Environmental Engineering Reference
In-Depth Information
5.4.3
Numerical Solution Approaches
To solve the general system of equations describing the EK transport pro-
cesses, usually three different approaches are used. These include:
1. Differential and algebraic equations approach that provides
a solution to the mixed differential and algebraic equations
in which the transport equations and chemical equilibrium
reactions are solved simultaneously as a system (Miller and
Benson, 1983; Lichtner, 1985).
2. he direct substitution approach, which consists of direct
substitution of the algebraic chemical equilibrium equations
into the differential transport equations to form a highly
nonlinear system of partial differential equations (Jennings
et al., 1982; Lewis et al ., 1987).
3. The sequential iteration approach, which consists of iterat-
ing between the sequentially solved differential and algebraic
equations (Kirkner et al. , 1985; Yeh and Tripathi, 1991).
5.5
EK Mass Transport Models
Many researchers have attempted to model mass transport under applied
electrical gradient (Acar et al., 1989; Shapiro et al., 1989; Acar et al., 1990;
Mitchell and Yeung, 1991; Eykholt, 1992; Alshawabkeh and Acar, 1992;
Shapiro and Probstein, 1993; Jacobs et al., 1994; Jacobs and Probstein,
1996; Alshawabkeh and Acar, 1996; Haran et al., 1997; Acar, et al., 1997;
Cao, 1997; Al-Hamdan and Reddy, 2008, 2011). Shapiro et al., (1989) and
Shapiro and Probstein (1993) described a 1-D model accounting for ion
diffusion, migration, and EO advection in predicting the species transport
rate. The model solves the charge flux equation in order to evaluate the
nonlinear electric field distribution. The governing convective-diffusion
equation in their model is given as:
C
t
DC
x
2
[
]
i
i
i
+
=
Cu
+
u
R
(5.40)
2
2
i
ei
c
i
t
x
where, t describes the tortuosity of the porous medium, and u c is the
convection velocity (the bulk EO velocity) given as:
1 2
t
ex
m
Φ
(5.41)
u
=
x
c
x
 
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