Environ. Sci. Technol. 1998, 32, 3778-3784
Electrophoretic Repair of Impoundment Leaks: Analysis and Verification with Experimental Data M. YAVUZ CORAPCIOGLU,* KIRAN K. R. KAMBHAM, AND KAGAN TUNCAY Department of Civil Engineering, Texas A&M University, College Station, Texas 77843-3136
Sealing leaks which have developed in liquid surface impoundments with geomembrane liners may be difficult due to aged liner material. A technique utilizing electrophoresis which causes clay particles to be attracted toward leaks when an electric field is applied provides a costeffective method for repair. This study presents an experimentally verified methodology to predict electrophoretic sealing of in-service geomembrane liners. The methodology includes a procedure to simulate axis-symmetric electrophoretic cake formation and a numerical technique to solve the electric field for voltage gradients. Path lines of solid particles are generated by superposing electrophoretic and Stokes’ settling velocities. A numerical method to obtain a steady-state cake profile by conserving solids mass and an approach which uses path lines to simulate transient cake formation are described. For an initially uniform suspension, final and transient cake profiles are obtained under varying conditions. The effects of voltage difference, surface electrode size, and initial bentonite concentration on cake formation are discussed. In general, a higher voltage difference or a wider surface electrode accelerates the cake formation process. For efficient cake formation, the surface electrode should be located close to the water surface over the leak. A comparison of theoretical predictions with previously obtained experimental data shows a reasonable agreement. The analysis presented in the study provides a relatively inexpensive and useful tool in the implentation of an in situ field operation.
Introduction Synthetic geomembrane liners, which have been installed in liquid waste impoundments to limit the seepage of contaminants into the subsurface, may develop leaks. Liner leaks can be detected by an electrical leak detection method which utilizes the high electrical conductivity of water seeping through the leak. Usual repair methods for in-service geomembranes require the removal of liquid from the impoundments. Aside from prohibitive cost, these repairs pose dangerous risks to workers as well as the possibility of additional damage to aged liner material. Hence, a costeffective and reliable repair method is needed for leaking geomembranes. An electrophoretic sealing method to repair in-service geomembrane leaks by attracting clay particles toward the leaks was proposed by Darilek et al. (5) * Corresponding author phone: (409)845-9782; fax: (409)862-1542; e-mail:
[email protected]. 3778
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Electrophoresis is an electrokinetic phenomena involving the migration of electrically charged suspended particles under the influence of an externally applied electric field. When a direct current (DC) voltage is impressed across the electrically insulated geomembrane liner by placing an electrode (cathode) in the liquid and another electrode (anode) in the earth outside the impoundment, current flowing through the leaks in the plastic liner establishes an electric field at the leaks (Figure la). When a clay suspension (e.g., bentonite) is introduced into the impoundment, the negatively charged clay particles migrate toward the anode under the influence of the applied electric field. These particles deposit over the leak and form a clay cake when the subgrade soil is fine enough to strain particles. Darilek et al. (5) noted that the positions of the electrodes are not critical in most cases where the leaks are not very large. It should be noted that the electrophoretic forces are much smaller than the hydraulic force of water flowing freely through a leak. Therefore, the electrophoretic leak-sealing process must be applied with little or no water flow through the leaks. Prior to electrophoretic treatment for double liner systems, the leak detection zone between the liners should be flooded to the same level as the water in the impoundment, so that water will not flow through the leaks. For single geomembrane liners, the composition of subgrade soil is usually compacted clay or another relatively impermeable soil. The clay will contain the leakage, and the leak rate will diminish, thereby allowing application of the method (5). A geotextile placed under the geomembrane liner improves electrophoretic sealing. Sand under the liner also provides a strong base for a tighter clay cake. The clay incrustation penetrates the geotextile or sand serving as a matrix to further strengthen the