Modelling of transport and reaction processes in a porous medium in an electrical field

Pergamon

Chemical Engineering Science. Vol. 51, No. 19, pp. 4355 4368. 1996 Copyright ~i" 1996 Elsevier Science kid Printed in Great Britain. All rights reserved P I I : S0009-2509(96)00283-7 0009 2509/96 $15.00 + 0.00

MODELLING OF TRANSPORT A N D REACTION PROCESSES IN A POROUS M E D I U M IN AN ELECTRICAL FIELD JI-WEI YU* and IVARS NERETNIEKS Department of Chemical Engineering and Technology, Chemical Engineering, Royal Institute of Technology, S 100 44 Stockholm, Sweden (Received 27 November 1995; revision received 5 March 1996 and accepted l April 1996)

Abstraet--A numerical model has been developed to describe the transport and reaction processes in a porous medium in an electrical field. The model discretizes the one-dimensional porous medium by a number of compartments. The governing equations are formulated by material balance over each compartment. The model describes transport processes of advection, dispersion, ionic migration and electroosmosis. Various reactions such as aqueous complexation, precipitation/dissolution, and electrochemical reactions are treated by kinetic approaches. For fast fluid-phase reactions such as complexation, local equilibrium assumption may be applied in the model, which reduces the number of primary components and hence the number of governingequations. The sorption processes occurring at the surfaces of the porous medium are at the moment treated only by a single-componentlinear isotherm. Numerical solution to the model gives concentration and electrical potential distributions in the porous medium at different times and gives as well the electrical current history. Modelling results for three sample cases are reported to demonstrate the application of the model. In the first sample case, the removal of copper from sand by an electrical field is simulated. The second case concerns an electrokinetic soil remediation process with cathode rinsing. The third case models the selective filtering function of the ion exchange membranes. Copyright © 1996 Elsevier Science Ltd Keywords: Compartment model, ionic migration, electroosmosis,electroreclamation.

1. INTRODUCTION In chemical engineering and environment engineering, it is necessary in some circumstances to describe the transport and reaction processes in a porous medium which is under the action of an electrical field. One of the examples is electrokinetic soil remediation (Probstein and Hicks, 1993). Electrokinetic remediation is an innovative technology for soil remediation. This technology applies a low-level direct current to the polluted soil by electrodes placed in the ground to remove inorganic or organic contaminants from the soil. It is also conceivable to dig up the soil and to treat it by electrokinetic technology. The presence of an electrical field provides a driving force for the charged species to migrate in addition to other driving forces, such as pressure gradient, concentration gradient and thermal gradient. The ions in the pore fluid will migrate by the electric field, which is called ionic migration. The electrostatic force will also cause electrophoresis, i.e. the migration of small charged colloids that are suspended in the pore fluid. Another phenomenon that occurs in a porous medium in an electrical field is electroosmosis. This is caused by the drag interaction between the bulk of the liquid in the pore and the electric double layer that, like a single ion, is moved under the action of the electric field in a direction parallel to it. Electroosmo-

* Corresponding author.

sis produces a rapid advective flow of water in low permeability soils compared to a hydraulic flow (Pamukcu and Wittle, 1992). The electrical field is created by an external DC source and electrodes: the anode and the cathode. The application of the DC voltage to the electrodes results in electrode reactions. The electrode reactions involve transfer of electrons between the solution and the electrode material. Also, other homogeneous and heterogeneous reactions may occur in a porous medium even without an electrical field. Modelling studies concerning electrokinetic processes have been reported in the literature. Alshawabkeh and Acar (1992) have developed a generalized theoretical model describing reactive solute transport, and solutions of the model have been demonstrated for special cases where a constant current and zero hydraulic gradient are assumed. Association of hydronium and hydroxide ions is not incorporated in the solution. Shapiro and coworkers (Shapiro et al., 1989; Shapiro and Probstein, 1993) modelled the removal of contaminants from saturated clay by electroosmosis. The model equations were solved numerically using a low-order (linear) finite element method in the spatial domain and third-order Adams-Bashforth integration in time. These studies provide valuable understanding and experience on the modelling of electrokinetic processes. Romero et al. (1991) developed a compartment model for release of radionuclides by diffusion and

4355

JI-WEl Yu and I. NERETNiEKS

4356

advection. The compartment model assumes that the system to be modelled is built up by a number of compartments in contact with each other according to the geometry of the system. The concentration of the species in each compartment is assumed to be represented by an average concentration. It is essentially a flexible finite difference or integrated finite difference model. We have adopted the concept of the compartment model to the model developed in this study to simulate one-dimensional transport and reaction processes in a porous medium in an electric field. In this paper, we discuss the mechanisms involved in the transport and reactions in a porous medium in an electrical field. The numerical formulation of the model is presented in the paper. Simulations in three sample cases are reported to demonstrate the easy application and flexibility of the model. 2. TRANSPORT AND REACTION PROCESSES IN A POROUS M E D I U M IN AN ELECTRICAL FIELD

When the fluid in the pore volume of a porous medium is under the influence of an electrical field, there will be three transport mechanisms: migration of ionic species, electroosmosis, and electrophoresis. The ionic migrational mass flux is proportional to the ion concentration and the electric field strength, expressed as minus the gradient of the electrical potential 2i J/" = :=, u , C , V ( Izd

- 4~)

11)

where J~ is the migrational mass flux, zi the ionic charge number, Ci the concentration, ~b the electrical potential, and u~ the ionic mobility which is related to diffusion coefficient by Einstein relation (Probstein and Hicks, 1993): ui -

DilzilF - RT

(2)

where D~ is the diffusion coefficient, F the Faraday's constant, R the universal gas constant, and T absolute temperature. The electroosmotic mass flux is also proportional to the ion concentration and to the applied electric field strength. This flux is given by (Probstein and Hicks, 1993): Jf = keoC,V(-

O)

(3)

where Jf is the electroosmotic mass flux, and keo the electroosmotic permeability which depends on the ionic strength, viscosity and permittivity of the fluid and the properties of the porous medium (Alshawabkeh and Acar, 1992). It should be noted that the sign of keo depends on the surface charge of the porous medium. For clay surfaces, for example, that are usually negatively charged, k~o has a positive value. In the present study, we assume electrophoresis of limited importance in a porous medium size since the

colloids are restrained from movement by the stationary phase. Besides the above-mentioned mechanisms, significant mass transport processes in a porous medium may also include hydraulic advection and dispersion. These transport processes occur even without the electrical field. The dispersive mass flux of a species under a concentration gradient is expressed by the equation similar to Fick's law: (4)

