Transport - Transport and Fate of Chemicals in Soils: Principles and Applications - page 79

three exact solutions to the CDE (Equation 3.24) that are widely cited in the
literature. All three solutions have the form:

+− ( ) Az t (,) 0 << t

CCC t i o i p

Czt (,) =

(3.48)

+− − ( ) Azt (,)

C Az t

( , − ) t >

CCC t t

i o i o p p

where we assume a simple initial condition ( C = C i ) representing uniform
solute concentration distribution in the soil at t = 0. The form of the solution
in Equation 3.33 is applicable for conditions representing continuous solute
application or a pulse-type input having duration t p . For the three exact solu-
tions, the appropriate expressions for A ( z , t ) are given below.

3.5 Brenner Solution
Brenner (1962) considered the case of a finite soil column with the more pre-
cise third-type boundary condition at z = 0 that accounts for dispersion and
advection across the upper surface (Equations 3.29 and 3.30). By defining the
Peclet number defined in Equation 3.31

P= vL
D

(3.50)

Brenner's solution can be expressed as:

2

ββ β

z

+

P β z zP

L

Pvt
LR

− β

vt
PLR

2 P cos

m

sin

m

exp −

m

∞ mm

L 2 L 24

∑

Azt (,)1 =−

2 2

P P 2 2

β ++ P β +

m = 1

m m

4 4

(3. 51)

where the eigenvalues β m are the positive roots of

2

β − β

P
4

m

β cot () P + =0

(3.52)

m m

Brenner's solution describes volume-averaged concentrations within the
column. Because of the zero concentration gradient at z = L , this solution also
defines a flux concentration at the lower boundary. Hence, Brenner's solution
correctly interprets effluent concentrations as representing flux-averaged