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In-situ electrokinetic remediation of groundwater contaminated by heavy metal
883
In-situ electrokinetic remediation of g r o u n d w a t e r c o n t a m i n a t e d by h e a v y metal S. Shiba a and Y. Hirata ~ a Department of Chemical Science & Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
One of the most cost-effective in-situ technologies for groundwater remediation is electrokinetic method, which utilizes electromigration of pollutants in pore water. Numerical simulations by a mathematical model have been done, in order to make clear the changes of the pollutant distributions in aquifers and the effects of the operational variables on the removal efficiency. The results of the numerical simulations show that: (1) electromigration is effective especially for the local removal of heavy metals (rebuild of distribution of heavy metal between the anode and the cathode); (2) heavy metal is removed from the upstream anode region and accumulated in the downstream cathode region; and (3) to carry away the heavy metal from the aquifer, the hydraulic flow by purge water is essential.
1. INTRODUCTION The need for water resources is increasing more and more these days not only in developed countries but also in developing countries. Because surface water resources have become more fully developed, interest has increased in the development of groundwater. It is generally recognized that both aspects of water resources, quantity and quality, are important and interdependent. However, in many aquifers groundwater has been seriously contaminated by the migration of various hazardous organic and inorganic chemicals from the disposal of municipal and industrial wastes. Many new techniques have been presented for remediation of groundwater and soil. Among them, one of the most cost-effective in-situ technologies is electrokinetic remediation, which utilizes electromigration of pollutants (especially heavy metals) in pore water of aquifers. However, understanding of electromigration of pollutants in pore water is not only experimentally but also theoretically quite insufficient. This is because the technique is relatively new and innovative, although for dewatering of soils civil engineers have practiced the electrokinetic method which utilizes electroosmotic flow in pore water. This paper aims to develop a mathematical model based on physicochemical mass transport theory and to simulate the non-steady characteristics of the electromigration of heavy metals in pore water of aquifers. However, only a few studies (Shapiro [1 ], Alshawabkeh & Acar [2], Bruell et al. [3], and Taha et al. [4]) can be utilized for construction of the mathematical model which takes the electromigration of heavy metals into consideration.
884
4 Purge Water Q [
..........
...........
!
1
i
I
,
r
I
.,
,
I
!
I
' l
i
I
i
Inaow=
<=o
[
o =3,.
9
i : :,L_ _~-~_teAnion(-) x
~
' , i
<=o
o
Irlowl--> o
/
Cation(+)o=~[--
Anode
',
,
I
n
-~Outflow _ :,
Cathode
'i : " x
,
x
,,
Figure 1. Electrokinetic remediation. Chemicals, i.e., cations (heavy metals) and anions are transported by the hydraulic flow caused by purge water in conjunction with electromigration.
2. E L E C T R O M I G R A T I O N OF HEAVY METAL Heavy metals get ionized in groundwater (e.g. Cu 2+, Pb 2+, and so on), which is an essential condition for the application of electrokinetic remediation technique. This technique utilizes electromigration of the charged species. It consists of: (1) installing electrodes (anode and cathode) into the aquifer saturated with the groundwater or the purge water, (2) injection of the purge water into the anode well, and (3) applying a fixed low voltage direct current between the electrodes, as is shown in Figure 1. Both the injection of the purge water and the drainage of the polluted water cause a hydraulic flow from the anode well to the cathode well in the aquifer. The electric force induced by the applied voltage accelerates heavy metal migration in pore water, co-working with the hydraulic flow caused by the injection of purge water. The electrokinetic driving force, which acts on the ionized heavy metals (cations) and causes the electromigration velocity, is directed towards the cathode and superposed on the hydraulic flow so that heavy metals (cations) are accelerated towards the cathode well, where they are swept out through the drainage of waste water. On the other hand, because electrokinetic driving force on anions is directed toward the anode, i.e., in the opposite direction to the hydraulic flow, the anion migration is retarded or even driven in opposite to the hydraulic flow. Then, the heavy metals (cations) are separated from anions and transported with increasing speed to the downstream well.