J i° = - D L i V C i

where J~ is the dispersive mass flux and Du the longitudinal dispersion coefficient. Several compositional and structural variables will affect the contribution of the mass flux caused by each mechanism to the total mass flux: mineral composition of the medium, pore fluid composition and conductivity, electrochemical properties of the species in the pore fluid, and pore size, particle size and tortuosity of the porous medium. Common reactions occurring in a porous medium include complexation and redox reactions between the dissolved species, acid-base reaction, precipitation and dissolution. Aqueous complexation is the chemical process of ionic association, characterized by the stoichiometric coefficient and stability constant, which depends on the thermodynamic equilibrium constant, ionic strength, and the temperature. Redox reactions involve the transfer of electrons between the species and the change of oxidation states. This type of reaction is also characterized by the stoichiometric coefficient and equilibrium constant. Acid-base reaction involves transfer of protons and determines the pH in the pore fluid. This process is described by the stoichiometric coefficient and equilibrium constant of the acid-base reaction. Precipitation and dissolution processes are characterized by the solubility product. These reactions involve transfer of solutes between the solution and the solid phase. Solid surfaces of the porous medium will adsorb species, and ion exchange may occur at the solid surface. Both adsorption and ion exchange involve mass transfer between the solution phase and the solid surface. Adsorption is characterized by isotherms, which depend on properties of the adsorbates and the solid surface. Ion exchange is characterized by the ion exchange capacity and selectivity of the solid. Electron transfer takes place between the electrodes and the species in the pore solution by electrode reactions. Anode reactions involve oxidation of the species, in which the species lose electrons to the anode. Cathode reactions involve reduction of the species, in which the species gain electrons from the cathode. For example, the electrolysis of water involves the evolution of oxygen at the anode and the evolution of hydrogen at the cathode: 2 H 2 0 - 4e ~ O 2 ( g ) + 4H + (anode) 2H20 + 2e- ~ H2(g ) + 2 O H -

(cathode).

(5) (6)

4357

Transport and reaction models in a porous medium

i

k-I k k+l Fig. 1. Illustration of the compartment model.

The electrode reactions are controlled by the electrode potential of the reaction. Specially, the electrode potential when the electrode reaction is in equilibrium can be calculated by using the Nernst equation (Atkins, 1990). 3. NUMERICALMODEL In a compartment model, the system is considered to consist of a number of compartments. Each compartment can be a porous or non-porous medium characterized by a porosity. The composition in the compartment is assumed to be uniform. Thus, a onedimensional column can be considered as a chain of compartments. Figure 1 illustrates the one-dimensional compartment model. The anode is placed at one end of the chain, while the cathode is placed at the other end. The choice for the number of compartments involved is made with a compromise between the computation effort and the accuracy required. A material balance over compartment k gives a differential equation: dCik ~'k V k

dt

(7)

= W i k "~- Rik

where •k is the porosity of the compartment, V k the volume of the compartment, Cik the fluid phase concentration of species i in compartment k, W i g the net transfer rate of species i into the compartment, and Rik the production rate of species i in the compartment due to reactions (including electrode reactions if the compartment contains the electrode). The material balance for a precipitated species does not include the transport term (neglecting electrophoresis): dCmk ek V k ~ -

(8)

= Rink

where C,.k is the concentration of precipitated species m in compartment k, based on void volume, R,,k the production rate of precipitated species m in the compartment.

tion, one by dispersion/diffusion, one by ionic migration and one by electroosmosis. W i k = W F i k Jr- W D i k ~- W M i k -~ W E i k.

The net transfer rate into the compartment caused by pressure-driven advection is expressed as W v i k = A " t)'(Ciu -- Cik)

WDi k = A(Ji~ 1,k -- J i Dk . k + l )

J i k - l,k

OAx k +

(11)

where the flux is approximated by a linear-drivingforce expression, which is derived from Fick's law for steady-state diffusion: D = 2 ( C i k - 1 -- Cik) Jig 1,k AXk-1/DLik-1 + AXk/DLik

(12)

where Ax is the length of the compartment. For pure molecular diffusion, DLik c a n be calculated by (13)

DLi k = F.k"CkDmi

where D,.i is the molecular diffusion coefficient of species i, and rk tortuosity factor in compartment k. The value of the tortuosity factor is less than or equal to unity, depending on the property of the porous material. The flux Ji~.k+ 1 is also calculated by eq. (12) with k replaced by k + 1. The net transfer into the compartment caused by ionic migration is expressed by the following equation: A (J~k- l,k -- Ji~,k + l)

(14)

where the ionic migrational flux is calculated by the following equation based on eq. (1):

O A X k e k - 1 r k - 1Uik- 1 C i k - 1 + (1 -- O ) A x k_ I ek rkUik Cik 2(~bk-1 -- qSk) m

(10)

where A is the cross-sectional area of the compartment, v the superficial velocity through the compartment due to pressure gradient, Ci, the fluid-phase concentration of species i in the upstream compartment, i.e. compartment k - 1 or k + 1, depending on the fluid flow direction. The dispersion/diffusion rate into compartment k is calculated by the difference in the in- and out-fluxes averaged over the cross-sectional area:

W Mik :

3.1. M a s s t r a n s p o r t The mass transfer rate can be divided into four components. One is caused by pressure-driven advec-

(9)

(1 - O ) A x k_ ~

A X R - 1 + AXk

( 1 -- O) A x k e k _ 1 ZR - XUik - 1 Cik - x + OAXR 1 ek Zk Uik Cik 2 (C~k_ I -- C~R)

(1 -- O ) A x k + O A x k 1

A X k - 1 + Axk

for cations (15)

for anions

4358

JI-WEI YU and I. NERETNIEKS

where U~kis the ionic mobility for species i in compartment k, Ck the electrical potential in compartment k. In eq. (15), 0 denotes the upgradient weighting factor. This factor is used for the stability of the numerical solution when a great concentration or potential gradient exists. When 0 = 1, the variables are fully upgradient weighted, i.e. the variables are represented by their values in the upgradient compartment. When 0 = 0, the variables are fully downgradient weighted. There is no weighting if 0 = 0.5 and the variables are represented by the values that are resulted from linear interpolation. The flux J~.k+ ~ is also calculated by eq. (15), with k replaced by k + 1. The net transfer rate into the compartment caused by electroosmosis is expressed by the following equation: WEik =

