Second-Order Transport Modeling - Transport and Fate of Chemicals in Soils: Principles and Applications - page 163

For the purpose of simulation and model evaluation, the appropriate ini-
tial and boundary conditions associated with Equations 6.23 to 6.25 were
as follows. We chose uniform initial solute concentration C i in a finite soil
column of length L such that:

cc T = ( = 0and0 << X 1)

(6.29)

i

We also assume that an input solute solution pulse having a concentration
C o was applied at the soil surface for a (dimensionless) time T p and was then
followed by a solute-free solution. As a result, at the soil surface, the follow-
ing third-type boundary conditions were used (Selim and Mansell, 1976):

=− ∂
∂

1 c

X

1 c ( XT T p =< , )

(6.30)

P

=− ∂
∂

1 c

X

0 c ( XT T p => , )

(6. 31)

p

At x = L , we have:

∂
∂

c
X

= 0( XT = 1, > 0)

(6.32)

The differential equations of the second-order model described above are
of the nonlinear type and analytical solutions are not available. Therefore,
Equations 6.21 to 6.23 must be solved numerically. A finite-difference
approximation (explicit-implicit) subject to the above initial and boundary
conditions can be derived as was carried out by Selim and Amacher (1988)
and documented in Selim, Amacher, and Iskandar (1990).

6.2 Sensitivity Analysis
Several simulations were performed to illustrate the kinetic behavior of solute
retention as governed by the proposed second-order reaction. We assumed
a no-flow condition to describe time-dependent batch (sorption-desorption)
experiments. The problem becomes an initial-value problem where closed-
form solutions are available.
The retention results shown in Figure 6.1 illustrate the influence of the
rate coefficients ( k 1 and k 2 ) on the shape of sorption isotherms ( S versus
C ). The parameters chosen were those of a soil initially devoid of solute