRd k ; Kd k Þ ¼ 1 þ ð3Þ n BCk where q = density of soil solids (g cm3); Kd k = distribution coefficient (lg1); and s k = absorbed species per unit mass of soil solid (mol g1). Kd k is estimated by the adsorption isotherm. The adsorption isotherms (e.g., Langmuir isotherm) (Krauskopf and Bird, 1995) between s k and C k are generally expressed by a linear function at low concentrations of the absorbing species. 2.2. Electromigration velocity Electromigration of chemical species constitutes the kernel of the model developed here. Electromigration velocity u ezk , which is acquired by charge species in pore water under electric field, plays an important role in the remediation of aquifers. Electromigration velocity is dependent on the potential gradient caused by the electric potential applied between the electrodes and is given by (Probstein, 1994): uezk ¼ 

where s = tortuosity obtained experimentally (); Dk = diffusion coefficient of chemical species k in pore water (cm2 s 1); and u ezk = electromigration velocity of chemical species k (cm1 s) defined by Eq. (4), mentioned later. It should be noted that electroosmotic flow does not appear in 1-D governing equation as Eq. (1). This is because the advective flow velocity in 1-D treatment is the cross-sectional averaged value, which is estimated by equation of continuity and is equal to the

307

F B/ zk D k RT Bz

ð4Þ

where u ezk = electromigration velocity (cm s1); F = Faraday’s constant (C mol1); R = universal gas constant (J K1 mol1); T = pore water temperature (K); z k = charge number of species k (); / = electric potential (V); and B//Bz = potential gradient (V cm1). B//Bz is given by: B/ RT ¼  f ðt; zÞ Bz F

ð5Þ

308

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

where f (t, z) is a function of C k and the electric current density i *s . f(t, z) is obtained as:   4 X P BCk 3 is zk D4  us ðzk Ck Þ þ10 k Fn Bz  f ðt; zÞ ¼ P 2 4 zk Dk Ck ð6Þ The factor 103 multiplying i *s is introduced, since concentration of species k, C k , is specified in moles per liter as usual practise. As i s(t) = I(t)/A, i *s is calculated from the electric current as: is4ðt Þ ¼

n n I ðt Þ i s ðt Þ ¼ 2 2 s s A

ð7Þ

where I(t) = electric current from power unit (A); and A = cross-sectional area of soil column (cm2). Integrating Eq. (5), electric potential /(t, z) along the soil column is estimated by: Z z RT /ðt; zÞ ¼ /ðt; 0Þ  f ðt; zÞdz ð8Þ F 0 Then the applied power voltage E (V) between the anode (z = 0) and the cathode (z = L) is obtained as follows as: Eðt Þ ¼ /ðt; 0Þ  /ðt; LÞ

ð9Þ

In the operation of electric power supply to the remediation system, control of applied voltage E(t) is supposed to be much easier than that of current I(t). However, numerical calculation of the function f(t, z) for given E(t) is somewhat complicated. 2.3. Electroosmotic flow Apart from the hydraulic flow due to injection of purge water, electroosmotic flow arises even in the stationary pore water, dragged viscously by electromigration of charged species under the electric field. Electroosmotic flow velocity is dependent on zeta potential f and potential gradient B//Bz and is given as (Probstein, 1994; Levich, 1962): uos ¼  f

e B/ l Bz

ð10Þ

where f = zeta potential (V); e = permittivity of water ( = 6.931010 C V1m1 = 6.93105 erg V2 cm1 at 25 8C); and l = viscosity of water ( = 8.97104 kg

m1 s1 =8.97103 g cm1 s1 at 25 8C). Zeta potential f is caused by the resultant charged species absorbed on the soil surface of flow channel and varies from negative to positive via the point of zero charge depending upon the pH of pore water. Electroosmotic flow velocity u os is dependent on the resultant zeta potential f and behaves in a complicated manner according to the sign and magnitude of f. However, f does not become so P large because f~ z C and because the quantity i i P z i C i is apt to be zero due to electroneutrality P condition ( z i C i =0). If electroneutrality condition could strictly hold true in the diffusion layer, f would be completely zero. This suggests that generally electroosmotic flow velocity u os is smaller than the individual electromigration velocity u ezk . This fact is shown by calculating the ratio u os/u ezk . Because RT/ F = 2.57102 (V) and e/l = 7.73103 (cm2 s1 V2), u os/u ezk is given as follows as: uos RT e f f ¼ ¼ 1:99  104 F l zk Dk zk D k uezk