A(d~'k

1,k - - Ji~,k+ 1)

( OAXk~k Jig 1,k

:

,

~

1

1 Zk 1 keok 1

-

1Cik 1

1

Q,k = 1-I a,kv,

+(1 -- O)AXk

~ l T L .x~ A k ~ _ [ l - ~ f f ) A - - X -k _ l-

where a~k = activity of species i in compartment k. For an aqueous species, its activity may be related to its concentration by an activity coefficient 7:

Cik

where Cst is the concentration at standard state. The activity of a precipitate is unity. It should be noted that for precipitation/dissolution reactions, the reaction rate constant incorporates the surface factors that actually vary with time and hence is not really constant. 3.3. Electric current and electrode reactions The Poissons equation for electrostatic potential or its equivalent Gauss' law reveals that any departure

1 ekZkkeokCik - - - l A X k

where keo k is the electroosmotic permeability in compartment k. The flux Ji~k.k+ 1 is also calculated by eq. (17), with k replaced by k + 1. 3.2. Reactions The total reaction production rate is the sum of the production rate contributed by each individual reaction. (18)

where vit is the stoichiometric coefficient of species i involved in reaction l, the sign of which depends on whether species i is a reactant or a product in the reaction, positive for reactant and negative for product, and Rtk reaction rate of reaction I in compartment k. The reaction rate expression depends on the kinetics of the reaction involved, which varies from reaction to reaction. To the sample cases in this paper, an empirical kinetic expression (Nyman, 1993) is adopted in the model for reaction rates. For a reaction / as follows: viIAi! :

0

(19)

i

where A.s are reactants or products, depending on the sign of the stoichiometric coefficient, the reaction rate is expressed as

Rtk = ~k Vk kl (Q~kKI -- 1)

(22)

aik = 7ik C St

2(qSk- 1 -- ~k) 1

~((1 -- O)AXkek l'Ck l keok-tCik-1 + OAxk lekZkk~okCik 2(¢k I -- Ck), (1 - O)AXR + O A X k - 1 A X k - 1 q- A X k

Rik = - - ~ v i l R l k l

(21)

i

(16)

where the electroosmotic flux is calculated by the following equation based on eq. (3):

e

compartment k, defined as

(20)

where kt is the reaction rate constant of reaction l, and K t the reaction equilibrium constant of the reaction. Qtk is the inverse reaction quotient of reaction I in

keo >1 0 (17)

keo < 0

from electroneutrality results in very large electrical forces. These large forces may restore charge balance on a time scale much faster than any other in the system, which suggests the electroneutrality being a good approximation. By assuming that the solution outside the electric double layer is electrically neutral, which means no net charge accumulation in each compartment, the electric current keeps constant throughout the one-dimensionalmedium. The electric current is caused by charge transfer by both dispersion and ionic migration of all charged species, assuming that the solid phase is not electrically conductive and assuming negligible current of double layer ions caused by the electric field or by a hydraulic flow. The advective transport of an electrically neutral solution gives no contribution to the electric current. With the notion of the compartment model, the electric current can be expressed as

It might be of interest to point out that in many multicomponent transport models a single value of diffusion coefficient should be used for different charged species in order to keep charge balance, while in this model individual values of diffusion coefficient for different species can be used. As long as the current is constant through out the column, as constrained by eq. (23), charge balance is held. At present, the model assumes that the anode reaction is dominated by a single electrode reaction, i.e. no multiple electrode reactions occur at the anode. Also the model assumes that the cathode reaction is dominated by a single electrode reaction, i.e. no multiple

Transport and reaction models in a porous medium electrode reactions occur at the cathode. The electrode reaction rates are related to the electric current through the medium by Ra = Rc = -

I F

(24)

where R, is the anode reaction rate, and Rc the cathode reaction rate. The expression for an electrode reaction depends on the specific electrode reaction. For example, a kinetic expression (Tarasevich et al., 1983) may be used for oxygen evolution reaction rate at the anode: R~ = 4A,k,C{~?;, 5 exp FE___& 2RT

-

FE,,

R,, = 2A
(26)

where A< is the contact area of the cathode, kc the cathode reaction rate constant, Cn+< the concentration of hydrogen ion in the solution adjacent to the cathode, and E< the cathode potential, i.e. the difference between the potential of the cathode and the potential of the solution adjacent to the cathode. The externally applied potential difference is equal to the electrode potential difference plus the potential drop through the medium: Eex = E, - g~ + A~b

(27)

where A~b = electrical potential drop through the medium, i.e. the potential of the solution adjacent to the anode minus the potential of the solution adjacent to the cathode. If the overpotential is negligible, the electrode potential may be simply replaced by the zero current electrode potential, which can be calculated according to thermodynamics regarding the solution composition, and thus the kinetic rate expression for the electrode reaction is not needed. 3.4. Fast reactions For some aqueous reactions, the reaction rate may be so high that local equilibrium is reached instantly in practice. For such fast reactions, the model adopts the local equilibrium assumption. For a reaction x as follows: 2 vixAix = X

by the mass action law: Cx =

(28)

i

where AixS are reactants and X is the reaction product. The concentration of the product can be calculated in terms of the concentrations of the reactants

Kx C 2, I]ia~'x

(29)

The kinetic expression is thus not needed in the model for such a reaction. However, the material balance for the reactants should account for the transport and the change in the concentration of the product. The resulting material balance for species i becomes (dCik

--

Cxjk d C l k \

d,

(25)

where Aa is the contact area of the anode, k. the anode reaction rate constant, CH+, the concentration of hydrogen ion in the solution adjacent to the anode, and Ea the anode potential, i.e. the difference between the potential of the anode and the potential of the solution adjacent to the anode. A similar kinetic expression (Gileadi, 1993) may be used for hydrogen evolution reaction rate at the cathode:

4359

= Wik + ~ v i i W x j k + Rik. J

(30)

In the above equation, the species are divided into two categories: primary species and secondary species, which is a concept adopted by a number of coupled transport reaction models (Yeh and Tripathi, 1991). The reactants as well as the products of slow reactions can be treated as primary species, while the secondary species are the products of fast reactions. In eq. (30) Cxj k is the concentration of secondary species j in compartment k and Wxjk is the net transfer rate of secondary species j into compartment k, which is defined in the same way as for primary species in eq. (9). The electric current can be expressed accordingly as I = AF