cake (5). The effectiveness of the in situ electrophoretic sealing method was demonstrated through laboratory-scale experiments (5, 17) and a full-scale test (5). In a laboratory test with an 8 by 10 mm size elliptic leak, electrophoretic sealing with bentonite clay reduced the leakage rate by a factor of 1667 (5). In the field test conducted by Darilek et al., a 10 mm-diameter hole was drilled in the geomembrane liner of a 2500 square meter impoundment with 60 cm of deep water. A slurry prepared with 227 kg of bentonite was uniformly sprayed on the water surface. Before the application of electric current, a leak rate of 1120 mL/min was recorded from the impoundment. After a 34-day electrophoretic treatment under a constantly applied DC voltage of 50 V, the leakage was reduced more than 500-fold to 2.1 mL/min. The details of the field and laboratory experiments are discussed in Darilek et al. (5). Yeung et al. (17) and Chung (2) conducted bench-scale tests to study the growth rate of bentonite clay cake under controlled laboratory conditions. Laboratory experiments also demonstrated the effectiveness of the technique in forming clay cakes under varying conditions. Yeung et al. (17) also measured the electrophoretic mobilities of bentonite particles in different chemical environments to evaluate the feasibility of the technology in practical situations. In ionic solutions, although some flocculation of bentonite was observed, the efficiency of the technique was not affected. Kambham et al. (10) presented a theoretical analysis of one-dimensional electrophoretic cake formation and compression. Since the cake filtration process is used in various water treatment and industrial processes to remove suspended solids as well as in well-drilling operations, studies on cake filtration are found scattered throughout the literature in various disciplines (3, 4). A filtration process is referred to 10.1021/es980009p CCC: $15.00
1998 American Chemical Society Published on Web 10/02/1998
as unidimensional if the filter cake grows in such a manner that its slurry interface is parallel to the filter septum (12). On the other hand, filtration through a flat circular leaf is an example of a nonunidimensional filtration process. Shirato et al. (16) noted that the prediction of nonunidimensional cake formation is an important aspect of industrial filtration. Most of the nonunidimensional filtration problems are solved using the orthogonal curvilinear coordinates (7). Brenner (1) presented a mathematical model to predict the formation of an incompressible oblate-spheroidal cake on a flat circular filter cloth using an orthogonal curvilinear coordinate system. Gregor and Scarlett (6) presented a semianalytical procedure to simulate the formation of an incompressible oblatespheroidal cake. In this study, we present a numerical analysis of the electrophoretic sealing method. A numerical approach is developed to describe the formation of axis-symmetric cake on a circular leak without limiting the cake geometry to any ideal shape. This accounts for the gravity effect on the migration of solids during electrophoresis. As in most filter cake studies, it is assumed that the cake is incompressible and the solid particles cannot pass through the cake and leak. The compressibility of the cake can be assumed negligible for all practical purposes of a sealing operation. The field application of the technique and the relative dimensions of the cake (a few centimeters) and the impoundment (tens or hundreds of meters) justify these assumptions. Path lines of solid particles generated by superposing the electrophoretic and Stokes’ settling velocities are used to simulate the transient and steady-state cake profiles by mass conservation principle. Sensitivity of the model to applied voltage and the size of the surface electrode on cake formation are investigated. Comparison of simulation results with experimental data shows a favorable match.
Analysis of Electric Field The mobility of a solid particle in a suspension depends on its surface charge and the electrochemical properties of the dielectric fluid. When an electric field is applied, the electrophoretic velocity of a discrete particle vse is calculated by
vse ) m∇V
(1)
where V is the electric potential (voltage) and m is the electrophoretic mobility of the solid particle. The value of m is obtained either experimentally or from theoretical expressions available in the literature for a spherical particle (e.g., refs 8, 9, 13). An experimental determination would yield more realistic values for bentonite solutions. For a medium of uniform dielectric constant, the electric potential V caused by an external DC field is governed by Poisson’s equation
∇ 2V ) -
4πF
