Agriculture Reference
In-Depth Information
treatment is given by Arah and Kirk (2000) in a general transport-reaction model,
which they simplify to focus on emissions fuelled by root exudation and death
and transmission through the plant. This model is now outlined.
Following Equation (2.6), the concentration profile with depth
z
of any non-
adsorbed substrate in an areally homogeneous system is given by:
D
∂C
∂z
−
vC
L
∂C
∂t
=
∂
∂z
+
O
+
P
−
Q
−
R
−
S
(
8
.
1
)
where
O
is the root-mediated influx,
P
is production,
Q
is consumption,
R
is
the root-mediated efflux and
S
is ebullition. The terms
D
,
v
,
O
,
P
,
Q
,
R
,
S
and
C
are effective areal averages at depth
z
and time
t
: they subsume within
themselves any areal heterogeneity in the real system. Temperature is an implicit
variable in Equation (8.1), influencing the instantaneous rates of all transport and
reaction processes.
Diffusion depends on the bulk concentration
C
; leaching and consumption
on the solution-phase concentration
C
L
; and root-mediated efflux and ebullition
on the gas-phase concentration
C
G
. Root-mediated influx and production are
independent of
C, C
L
and
C
G
, though they may depend on other properties of
the system. The concentrations
C, C
L
and
C
G
are easily inter-converted assuming
equilibrium between solution and gas phases:
C
L
=
HC
G
(
8
.
2
)
where
H
is a dimensionless Henry's law constant. The bulk concentration is
given by
C
=
θ
G
C
G
+
θ
L
C
L
(
8
.
3
)
where
θ
G
is the air-filled porosity and
θ
L
is the volumetric water constant. Hence
C
θ
G
+
Hθ
L
,
i.e.
∂θC
G
∂C
1
θ
G
+
Hθ
L
C
G
=
=
(
8
.
4
)
and
C
θ
G
+
Hθ
L
,
i.e.
∂θC
L
∂C
H
θ
G
+
Hθ
L
C
L
=
H
=
(
8
.
5
)
The diffusion coefficient allows for both gas and liquid phase diffusion. It is
given by (Stephen
et al
., 1998a,b):
D
G
θ
G
+
D
L
Hθ
L
(θ
G
+
Hθ
L
)f
D
=
(
8
.
6
)
where
f
is a tortuosity factor, approximately equal to unity in a well puddled soil.
Root-mediated influx may be represented as an exchange process in which
only the gas phase moves. Most simply this can be expressed (Stephen
et al
.,
1998a,b):
O
=
κD
G
C
G0
(
8
.
7
)