ZiJ°

l+k ~-

~ZxJxk

1,k -[-

x

l.k[

i

+ ~lZxJ~"k-,.kl). x

(31)

/

By introducing secondary species, the differential equations for material balance of fast reaction products may be omitted and thereby the computation effort for the numerical solution of the model may be reduced. 3.5. Adsorption with a single component linear isotherm In a variety of porous media, adsorption is a significant process that may influence the chemical speciation and transport. Adsorption with a single component linear isotherm has been taken into account in the model. The application of a linear isotherm should be restricted to the linear region of the adsorption isotherm obtained from experiments. The linear isotherm is often a good approximation to adsorption equilibria when the adsorbate concentration is very low, as in the case of trace contaminants in soils. A single component linear adsorption isotherm is expressed as q = KdC

(32)

where q is the concentration in adsorbed phase, based on void volume, and Kd the adsorption equilibrium constant.

4360

JI-WEI YU and I. NERETNIEKS

When adsorption with a single-component linear isotherm is considered in the model, with the assumption of local adsorption equilibrium, the resulting material balance equation becomes

The externally applied DC voltage is 30 V. There is no advective flow through the medium. In the simulation, the following aqueous reactions are considered: H ÷ + O H - = H20

dCik

ekVk (1 + Kdi~)--dT- + ~vij(1 + K~jk) J

Cxjk dClk~ =

Y ×~vtj Ctk dt J

C u 2+ ~-

Wik q- 2vijWxjk d- Rig

(33)

3.6. Solution to the model The ordinary differential equations in the numerical model are solved by using the standard ODE stiff problem solver DDRIV based on Gear's method (Kahaner et al., 1988). At each time step, the current, the electrical potentials, the electrode potentials, and the electrode reaction rates are calculated by the algebraic eqs (23)-(27) and, together with the equations for transport rates and reaction rates, eqs (9)-(22), they are used to evaluate the right-hand side ofeqs (7) and (8), used by DDRIV to integrate. When secondary species are involved, eq. (7) is replaced by eq. (30) or eq. (33); and eq. (23) is replaced by eq. (31). 4. SIMULATIONS F O R SAMPLE CASES

4.1. Removal of copper from sand by an electrical field In this sample case, the experiments of removal of copper from sand by an electrical field in a previous study (Li et al., 1996) are simulated by the model. The sand contains copper nitrate, with a copper content of 300mg/kg sand. To avoid precipitation of copper hydroxide in the sand due to high pH at the cathode, the sand is not directly connected to the cathode. Instead, a conductive solution column is inserted between the sand and the cathode, as shown in Fig. 2. The conductive solution contains 0.01 M potassium nitrate. The length of the sand column is 30 cm. The length of the conductive solution column is 60 cm.

= CuOH +

(36)

Cu 2+ + 3 O H - = Cu(OH);

(37)

Cu 2 + + 4 O H - = Cu(OH)~ .

(38)

The evolution of oxygen at the anode and the evolution of hydrogen at the cathode are assumed to be the dominant electrode reactions, as defined in eqs (5) and (6). The precipitation of copper hydroxide is considered in the simulation: Cu 2+ + 2 O H - = Cu(OH)2(s ).

I

anode

electric wire

t~e \

sand

\

glass filter

(39)

Because of lacking reaction kinetics data, the reactions are assumed to be instant. The activity coefficients are approximated by unity. In the simulation, the sand column is composed of nine equally sized compartments and the solution column is composed of 19 equally sized compartments. The porosity in the sand compartments is assumed to be 0.4. The reaction equilibrium constants, diffusion coefficients and ionic mobilities are listed in Table 1. The removal efficiencies resulted from both the simulation and the experiments (Li et al., 1996) are compared in Fig. 3. The simulation results shown in Fig. 4 exhibit the accumulation of copper near the pH jump. As shown in Fig. 5, the pH value jumps at the location about 0.5 m from the anode. The whole system is divided into a low pH region and a high pH region. Copper is mainly present in the form of cations in the low pH region, and these cations migrate toward the cathode and hence toward the pH jump. In the high pH region, copper is mainly present in the form of neutral Cu(OH)2 or anions and these anions migrate toward the anode and hence also toward the pH jump. The overall effect is that copper gathers at the pH jump. The insertion of the conductive solution between the

ampere meter

glass

(35)

Cu 2+ + 2 O H - = Cu(OH)2(aq)

J

where Kdi k is the adsorption equilibrium constant for primary species i in compartment k, and Kdxjk the adsorption equilibrium constant for secondary species j in compartment k.

OH

(34)

\ conductive solution

\ cathode

Fig. 2. Illustration of the experiments of removal of copper by an electrical field (Li et al., 1996).

436l

Transport and reaction models in a porous medium

s a n d a n d t h e c a t h o d e shifted t h e a c c u m u l a t i o n location o u t o f t h e s a n d a n d t h u s a h i g h e x t e n t o f c o p p e r r e m o v a l c o u l d be a c h i e v e d . F i g u r e 6 s h o w s t h e electrical p o t e n t i a l profile at t h r e e different times. It c a n be s e e n t h a t t h e p o t e n t i a l g r a d i e n t i n c r e a s e s w i t h t i m e at t h e p H j u m p . T h i s p o t e n t i a l g r a d i e n t c o u l d b e c o m e very s h a r p a n d a n y i o n s u n d e r this p o t e n t i a l g r a d i e n t w o u l d m i g r a t e o u t of t h e r e g i o n at a h i g h s p e e d t h a t w o u l d f u r t h e r l o w e r the conductance and hence make the potential gradie n t e v e n s h a r p e r . Finally, t h e w h o l e p o t e n t i a l differe n c e w o u l d fall w i t h i n a very n a r r o w region. O u t s i d e t h e n a r r o w r e g i o n t h e p o t e n t i a l g r a d i e n t w o u l d be very s m a l l a n d h e n c e t h e i o n i c m i g r a t i o n b e c o m e s insignificant. T h i s ' s w i t c h - o f f ' p h e n o m e n o n h a s b e e n i n d i c a t e d in t h e l i t e r a t u r e ( H i c k s a n d T o n d o r f , 19941. H o w e v e r , n o ' s w i t c h - o f f ' w a s o b s e r v e d in t h e experim e n t s by Li e t al. (1996). In t h e b e g i n n i n g , w h e n we did t h e s i m u l a t i o n of this s a m p l e case, we also e n c o u n t e r e d t h e ' s w i t c h - o f f ' after o n e day. W h e n we