3. CHEMISTRY IN E L E C T R O K I N E T I C REMEDIATION Three types of chemical reactions appear in this model. The first is the homogeneous liquidphase reaction, which generates the molar rate R~q and is included in the governing equation as the reaction term. The second is the heterogeneous interface reaction (adsorption and desorption) between the pore water and the soil surface, which generates the molar rate R~p and is taken into account by introducing the retardation factor. The third is the electrode reaction,
885
which generates the flux J~*kdue to the production of H + and OH-. J~k is combined with the flux due to the advection and the diffusion. 3.1. Liquid-Phase reaction The contaminant treated here is copper sulfate (CuSQ) as a typical compound of heavy metal. In this case the homogeneous liquid-phase chemical reactions in the pore water can be expressed as follows:
CuSO4 ~ Cu 2+ -k- SO~HSO~- ~- H + + SO]-
(K1); (K4);
H2SO4 ~ H + + HSO4 H20 ~ H + + OH-
(K2)
(1), (2)
(Kw)
(3), (4)
where K1, K2, K4 and Kw are the dissociation constants. Because dissociation constant K2 is large (>> 1), H2SO4 is dissociated almost completely into H + and HSO4. This means that Equation 2 for H2SO4 can be omitted, in other words, it can be assumed that [H2SO4] ~ 0. Then the chemical species treated here are CuSO4, Cu 2+, SO]-, HSO4, H +, OH-. The concentrations of chemical species, which appear in the above reactions and is treated here, are defined as follows: (C1, C2, C3, C4, C5, C6) -- ([CuSO4], [Cu2+], [SO2-], [HSO~-], [H+], [OH-])
(5)
where Ck = liquid-phase concentration of chemical species k (mol/L). From the liquid-phase chemical reactions described by Equations 1, 3 and 4, the reaction rates for species k, R~ q, are given as" /~q :
--( kl+C1 - k l- C 2 C 3 ) ;
/ ~ q "- kl+C1 - kl-C2C3
t ~ q : kl+C1 - k1-C2C3 -+- k4+C4 - k4-C3C5 ;
/~q m_ k4+C1 - k4_C3C5 + kw+ - kw_C5C6 ;
/~q = k4+C4 - k4-C3C5 /~q = kw+ - kw_C5C6
(6), (7) (8), (9) (10), (11)
where kl+/kl_ = K1; k4+/k4_ = K4; and kw+/kw_ = Kw. K1, K4 and Kw are easily obtained, although it is not so easy to estimate the reaction rate constants ki+ and ki_. 3.2. Interface reaction Adsorption and desorption of chemical species k at soil surface is represented by:
X~q + Soil site ~ X~~
(Kdk)
(12)
where X~,q = chemical species k in liquid phase; X~~ = chemical species k in soil phase; and Kdk = distribution coefficient (L/g). 3.3. Electrode reaction At the electrodes, electrode reactions occur by passage of the electric current which is caused by the application of electric potential between the electrodes. At the anode, water gives electrons to the electrode and generates oxygen gas and H + ion. At the cathode, water takes electrons from the electrode and releases hydrogen gas and O H - ion. These reactions are described by:
2H20 - 4e- ~ O2(g) + 4H + ; where e- = electron.
2H20 + 2e- ~ H2(g) + 2OH-
(13),(14)
886 4. M A T H E M A T I C A L F R A M E W O R K The mathematical model developed here is the distributed type mass trasport model which predicts contaminants in the saturated zone. The model is composed of: (1) a kind of advective diffusion equation with homogeneous and heterogeneous chemical reactions and (2) its boundary conditions with electrode reactions. 4.1. Governing equation
Under steady uniform forced flow, the equation of conservation of mass (diffusion equation), which considers the advection, dispersion, and sorption for chemical species k in saturated homogeneous isotropic media, can be described as follows (Shapiro et al. [1 ]): c3 (nRdkCk) +
0"~
0 c~2Ck {(Us + * )Ck} = 79~ + nR~ q ~ZZ Uezk 0Z 2
(15)
where Ck = concentration of species k (mol/L); Us = pore flow velocity (cm/s); n = porosity of soil (-); Rdk = retardation factor (-) defined by Equation 17 mentioned later; R~q = molar rate due to liquid-phase chemical reactions (mol/L/s); and T~, and U~zk = effective diffusion coefficient (cm2/s) and effective electromigration velocity (cm/s), respectively, defined by:
(~)~, Uezk 13_ n 9 ) --" (~-~T)k, 7Uezk)
(16)
where 7- = tortuosity obtained experimentally (-); 9k = diffusion coefficient of chemical species k in pore water (cm2/s); and Ue,.k = electromigration velocity of species k (crn/s) defined by Equation 18 mentioned later. 4.2. Retardation factor
In order to take account of the heterogeneous reaction between pore water and soil (adsorption and desorption ), which generates the molar rate R~p, the retardation factor Rdk is introduced satisfactorily (Inoue & Kaufman [5]). The retardation factor is defined by: (1 - n)p
Rdk = 1 +
n
Kdk = 1 +
(1 - n)p C3Sk n
(17)
OCk
where p = density of soil solids (g/cm3); Kdk = distribution coefficient of species k (L/g); and Sk = absorbed concentration of species k per unit mass of soil solid (mol/g). The adsorption isotherms between Sk and Ck are generally expressed by a linear function. 4.3. Electromigration velocity
The electromigration is the core of this model. The electromigration velocity Uezk, which is proportional to potential difference, is given by: F
0r
Uezk = --~-'~ Zk ~)k ~ZZ -- Zk ~)k f (t, Z)
(18)
where F = Faraday's constant (C/mol); R = universal gas constant (J/K/mol); T = water temperature (K); Zk = charge number of species k (-); r = electric potential (V); and O r = potential gradient (V/cm); and f(t, z) = function of Ck. f(t, z) and effective current density are given by: ,
~2 (Zk~)kCk) 2 9
, T2 A
(19)
887
where i*(t) = effective current density (AJcm2); I(t) = electric current (A); and A = cross sectional area of soil column (cm2). The factor 103 to i s is due to the fact that units of Ck is mol/L.