(2)
where F and are the electric charge density and dielectric constant of the liquid, respectively. A clay particle placed in an electrolytic solution adsorbs electric charges, resulting in the formation of an electric double layer. Since the fluid outside the thin double layer remains neutral, eq 2 reduces to Laplace’s equation (14). For an axis-symmetric problem in cylindrical coordinates r and z, it reduces to
1 ∂ ∂V ∂2V r + 2 )0 r ∂r ∂r ∂z
( )
(3)
where we assume that the streaming potential is negligible because of particle migration. Furthermore, changes in the
concentration of solids in the suspension and the formation of cake on the leak are assumed to have negligible effect on the conductivity of the medium. Thus, the electric field in the domain is independent of time. In this study, a constant DC voltage is applied to seal a circular hole at the bottom of a geomembrane liner. The geometry and location of the leak and the surface electrode are shown in Figure 1b. The leak at the bottom of the geomembrane and the plane electrode at the water surface are assumed to be circular, with leak and electrode centers aligned on a vertical line. This allows us to analyze the problem in axis-symmetric coordinates. Replicate experiments by Yeung et al. (17) demonstrate hemispherical bentonite cake formation which justifies the use of axis-symmetric coordinates. Furthermore, the shape and the location of the electrode can be easily handled in a field operation (5). In Figure 1b, H, R, R0, and RH represent the depth of water, radii of the lateral boundary, leak, and surface electrode, respectively. V+ and VH denote the electric potentials (voltages) at the anode and the cathode, respectively. The lateral boundary for a large impoundment can be either the geomembrane liner or the radius of influence of the electric field. In general, when the anode is not located under the leak, the voltage at the leak is reduced by a factor f *, where f * e1 depends on the electrical properties of the medium underlying the liner. Then, the voltage at the leak is given by V0 ) f *V+, and the effective voltage difference is defined as ∆V ) V0 - VH. For a constant, externally applied voltage difference, the solution for electric potentials is obtained using the finite element method. When finite element discretization is employed, the approximation of voltage distribution in terms of nodal values of V results in the following matrix equation [K]{V} ) [f] where [K] ) ∑[Ke] [f] ) ∑[fe]. Here the summation operator denotes the assemblage of element matrices. The element matrices are
Keij )
∫
Ωe
(
)
∂Ψie∂Ψje ∂Ψie∂Ψje + r dr dz; ∂r ∂r ∂z ∂z fie ) IΓerΨie
∂V ds (4) ∂n
where Ωe and Γe denote the domain and external boundaries of an element, respectively, and Ψje is the jth shape function and n denotes the outward normal to the element boundaries. In this study, we used four-node isoparametric elements. Once the solution for electric potential is obtained, the derivatives of V in r- and z-directions are computed at four Gaussian points of each element. The derivatives are used to calculate the electrophoretic velocity of solid particles. Because of the axis-symmetric nature of the problem, eq 3 is solved for V in the domain shown in Figure 1b,c. The leak opening and the surface electrode represent constant voltage boundaries. Because of the symmetry, the normal derivative of the electric potential on the z-axis is zero. The geomembrane liner at the bottom, air-water interface at the top, and the lateral boundary form no-flux boundary conditions for electricity. Generation of Path Lines. When the electric field is applied, clay particles in the suspension migrate as a result of electrical and gravitational forces. As noted earlier, if proper precautions are taken to reduce or eliminate the flow of water through the leak, the particle migration of the water flow generated by the leak can be neglected in comparison to electrophoretic and gravitational components (5). Furthermore, the positions of the electrodes are not critical in most cases where the leaks are not very large. With no large leaks, the current flow in the impoundment is small; consequently, the voltage drop in the water will be small, and the voltage across the liner will be relatively uniform. This is indicated by the low potential gradient at larger distances from the leak as shown by Darilek et al. (5). VOL. 32, NO. 23, 1998 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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FIGURE 1. Principle of the electrophoretic sealing of leaks in geomembrane liners [after Darilek et al., 1996]. Schematic diagram of (B) axis-symmetric electrophoretic cake formation. (C) Boundary conditions for electric field. Then, assuming that the electrophoretic and gravitational velocities are additive, we write radial and vertical velocity components of a solid particle as