Table 1. Reaction equilibrium constants, diffusion coefficients and ionic mobilities used in the simulation Reaction

lg K

H20 = H + + OH Cu 2+ + HzO = C u O H + + H + Cu 2+ + 2 H 2 0 = Cu(OH)2 + 2H + Cu 2+ + 3 H 2 0 = Cu(OH) 3 + 3H + Cu 2+ + 4 H 2 0 = C u ( O H ) ] - + 4H + Cu 2+ + 2 H , O = Cu(OH)z(s) + 2H +

Species

14 - 8.00 - 13.7 - 26.9 39.6 7.67

Diffusion coefficients (m2/s}

Ionic mobilities (mZ/s V)

9.31x10 9 7 . 1 4 x 1 0 10 1.90x 10 - ° 1.96 x 10 - ° 5.30x10 9 1.43 x 10 -9 1.43 x 10 9 1.43 x 10 -9 7 . 1 4 x 1 0 l0

3.62x10 ; 5.56x10 s 7.40x 10 s 7.62 x 10 -8 2.06x10 s 5.56 x 10 - s 0 5.56 x 10 8 5.56x10 s

H+ C u 2+

NO 3 K+ OH CuOH + Cu(OH)2 Cu(OH)~ Cu(OH) 2

i~ simulation

100 90u

z

uJ ~D m [A. kl. l.lJ

< > 0 ktJ

o,-

80706050403020100

98

[] experiment

98

100

6 day

11 day

8 68 67

1 day

2 day

3 day

DECONTAMINATION TIME Fig. 3. Removal efficiencies from the simulation, compared with the experimental results from Li et al. (1996).

-2

2.0x10

-

1 .5-

z

_o n.lz

1.0-

U.l

z oo

0.5-

0.0....

0.0

I ....

I ....

I ....

0.2

I .... 0.4

DISTANCE

I ....

I .... 0.6

FROM ANODE

i ....

I .... 0.8

(m)

Fig. 4. Concentration profiles of copper hydroxide precipitate from the simulation after 1 day, 2 days, and 3 days.

4362

JI-WEI Yu and I. NERETNIEKS .................

10

10s.

r-

'<

-10

n'-lO

~

Icul°H/4--I

0-15_

10

2

:::::::::::::::::::::::::::::::: _

o°-O'°°°'""

Z

1

[]

Z O 1 O

~

a ..~ -6

7-

-4

-20 I

,,,i

0.0

....

I ....

0.2

i ....

I . . . . . . . .

0.4

I ....

i ....

0,6

I

....

-2

0,8

DISTANCE FROM A N O D E (m)

Fig. 5. The pH profile and the concentration profiles of dissolved species from the simulation after 1 day.

30-

~

20-

0

-';~'%

I .... 0.0

I ....

I .... 0.2

I ....

I .... 0.4

I ....

I .... 0.6

I ....

I .... 0.8

DISTANCE FROM A N O D E (m)

Fig. 6. Electrical potential profiles from the simulation after 1 day, 2 days, and 3 days.

introduced a dispersion effect with dispersion coefficients of 8 x 10-8 m2/s plus the diffusion coefficients, the 'switch-off did not occur anymore and the copper removal was able to continue in the simulation. A lower value of the dispersion coefficient may also prevent the 'switch-off', but the removal efficiency in the later period will be much lower in this case. The dispersion effect should be present in any transport process in an electrical field. If there is any flow in the system, the flow will cause dispersion. Even in a stagnant fluid, the ionic migration will cause dispersion because a plug pattern of migration is only an ideal pattern. In a real system, the residence time of the ions migrating in the electrical field is more or less dispersed. When the dispersion effect gets large, the 'switch-off' effect becomes small. The current density history and the pH history at the location 45 cm from the anode resulted from the simulation are compared with the experimental results from Li et al. (1996), shown in Figs 7 and 8, respectively. The pH value shown in Fig. 8 increases to over 11 after a certain period, which reveals a shift

of the pH jump toward the anode. The shift of the pH jump could be caused by the adsorption of hydrogen ions on the sand. Because the hydrogen concentration in the sand increases with time, the adsorbed amount would also increase with time, which would retard the hydrogen ion migration toward the pH jump. Thus, the pH jump moves toward the anode. The concentration profiles of other ions in the solution would also influence the position of the pH jump. To simulate this shift of the pH jump, adsorption of H +, K +, NO3 and Cu 2+ ions on the sand is assumed. The values of the adsorption equilibrium constants were adjusted as listed in Table 2 and the tortuosity factor of the sand was adjusted to 0.6 in the simulation so that the simulated pH history fits the experimental data closely. The different parameters influence different parts of the simulation results. The fitting could thus be made by 'manually' adjusting one parameter at a time. To reduce the computation effort, the adsorption equilibrium constants and the tortuosity factor were roughly adjusted and may not result in a best fit. The adsorption of hydrogen ions and other ions may

Transport and reaction models in a porous medium

4363

1.0 E ~0.8

g

~0.6

z iii ao.4 tz ILl wO.2 n.(O 0.0

#" i

i

i

0

i

20

' ' I' 40

' ' I ' ''1 ' ' ' [ ' ' '1 ' ' ' I ' 60 80 100 120 140 TIME

(hr)

Fig. 7. Electrical current density history from the simulation ( - ), compared with the experimental results (e) from Li et al. (1996).

1412 10 -IQ,.

8 6

IlL

o

ooo o •

4

2 0

•

7

o

i,

,

,,

i

, , , , i ,

0

, , , i , , , , i , ,

20

, , i , , ,

40 TIME

, i

60

,

t

i

l

l

*

l

|

8O

(hr)

Fig. 8. pH history at the location 45 cm from the anode from the simulation - ) , compared with the experimental results (e) from Li et al. (1996).