4.4. Applied voltage and electric potential Integrating Or the applied voltage E, i.e., the potential difference between the anode (z = 0) and the cathode (z = L), is estimated by: E ( t ) = r (t, 0) - r (t, L) = ~
(20)
f (t, z) dz
As is seen from the above equation, the applied voltage E is usually time varying. However, it can be controlled to be constant.
4.5. Initial condition The initial condition for the governing equation, Equation 15, is simply given by: Ck = Ck0
at
t = 0
(21)
This equation specifies the distribution of the concentrations of chemical species k in pore water at t = 0. Ck0 is not necessarily uniform.
4.6. Boundary conditions The boundary conditions for the governing equation are given at the anode and the cathode to satisfy the continuity of the fluxes of chemical species k, i.e., J~. The production rates of H + and OH- can be expressed in terms of their fluxes J~H and J~ou, respectively, as follows: i* (J~*H, J~*OH)= ( 103 ~
103 '
(22) ZoHF/
where i* = effective electric current density (AJcm2). J*H and J:OH are superposed on di~ (for H +) and J ; H (for O H - ) , respectively. As i* = i~/n; ZH = +1; and Zon = --1, the boundary conditions in terms of fluxes of chemical species J~ can be stated as:
j~
=
at z = 0
(23)
0(7)~Ck) ) C k = {UsCk --103J~for O H - } Oz + (us 2r- u;zk UsVk , Fn ' for others ' at z = L
(24)
(9(~kCk) 2f_ (u s + --
J~ = -
OZ
* )Ck -Uezk
s k ?-tsGIN
Fn'
for others
Equation 15 can be solved numerically using the initial condition, Equation 21, and the boundary conditions, Equations 23 and 24.
5. NUMERICAL SIMULATION Using the finite element method, numerical simulations were carried out under the conditions shown in Table 1 (for the bench scale experiments under planning). Technical difficulty in numerical calculations arises from: (1) the advection in the governing equation (cause of unstable solution) and (2) the problematic elecroneutrality for the charged species (excess condition).
888 Table 1. Properties of soil column and operational variables of remediation. Property of soil column Length (L) 40 cm Cross sectional area (A) 100 cm 2 Porosity of soil (n) 0.4 Tortuosity of pore (7-) 1.5
Operational variable Flow rate of purge water (Q) 0.0010 and 0.0015 cc/s Applied Voltage (E) 2 and 3 V Acidity of purge water (pH) 7.0 Initial acidity of pore water (pH) 6.0
5.1. Practical procedure In the practical calculation procedure adopted here, the estimation of unknown constants ki+ is omitted by eliminating ki• from the chemical reaction term R~q in the governing equation. This makes a new set of differential equations about reduced number of new dependent variables (virtual concentrations). The new dependent variables are: CN1 "-- C 1 - [ - C 2 ;
CN2 -- C1-Jr- C3 - C4 ;
CN3 : C 4 - [ - C 5 - C 6
(25), (26), (27)
Seven governing equation for Ck are reduced to three for CNi. From the estimated virtual concentrations CN1, CN9 and CNa, the real concentrations Ck can be obtained at every time step of the numerical integration, using the equations from the dissociation reactions given by Equations 1, 3 and 4. The dissociation equations are: K l C l = C2 Ca ;
K4 C4 = C5 Ca ;
Kw = C5 C6
(28), (29), (30)
5.2. Electroneutrality Electroneutrality condition, which is said to hold closely in aqueous electrochemical solutions, is a troublesome problem, because it brings about an excess condition into the mathematical model. Unless in lumped models without advection and diffusion, the condition is thought to be held in the equilibrium (or steady) state in which concentrations are uniform. There seems to be no clear answer to the question that in what case the electroneutrality could be dropped or could not. However, here the condition is taken into account by adjusting C5 and C6, which are changeable to and from H20, at every time step of the integration of governing equations.
5.3. Distribution of total copper (X1) and acidity (pH) Electrokinetic driving force acts on copper ions (Cu 2+) resulting their migration toward the cathode. This phenomenon increases the copper ion concentration in the cathode. Consequently, the concentration in anode decreases. As a result, the gradient of the concentration curve, assumed to be approximately linear, increases as the process continues. The time variations of the distribution of total copper in the soil column are plotted in Figure 2 (left). As assumed, at any time the distributions of X1 are almost linear with respect to the distance from the anode. The distribution curves have ascending gradients except for the initial distribution (horizontal dotted line). This implies that copper is removed from the anode (upstream end), and transported and accumulated near the cathode (downstream end). That is, the remediation (local removal) in the soil column advances from the anode to the cathode with the passage of time. This tendency of distributions of the copper concentration accords qualitatively with the experimental results of heavy metal removal by Nekrasova and Korolev [6] using onedimensional soil column. The gradients of these curves become steeper and steeper with the
889
1.25
,_, 1.00
020106-F1
,
I Q = 0.0010 cc/s
t=0h t=40h t= 320h
. .<.; .......
E=3V
.... ,r
0.75
I
v
. . - :7.
r
s
,
l I r S !