∂V vr ) m ; ∂r
∂V vz ) m + vst (1 - φ)4.65 ∂z
(5)
where vst is Stokes’ settling velocity of a discrete particle. 3780
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Then, the magnitude of resultant velocity of solid particles is vs ) xv2r + v2z . The volume fraction of solids in the suspension is denoted by φsus (volume of solids/volume of suspension). The last term (1 - φ)4.65 in eq 5 accounts for the effect of particle concentration on the electrophoretic settling velocity of a particle in the suspension zone (15). The effect of concentration on the velocity of solids can be
FIGURE 2. Path lines for ∆V ) 10 V on a 50 mm diameter leak with a point electrode at the top. determined either experimentally or by using an experimentally validated empirical relationship like the one proposed by Richardson and Zaki (15). For concentrations such as the one used in this study (13.5 Kg/m3), φsus ) 13.5/ 2700 ) 0.005, (1 - φsus)4.65 = 1. The finite element grid used to solve for voltages and voltage gradients is also used to track the migration paths of solid particles in the suspension. Each cell is divided into four quadrants, and each quadrant is assigned a set of ∂V/∂r and ∂V/∂z values that correspond to the Gaussian point located in that quadrant. Then, r- and z-velocities are computed using eq 5. Initially, four particles placed in each cell are distributed in a geometrically uniform rectangular pattern. After each time step, every particle moves a distance equal to the product of the time increment and velocity. Then, the new position of a particle j is calculated by
rj(i + 1) + rj(i) + drj ) rj(i) + (ti+1 - ti)vjr(i) zj(i + 1) ) zj(i) + dzj ) zj(i) + (ti+1 - ti)vjz(i)
(6)
vj
where t is time, i is the index number to time, and r(i) and vjz(i) are the components of velocity at the previous position in r- and z-directions, respectively. drj and dzj are the distances traveled by the particle in r- and z-directions, respectively. Since the electric field is independent of time, at a given location vr and vz are constants as given by eq 5. Because vr and vz are not time dependent, path lines can be generated by tracking the particles that are initially located close to the water surface. Since the path followed by a particle is a single-value function of z, a path line can be represented by function R(z). For the sake of brevity, we denote a path line generated by the particle j as Rj and refer to it as “path line j”. To generate smooth path lines, a restriction is imposed on the size of the time increment in such a way that neither drj nor dzj exceed one-fourth of the grid spacing in r- and z-directions, respectively. When a particle crosses the lower boundary, its z-coordinate is set to zero at the new time and this will be the last point on that path line. Some of the path lines generated for H ) 50 cm, R ) 125 cm, and ∆V ) 10 V for a leak of 50 mm in diameter with a point electrode at the water surface are shown in Figure 2.
Steady-State Cake Formation The mass conservation principle is implemented numerically to analyze the steady-state cake formation. At any instance, the total mass of solids between any two path lines is constant. For an initially uniform volume fraction of solids φsus and total volume of the suspension Vsus, the final volume of incompressible cake (Vcake) is given by
FIGURE 3. Schematic of (A) steady-state cake formation and (B) transient cake formation.
Vcake ) Vsusφsus/φ0
(7)
where φ0 is the uniform volume fraction of solids in the incompressible cake and Vsus ) πR2H. In this study, φ0 is assumed to be equal to the volume fraction of solids at zero effective stress or the “stress-free state” (11). Equation 7 is discretized and employed between two adjacent path lines (j - 1) and j as (Figure 3a)
Vjcake )
Vjsusφsus ; Vjsus ) 2π φ0
∫∫ H
0
Rj
Rj-1
r dr dz;
∫ ∫
Vjcake ) 2π
zc(j)1
0
Rj
Rj-1
r dr dz (8)
In eq 8, zc(j)1 corresponds to the thickness of cake surface between path lines (j - 1) and j. Similarly, conserving the mass of solids between path lines j and (j + 1) gives the value of zc(j)2. The mean of zc(j)1 and zc(j)2, which is represented by zcj, on Rj is a point on steady-state cake profile. The same procedure is followed to obtain points on a set of selected path lines, and cubic spline interpolation is used to plot smooth cake profile through these points. As a check, the total volume of the final cake is calculated by numerical integration for comparison with Vcake, which is the actual volume of the cake [eq 7]. In all cases, the maximum mass balance error is found to be less than 1%.