ð11Þ

Therefore, it is shown that, at room temperature, the ratio u eo/u ezk c101, assuming fc102 V; Dk c105 cm2 s1 ; and z k = 2. The more rigorously the local electroneutrality condition is satisfied, the smaller f becomes, and thus becomes electroosmotic flow. However, in this context, it is not intended to show that u os is much smaller than u ezk to be neglected, but merely intended to clear the characteristics or role of electroosmotic flow. Let us consider the reason why electroosmotic flow u os does not appear in the governing equation for charged species k (i.e., C k ) as is seen in Eq. (1). This is not because u os is much smaller than u ezk to be neglected. This is because u os should not be included in the governing equation in view of its origin, regardless of its local magnitude. It should be noted that electroosmotic flow is caused interactively inside the pore water by the resultant viscous fluid movement dragged by the individual electromigration of the charged species. In other words, electroosmotic flow is an adjunct to electromigration of various charged species. From the viewpoint that fluid flow is the carrier of charged species, electroosmotic flow caused adjunctly

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

by electromigration cannot be a real carrier and is quite different from hydraulic flow caused independent of the electric field and externally by pumped purge water as a carrier. It is not electroosmotic flow but hydraulic flow that should be superposed on electromigration as a real carrier of charged species. Therefore, electroosmotic flow is not included in the governing equation, being buried under hydraulic flow in a sense. However, if the local electroosmotic flow direction, which is dependent on the polarity of zeta potential f, is the same as hydraulic flow, the electroosmotic flow cooperates with hydraulic flow in saving the power consumption for pumping purge water. This is because, in this case, the local electroosmotic flow causes slip velocity in the vicinity of the channel wall (soil surface) to reduce the drag of soil wall on the fluid flow. Anyhow, regardless of the above exaggeratedly microscopic consideration on the local electroosmotic flow, if there is no purge flow in 1-D governing equations, we can dispose of the electroosmotic flow more simply by reason that the cross-sectional averaged electroosmotic flow velocity is estimated at zero with the equation of continuity. 2.4. Boundary conditions

309

* and J eOH * , respectively, as of their fluxes J eH follows:   * * 3 iz 3 iz * * ; 10 ðJeH ; JeOH Þ ¼ 10 ð14Þ zH F zOH F where i*z = effective electric current density (A cm2). * and J eOH * are superposed on J H* for H+ and on J eH * J OH for OH, respectively. As i*z = i*s /n, z H = +1, and z OH = 1, the boundary conditions for Eq. (1) in terms of fluxes J k* can be stated as: BðDk*Ck Þ * ÞCk þ ðus þ uezk Jk* ¼  Bz 8 9 * < = þ 3 is IN us Ck þ 10 ; for H at z ¼ 0 ¼ Fn : ; us CkIN ; for others ð15Þ BðDk*Ck Þ * ÞCk þ ðus þ uezk Jk* ¼  Bz 8 9 * <  = 3 is at z ¼ L ¼ us Ck  10 Fn ; for OH : ; us C k ; for others ð16Þ where C kIN = inflow concentration of species (usually C kIN = 0 except for H+ and OH).

k

2H2 O  4e ! O2 ðgÞ þ 4Hþ

ð12Þ

The second term in the right-hand side of the above equations for H+ and OH represents the contribution from the electrode reaction. We would like to emphasize that electrokinetic remediation is performed in liquid phase and that such gas-phase species as O2 (g) and H2 (g) are not concerned in liquid-phase equation. Applying the model to 1-D column experiment in preparation and integration of the governing equations for chemical species (as an example of heavy metal compound, CuSO4, and its dissociations) with respect to time t and space z, the pollutant (heavy metal) distribution C k (t,z) between the electrodes and the removal from the treatment area through the effluent (area averaged value) is numerically simulated by utilizing the finite-element method.

