Page 157 - Biosystems Engineering

136 Cha pte r F o u r

space. When the soil is wet, water acts as a continuous medium in
which air granules and solid particles are arranged. The soil solid
phase is made up of an organic fraction and a mineral fraction. Each
fraction makes its contribution according to its characteristic value of
conductivity. Under these assumptions, the thermal conductivity of
each constituent is weighted by a factor that is dependent on the vol-
ume fraction of each constituent, on the ratio of the thermal conduc-
tivities of the particles, and on the continuous phase and shape of the
particles. The weighting factors are calculated taking into consider-
ation the influence of three shape factors, one for each type of particle
according to the phase to which it belongs.
For the solid phase, a shape factor of 0.125 has been determined
for randomly arranged ellipsoidal granules despite the changes in
scale caused by different particle sizes. This shape factor has proved
valid for sandy and clay soils. To determine the shape factors for the
discontinuous fluid phase, water or air, the model has been subdi-
vided into four moisture regions (Hopmans and Dane 1986):

1. Dry soil: Air is a continuous medium. The ratio between the
conductivities of the solid phase and the continuous phase is
very high and the values reported by de Vries (1963) must be
increased by 25 percent.
2. Soil moisture content from permanent wilting point (PWP) to satu-
ration: Water is a continuous medium. Because the air shape
factor ranges from 0.035 to 0.333 for PWP and saturation
moisture contents, respectively, the shape factor suggested
for a given moisture content is a linear approximation between
both values.
3. Soil moisture content below PWP: Progressive drying below the
permanent wilting point causes a decrease in the air shape
factor from a value of 0.013 to zero moisture content.
4. Soil moisture content below the critical water content: The air
shape factor is very low and is dependent on soil texture.
The model predicted values of soil effective thermal conductivity
with errors usually less than 5 percent, except for interpolation inter-
vals, with errors of approximately 10 percent (de Vries 1975). Johan-
sen (1975) developed another method for determining soil effective
thermal conductivity and proposed three submodels according to
soil moisture content:
1. Dry soil: The conductivity of a dry soil depends mainly on its
bulk density. To estimate the thermal conductivity of a dry soil,
we can use the expression developed by Johansen (1975), which
requires three dimensionless constants similar to the expression
developed by Kersten (1949). The dimensionless constants
required have been empirically tabulated for some soils.