Q
020106-F2 |
|
s
i; 0.50
..o-
t=0h t=80h t = 160h t = 320h
Q
....
t--, 0.25 0.00 0.00
f B4' Is d,r r II'
,,-
'
'
0.25
'
0.50
~
"~"~ -'-'~ ~ " ~
'
0.75
1.00
Q = 0.0010 cc/s
~"~-"
E=
I
i
0.00
3 V
I
0.25
I
0.50
l
0.75
1.00
Dimensionless Distancez (-)
Dimensionless Distancez (-)
Figure 2. Distribution of total copper X1 (left) and acidity pH (right) at various time. 0.0000 , , ~ . . , . . .
0.08
,
,
,
020106-F3
t= 300h 0.0025
0.04 ~
Q
..,,r
,--r
0.0050
0.00
,-, -0.04 9
......" " "
Q .,..~ .,..,
>
-0.08 0.00
|
i
I
AE=+I V (from 2 V) AQ = +0.0005 cc/s (from 0.0010 cc/s) |
i
0.25 0.50 0.75 Dimensioless Distance z (-)
m
0.0075
i~ AQ = +0.00005 cc/s (+50 %) AE = +1 V (+50 %)
0.0100
i
1.00
,
0
/
50
,
I
100
,
t
,
I
~ . ,
150 200 Time t (h)
Figure 3. Variation of total copper AX1 (left) and average total copper
,
250
300
AAX1 (right).
passage of time. However, the rate of increase in gradients become smaller and ultimately it approaches to zero. Temporal variation of pH (= - lOgl0([H+])) distribution between the electrodes is shown in Figure 2 (right). As discussed in the earlier section, electrode reactions produce hydrogen ions and hydroxyl ions at anode and cathode respectively (Equations 13 and 14). Therefore, it is reasonable to assume the decrease in pH value at anode and increase at cathode. As a result, one can expect an ascending gradient of a pH distribution curve. As depicted in Figure 2 (right), there is a sharp increase in pH values (pH jump) towards the cathode as simulation progresses. The increasing portions of the distribution curves are known as acid fronts. On the other hand, it decreases (pH drop) towards the anode. The decreasing portions of the distribution curves are known as base fronts. Based on these results, it may be inferred that the numerical simulations using the mathematical model developed here can well reproduce the pH distribution in the soil columns.
5.4. Operation of purge flow (Q) and applied voltage (E) The effects of the increments of the applied voltage E and the purge flow rate Q on the distributions of copper concentration X1 are demonstrated in Figure 3 (left). The effects are estimated by the variation of total copper concentration AX1 with the increments, A E and AQ. The ap-
890 plied voltage E and the purge flow rate Q are increased by 50 % from their standard values. The solid curve is for the variation with the applied voltage increment (AE = + 1 V) and the dotted curve is for the variation with the flow rate increment (AQ = +0.0005 cc/s). It distinctly clear from this figure that the applied voltage operation is more effective to separate and accumulate copper ions than the purge water operation. The local removal at the anode increases by 11% due to the applied voltage operation, where as this removal increases by 6.9 % due to the purge flow rate operation. This result means that the applied voltage is the appropriate operational variable for the local separation of heavy metals. The decreases in average total copper AX1 in the column for 50 % increment in the purge flow rate Q and applied voltage E are shown in Figure 3 (right). The solid curve represents the reduction of copper due to the operation on the purge flow rate Q. The dotted curve represents the reduction of copper due to the operation on the applied voltage E. At t = 300 hours, the increment of purge flow rate reduces AXI by 2.9 %. On the other hand, the increment of applied voltage reduces AX1 by only 0.23 %. Figure 3 (right) signifies the importance of the purge water in the electrokinetic remediation. The total copper is globally carried away from the column with purge water effluent, where as electromigration greatly contributes to local removal from the anode region.
6. CONCLUSIONS The results of the numerical simulations show that: (1) electromigration is effective especially for the local removal of heavy metals (rebuild of distribution of heavy metal between the anode and the cathode); (2) heavy metal is removed from the upstream anode region and accumulated in the downstream cathode region; (3) to carry away the heavy metal from the aquifer, the hydraulic flow by purge water is essential; and (4) the aquifer pH distribution, which shows low pH in the anode and high in the cathode with sharp increase, is well reproduced. At the moment, it may be fair to say that the model developed here is adequate for obtaining first-approximation of heavy-metal movement.
REFERENCES
1. A. E Shapiro, E C. Renaud and R. E Probstein, Physico Chemical Hydrodynamics, 11(5/6) (1989) 785-802. 2. A. N. Alshawabkeh and Y. B. Acar, Environ. Science Health, A27(7) (1992) 1835-1861. 3. C. J. Bruell, B. A. Segall and M. T. Walsh, J. Environmental Engineering ASCE, 118(1) (1992) 68-83. 4. M. R. Taha, Y. B. Acar and R. J. Gale, In E G. Marinos et al. (eds), Engineering Geology and the Environment, Balkema, Rotterdam, 2 (1997) 2209-2214. 5. Y. Inoue and W. J. Kaufman, Health Physics, 9(7) (1963) 705-715. 6. M. A. Nekrasova, M. A. and V. A. Korolev, In E G. Marinos et al. (eds), Engineering Geology and the Environment, Balkema, Rotterdam, 2 (1997) 2047-2052.