2H2 O þ 2e ! H2 ðgÞ þ 2OH

ð13Þ

2.5. Homogeneous liquid-phase reaction in pore water

The production rate of H+ at the anode and that of OH at the cathode can be expressed in terms

As the heavy metal treated here is copper sulfate CuSO4, the homogeneous liquid-phase

At electrodes, the boundary conditions for Eq. (1) are given as the continuity of the fluxes of chemical species J k*. Because electrode reactions occur by passage of the electric current, the fluxes due to * must be taken into account in electrode reactions J ek the boundary conditions in addition to the convective fluxes. At the anode, H2O gives electrons to the electrode and generates oxygen gas and H+ ion. At the cathode, H2O receives electrons from the electrode and releases hydrogen gas and OH ion. These electrode reactions are described by:

310

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

chemical reactions in pore water can be expressed as follows:

3. Numerical simulation

CuSO4 WCu2þ þ SO2 4 ðK1 Þ

ð17Þ

H2 SO4 WHþ þ HSO 4 ðK2 Þ

ð18Þ

þ 2 HSO 4 WH þ SO4 ðK4 Þ

ð19Þ

H2 OWHþ þ OH ðKW Þ

ð20Þ

For the abovementioned reason, no ready-made program for the numerical simulation can be utilized and we have to prepare the simulation program at our own expense. Numerical simulations were done under the conditions shown in Tables 1–3. Table 1 represents the assumed values for the bench-scale soil column experiments. Table 2 shows the values of the operational variables for electromigration. Physical properties used for numerical simulations are shown in Table 3.

where K 1, K 2, K 4, and K W are the dissociation constants. The concentrations of chemical species are defined as: ðC1 ; C2 ; C3 ; C4 ; C5 ; C6 Þ ¼ ð½CuSO4 ; ½Cu2þ ; ½SO2 4 ; þ  ½HSO 4 ; ½H ; ½OH Þ

ð21Þ

From Eqs. (17)–(20), the reaction terms R kaq for homogeneous liquid-phase reaction in the governing equation are given as: Raq 1 ¼ðk1þ C1  k1 C2 C3 Þ

ð22Þ

Raq 2 ¼ k1þ C1  k1 C2 C3

ð23Þ

Raq 3 ¼ k1þ C1  k1 C2 C3 þ k4þ C4  k4 C3 C5

ð24Þ

Raq 4 ¼ðk4þ C4  k4 C3 C5 Þ

ð25Þ

Raq 5 ¼ k4þ C1  k4 C3 C5 þ kWþ  kW C5 C6

ð26Þ

Raq 6 ¼ kWþ  kW C5 C6

ð27Þ

where k 1+, k 1, k 4+, k 4, k W+, and k W are reaction rate constants satisfying the next equation: ðK1 ; K4 ; KW Þ ¼ ðk1þ =k1 ; k4þ =k4 ; kWþ =kW Þ

ð28Þ

As usually the precise values of k 1+, k 1, k 4+, k 4, k W+, and k W are not obtained, we have to do without them; in other words, we have to be content with K 1, K 4, and K W. Then, with a somewhat sophisticated technique, it is necessary to eliminate k 1+, k 1, k 4+, k 4, k W+, and k W in the reaction terms of the governing equations, although this makes even the numerical calculation of the simple homogeneous liquid-phase reactions so complicated as to describe the process here (see Appendix A).

