Thermal Properties of Soils: Volumetric Heat Capacity

8.2 Thermal Properties of Soils: Volumetric Heat Capacity

In order to compute damping depths, admittances, and soil temperature profiles, the thermal diffusivity of the soil needs to be known. This, in turn, requires a knowledge of the thermal conductivity and specific heat of the soil. In this section we tell how to find these quantities.

The volumetric heat capacity of a soil is the sum of the heat capacities of the soil components. Soil typically is made up of minerals, water, and organic matter. The soil heat capacity is therefore computed from

where is water content (volume fraction of water), and are volume fractions of minerals and organic material, and c and are the specific heat and density. While air is almost always present, its contribution to the soil heat capacity is negligible. Other constituents, like ice, are added to Eq. (8.12) when present. Table 8.2 lists thermal properties for a num- ber of soil constituents. Thermal properties with significant temperature

dependence are indicated. Figure 8.2 shows the variation in heat capacity of four typical soils when the water content varies from zero to saturation. As indicated by Eq.

the change is linear, and values range from less than 0.5 to

Heat Flow in the Soil

T A BLE 8.2. Thermal properties of typical soil materials.

Material

Specific

Thermal Volumetric

Heat

Conductivity heat capacity

Soil minerals

Organic matter

Ice

2.22 - 0.01 T

Air

about 3.5 MJ The slope of all lines is the same and is deter- mined by the heat capacity of water. The intercepts differ because of the differences in solid fractions in the different soils.

Example 8.1. Find the volumetric heat capacity of loam soil with a water content of 0.2 m 3 and a bulk density of 1.3

Assume the organic fraction is zero.

Solution. The mineral fraction is the ratio of the bulk density to the min-

eral density. The mineral density of soil, from Table 8.2, is 2.65

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume Fraction of Water

8.2. Volumetric heat capacity of organic and mineral soils. Differences are mainly due to differences in soil bulk density.

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Thermal Properties of Soils: Thermal Conductivity

0.49. Using Eq.

Thermal Properties of Soils: Thermal Conductivity

The thermal conductivity of soil depends on the conductivities andvolume fractions of the soil constituents. The heat flows through a complicated network of mineral, water, and air paths and the quantity and conductivity of each strongly influences the effectiveness of the others. In addition, a substantial quantity of heat is carried by evaporation and condensation in the soil pores, and this is both water content and temperature depen- dent.

(1963) proposed that the thermal conductivity of soil be computed as a weighted sum of the conductivities of the constituents:

where is the volume fraction, is a weighting factor, k is the thermal conductivity of the constituent, and subscripts w, g, and m indicate the water, gas, and mineral fractions.

The apparent thermal conductivity of the gas phase is the sum of the thermal conductivity of air, given in Table 8.2, and an apparent conductiv- ity resulting from latent heat transport within the pores of the soil. Water evaporates on one side of the pore, diffuses across the pore in the air space, and then condenses on the other side of the pore. The latent heat of evaporation is carried with the water across the pore. After the water

condenses, it can flow back to the hot side pore and evaporate again. Engineers have used this same idea in highly effective heat exchangers called heat pipes. The pipes are tubes with a volatile liquid and a wick

sealed inside. The liquid evaporates on the hot end of the tube, diffuses to the cold end, condenses, and then moves back to the hot end through the wick. The heat pipe is sealed so there is always plenty of liquid, but the soil can dry out. As the soil water content decreases, the water films

become thinner, and the return flow of liquid water in the soil pores is increasingly impeded until there is no contribution of latent heat to the

overall heat transport in soil pores. Fick's law can be used to compute the latent heat flow in a pore. Using Eq. (6.5) gives:

where is the molar density of air, is the latent heat of vaporization of water, D, is the vapor

for soil, and is the vapor mole fraction given by the

of vapor pressure divided by total atmospheric

Heat Flow in the Soil

pressure 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 = (T) from Ch. 3 where is the relative humidity in the soil. Since is not temper- ature dependent, it can be taken out of the derivative. Now, using another definition from Ch. 3: =

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 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

for where A is the slope of the saturation vapor pressure function. From Ch. 3, = A At typical environmental temperatures

>> this

substitution will have very little effect. If a moist soil is heated by a at the surface, however, the Stefan correction becomes very large, and the soil becomes an excellent conductor of heat because

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 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 is from around 0.05 for coarse sand to 0.25 for heavy

Thermal Properties of Soils: Thermal Conductivity 121

clay. The range for q is roughly 2 to 6 with coarser materials generally having lower values, but the pattern is not as clear as for the cutoff water

content. The complete expression for vapor phase apparent conductivity is

where is the thermal conductivity of weighting factors are deter- mined by the shapes, conductivities, and volume fractions of the soil con- stituents. Campbell et (1994) defined a fluid conductivity for the soil as

In dry soil the fluid conductivity is the value for dry air and in saturated soil it is the value for water. The same

function used for the liquid return flow is used in Eq. (8.19). Using Eq. (8.19) the weighting functions can now be computed:

In these equations, is from Eq.

while

and are from Table

8.2. The shape factors, and g,, depend on the shape of the soil particles.

For mineral soils has a value around 0.1. For organic soils it is 0.33. These equations are most useful as part of a computer program as they are quite long for hand calculations. They do, however, include all of the effects of temperature, moisture, density, and soil composition. The interaction among these factors is complex and

One can compute from g, = 1 -

and, at present, no simpler approach is apparent. Figure 8.3 shows thermal con- ductivity computed using Eq. (8.13) for the soils in Fig. 8.2. The mineral conductivities of the clay and loam samples are 2.3 and 2 W

The organic is 0.3 and the sand

is 5 W

. The sand curve is

meant to represent a sample with high quartz content. Example 8.2. For the soil in Example 8.1, find the thermal conductivity.

Assume T

2.5 W , q = 4, and = 0.15.

C,

Solution. To do the calculation, values are needed for A, A, D,, and

From Table A.3, A 145 and e

e. From Table A.2,

2.42 x m 2 /s. From Table 8.2,

2340 Pa. From Table A. 1, D,

0.025 W

and

0.60 w

. There is no easy

way of calculating but a loam soil at 0.2 water content is well above

Heat Flow in the Soil

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Volume Fraction of Water

8.3. Thermal conductivity of mineral and organic soils from Eq. (8.1 1). Temperature is 20" C.

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permanent wilting point, so the humidity must be nearly 1 .O. The volume fraction of minerals is 0.49, from example 1, and the water fraction is 0.2. The gas fraction is

1 - 0.2 - 0.49 0.31.

From Eq.

the dimensionless flow factor is

The thermal conductivity of the gas phase is (Eq. (8.18)):

Note that the contribution from latent heat transport is about twice that for conduction through the dry air. Using Eq. (8.19) the fluid conductivity can be computed. It is

Thermal and Admittance of Soils 123

Now the weighting factors can be computed using Eq. (8.20):

where we have assumed

0.1. Equation (8.13) is now used to find the thermal conductivity:

Even though the air has a very low thermal conductivity, it profoundly influences the conductivity of the soil when the gas fraction is high. Most of the heat has to flow through the air spaces, so they exert a controlling influence on overall heat flow. The model accounts for this through the fact that the weighting factor for the gas phase is larger than the other two factors.

The slope of the saturation vapor pressure function is strongly temper- ature dependent, so the apparent thermal conductivity of the gas phase increases rapidly with temperature. In the example just described, the gas phase conductivity is only a little over 10 percent of the water conduc- tivity, but as temperature increases they become more similar. At about

C, the gas and water phase conductivities are equal, so for moist soil (

the conductivity becomes independent of water content.