Page 158 - Biosystems Engineering

Models for Heat Transfer in Heated Substrates 137

2. Saturated soil: The solid and liquid phases increase the weight
of their volume fractions. The effective thermal conductivity
is a function of (a) the value of water conductivity raised to a
power equivalent to the value of porosity and (b) the value of
the conductivity of solid particles raised to a power equiva-
lent to its volume fraction.
3. Unsaturated soil: Thermal conductivities can be obtained by
linear interpolation between the value of thermal conductiv-
ity for dry soils and the value of thermal conductivity for
saturated soils.

Campbell et al. (1994) modified the equation proposed by de
Vries (1963) to model the variation in soil thermal conductivity for
temperatures of up to 600°C. They defined a new term for fluid ther-
mal conductivity, which allowed them to use a single expression for
estimating conductivity in the gaseous phase without determining
moisture ranges. This model required soil physical properties that
were not needed in other models, such as the power for liquid recir-
culation and the mean geometric diameter of the solid particles. Some
of the empirical factors proposed by Campbell et al. (1994) were
aimed at characterizing the influence of water content and air on
thermal conductivity, and the moisture content at which water or air
gained relevance in the heat-transfer process.
Water conductivity is a function of temperature, whereas air con-
ductivity is dependent on the temperature, pressure, and amount of
water vapor, which in turn depends on temperature and pressure.
Air conductivity is considered to be an apparent thermal conductiv-
ity, which is the sum of the actual conduction through the air and the
latent heat of distillation across pores in the soil. For vapor diffusivity
in air, Campbell et al. (1994) used the expression proposed by Fuller
et al. (1966). The slope of saturation vapor pressure according to tem-
perature corresponded to the slope developed by Richards (1971).
The conductivity of solids was assumed constant with temperature.
Campbell and Norman (1998) presented a different version of the
method by de Vries (1963) for determining specific heat. They suggested
modeling the soil heat capacity per unit mass as the weighted sum of
the heat capacities of the soil components. The weighting was a func-
tion of the density of solid particles, water density, and a dimensionless
factor that related the bulk densities of the soil and of the particles.
Ochsner et al. (2001) developed a method to measure the thermal
properties of medium-textured soils. They applied the method to
four soil types and suggested relationships between the soil thermal
properties and factors such as the proportion of air-filled pores, soil
water content, or the relationship between the bulk density of the soil
and the bulk density of the solid particles. Unlike de Vries (1963),
they found a linear relationship between the thermal conductivity
and the volume fraction of air in pores.