3.1. Distribution of total copper Electrokinetic driving force acting on Cu2+ is directed toward the cathode. This makes Cu2+ and CuSO4 concentrations of the cathode higher and that of the anode lower, even if the hydraulic flow is very small. To confirm numerically this supposition, the total copper concentration X 1 is defined as: C1 C2 X1 ¼ Cˆ 1 þ Cˆ 2 ¼ þ ð29Þ C0 C0 where Cˆ 1 and Cˆ 2 =normalized copper sulfate concentration and normalized copper ion concentration, respectively; and C 0 = standard concentration for normalization. Considering the electromigration of Cu2+, it is supposed that the distribution curve of normalized total copper concentration X 1 in the soil column has an ascending gradient with respect to the distance from the anode. The time variations of the total copper distribution in the soil column are plotted in Fig. 2. The abscissa is the normalized distance zˆ , which is given as z divided by the column length L. The ordinate is the dimensionless concentration (normalized by C 30, i.e., the initial concentration of sulfate ion C 3). These curves are shown parametrically in time (in hours). As Table 1 Characteristics of soil column Property

Value

Length of soil column (L) [cm] Cross-sectional area of soil column (A) [cm2] Porosity of soil (n) Tortuosity of pore (s) Initial acidity of pore water (pH)

40 100 0.4 1.5 6.0

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

311

Table 2 Operational variables of electrokinetic remediation Variable

Value

Flow rate of purge water ( Q) [cm3 s1] Applied voltage (E) [V] Acidity of purge water (pH)

0.001 and 0.0015 2 and 3 7.0

is supposed, at any time, the distributions of total copper concentration X 1 in the soil column are almost linear with respect to the normalized distance from the anode zˆ . Curves have an ascending gradient, except for the initial distribution (horizontal dotted line). These plots suggest that copper is removed from the anode (upstream end) and that it is transported and accumulated near the cathode (downstream end). That is, the remediation (local removal of copper) in the soil column advances from the anode to the cathode with the passage of time. This tendency of distribution of copper concentration accords qualitatively with the results for the concentration distributions of heavy metals observed by Nekrasova and Korolev (1997) in a 1-D column experiment. From Fig. 2, it can be seen that the electromigration contributes much to the alteration of the desirable distribution of copper between electrodes [i.e., low concentration (local removal) near the anode and high concentration (local accumulation) near the cathode].

Fig. 2. Distribution of total copper (Cu2++CuSO4) concentration X 1 between anode and cathode. Initial distribution of X 1 is uniform.

Property

Value

Diffusion coefficient (Dk ; k ¼ 16Þ Equilibrium constant (K i ; i = 1, 4, and W) Retardation factora (Rd k ; k = 1–6)

1.0105 cm2 s1

time, pH at the anode should decrease and pH at the cathode should increase in their values. Then, pH distribution curves are expected to have an ascending gradient, and the gradient is considered to become steeper and steeper with the passage of time. Time variation of the distribution of pH between the anode and the cathode is shown in Fig. 3. The abscissa is the dimensionless distance from the anode zˆ (normalized z by column length L) and the ordinate is pH value. As usual, pH is defined as log([H+]). Curves are potted parametrically in time (in hours). The initial pH of pore water is 6.0 and is represented by the horizontal line. As is supposed from the electrode reaction, pH distribution curves obtained by the model have an ascending gradient with respect to the distance from the anode, although the gradient varies greatly with the distance. Furthermore, they have a sharp increase of pH value (pH jump; in other words, acid front) in the distributions with respect to the anode. On the other hand, with respect to the cathode, they have a sharp decrease of pH value (pH drop; in other words, base front) in the distribution. Fig. 3 proves that the numerical simulations using the mathematical model developed here can well reproduce the pH distribution, which exhibits so-called acid and base fronts as is observed in the experiments, in the soil column.

4.37103 M, 1.20102 M, and 1014 M2 3, 4, 1, 2, 5, and 2

3.3. Effect of applied voltage and purge flow on distribution of total copper

3.2. Distribution of pH As is mentioned earlier, by the electrode reactions at the anode and the cathode, hydrogen ion H+ is produced at the anode and hydroxyl ion OH is generated at the cathode (see Eqs. (12) and (13)). Therefore, it is supposed that, with the passage of

Table 3 Values of physical properties

a The values of retardation factor (N1) are adjustable according to the properties of remediation site.