In-situ electrokinetic remediation of g r o u n d w a t e r c o n t a m i n a t e d by h e a v y metal S. Shiba a and Y. Hirata ~ a Department of Chemical Science & Engineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.
One of the most cost-effective in-situ technologies for groundwater remediation is electrokinetic method, which utilizes electromigration of pollutants in pore water. Numerical simulations by a mathematical model have been done, in order to make clear the changes of the pollutant distributions in aquifers and the effects of the operational variables on the removal efficiency. The results of the numerical simulations show that: (1) electromigration is effective especially for the local removal of heavy metals (rebuild of distribution of heavy metal between the anode and the cathode); (2) heavy metal is removed from the upstream anode region and accumulated in the downstream cathode region; and (3) to carry away the heavy metal from the aquifer, the hydraulic flow by purge water is essential.
1. INTRODUCTION The need for water resources is increasing more and more these days not only in developed countries but also in developing countries. Because surface water resources have become more fully developed, interest has increased in the development of groundwater. It is generally recognized that both aspects of water resources, quantity and quality, are important and interdependent. However, in many aquifers groundwater has been seriously contaminated by the migration of various hazardous organic and inorganic chemicals from the disposal of municipal and industrial wastes. Many new techniques have been presented for remediation of groundwater and soil. Among them, one of the most cost-effective in-situ technologies is electrokinetic remediation, which utilizes electromigration of pollutants (especially heavy metals) in pore water of aquifers. However, understanding of electromigration of pollutants in pore water is not only experimentally but also theoretically quite insufficient. This is because the technique is relatively new and innovative, although for dewatering of soils civil engineers have practiced the electrokinetic method which utilizes electroosmotic flow in pore water. This paper aims to develop a mathematical model based on physicochemical mass transport theory and to simulate the non-steady characteristics of the electromigration of heavy metals in pore water of aquifers. However, only a few studies (Shapiro [1 ], Alshawabkeh & Acar [2], Bruell et al. [3], and Taha et al. [4]) can be utilized for construction of the mathematical model which takes the electromigration of heavy metals into consideration.
884
4 Purge Water Q [
..........
...........
!
1
i
I
,
r
I
.,
,
I
!
I
' l
i
I
i
Inaow=
<=o
[
o =3,.
9
i : :,L_ _~-~_teAnion(-) x
~
' , i
<=o
o
Irlowl--> o
/
Cation(+)o=~[--
Anode
',
,
I
n
-~Outflow _ :,
Cathode
'i : " x
,
x
,,
Figure 1. Electrokinetic remediation. Chemicals, i.e., cations (heavy metals) and anions are transported by the hydraulic flow caused by purge water in conjunction with electromigration.
2. E L E C T R O M I G R A T I O N OF HEAVY METAL Heavy metals get ionized in groundwater (e.g. Cu 2+, Pb 2+, and so on), which is an essential condition for the application of electrokinetic remediation technique. This technique utilizes electromigration of the charged species. It consists of: (1) installing electrodes (anode and cathode) into the aquifer saturated with the groundwater or the purge water, (2) injection of the purge water into the anode well, and (3) applying a fixed low voltage direct current between the electrodes, as is shown in Figure 1. Both the injection of the purge water and the drainage of the polluted water cause a hydraulic flow from the anode well to the cathode well in the aquifer. The electric force induced by the applied voltage accelerates heavy metal migration in pore water, co-working with the hydraulic flow caused by the injection of purge water. The electrokinetic driving force, which acts on the ionized heavy metals (cations) and causes the electromigration velocity, is directed towards the cathode and superposed on the hydraulic flow so that heavy metals (cations) are accelerated towards the cathode well, where they are swept out through the drainage of waste water. On the other hand, because electrokinetic driving force on anions is directed toward the anode, i.e., in the opposite direction to the hydraulic flow, the anion migration is retarded or even driven in opposite to the hydraulic flow. Then, the heavy metals (cations) are separated from anions and transported with increasing speed to the downstream well.
