Reduction of Contaminants in Soil and Water By Direct Electric Current - Electrokinetics for Petroleum and Environmental Engineers - page 44

chemical reactions, and small double layer thickness were considered.
These models were all based on the Nernst-Planck equation, and each was
calibrated to the experimental findings (Shapiro and Probstein (1993);
Mitchell and Yeung (1991); Denisov and Probstein (1996); Alshawabkeh
and Acar (1996); and Cao (1997)). Cao in 1997 modeled the ion trans-
port considering the effect of changing electrical fields due to the re-dis-
tribution of charge concentration. The effect was later shown in a rigorous
mathematical analysis by Chu, (2005) where concentration gradients give
rise to spatial variation of conductivity to overcome the violation of elec-
troneutrality of the bulk fluid.
In the basic governing equation of advection-diffusion, dispersion refers
to the movement of species under the influence of gradient of chemical
potential, whereas advection is the stirring or hydro-dynamic transport
caused by density gradient or forced convection. A general one-dimen-
sional mass transfer to an electrode is governed by Nernst-Planck equation:

( ) ( )

∂ Cx

x

∂ Φ x

( ) =− ( )

* i

(2.9) Jx D − uzFC − Cv x

i i ii i i

∂ ∂ x

where, J i = total flux of species i , [ Mt -1 l ] ; u i = D * i /RT = mobility of species
i (Nernst-Einstein relation) v(x) = advective velocity (= v eo (x) , the electro-
osmotic velocity)
The mobility of an ion u i is defined as the limiting velocity of an ion
moving in an electric field of unit strength. The minus sign arises because
the direction of flux opposes the direction of increasing C i . Applying Fick's
second law, we arrive at another form of Nernst -Planck equation (also
given earlier in equation 2.1) with the added advection term:

∂
∂

C
t

( ) +

( ) ∇

*

(2.10)

i

=∇ D ∇ C + uzFC ∇ Φ uzF ∇+ Φ v C

i i i i i i i i

Equation 2.10 is the basic mass transfer equation for an electrochemical
system under an electric field. . In equation 2.10 the first term is the diffu-
sion; the second term is the migration; and the third term is the advection
contribution to the total mass transport of the species i in an electrochemi-
cal system. In clay soils where the hydraulic advection is negligible com-
pared to electroosmotic advection, the velocity term v is simplified as the
electro-osmotic velocity ( v eo ).
The two important system parameters that contribute to the ion
flux, and hence the distribution of current density in the system are the