The local separation of copper sulfate and copper ions near the anode and their accumulation near the

312

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

Fig. 3. Distribution of pH in soil column. Acid fronts from the anode meet base fronts from the cathode at the point where zˆ c0.6.

cathode due to electrokinetic forces in the soil column are demonstrated in Fig. 4. Apart from the electrokinetic driving force, there is hydraulic flow in the pore due to purge water, which may also affect copper distributions. The applied voltage E and the purge flow Q are increased by 50% from their values in the standard condition of numerical simulations (E = 2 V and Q = 0.0010 cm3 s1) to 3 V and 0.0015 cm3 s1 in other operations, respectively. The effects of the increments on the distributions of total copper concentration X 1 are shown in Fig. 4. The effects are described by the variation of total copper concentration DX 1 against operations of E and Q.

Fig. 4. Effect of applied voltage and purge flow on distribution of total copper. DX 1 = variation of total copper concentration. DX 1 b 0 means that copper is locally removed and DX 1 N 0 means that copper is locally accumulated.

The solid curve is for the variation with the applied voltage increment (DE = +1 V) and the dotted curve is for the variation with the flow rate increment (DQ = +0.0005 cm3 s1). It is distinctly clear from the figures that the applied voltage operation is more effective in separating and accumulating copper ions than the purge water operation. The local removal at the anode increases by 11.2% for 1-V increment in the applied voltage, whereas this increment is 6.88% when purge flow rate is increased from 0.0010 to 0.0015 cm3 s1. This result means that the applied voltage is the appropriate operational variable for the local separation and removal of heavy metals from the anode region. 3.4. Effect of purge flow and applied voltage on global reduction of total copper In order to evaluate the copper amount taken out from the aquifer, average total copper concentration in the column is defined as follows as: Z 1 Z 1   AX1 ¼ ð30Þ X1 dˆz ¼ Cˆ 1 þ Cˆ 2 dˆz 0

0

The decrease in average total copper AX 1 for 50% increment in the purge flow rate Q and applied voltage E is shown in Fig. 5. The solid curve is for the variation with the flow rate increment (DQ = +0.0005 cm3 s1) and the dotted curve is for the variation with the applied voltage increment (DE = +1 V). A 50% increment of purge water flow reduces AX 1 by 2.94%.

Fig. 5. Effect of applied voltage and purge flow on global reduction. DAX 1 = variation of average total copper concentration. DAX 1 b 0 means that copper is globally removed from the aquifer.

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

On the other hand, a 50% increment of applied voltage reduces AX 1 by only 0.230%. Fig. 5 signifies the importance of the purge water in electrokinetic remediation. The total copper in the soil column is carried away with purge water effluent (evaluated by the reduction in AX 1), whereas electromigration greatly contributes to local removal from the anode region. It should be noted here that injection of purge water is essential to carry away the locally separated heavy metal.

313

value by more than one order of magnitude). To overcome this problem, a somewhat sophisticated technique is introduced in the model calculation. Those who are interested in the technical aspects of the numerical calculations may be patient with reading this section. This technique consists of two procedures. The first is the transformation of the governing equation on C i to the reduced one, which is free from the reaction term, by employing the new variable X k . The second is the inverse transformation of the new variable X k to the original variable C i to obtain the real concentrations of chemical species.

4. Conclusions From the results of the numerical simulations with the mathematical model based on physico-chemical consideration, it is concluded that: (1)

Applied voltage to electrode E and the purge flow rate injected into anode well Q act as the control variables for electrokinetic remediation of aquifer. (2) Heavy metal (copper) is removed from the anode region (upstream region) and transported for accumulation in the cathode region (downstream region). (3) Electromigration is effective especially for the local removal (redistribution of heavy metal between electrodes) of heavy metals near the anode because electromigration redistributes heavy metals between the electrodes. (4) For the carrying away of the heavy metal through the effluent, the hydraulic flow by purge water is essential because electromigration contributes mainly to local removal. (5) The pH distribution, which is characterized by the pH jump with acid front and pH drop with base front, is well reproduced by the mathematical model.