3. CHEMISTRY IN E L E C T R O K I N E T I C REMEDIATION Three types of chemical reactions appear in this model. The first is the homogeneous liquidphase reaction, which generates the molar rate R~q and is included in the governing equation as the reaction term. The second is the heterogeneous interface reaction (adsorption and desorption) between the pore water and the soil surface, which generates the molar rate R~p and is taken into account by introducing the retardation factor. The third is the electrode reaction,
885
which generates the flux J~*kdue to the production of H + and OH-. J~k is combined with the flux due to the advection and the diffusion. 3.1. Liquid-Phase reaction The contaminant treated here is copper sulfate (CuSQ) as a typical compound of heavy metal. In this case the homogeneous liquid-phase chemical reactions in the pore water can be expressed as follows:
CuSO4 ~ Cu 2+ -k- SO~HSO~- ~- H + + SO]-
(K1); (K4);
H2SO4 ~ H + + HSO4 H20 ~ H + + OH-
(K2)
(1), (2)
(Kw)
(3), (4)
where K1, K2, K4 and Kw are the dissociation constants. Because dissociation constant K2 is large (>> 1), H2SO4 is dissociated almost completely into H + and HSO4. This means that Equation 2 for H2SO4 can be omitted, in other words, it can be assumed that [H2SO4] ~ 0. Then the chemical species treated here are CuSO4, Cu 2+, SO]-, HSO4, H +, OH-. The concentrations of chemical species, which appear in the above reactions and is treated here, are defined as follows: (C1, C2, C3, C4, C5, C6) -- ([CuSO4], [Cu2+], [SO2-], [HSO~-], [H+], [OH-])
(5)
where Ck = liquid-phase concentration of chemical species k (mol/L). From the liquid-phase chemical reactions described by Equations 1, 3 and 4, the reaction rates for species k, R~ q, are given as" /~q :
--( kl+C1 - k l- C 2 C 3 ) ;
/ ~ q "- kl+C1 - kl-C2C3
t ~ q : kl+C1 - k1-C2C3 -+- k4+C4 - k4-C3C5 ;
/~q m_ k4+C1 - k4_C3C5 + kw+ - kw_C5C6 ;
/~q = k4+C4 - k4-C3C5 /~q = kw+ - kw_C5C6
(6), (7) (8), (9) (10), (11)
where kl+/kl_ = K1; k4+/k4_ = K4; and kw+/kw_ = Kw. K1, K4 and Kw are easily obtained, although it is not so easy to estimate the reaction rate constants ki+ and ki_. 3.2. Interface reaction Adsorption and desorption of chemical species k at soil surface is represented by:
X~q + Soil site ~ X~~
(Kdk)
(12)
where X~,q = chemical species k in liquid phase; X~~ = chemical species k in soil phase; and Kdk = distribution coefficient (L/g). 3.3. Electrode reaction At the electrodes, electrode reactions occur by passage of the electric current which is caused by the application of electric potential between the electrodes. At the anode, water gives electrons to the electrode and generates oxygen gas and H + ion. At the cathode, water takes electrons from the electrode and releases hydrogen gas and O H - ion. These reactions are described by:
2H20 - 4e- ~ O2(g) + 4H + ; where e- = electron.
2H20 + 2e- ~ H2(g) + 2OH-
(13),(14)
886 4. M A T H E M A T I C A L F R A M E W O R K The mathematical model developed here is the distributed type mass trasport model which predicts contaminants in the saturated zone. The model is composed of: (1) a kind of advective diffusion equation with homogeneous and heterogeneous chemical reactions and (2) its boundary conditions with electrode reactions. 4.1. Governing equation
Under steady uniform forced flow, the equation of conservation of mass (diffusion equation), which considers the advection, dispersion, and sorption for chemical species k in saturated homogeneous isotropic media, can be described as follows (Shapiro et al. [1 ]): c3 (nRdkCk) +
0"~
0 c~2Ck {(Us + * )Ck} = 79~ + nR~ q ~ZZ Uezk 0Z 2
(15)
where Ck = concentration of species k (mol/L); Us = pore flow velocity (cm/s); n = porosity of soil (-); Rdk = retardation factor (-) defined by Equation 17 mentioned later; R~q = molar rate due to liquid-phase chemical reactions (mol/L/s); and T~, and U~zk = effective diffusion coefficient (cm2/s) and effective electromigration velocity (cm/s), respectively, defined by:
(~)~, Uezk 13_ n 9 ) --" (~-~T)k, 7Uezk)
(16)
where 7- = tortuosity obtained experimentally (-); 9k = diffusion coefficient of chemical species k in pore water (cm2/s); and Ue,.k = electromigration velocity of species k (crn/s) defined by Equation 18 mentioned later. 4.2. Retardation factor
In order to take account of the heterogeneous reaction between pore water and soil (adsorption and desorption ), which generates the molar rate R~p, the retardation factor Rdk is introduced satisfactorily (Inoue & Kaufman [5]). The retardation factor is defined by: (1 - n)p
Rdk = 1 +
n
Kdk = 1 +
(1 - n)p C3Sk n
(17)
OCk
where p = density of soil solids (g/cm3); Kdk = distribution coefficient of species k (L/g); and Sk = absorbed concentration of species k per unit mass of soil solid (mol/g). The adsorption isotherms between Sk and Ck are generally expressed by a linear function. 4.3. Electromigration velocity
The electromigration is the core of this model. The electromigration velocity Uezk, which is proportional to potential difference, is given by: F
0r
Uezk = --~-'~ Zk ~)k ~ZZ -- Zk ~)k f (t, Z)
(18)
where F = Faraday's constant (C/mol); R = universal gas constant (J/K/mol); T = water temperature (K); Zk = charge number of species k (-); r = electric potential (V); and O r = potential gradient (V/cm); and f(t, z) = function of Ck. f(t, z) and effective current density are given by: ,
~2 (Zk~)kCk) 2 9
, T2 A
(19)
887
where i*(t) = effective current density (AJcm2); I(t) = electric current (A); and A = cross sectional area of soil column (cm2). The factor 103 to i s is due to the fact that units of Ck is mol/L.
