Table 5. Total length of cotton
the models to describe radial flux to the root by
as affected by plant age and
massflow and diffusion as affected by uptake by
root medium; hydroponic or
the root is
sand (after Brar et al. 1990b).
ci 1
rvc
c
=
rDc i + o o i
Root length
(12)
t r r
b
r
Plant
Hydroponic
Sand
age*
(m)
(m)
where ci = ion concentration in solution
3.3†
2.6a
12
(1.0)**
(0.7)
r = radial distance from the root axis
13.4b
14.4b
24
(2.4)
(2.9)
r0 = root radius
28.8c
17.4b
36
(1.5)
(4.9)
Dc = apparent diffusion coefficient
93.9d
17.1b
48
(6.2)
(2.6)
v0 = flow velocity of soil solution to root
* Days after planting.
b = buffer capacity = ( ci/ t)/( ci/ r)
† Means with the same letter are
t = uptake time.
not significantly different (P <
0.05).
** Standard error of mean.
Boundary conditions are required to find the
solution to eq 12. The initial boundary condition
take (Nye and Tinker 1969, Barber and Silberbush
of the equation is:
1984). The relationship between solution concen-
tration (ci ) of a given nutrient and its uptake by
r > r0,
t = 0,
ci = cii
(13)
plant roots intact with the solution was proposed
by Bouldin (1961) and modified by Nye (1966)
where cii = initial nutrient concentration in the
J = αci
(9)
soil solution.
The first boundary condition is
where J = rate of nutrient uptake (flux) expressed
(c c )
in terms of quantity per unit root area per unit
J
ci
+ voci = max i min at r = r0 , t > 0
time, and α = the proportionality term describing
Dcb
Km + (ci cmin )
t
the root absorbing power.
(14)
If the term "uptake per unit root area" in the
above equation is replaced with "root length,"
where cmin = ci when Jn = net flux is zero, Km =
MichaelisMenten constant.
J = 2Παrci
(10)
The second boundary condition, added by
Cushman to account for competition between ad-
where r = root radius. Nye and Tinker (1969) com-
jacent roots, is
bined the terms α and r and provided a new term
ci
called root demand coefficient. For the evaluation
+ voci = 0,
r = ri ,
t>0
(15)
Dcb
of total uptake, they incorporated the total root
t
length (L) in eq 10
where ri = mean half-distance between root axes.
In the absence of competition, ci = cii at r = ri and t
J = 2ΠαrciL.
(11)
> 0.
The soil and plant factors affecting the root de-
Assumptions made in the Claassen-Barber
mand coefficient are
model are
1. Nutrient demand for plant growth;
1. Soil is homogeneous and isotropic;
2. Operation of metabolically driven uptake
2. Root zone has constant soil water content
and transport mechanisms;
that is maintained at field capacity;
3. Functioning of physical and chemical pro-
3. Plant or root age has no impact on nutrient
cesses in the root environment.
flux;
Mathematical models have been developed for
4. Nutrient concentration has no impact on
nutrient uptake by plant roots (Nye and Marriott
convective component of flux;
1969, Nye et al. 1975, Claassen and Barber 1976,
5. Nutrients move to the root by diffusion and
Barber and Cushman 1981, Tillotson and Wagen-
massflow processes; roots are considered
et 1982). These models help in understanding the
as smooth cylinders that absorb nutrients
complex soilrhizosphererootplant system and
from the soil solution as per Michaelis
in determining the significance of the parameters
Menten kinetics;
involved. The basic transport equation used in
6. Apparent diffusion coefficient and buffer
9
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