Agriculture Reference
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is proportional to the concentration gradient, and can be described (in the z
dimension) by
C
z
=
(3.14)
U
D
d
o
where D o is the coefficient of molecular diffusion for a solute in a bulk or
free solution (cm 2 h -1 ). Values for D o for several solutes are available in the
literature.
To characterize diffusion of ions in soils, the diffusion coefficient given in
Equation 3.14 must account for the presence of the matrix or soil particles.
Specifically, the presence of soil particles results in the reduction of the vol-
ume of solution and, more importantly, the increase in the flow length of the
solute. This increase in flow or path length is often referred as tortuosity. As
a result, the diffusion equation for a soil system is now modified as follows
(Olsen and Kemper, 1968; Nye 1979):
C
z
=
θ
(3.15)
U
D
d
m
This new term D m is the apparent diffusion coefficient and takes into
account the effect of the solid phase of the porous medium on the diffusion.
In Equation 3.15, the flux also accounts for the soil moisture content Θ . Va n
Schaick and Kemper (1966) and Epstein (1989) related D m to molecular diffu-
sion of a solute species in pure water ( D o ) according to:
= D
o τ
D
(3.16)
m
where τ is a tortuosity factor (dimensionless) that is defined as:
2
τ= τ
() where
τ =
LL
/
(3.17)
o
a
and L a and L are the actual and shortest path lengths for diffusion and τ is the
t for t uo is it y. τ is an apparent tortuosity for porous media. This factor τ takes
a value less than 1, with a range of 0.3 to 0.7 for most soils. For unsaturated
soils, various forms were proposed in order to account for the soil water con-
tent Θ. Examples for expressions D m versus D o are presented in the classical
work of Millington (1959) and Millington and Quirk (1961).
To illustrate the transport of a solute in porous medium due to diffusion,
one may consider two blocks of soils, one with high and one with low con-
centration. At some time t , the two blocks are brought in contact with each
other, and after sufficient time has elapsed, the blocks are sectioned and
solute concentrations obtained. This approach yields a concentration profile
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