4.4. Applied voltage and electric potential Integrating Or the applied voltage E, i.e., the potential difference between the anode (z = 0) and the cathode (z = L), is estimated by: E ( t ) = r (t, 0) - r (t, L) = ~
(20)
f (t, z) dz
As is seen from the above equation, the applied voltage E is usually time varying. However, it can be controlled to be constant.
4.5. Initial condition The initial condition for the governing equation, Equation 15, is simply given by: Ck = Ck0
at
t = 0
(21)
This equation specifies the distribution of the concentrations of chemical species k in pore water at t = 0. Ck0 is not necessarily uniform.
4.6. Boundary conditions The boundary conditions for the governing equation are given at the anode and the cathode to satisfy the continuity of the fluxes of chemical species k, i.e., J~. The production rates of H + and OH- can be expressed in terms of their fluxes J~H and J~ou, respectively, as follows: i* (J~*H, J~*OH)= ( 103 ~
103 '
(22) ZoHF/
where i* = effective electric current density (AJcm2). J*H and J:OH are superposed on di~ (for H +) and J ; H (for O H - ) , respectively. As i* = i~/n; ZH = +1; and Zon = --1, the boundary conditions in terms of fluxes of chemical species J~ can be stated as:
j~
=
at z = 0
(23)
0(7)~Ck) ) C k = {UsCk --103J~for O H - } Oz + (us 2r- u;zk UsVk , Fn ' for others ' at z = L
(24)
(9(~kCk) 2f_ (u s + --
J~ = -
OZ
* )Ck -Uezk
s k ?-tsGIN
Fn'
for others
Equation 15 can be solved numerically using the initial condition, Equation 21, and the boundary conditions, Equations 23 and 24.
5. NUMERICAL SIMULATION Using the finite element method, numerical simulations were carried out under the conditions shown in Table 1 (for the bench scale experiments under planning). Technical difficulty in numerical calculations arises from: (1) the advection in the governing equation (cause of unstable solution) and (2) the problematic elecroneutrality for the charged species (excess condition).
888 Table 1. Properties of soil column and operational variables of remediation. Property of soil column Length (L) 40 cm Cross sectional area (A) 100 cm 2 Porosity of soil (n) 0.4 Tortuosity of pore (7-) 1.5
Operational variable Flow rate of purge water (Q) 0.0010 and 0.0015 cc/s Applied Voltage (E) 2 and 3 V Acidity of purge water (pH) 7.0 Initial acidity of pore water (pH) 6.0
5.1. Practical procedure In the practical calculation procedure adopted here, the estimation of unknown constants ki+ is omitted by eliminating ki• from the chemical reaction term R~q in the governing equation. This makes a new set of differential equations about reduced number of new dependent variables (virtual concentrations). The new dependent variables are: CN1 "-- C 1 - [ - C 2 ;
CN2 -- C1-Jr- C3 - C4 ;
CN3 : C 4 - [ - C 5 - C 6
(25), (26), (27)
Seven governing equation for Ck are reduced to three for CNi. From the estimated virtual concentrations CN1, CN9 and CNa, the real concentrations Ck can be obtained at every time step of the numerical integration, using the equations from the dissociation reactions given by Equations 1, 3 and 4. The dissociation equations are: K l C l = C2 Ca ;
K4 C4 = C5 Ca ;
Kw = C5 C6
(28), (29), (30)
5.2. Electroneutrality Electroneutrality condition, which is said to hold closely in aqueous electrochemical solutions, is a troublesome problem, because it brings about an excess condition into the mathematical model. Unless in lumped models without advection and diffusion, the condition is thought to be held in the equilibrium (or steady) state in which concentrations are uniform. There seems to be no clear answer to the question that in what case the electroneutrality could be dropped or could not. However, here the condition is taken into account by adjusting C5 and C6, which are changeable to and from H20, at every time step of the integration of governing equations.
5.3. Distribution of total copper (X1) and acidity (pH) Electrokinetic driving force acts on copper ions (Cu 2+) resulting their migration toward the cathode. This phenomenon increases the copper ion concentration in the cathode. Consequently, the concentration in anode decreases. As a result, the gradient of the concentration curve, assumed to be approximately linear, increases as the process continues. The time variations of the distribution of total copper in the soil column are plotted in Figure 2 (left). As assumed, at any time the distributions of X1 are almost linear with respect to the distance from the anode. The distribution curves have ascending gradients except for the initial distribution (horizontal dotted line). This implies that copper is removed from the anode (upstream end), and transported and accumulated near the cathode (downstream end). That is, the remediation (local removal) in the soil column advances from the anode to the cathode with the passage of time. This tendency of distributions of the copper concentration accords qualitatively with the experimental results of heavy metal removal by Nekrasova and Korolev [6] using onedimensional soil column. The gradients of these curves become steeper and steeper with the
889
1.25
,_, 1.00
020106-F1
,
I Q = 0.0010 cc/s
t=0h t=40h t= 320h
. .<.; .......
E=3V
.... ,r
0.75
I
v
. . - :7.
r
s
,
l I r S !