Transient Cake Formation The cake growth is directly proportional to the velocity of solids at the cake surface. We describe the transient movement of cake surface along a path line by conserving the mass of solids between two adjacent path lines. At time t, consider a horizontal cake surface between path lines (j - 1) and j as shown in Figure 3b. On the path line Rj, a particle in the suspension at [rj(m), zj(m)] close to the cake surface is identified as q1. At the same time, a particle on the cake surface at [rjc(t), zjc(t)] is identified as q2. If particle q1 reaches the moving cake surface at time t + ∆t, then all the particles between q1 and q2 will become part of the cake at time t + ∆t. Here, the time step ∆t, which is an unknown, is the time required for particle q1 to reach the cake surface. VOL. 32, NO. 23, 1998 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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Then, the volume of fresh cake formed in time increment ∆t is given by
∫
∆Vjcake ) 2π
zcj(t+∆t)
zcj(t)
∫
Rj
Rj-1
r dr dz
(9)
where rcj(t) and rcj(t + ∆t) are the r-coordinates and zcj(t) and zcj(t + ∆t) are the z-coordinates of the cake surface at times t and t + ∆t, respectively. By conserving the mass of solids, the increase in cake volume because of the deposition (∆Vjcake) is approximated as
2π ∆V cake ) φ0 j
∫
∫
zcj(t+∆t)
Rj
Rj-1
zcj(t)
r dr dz
(10)
where φ(r,z,t) is the volume fraction of solids in suspension. If ∆Vjcake is evaluated from eq 10, eq 9 can be solved for zcj(t + ∆t). Because of the geometry of the problem where the volume fraction of solids (φ) inversely varies with the radial distance, it is difficult to evaluate the integral on the right-hand side of eq 10. Since the initial volume fraction of solids is assumed to be uniform, the integral must be transformed in such a way that the variable φ can be replaced by φsus. This is accomplished by changing the integration limits. In other words, the mass conservation principle indicates that the volume of solids enclosed by Rj-1 and Rj between q1 and q2 is equal to the initial volume of solids volume between these two particles when they were at their respective initial locations. Therefore, for an initially uniform suspension, ∆Vjcake is approximated as
∆Vjcake )
2πφsus φcake
∫
zjq1(0)
zj
q2(0)
∫
Rj
Rj-1
r dr dz
(11)
where zjq1(0) and zjq2(0) are initial z-coordinates of particles q1 and q2, respectively. If the initial locations of q1 and q2 are determined, eqs 9 and 11 can be solved for zcj(t + ∆t). The next step is the development of a numerical procedure to locate the initial position of a particle traveling along a given path line. The path line generated by particle j is represented by Rj(z). Let the coordinates of the particle j at time tk be [rj(k),zj(k)] and 0 e k e n, where n refers to the nth time step at which the particle reaches the lower boundary. At initial time t0 ) 0, the initial z-coordinate is zj(0) ) H, and the final z-coordinate is zj(n) ) 0. If a particle p1 traveling along this path line is at [rj(m),zj(m)] at time t* (0 e t* e tm), then the initial location of p1, i.e., [rjp1(0),zjp1(0)], is obtained from
ti+1 - (tm - t*) rjp1(0) - rj(i + 1) zjp1(0) - zj(i + 1) ) j ) j ti+1 - ti r (i) - rj(i + 1) z (i) - zj(i + 1) (12) where ti e (tm - t*) e ti+1, 0 e i e m. Here, ti and ti+1 are evaluated numerically from the travel time of particle j. It should be noted that when t* ) tm, the last particle on the path line, which is the surface particle, reaches the location [rj(m),zj(m)]. Similarly, the initial location of a particle p3 at time t*, i.e., [rjp3(0),zjp3(0)] at [rj(m + 1),zj(m + 1)], can be computed. Then, the initial location of a particle on the portion of the path line from [rj(m),zj(m)] to [rj(m + 1),zj(m + 1)] can be computed by linear interpolation between [rjp1(0),zjp1(0)] and [rjp3(0),zjp3(0)]. This numerical procedure is used to evaluate the initial position of particles q1 and q2 for the integral in eq 11. Then, eqs 11 and 9 are solved for the new cake thickness zcj(t + ∆t). Time step ∆t is solved next. Since, the velocity of the depositing solids determines the growth rate of the cake, ∆t is obtained from 3782
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TABLE 1. Material Parameters for Axis-Symmetric Electrophoretic Cake Formation parameter
value
depth of suspension radial distance of the lateral boundary initial concentration of suspension specific gravity of solid particle Stokes’ settling velocity of particles electrophoretic mobility
0.50 m 1.25 m 13.5 Kg/m3 2.7 1.0 × 10-10 m/s 4.0 × 10-8 m2 V-1 S-1 1, 10, 100 V 0.05 50 mm
effective voltage difference volume fraction of solids in cake leak diameter
∆t ) [x[rj(m) - rjc(t)]2 + [zj(m) - zjc(t)]2 -
x[rjc(t + ∆t) - rjc(t)]2 + [zjc(t + ∆t) - zjc(t)]2]/vjs(m) (13)
where vjs(m) is the magnitude of resultant particle velocity at [rj(m),zj(m)]. To allow the cake surface to rise above zj(m), a criterion for the minimum value of d should be set in order to focus on a particle at [rj(m-1),zj(m-1)]. Otherwise, the cake surface never passes [rj(m),zj(m)] on Rj. The criterion used in this study to shift to the upper location is
d e 1/2 x[rj(m) - rj(m + 1)]2 + [zj(m) - zj(m + 1)]2 This procedure, which allows us to follow the transient movement of cake surface along a path line, is employed until the last particle on the path line reaches the cake surface. Similar points are obtained for a set of selected path lines in the domain. Since these points are a function of time, to obtain the cake profile at a given time, the points on the path lines are linearly interpolated and connected using cubic spline. Later, it is shown that the final cake profile generated with the transient cake formation procedure is very close to that of the steady-state cake formation approach.