Table 2. Adsorption equilibrium constants used in the simulation Ion

H+

K+

NO 3

Cu 2+

Ka

10

5

l

0.5

also cause the desorption of copper ions from the sand particles, which, however, was not considered in the simulation. To improve the simulation, the model needs to include a m u l t i c o m p o n e n t a d s o r p t i o n a n d / o r ion exchange model. In the simulation, the a d s o r p t i o n equilibrium constant of Cu 2 + ions on the sand was adjusted to 0.5 so that the copper removal efficiencies after 1 day a n d 2 days o b t a i n e d from the simulation are close to the experimental results. However, when using this value, the copper removal efficiency after 3 days deviates m u c h from the experimental result. It is not clear why the copper removal accelerated d u r i n g the third day in the experiment, a n d therefore the model is not able to simulate it.

It should be noted that the a d s o r p t i o n constants listed in Table 2 are just model parameters that m a y not agree with the real a d s o r p t i o n b e h a v i o r of the ions involved. T h a t the simulation result is sensitive to these parameters indicates that the soil-water interaction is i m p o r t a n t to the c o n c e n t r a t i o n a n d electric potential distribution a n d must be further studied. 4.2. Electrokinetic soil remediarion with cathode rinsing Hicks a n d T o n d o r f (1994) reported their experimental work where zinc was removed from Georgia kaolin with a particle size smaller t h a n 60/1m. As a measure of efficiency enhancement, the hydroxyl ions were prevented from entering the soil by rinsing the cathode with tap water so t h a t precipitation of zinc hydroxide could be eliminated. This process has been modelled by Jacobs et al. (1994). To model the rinsed cathode, however, they did not model the rinsing but instead they used a fixed pH at 7.0 in the cathode well. To d e m o n s t r a t e the flexibility of the c o m p a r t m e n t model, we simulate this process as

4364

Jl-WEl Yu and

], NERETNIEKS

a sample case, where the rinsing of the cathode with tap water is actually simulated and the pH of the cathode well is not fixed. In the simulation, the 20 cm long soil column was composed of 25 equally sized compartments, and the anode well and the cathode well, both 1.5 cm long, are each represented by a compartment. The diameter of the column is 3.2 cm. The tortuosity factor is 0.7. The externally applied DC voltage is 20 V. The rinsing water is assumed to contain 0.001 M sodium nitrate to represent the ions in the tap water. These parameters are the same as those of Jacobs et al. (1994). The porosity is assumed to be 0.4. To model the cathode rinsing, the material balance equation for the compartment representing the cathode has an extra flow term. That is, one term is added to the mass transfer rate defined in eq. (9):

model with a variable composition at the cathode. The peak of total zinc concentration was wellsimulated by the model. To fit this peak to the experimental data from Hicks and Tondorf (1994), two model parameters were adjusted: the dissolution/precipitation rate constant of Zn(OH)2, and the rinsing flowrate. The height of the peak is sensitive to the dissolution/precipitation rate constant of Zn(OH)2, while the position of the peak is sensitive to the rinsing flowrate. The pH resulted from the simulation did not agree well with the experiment data. One explanation could be that the electrode reactions would not be as fast as the simulation assumed. However, when slow electrode reactions were assumed, the pH results varied little, which indicates that pH is not sensitive to the electrode reaction kinetics. The soil may contain some minerals that have a pH buffering effect. To make a simulation considering this buffering effect, the model should include the reactions involving the buffer minerals. The model should be capable of including such reactions if the information about the buffer minerals would be available. Sorption may also have a buffering effect on the pH of the void solution, which, however, has not been included by the simulation. It could also be attributed to the assumption that the ionic mobilities of the species would not be corrected by the ionic strength. Figure 9 shows that in most part of the soil, pH is below 2, which corresponds to an ionic strength of order of magnitude 10-2 M. At high ionic strength, the ionic mobility will be reduced by the relaxation effect and the electrophoretic effect (Atkins, 1990). The reduction of ionic mobility decreases the electric current and hence the generation rate of hydrogen ions at the anode and hydroxide ions at the cathode. Consequently, the pH should be higher at the anode region and lower at the cathode region than that from the simulation. However, this will not be the major cause for pH deviation, as ionic mobilities do not vary so much with the ionic strength, while the experimental pH deviates the

(40)

W R i c = Fr'(Cg, - Cic)

where WRic is the net transfer rate into the cathode compartment caused by rinsing flow, Fr the volumetric flowrate of the rinsing flow, Ci, the concentration of species i in the rinsing flow, and Cic the concentration of species i in the cathode compartment. The rinsing flowrate is assumed to be 1 ml/s. Complexes ZnOH +, Zn(OH)2, Zn(OH)3 and Zn(OH) 2 are included in the simulation and their mobilities are assumed to the same as Zn 2 +, except that the mobility of Zn(OH)2 is zero. The initial composition of the system used in the simulation is the same as that of Jacobs et al. (1994). The aqueous-phase reactions are assumed to be instant, while the reaction rate constant for the dissolution/precipitation of Zn(OH)2 is 2.2 × 10 -5 mol/m3s. The simulation assumes dispersion coefficients which are 10 -9 m2/s plus the diffusion coefficients. The pH profile and the total zinc concentration profile after 9 days from the simulation were shown in Fig. 9, compared with the experimental results from Hicks and Tondorf (1994). This sample case illustrates that the compartment model easily incorporates the cathode rinsing into the

•

-10

0.4-. total Zn (sim.) .......... pH (sim.) [ ] total Zn (exp.) O pH (exp.)

z" _O 0.3-

W

o z

0.2-

o z

0.1

8

/

<

oo

o"

N _.,.I

0

~

0

0

o . ......... ..,,..

....g,

N.

rl

0.0-

.... 0

6 -r

4

o

2

' ......

_

....

0.0

................

I , ' ;"i 1 . . . . 0.2

U

I ....

I

jll,~Jllll

0.4

I ....

0.6

I ¥'

''1

illl

0.8

I

0

.... 1.0

NORMALIZED DISTANCE FROM ANODE

Fig. 9. pH profile and total zinc concentration profile of dissolved species after 9 days. Comparison between the simulation results in this study and the experimental results from the literature (Hicks and Tondorf, 1994).

Transport and reaction models in a porous medium

4365

11.0-

10.5-

10.0-

9.5

9.0

._1

....

i ....

I .... 2

0

t .... I ....

i ....

4

I ....

J ....

6

I ....

i ....

8

TIME (day)

Fig. 10. Simulated pH history in the cathode compartment.

pulse pump

soil

an g e

membrane

tl.