Q
020106-F2 |
|
s
i; 0.50
..o-
t=0h t=80h t = 160h t = 320h
Q
....
t--, 0.25 0.00 0.00
f B4' Is d,r r II'
,,-
'
'
0.25
'
0.50
~
"~"~ -'-'~ ~ " ~
'
0.75
1.00
Q = 0.0010 cc/s
~"~-"
E=
I
i
0.00
3 V
I
0.25
I
0.50
l
0.75
1.00
Dimensionless Distancez (-)
Dimensionless Distancez (-)
Figure 2. Distribution of total copper X1 (left) and acidity pH (right) at various time. 0.0000 , , ~ . . , . . .
0.08
,
,
,
020106-F3
t= 300h 0.0025
0.04 ~
Q
..,,r
,--r
0.0050
0.00
,-, -0.04 9
......" " "
Q .,..~ .,..,
>
-0.08 0.00
|
i
I
AE=+I V (from 2 V) AQ = +0.0005 cc/s (from 0.0010 cc/s) |
i
0.25 0.50 0.75 Dimensioless Distance z (-)
m
0.0075
i~ AQ = +0.00005 cc/s (+50 %) AE = +1 V (+50 %)
0.0100
i
1.00
,
0
/
50
,
I
100
,
t
,
I
~ . ,
150 200 Time t (h)
Figure 3. Variation of total copper AX1 (left) and average total copper
,
250
300
AAX1 (right).
passage of time. However, the rate of increase in gradients become smaller and ultimately it approaches to zero. Temporal variation of pH (= - lOgl0([H+])) distribution between the electrodes is shown in Figure 2 (right). As discussed in the earlier section, electrode reactions produce hydrogen ions and hydroxyl ions at anode and cathode respectively (Equations 13 and 14). Therefore, it is reasonable to assume the decrease in pH value at anode and increase at cathode. As a result, one can expect an ascending gradient of a pH distribution curve. As depicted in Figure 2 (right), there is a sharp increase in pH values (pH jump) towards the cathode as simulation progresses. The increasing portions of the distribution curves are known as acid fronts. On the other hand, it decreases (pH drop) towards the anode. The decreasing portions of the distribution curves are known as base fronts. Based on these results, it may be inferred that the numerical simulations using the mathematical model developed here can well reproduce the pH distribution in the soil columns.
5.4. Operation of purge flow (Q) and applied voltage (E) The effects of the increments of the applied voltage E and the purge flow rate Q on the distributions of copper concentration X1 are demonstrated in Figure 3 (left). The effects are estimated by the variation of total copper concentration AX1 with the increments, A E and AQ. The ap-
890 plied voltage E and the purge flow rate Q are increased by 50 % from their standard values. The solid curve is for the variation with the applied voltage increment (AE = + 1 V) and the dotted curve is for the variation with the flow rate increment (AQ = +0.0005 cc/s). It distinctly clear from this figure that the applied voltage operation is more effective to separate and accumulate copper ions than the purge water operation. The local removal at the anode increases by 11% due to the applied voltage operation, where as this removal increases by 6.9 % due to the purge flow rate operation. This result means that the applied voltage is the appropriate operational variable for the local separation of heavy metals. The decreases in average total copper AX1 in the column for 50 % increment in the purge flow rate Q and applied voltage E are shown in Figure 3 (right). The solid curve represents the reduction of copper due to the operation on the purge flow rate Q. The dotted curve represents the reduction of copper due to the operation on the applied voltage E. At t = 300 hours, the increment of purge flow rate reduces AXI by 2.9 %. On the other hand, the increment of applied voltage reduces AX1 by only 0.23 %. Figure 3 (right) signifies the importance of the purge water in the electrokinetic remediation. The total copper is globally carried away from the column with purge water effluent, where as electromigration greatly contributes to local removal from the anode region.
6. CONCLUSIONS The results of the numerical simulations show that: (1) electromigration is effective especially for the local removal of heavy metals (rebuild of distribution of heavy metal between the anode and the cathode); (2) heavy metal is removed from the upstream anode region and accumulated in the downstream cathode region; (3) to carry away the heavy metal from the aquifer, the hydraulic flow by purge water is essential; and (4) the aquifer pH distribution, which shows low pH in the anode and high in the cathode with sharp increase, is well reproduced. At the moment, it may be fair to say that the model developed here is adequate for obtaining first-approximation of heavy-metal movement.
REFERENCES
1. A. E Shapiro, E C. Renaud and R. E Probstein, Physico Chemical Hydrodynamics, 11(5/6) (1989) 785-802. 2. A. N. Alshawabkeh and Y. B. Acar, Environ. Science Health, A27(7) (1992) 1835-1861. 3. C. J. Bruell, B. A. Segall and M. T. Walsh, J. Environmental Engineering ASCE, 118(1) (1992) 68-83. 4. M. R. Taha, Y. B. Acar and R. J. Gale, In E G. Marinos et al. (eds), Engineering Geology and the Environment, Balkema, Rotterdam, 2 (1997) 2209-2214. 5. Y. Inoue and W. J. Kaufman, Health Physics, 9(7) (1963) 705-715. 6. M. A. Nekrasova, M. A. and V. A. Korolev, In E G. Marinos et al. (eds), Engineering Geology and the Environment, Balkema, Rotterdam, 2 (1997) 2047-2052.