Heat Flow in the Soil
pressure ( e l p a ) . The second equation is obtained by applying the chain
rule of calculus. The derivative of water concentration with respect to
temperature can be expanded using the relationship C , = h, C, ( T ) from
Ch. 3 where h, is the relative humidity in the soil. Since h, is not temperature dependent, it can be taken out of the derivative. Now, using another
definition from Ch. 3: s = d C , ( T ) / d T , gives the slope of the saturation
mole fraction function for water; which is simply related to the slope of
the saturation vapor pressure versus temperature. Substituting these into
Eq. (8.14) gives
The apparent thermal conductivity for distillation across a pore is made
of all the terms which multiply the temperature gradient.
Equation (8.15) is adequate for moist soils at low temperature, but
requires two corrections for it to work at high temperatures or for dry
soils. When water evaporates from a surface, mass in the vapor phase is
created at the liquid-gas interface which causes the entire gas phase to
flow away from the surface. At low temperature this mass flow effect is
negligible, but at boiling point its effect is far greater than the diffusive
flux from Fick's law. The correction to the equation is called the Stefan
correction. It can be inserted into Eq. (8.15) by substituting A / ( p a - ea)
for s where A is the slope of the saturation vapor pressure function. From
Ch. 3, s = A / p a . At typical environmental temperatures pa >> ea this
substitution will have very little effect. If a moist soil is heated by a fire at
the surface, however, the Stefan correction becomes very large, and the
soil becomes an excellent conductor of heat because pa - ea becomes
small:
The second correction was mentioned previously relating to the return
flow of water. Even before the humidity in the soil drops significantly
below one (remember from Ch. 4 that the humidity in moist soil is always
close to one) the return flow of liquid water in the soil pores has dropped
sufficiently to render the latent heat component of the pore conductivity
negligible. No fundamental theory has been developed yet to account
for this. Campbell et al. (1994) give a dimensionless flow factor which
depends on the soil water content. This factor multiplies Eq. (8.16) to
give the actual latent heat flux. The factor is
The constant 8, determines the water content where return flow cuts
off and q determines how quickly the cutoff occurs. Both constants are
correlated with soil texture and tend to increase as textures become finer.
The range for 8, is from around 0.05 for coarse sand to 0.25 for heavy
pressure ( e l p a ) . The second equation is obtained by applying the chain
rule of calculus. The derivative of water concentration with respect to
temperature can be expanded using the relationship C , = h, C, ( T ) from
Ch. 3 where h, is the relative humidity in the soil. Since h, is not temperature dependent, it can be taken out of the derivative. Now, using another
definition from Ch. 3: s = d C , ( T ) / d T , gives the slope of the saturation
mole fraction function for water; which is simply related to the slope of
the saturation vapor pressure versus temperature. Substituting these into
Eq. (8.14) gives
The apparent thermal conductivity for distillation across a pore is made
of all the terms which multiply the temperature gradient.
Equation (8.15) is adequate for moist soils at low temperature, but
requires two corrections for it to work at high temperatures or for dry
soils. When water evaporates from a surface, mass in the vapor phase is
created at the liquid-gas interface which causes the entire gas phase to
flow away from the surface. At low temperature this mass flow effect is
negligible, but at boiling point its effect is far greater than the diffusive
flux from Fick's law. The correction to the equation is called the Stefan
correction. It can be inserted into Eq. (8.15) by substituting A / ( p a - ea)
for s where A is the slope of the saturation vapor pressure function. From
Ch. 3, s = A / p a . At typical environmental temperatures pa >> ea this
substitution will have very little effect. If a moist soil is heated by a fire at
the surface, however, the Stefan correction becomes very large, and the
soil becomes an excellent conductor of heat because pa - ea becomes
small:
The second correction was mentioned previously relating to the return
flow of water. Even before the humidity in the soil drops significantly
below one (remember from Ch. 4 that the humidity in moist soil is always
close to one) the return flow of liquid water in the soil pores has dropped
sufficiently to render the latent heat component of the pore conductivity
negligible. No fundamental theory has been developed yet to account
for this. Campbell et al. (1994) give a dimensionless flow factor which
depends on the soil water content. This factor multiplies Eq. (8.16) to
give the actual latent heat flux. The factor is
The constant 8, determines the water content where return flow cuts
off and q determines how quickly the cutoff occurs. Both constants are
correlated with soil texture and tend to increase as textures become finer.
The range for 8, is from around 0.05 for coarse sand to 0.25 for heavy