I

stirrer

I

Fig. 11. Illustration of electrokinetic soil remediation with ion exchange membranes (Jensen et al., 1994).

simulated value by nearly one at the anode region. Although not accurate, the simulated pH history shown in Fig. 10 indicates that the pH value in the cathode compartment is not necessarily constant. It varies with time and depends on the rinsing flowrate. 4.3. Eleetrokinetie soil remediation with ion exchange membranes Jensen et al. (1994) used strong ion exchange membranes to separate the soil from the electrode chambers to ensure high effectiveness of the current with respect to removal of heavy metals. In their method, an anion exchange membrane is placed in front of the anode and a cation exchange membrane is placed in front of the cathode. Due to the cation exchange membrane, the negatively charged ions cannot penetrate from the cathode compartment into the soil, and the positively charged ions in the anode compartment are prevented from passing into the soil by the anion exchange membrane, as illustrated in Fig. l l

(Jensen et al., 1994). To keep the pH constant at the electrodes, the solution in the anode and solution in the cathode are circulated to a flask where they are mixed and neutralized with each other. The circulation is so arranged that electrical shortcut is avoided. Also, pH is kept constant by continuous addition of nitric acid (Jensen et al., 1994). This process has been simulated by the compartment model in this work. The specially interesting part in the modelling here is to simulate the selective filtering function of the ion exchange membranes and the circulation and mixing of the solutions in the electrode compartment. To model that the cations in the anode compartment cannot enter the soil, the term in eq. (11) is set to zero for cations which corresponds to dispersion/diffusion between the anode compartment and the soil compartment adjacent to it. Also the term in eq. (14) is set to zero for cations which corresponds to ionic migration between the anode compartment

4366

JI-WE1 Yu and I. NERETN1EKS

and the soil compartment adjacent to it. Similarly, to model that the anions in the cathode compartment cannot enter the soil, the term in eq. (11) is set to zero for anions which corresponds to dispersion/diffusion between the cathode compartment and the soil compartment adjacent to it. Also the term in eq. (14) is set to zero for anions which corresponds to ionic migration between the cathode compartment and the soil compartment adjacent to it. This can be easily done in the compartment model. Advection including electroosmosis is not considered in this simulation. To model the circulation and mixing of the solutions in the electrode compartments, an additional compartment is added, representing the solution in the mixing flask. Material balance over the compartment is made according to the circulation flow and the concentrations in the mixing flask and in the electrode compartments as well as the reactions involved: dfim

(41)

Vm---~-- = Fc(Cia 4- Cic - 2Cim) Jr- Ri,.

where V,, is the volume of the mixing flask compartment, Ci,. the concentration of species i in the mixing flask compartment, F+ the flowrate of the circulation to the anode, equal to that of the circulation to the cathode, Ci, the concentration of species i in the anode compartment, C~c the concentration of species i in the cathode compartment, and R~,, the production rate of species i in the mixing flask compartment due to reactions. Extra flow term due to the circulation is added to the mass transfer rate defined in eq. (9) for the electrode compartments: (42)

W c i = F c ' ( C i m - Ci)

where Wc~ is the net transfer rate into the anode/cathode compartment caused by the circulation flow, and Ci the concentration of species i in the anode/cathode compartment.

Table 3. Parameters used in the simulation Parameter

Assumed value

Soil length (cm) Cross-sectional area (cmz) Porosity of the soil Tortuosity factor Volume of the mixing flask (l) Circulation flowrate (1/min) Initial content of Zn (mg) Concentration of NaNO3 in the mixing flask (M) pH in the mixing flask Leakage of the membrane (%) Electric current (mA)

3 4.2 0.4 1 1 0.6 8 0.01 2.2 6 5

The parameters used in the simulation are listed in Table 3. The pH in the mixing flask is held constant. The electric current is held constant, while the voltage changes due to the change in the conductance, as it was in the experiments of Jensen et al. (1994). In addition to zinc, the soil is assumed to contain 230 mg of calcium, for which an adsorption equilibrium constant Kd of 25, as defined in eq. (32), is assumed. The reaction products considered in the simulation include H20, ZnOH +, Zn(OH)2, Zn(OH)3, Zn(OH)4z-, CaOH +, and Zn(OH)2 precipitate. The aqueousphase reactions are assumed to be instant, while the reaction rate constant for the dissolution/precipitation of Zn(OH)/ is 2.2 x 10-5 mol/m3s. The simulation assumes dispersion coefficients which are 10 -9 m2/s plus the diffusion coefficients. The removed fraction of zinc and the electrical potential difference over the electrodes as functions of time from the simulation were shown in Fig. 12, compared with the experimental results from Jensen et al. (1994). This sample case demonstrates that the compartment model can easily be modified to model the selective filtering of the ion exchange membranes and to model the circulation and mixing of the solutions in

100

-

Zn removed (sim.) .......... Voltage (sim.) Zn removed (exp.) • Voltage (exp.)

8O -

.

•

. •

•

3O

• ~ , ~ m

25~

•

60

2o~

~ 4o

......

rr

. .....

. ............

- ....... ~'+"'l

....

~ 2o "[

0

'

'

................................... .......................

5

•

'

'

I

50

'

15m tO~

""

• ..Y I~""-

0

35

*

'

'

'

I

. . . .

,

TIME

(hr)

1 O0

150

. . . .

I

200

. . . .

I

o

250

Fig. 12. The removed fraction of zinc and the electrical potential difference over the electrodes versus time. Comparison between the simulation results in this study and the experimental results from the literature (Jensen et al., 1994).

Transport and reaction models in a porous medium the electrode compartments. However, the simulation results do not agree with the experimental results. The experimental data from Jensen et al. (t994) showed that the removal of zinc increases rapidly within a certain period, which the model failed to simulate. The difficulty here is the lack of information on the right initial composition in the soil in the simulation. Zinc may be adsorbed or present in other minerals. The desorption and dissolution of zinc may be influenced by other components and the pH value. In the experiments ofJensen et al. (1994), there were small amounts of other metals in the soil, which are not included in our simulation. Calcium minerals may be present instead of simply adsorbed calcium. It was found during the simulation, that the ion exchange membranes should not prevent the passing of counter-ions to a 100%, otherwise the potential drop would have increased rapidly to switch off the process in an early stage. A 6% of leakage of the membrane was chosen to make the simulated voltage close to the experimental data.

that are important and have not yet been included in the simulations of the sample cases. Our attempt to compare simulated results with experimental results has shown that more elaborate experiments must be performed to identify the exact reasons for the not always acceptable agreement. New experiments are being devised.