Appendix A Theoretically, C i is obtained by integrating the governing equation given by Eq. (1). However, Eq. (1) includes the almost unknown k iF (no reliable values have been published yet because of the measuring difficulty and the published k iF varies in

A.1. Reduced governing equation without chemical reaction term The reaction term with unknown k iF in Eq. (1) is eliminated by introducing the new variables X 1, X 2, and X 3. Chemical species (hence, C i ) are grouped with respect to Cu(2), S(6), and H(1), respectively. Considering the aqueous-phase chemical reactions given by Eqs. (22)–(27), X k are defined as follows as: X1 ¼ Cˆ 1 þ Cˆ 2

ðA1Þ

X2 ¼ Cˆ 1 þ Cˆ 3 þ Cˆ 4

ðA2Þ

X3 ¼ Cˆ 4 þ Cˆ 5  Cˆ 6

ðA3Þ

where Cˆ i is the normalized value of C i (=C i /C30). From Eqs. (22) and (23), it can be seen that X 1 given by Eq. (A1) eliminates the reaction term (not BX 1/Bt but dX 1/dt; i.e., not accumulation rate but reaction rate) related to Cu(2) as follows as: dX1 dCˆ 1 dCˆ 2 Raq þ Raq 2 ¼ þ ¼ 1 ¼0 dt dt dt C30

ðA4Þ

where C 30 is the initial value of C 3 selected for normalization (other C i0 can be selected, although the maximum C i0 is the most suitable for normalization). X 2 and X 3 also eliminate the reaction terms related to S(6) and H(1), respectively. Reduced governing equations (free from reaction term) for X 1 is obtained by adding Eq. (1) of C 1 and that of C 2 side by side as follows as: 2 B B * B X1 ðnRd x1 X1 Þ þ ðux1 X1 Þ ¼ Dx1 Bt Bz Bz2

ðA5Þ

314

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

* are given as follows as: where nRd x1, u x1, and Dx1 nRd 1 Cˆ 1 þ nRd 2 Cˆ 2 X1

ðA6Þ

   * Cˆ 1 þ us þ uez2 * Cˆ 2 us þ uez1 ux1 ¼ X1

ðA7Þ

nRd x1 ¼ 

*¼ Dx1

D1* Cˆ 1 þ D2*Cˆ 2 X1

ðA8Þ

Values of Cˆ 1(t), Cˆ 2(t), and X 1(t) in the righthand side of Eqs. (A6)–(A8) are unknown variables; however, in the numerical calculation, we can use known values of Cˆ 1(tdt), Cˆ 2(tdt), and X 1(tdt) at the previous time (tdt: time step dt should be appropriately small) as good approximations. Reduced governing equations for X 2 and X 3 also easily can be taken with the same way. After evaluating the * , we can integrate the values of nRd xk , u xk , and Dxk reduced governing equations to obtain X k . Practically, time- and space-varying values of X 1, X 2, and X 3 are numerically estimated by such standard numerical method as finite-element method or finite-difference method.

From Eqs. (A3) and (A11), C 5 is obtained as follows as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðX3  C4 Þ þ ðX3  C4 Þ2 þ 4KW ðA12Þ C5 ¼ 2 Because it can be approximated that C 4c0 (K 4 is much greater than others; see Table 3), as the first approximation, C 5 is given as follows as: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X3 þ X32 þ 4KW C5 c ðA13Þ 2 From Eq. (A11), C 6 is obtained as: C6 ¼

To obtain six unknowns C i (or Cˆ i ) from X k , six equations must be prepared. Three of six are Eqs. (A1)–(A3). Another three are given from Eqs. (17), (19), and (20). HSO4 is almost perfectly dissociated from H2SO4 (i.e., it can be assumed that K 2cl). Therefore, Eq. (18) is omitted from the calculation. From Eqs. (17), (19), and (20), three equations of equilibrium are obtained as follows as: Cˆ 1 Kˆ 1 ¼ Cˆ 2 Cˆ 3

ðA9Þ

Cˆ 4 Kˆ 4 ¼ Cˆ 5 Cˆ 3

ðA10Þ

Kˆ W ¼ Cˆ 5 Cˆ 6

ðA11Þ

2 where Cˆ i = C i /C 30; Kˆ i = K i /C 30; and Kˆ W = K W/C 30 . For brevity of expression, the curette (the symbol 1) over C i and K i is omitted hereafter.

