In different configurations, the time required for remediation is a function of contaminant transport rates and electrode spacing. Some authors [67,71] have provided practical methods to estimate it in one-dimensional systems. Thus, neglecting the hydrodynamic dispersion and assuming, as usual, homogeneous soil and linear electric field, the linear velocity of cations through the soil with typical electroosmotic flow is:
Rt where u* (L2V-1T-1) is the effective ionic mobility of the species in soil ke (L2V-1T-1) is the coefficient of electroosmotic permeability
Oe (VL-1) is the voltage gradient
Rt (dimensionless) is a time-delay factor to account for the time required for contaminant desorption and dissolution
The value of Rt depends on soil type, pH, and type of contaminant. Sorption retardation factor (Rd) can be used to estimate Rt (Rt =1 for nonreactive contaminants). However, the values of Rt might be different from Rd because Rt should account for time delay due to all chemical reactions (solubilization, complexation, and desorption), but Rd values account only for sorption.
Time required to remediation TR (T) is calculated using this velocity and opposite-polarity electrode spacing, LE, as:
where o* (siemens L-1) is the effective conductivity in the soil medium and P (L3 C-1) is a lumped property of the contaminant and the soil that measures the rate of reactive transport of a species relative to the electric conductivity of a medium, given by:
Typical values of P for contaminated fine-grained soils are estimated to be in the range of 10-8 to 10-6 m3 C-1 . If time is to be calculated using current density, Equation 18.12 becomes
p Id where Id (amps L-2) is electric current density.
In treatments employing two-dimensional systems, it is assumed that radial electrical distribution, Rw is the radius of central well (usually cathode), RE, is the distance between central and peripheral electrodes, and Z is the depth of the site. The difference between this case and the one-dimensional case is that the current density in radial flow is a function of the radial distance (r); however, in both cases, it assumes operating with a constant total current. The electric current per unit depth for the radial transport is given by:
where Iz (amps L-1) is the current per unit depth and Or (VL-1) is the radial voltage gradient.
Contaminant transport rate depends on the voltage gradient, which is a nonlinear function of the radial distance. Ignoring dispersion and accounting for ion migration and electroosmosis, the radial velocity of ions transport is given by
Rt where v(r) (LT-1) is the radial velocity of reactive species transport. Substituting Or from Equation 18.15 into Equation 18.16 yields:
The velocity of contaminant transport is a nonlinear function of the radial distance even if the soil is homogeneous and isotropic. The time required for the contaminants to be transported from the outside electrodes (anodes) to the center electrode (cathode) is calculated integrating along the radius dt = dr /v(r) from Rw to RE, leading to:
In order to provide time evaluation as a function of the voltage, a transformed voltage expression, Or (V), which is constant along the radius, is given by:
Substituting the value of Iz from Equation 18.19 and simplifying because (RW)2 << (RE)2, Equation 18.18 yields:
The form of Equation 18.20 for radial transport is similar to the form of Equation 18.12 for one-dimensional transport. In both, p and o* are soil properties and Oe and Or are constants if the first ones are. However, the comparison of the two equations shows that, although TR is a function of the linear distance between the electrodes for the one-dimensional case, it is a function of square of the radial distance for two-dimensional configurations. This is important for selection of electrode spacing. Selection for radial spacing in radial systems is much more critical than for one-dimensional systems because time and cost remediation will significantly increase when radial spacing increases.
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