Transport - Transport and Fate of Chemicals in Soils: Principles and Applications - page 80

concentrations. Another feature of Brenner's solution is that the mass bal-
ance requirement for a finite column is met. That is, the amount of solute that
is entering the column minus the amount that is leaving that column equals
that which is stored in the column. Brenner's solution, which is a series, con-
verges only for relatively small values of P . In fact, Selim and Mansell (1976)
found that Brenner's solution requires as many as 100 terms to obtain con-
vergence for P > 20. Therefore, approximate solutions are recommended for
large values of P .

3.6 Lindstrom Solution
Lindstrom et al. (1967) considered the case for a semi-infinite medium with
a third-type boundary at the soil surface. Specifically, the boundary condi-
tions given by Equations 3.29, 3.30, and 3.32 were used. This case is similar to
that considered by Brenner (1962) except for a semi-infinite rather than finite
column lengths. Lindstrom's solution can be expressed as:

1/2

2

1
2

Rz − vt

+

v t
DR

2

exp- (

Rz vt

DRt

− )

Azt (,) = erfc

1/2

(4 DRt ) π 4

1
2

vz
D

v t
DR t

2

vz
D

Rz + vt

(3.53)

− 1 + + exp erfc

1/2

(4 DRt )

This solution does not suffer from convergence problems as does that of
Brenner. In the meantime, it provides accurate mass balance and describes
volume-averaged concentrations in the column.

3.7 Cleary and Adrian Solution
Cleary and Adrian (1973) considered a similar case to that of Brenner (1962)
except for a first-type boundary condition at the inlet ( z = 0). Their solution
is for finite column lengths having boundary conditions given by Equations
3.26, 3.27, and 3.31, and may be expressed as:

2

β z zP

L

Pvt
LR

− β

vt
PLR

2 in β

m

exp −

m

∞ m

L 24

∑

(3.54)

Azt (,)1 =−

2

P

P

2

β ++

m = 1

m

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