NOTATION

a A Aa Ac C C~t D DL E, E,. E,,x F Fc

5. DISCUSSION AND CONCLUSIONS

This paper has introduced a model with the compartment concept for transport and reaction processes in a porous medium in an electrical field. The sample cases have demonstrated the usage of the model. One of the advantages of the compartment model is its flexibility. It is also simple to use. The number of compartments may be increased to improve the discretization error at the cost of computation effort. The model can be a useful tool in the studies of the transport and reaction processes in a porous medium in an electrical field, The numerical model does not confine the reaction rates to the specific kinetic expressions defined in eqs (20), (25) and (26). If necessary, these equations can be replaced by any other appropriate kinetic expressions. It requires only minor work for modification of the computer program. Sorption processes can be significant mechanisms in the electrokinetic soil remediation. At present, the model considers only adsorption with a single cornponent linear isotherm. The model needs to be further improved to account for the multicomponent adsorption and ion exchange. It is necessary to include the dispersion mechanism in the model. The 'switch-off" phenomenon in the modelling of transport and reaction processes in an electrical field will diminish in the presence of dispersion effect. Dispersion is always present in the real transport processes in an electrical field because it can even be provoked by ionic migration. The model neglects the diffusion at the boundary of the system. This is not a problem to a closed system. The error could be significant when there is an open but very slow fluid flow and there is high concentration gradient at the boundary. Much of the work reported in this paper concerns the development of the model. There may be reactions

4367

F,

activity cross-sectional area of the compartment, m 2 contact area of the anode, m 2 contact area of the cathode, m 2 concentration, mol/m 3 concentration at standard state, mol/m 3 diffusion coefficient, m2/s longitudinal dispersion coefficient, m2/s anode potential, V cathode potential, V externally applied potential difference, V Faraday's constant, C/mol volumetric flowrate of circulation to anode/cathode, m3/s volumetric flowrate of the cathode rinsing flOW, m 3 / s

I dD d" jm k k,,

electric current, A dispersive mass flux, mol/mZ s electroosmotic mass flux, mol/m 2 s ionic migrational mass flux, mol/m2 s reaction rate constant, mol/m3 s anode reaction rate constant, mol~5/

k,. keo K Ka q Q

cathode reaction rate constant, m/s electroosmotic permeability, m2/s V reaction equilibrium constant adsorption equilibrium constant concentration in adsorbed phase, mol/m 3 inverse reaction quotient, defined in eq. (21) universal gas constant, J/K mol anode reaction rate, mol/s cathode reaction rate, mol/s production rate of species i in compartment k due to reactions, mol/s reaction rate of reaction l, mol/s absolute temperature, K ionic mobility, m2/s V volume of a compartment, m 3 net mass transfer rate into a compartment, mol/s ionic charge number

m 35 s

R R, Rc Rik R~ T u V W z

Greek letters 7 activity coefficient Ax length of the compartment, m A4, electrical potential drop through the medium, V e porosity 0 upgradient weighting factor

4368 ~il

JI-WEI YU and I. NERETNIEKS stoichiometric coefficient of species i involved in reaction l, positive for reactant and negative for product tortuosity factor superficial velocity of flow due to pressure gradient, m/s electrical potential, V

REFERENCES

Alshawabkeh, A. N. and Acar, Y. B., 1992, Removal of contaminants from soils by electrokinetics: A theoretical treatise. J. Environ. Sci. Hlth A27, 1835-1861. Atkins, P. W., 1990, Physical Chemistry, pp. 260-266. Oxford University Press, Oxford. Gileadi, E., 1993, Electrode Kinetics for Chemists, Chemical Enyineers, and Materials Scientists, pp. 164-166. VCH Publishers, New York. Hicks, R. E. and Tondorf, S., 1994, Electrorestoration of metal contaminated soils. Environ. Sci. Technol. 28, 2203-2210. Jacobs, R. A., Sengun, M. Z., Hicks, R. E. and Probstein, R. F., 1994, Model and experiments on soil remediation by electric fields. J. Environ. Sci. Hlth A29, 1933-1955. Jensen, J. B., Kubes, V. and Kubal, M., 1994, Electrokinetic remediation of soils polluted with heavy metals. Removal of zinc and copper using a new concept. Environ. Technol. 15, 1077-1082. Kahaner, D., Moler, C. and Nash, S., 1988, Numerical Methods and Software. Prentice Hall, Englewood Cliffs, NJ.

Li, Z., Yu, J.-W. and Neretnieks, I., 1996, A new approach to electrokinetic remediation of soils polluted by heavy metals. J. Contaminant Hydrolooy 22, 241-253. Nyman, C., 1993, Development of 'CHEMFRONTS', A Coupled Transport and Geochemical Proyram to Handle Reaction Fronts, pp. 16-21. Licentiate treatise, Royal Institute of Technology, Stockholm. Pamukcu, S. and Wittle, J. K., 1992, Electrokinetic removal of selected heavy metals from soil. Environ. Progress 11, 241-250. Probstein, R. F. and Hicks, R. E., 1993, Removal of contaminants from soils by electric fields. Science 2611, 498-503. Romero, L., Moreno, L. and Neretnieks, I., 1991, A compartment model for solute transport in the near field of a repository for radioactive waste. Paper presented at the MRS meeting, Strasbourg, 4-7 Nov. Shapiro, A. P. and Probstein, R. F., 1993, Removal of contaminants from saturated clay by electroosmosis. Environ. Sci. Technol. 27, 283-291. Shapiro, A. P., Renaud, P. C. and Probstein, R. F., 1989, Preliminary studies on the removal of chemical species from saturated porous media by electroosmosis. PhysicoChem. Hydrodynamics 11, 785-802. Tarasevich, M. R., Sadkowski, A. and Yeager, E., 1983, Oxygen Electrochemistry. Comprehensive Treatise of Electrochemistry, Vol. 7 (Edited by Conway, B. E., Bockris, J. O'M., Yeager, E., Khan, S. U. M. and White, R. E.), pp. 301 398. Plenum Press, New York. Yeh, G.-T. and Tripathi, V. S., 1991, A model for simulating transport of reactive multispecies components: Model development and demonstration. Water Resources Res. 27, 3075-3094.