Results and Discussions A realizable set of material parameters given in Table 1 is used to simulate axis-symmetric electrophoretic cake formation. Figure 4 illustrates the final cake profiles obtained by using the steady-state cake formation procedure presented earlier. Figure 4a presents the steady-state cake profiles for ∆V ) 1, 10, and 100 V on a 50 mm diameter leak when a point electrode is at the water surface. Though a higher voltage difference moves more solid mass toward the leak, the final cake thickness over the leak for the range of voltages analyzed remained almost the same. As seen in Figure 4a, when the applied voltage difference is small, the particle movement away from the leak is in a vertical direction only. It is noted that when ∆V is zero, i.e., for one-dimensional gravity settling, the final cake thickness is φsusH/φ0 ) 5 cm. In all three cases, because of the no-flux boundary condition employed at the far end, the final cake thickness close to the lateral boundary is equal to that of one-dimensional settling. In other words, the r-component of particle velocity is zero at the lateral boundary. As shown in Figure 4a, an increase in voltage difference from 1 to 100 V did not affect the shape of the final cake profile significantly. The influence of ∆V on the time scale of cake growth is presented in the latter part of this section. It was also shown that variations in leak sizes, which are small in comparison to the impoundment, do not significantly affect the electric flow field. For practical purposes, therefore, it can be concluded that the cake formation is not very sensitive to the leak size experienced
the trend followed by the cake profiles is similar to those obtained for a point electrode. Results indicate that the cake radius grows to 20 cm in about a million seconds (approximately 11.5 days), which is an acceptable thickness and extent for practical purposes. As shown in Figure 4c, the final cake profiles generated by the steady-state and transient cake formation procedures yield very close results. This implies that the accuracy of the numerical integration presented in the transient cake formation procedure is satisfactory. As noted earlier, the maximum error in mass of solids computed from the final cake profiles generated by steady-state cake formation procedure is less than 1%.
Comparison with Experimental Data
FIGURE 4. (A) Steady-state profiles for different voltage differences. (B) Effect of upper electrode size on steady-state cake profile. (C) Transient cake formation with a large surface electrode and a voltage difference of 10 V.
