Transport - Transport and Fate of Chemicals in Soils: Principles and Applications - page 81

where the eigenvalues β m are the positive roots of

P

β cot( β += ) 0

(3.55)

m m

2

This solution suffers from the same convergence problems as Brenner's,
which converges only for relatively small values of P . Moreover, Cleary
and Adrian's solution fails the mass-balance requirement, and also violates
mass balance for the effluent curve. Van Genuchten and Weirenga (1986)
recommended that this solution not be applied to solute transport column
experiments. The significance of using the precise boundary conditions is
illustrated by comparing the Cleary and Adrian (1973) solution with the con-
centration profiles calculated using the mathematical solutions of Brenner
(1962) and Lindstrom et al. (1967). Unlike the boundary condition used by the
other two solutions, Cleary and Adrian (1973) assumed a first-type bound-
ary condition ( C = C o at z = 0 ), which has been used by several investigators
(Gupta and Greenkorn, 1973; Kirda et al., 1973; Lai and Jurinak, 1972; Warrick
et al., 1971). The solution by Lindstrom et al. (1967) was developed for a semi-
infinite soil column but was later applied to finite soil columns by Davidson
et al. (1968) and Davidson and Chang (1972) among others.
The above solutions were used to calculate distributions of solute concen-
tration in a 30-cm soil column (Figure 3.8) for selected times during con-
tinuous application of a solute solution (see Selim and Mansell, 1976). These
concentration profiles were obtained for pore-water velocities v of 0.5, 1.5,
and 3.0 cm/h. Parameters used were those of Lai and Jurinak (1972), where
D = 1.5 cm 2 h -1 , ρ = 1.30 g cm -3 , Θ = 0.45 cm 3 cm -3 , K d = 2.5 cm 3 g -1 , and L =
30 cm with an R value of 8.22.
As was expected, the solution of Cleary and Adrian provided higher con-
centrations throughout the soil column at all times and for all three pore
velocities (Figure 3.8) in comparison with results obtained by using the other
two solutions (see Selim and Mansell, 1976). With decreasing pore water
velocity ( v ) the magnitude of deviation among the three solutions increased.
The higher concentrations are attributed to the assumption that C = C o at
z = 0 for all times. However, the concentration profiles obtained using the
solution of Lindstrom et al. (1967) are essentially identical to results obtained
using the Brenner (1962) solution except in the vicinity of the exit end of the
soil column ( z = L = 30 cm) at large times. Deviation of the Lindstrom et al.
(1967) solution in the vicinity of z = L is clearly due to forcing a semi-infinite
boundary condition to describe solute transport for the finite soil columns
presented here.
Solute breakthrough curves (BTCs) corresponding to pore water velocities
of 0.5, 1.5, and 3.0 cm h -1 for the same soil parameters as in Figure 3.8 are
shown in Figure 3.9 (see Selim and Mansell, 1976). BTCs are commonly used
in miscible displacement studies and represent relative solute concentration
C/C o at z = L versus relative volume of accumulated effluent V / V o , where V o is