ðA14Þ

From Eq. (A3), C 4 is obtained as: C4 ¼ X3  C5 þ

KW C5

ðA15Þ

To obtain more precise values of C 5, C 6, and C 4, iterate the calculations by Eqs. (A12), (A14), and (A15) until their values are recognized as convergent ones. Inserting Eqs. (A1) and (A2) into Eq. (A9) to eliminate C 2 and C 3, C 1 is obtained as: C1 ¼

A.2. Inverse transformation of X k to C i

KW C5

ðX1 þ X2 þ K1  C4 Þ 

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðX1 þ X2 þ K1  C4 Þ2  4X1 ðX2  C4 Þ 2

ðA16Þ

From Eq. (A1), C 2 is obtained as: C2 ¼ X1  C1

ðA17Þ

From Eq. (A2), C 3 is obtained as: C3 ¼ X2  C4  C1

ðA18Þ

In performing the time integral of the reduced governing equations, both the transformation of C i to X k by Eqs. and the inverse transformation of X k to C i by the above procedure are executed at every time step of dt. If the local electroneutrality condition can be assumed to be satisfied, X 3 is obtained without integrating Eq. (A5) (reduced governing equation for X 3) as follows as: X3 ¼  2X1 þ 2X2 þ a

ðA19Þ

S. Shiba et al. / Engineering Geology 77 (2005) 305–315

where a = constant determined by the initial values of X k . That is, a is given by: X a¼ zi Ci0 ðA20Þ ¼ 2C20  2C30  C40 þ C50  C60

References Alshawabkeh, A.N., Acar, Y.B., 1992. Removal of contaminants from soils by electrokinetics: a theoretical treatise. Environ. Sci. Health A 27 (7), 1835 – 1861. Eykholt, G.R., Daniel, D.E., 1994. Impact of system chemistry on electroosmosis in contaminated soil. ASCE J. Geotech. Eng. 120 (5), 797 – 815. Inoue, Y., Kaufman, W.J., 1963. Prediction of movement of radionuclides in solution through porous media. Health Phys. 9 (7), 705 – 715. Krauskopf, K.B., Bird, D.K., 1995. Introduction to Geochemistry, 3rd ed. McGraw-Hill, New York, pp. 143 – 144. Levich, V.G., 1962. Physicochemical Hydrodynamics. PrenticeHall, Englewood Cliffs, NJ.

315

Nekrasova, M.A., Korolev, V.A., 1997. Electrochemical cleaning of polluted soils. In: Marinos, P.G., et al. (Eds.), Engineering Geology and the Environment, vol. 2. Balkema, Rotterdam, The Netherlands, pp. 2047 – 2052. Probstein, R.F., 1994. Physicochemical Hydrodynamics, 2nd ed. Wiley, New York. Rødsand, T., Acar, Y.B., Breedveld, G., 1995. Electrokinetic extraction of lead from spiked Norwegian Marine Clay. In: Acar, Y.B., Daniel, D.E. (Eds.) Geoenvironment 2000, vol. 2. ASCE, New York, pp. 1518 – 1534. Segall, B.A., Bruell, C.J., 1992. Electroosmotic contaminant removal processes. ASCE J. Environ. Eng. 118 (1), 84 – 100. Shapiro, A.P., Renaud, P.C., Probstein, R.F., 1989. Preliminary studies on the removal of chemical species from saturated porous media by electroosmosis. Phys. Chem. Hydrodyn. 11 (5/6), 785 – 802. Taha, M.R., Acar, Y.B., Gale, R.J., 1997. Electrokinetic enhancement tests on TNT contaminated soil. In: Marinos, P.G., et al. (Eds.) Engineering Geology and the Environment, vol. 2. Balkema, Rotterdam, The Netherlands, pp. 2209 – 2214. Wan, T., Mitchell, J.K., 1976. Electro-osmotic consolidation of soils. ASCE J. Geotech. Eng. 102 (GT5), 473 – 491.