in the impoundments. This was also observed in the field by Darilek et al. (5). Since the electrode size can be controlled in the field, the effect of upper electrode size on the cake formation is investigated. For a voltage difference of 10 V, Figure 4b illustrates the steady-state cake profile when a 40 cm radius electrode is used to seal a 50 mm diameter leak. The comparison of final cake profiles formed with the point electrode and the large electrode infer that the upper electrode size has a notable effect on cake formation. In this case, the use of a larger electrode increases the cake thickness over the leak by 50%. Thus, a larger surface electrode directs more clay particles toward the leak. To direct more particles toward the leak for efficient sealing, the upper electrode must be located close to the water surface. Figure 4c illustrates the transient cake formation when the point electrode is replaced by a 40 cm radius electrode for ∆V ) 10 V. In this case, although a bigger cake forms,
A series of bench-scale experiments performed by Yeung et al. (17) and Chung (2) investigate electrophoretic cake formation under controlled laboratory conditions. A small circular hole was drilled at the center of a plastic container 130 mm deep with inside dimensions of 290 × 340 mm. This container was installed on a sand bed in a larger container. A bentonite clay suspension was introduced into the inner container, and the outer vessel was filled with deionized water to keep the liquid levels in both containers equal. At the water surface of the inner container, a point electrode, acting as a cathode, was placed over the leak in alignment with the center of the hole. A plate anode was placed at the bottom of the sand layer directly underneath the leak. The cathodehole and cathode-anode distances were measured as 66 and 100 mm, respectively. Replicate experiments were performed with uniform initial concentrations of 10 and 15 Kg/m3 on 1.5 and 4.5 mm-diameter leaks under two different constant voltage differences of 20 and 30 V between the anode and the cathode. The cake dimensions were monitored by an ultrasonic probe, and the image of clay cake displayed on a monochrome monitor was recorded on a videotape. The temporal variations of cake thickness and volume were recorded. After conducting one-dimensional electrophoretic settling experiments on bentonite suspension, Chung (2) reported the value of electrophoretic mobility between 0.6 × 10-8 and 1.1 × 10-8 m2 V-1 s-1. The porosity of a bentonite cake varied from 0.85 to 0.95. We chose two sets of experimental data corresponding to case (a) 20 V, 10 Kg/m3 initial bentonite concentration and case (b) 30 V, 15 Kg/m3 initial bentonite concentration over a 1.5-mm diameter leak. The other parameters inferred from the experimental data for use in the numerical analysis are H ) 66 mm, R ) 145 mm, R0 ) 0.75 mm, m ) 0.6 × 10-8 m2 V-1 s-1, φsus ) 0.06, and a point electrode (cathode) at the liquid surface. Specific gravity and Stokes’ settling velocity of solids were taken as 2.7 and 1.0 × 10-10 m/s, respectively. As suggested by Chung (2), the correction factor f * to calculate effective voltage is taken as 0.8. Therefore, for the applied voltages of 20 and 30 V, the effective voltage differences employed in the simulations were 16 and 24 V, respectively. The experimental data and corresponding numerical results are presented in Figure 5. Figure 5a compares the temporal variation in cake thickness at the center of the leak. Though the simulation results follow the trend of experimental data, the theoretical results underestimate the growth rate of cake, especially for case (b). During the cake formation process, the experimental data indicate intermittent collapse of the clay cake which is not considered in the present model. Figure 5b,c compares the experimental and theoretical cake profiles for cases (a) and (b), respectively. A reasonably good match between the measured and predicted cake profiles is observed at early times. For case (b), the simulation underestimates the cake thickness away from the leak at larger times. The gravitational settling of clay particles and/ or higher porosity at the edges of the cake can be the reasons for the discrepancy between model and experimental results VOL. 32, NO. 23, 1998 / ENVIRONMENTAL SCIENCE & TECHNOLOGY
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water at different parts of the cake. As the cake was formed by particles from a dilute bentonite suspension, its pores are initially fully water-saturated. It has been observed that the water content, i.e., porosity at the center of the cake is lower than at the cake surface. The observed nonuniformity in porosity distribution is probably due to settlement/collapse of bentonite particles after deposition. As the most recently deposited particles were on the surface, and the loading on these particles and the settlement time were smaller, they possessed higher porosity. As noted earlier, negligible compressibility assumption for all practical purposes excludes this phenomenon.
Acknowledgments This research was supported by the Texas Advanced Technology Program Grant 999903-056.
Literature Cited
FIGURE 5. Comparison of transient cake formation with experimental data. (A) Effect of voltage difference and bentonite concentration. (B) Comparison of transient cake profiles at 3, 5.5, and 7.5 days for case (a). (C) Comparison of transient cake profiles at 1, 3, and 5 days for case (b). at high concentrations. Yeung et al. (17) studied the internal cake structure by nuclear magnetic resonance (NMR) images nondestructively. NMR images indicate a larger quantity of
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Received for review January 8, 1998. Revised manuscript received August 14, 1998. Accepted August 26, 1998. ES980009P
