22
q tr
h hQ = dr = ( In ) (12)
4 irk a
where hQ = the potential at the outer radius of the hollow cylinder,
b = one half the distance between neighboring roots, and
h = the potential at the inner radius a.
If b = 2/Dt, then equation (12) is identical to equation (11)
except for the g term which is small compared to the logarithmic term
and may be ignored.
Gardner and Ehlig (1962) presented a macroscopic water uptake
equation in which the rate of water uptake is proportional to the
potential energy gradient and inversely proportional to the impedance to
water movement within the soil and the plant. The potential energy was
expressed as the difference between the diffusion pressure deficit and
the soil suction. The impedance was expressed as the sum of plant
impedance and soil impedance. They analyzed greenhouse experiments and
obtained results consistent with the uptake equation.
Molz and Remson (1970) developed a mathematical model describing
moisture removal from soil by plant roots. The model described one
dimensional water movement and extraction. The model used a macroscopic
water extraction term to describe moisture removal by plants. Their
formula for the extraction term was given as
R(z) D(e)
S(z,t) = T ( ) (13)
i R(z) D(e) dz
where D(e) = diffusivity,
R(z) = the effective root density,
T = the transpiration rate per unit area and
v = the root depth.