Heat Flow and Storage in Soil

8.1 Heat Flow and Storage in Soil

In analyzing heat flow in the soil or the atmosphere, it is to mentally

divide the medium into a large number of thin layers, and consider the heat flow and storage in each layer. The amount of heat stored in a layer of air is small compared to the amount of heat transferred through it. Within the first few meters of the atmosphere the heat stored in the air is generally ignored and heat transfer processes are assumed to be approximately steady. The results of these assumptions are the equations developed in Ch. 7.

In soil the storage term is much larger and cannot be ignored. The heat flow from one layer to the next is still computed using the Fourier law (Eq. (6.3)) but now the continuity equation must be solved simultaneously to find the temperature variation with depth and time. The continuity equation is:

where

is the density of the soil, c, is the soil specific heat,

is

the volumetric heat capacity, and G is the heat flux density in the soil (from Eq. (6.3)). The left-hand side of Eq. (8.1) represents the rate of heat storage in a layer of soil and the right-hand side represents the heat flux divergence, or rate of change of heat flux density with depth.

114 Heat Flow in the Soil

Combining eqs. 6.3 and 8.1 gives

If thermal conductivity is constant with depth, k can be taken outside the derivative. We can also divide both sides by

to obtain a more familiar form of the heat equation:

where

is the soil thermal

the location in the soil where temperature will change fastest with time is the location where the change with depth of the temperature gradient is largest.

According to Eq.

In principle, solutions to Eq. (8.2) can simulate the behavior of soil temperature in space and time. The conditions for which analytic solutions

can be obtained, however, are very restrictive, and do not represent real soil environments very well. Realistic conditions can be simulated by solving the equation numerically, but these solutions are not very useful

for understanding the behavior of the system. We now look at a couple of simple solutions to Eq. (8.3). These are useful for understanding, at least qualitatively, spatial and temporal patterns in soil temperature.

If the soil is assumed to be infinitely deep, with uniform thermal prop- erties, and a surface temperature that varies sinusoidally according to the equation:

then the temperature at any depth and time is given by:

where is a phase shift that depends on whether t is local time, universal time or some other time reference. In Eq. (2.4) local time was used and

8. Recall from Ch. 2 that is the average temperature over a temperature cycle,

is the amplitude of the temperature fluctuations (half the difference between minimum and maximum) and is the angular frequency, which is calculated from

where is the period of the temperature fluctuations. In Ch. 2 we were using time in hours, so was in hours, but here we need in seconds.

Heat Flow and Storage in Soil

We are now interested in diurnal and annual fluctuations so

2n

365 x 24 x 3600

The symbol D represents the damping depth, and is calculated from:

Referring to Eq. it can be seen that D determines how much the amplitude of the temperature variation is attenuated with depth and how much the phase is shifted in time. When z =

D the exponential in Eq. (8.6) has a value of

indicating that the amplitude of temperature fluctuations at that depth is 37 percent of the amplitude at the surface. At =

0.14, and at z the amplitude is

the amplitude is

= 0.05. The damping depth therefore gives useful informa- tion about the depth to which temperature fluctuations penetrate into the soil. Even though the surface temperature is not sinusoidal, the damping

depth still gives a good idea of how deep diurnal and annual temperature fluctuations will penetrate.

The damping depth also affects the phase. At the depth where = or =

D, the temperature reaches a maximum when the surface temperature is at its minimum. To get an overall picture of temperature variation with depth and time Eq. (8.6) can be plotted in three dimensions.

This is shown in Fig. 8.1. Note how the temperature fluctuations are attenuated with depth and are shifted in time. At the bottom of the graph,

amplitude is only about five percent of the amplitude at the surface and the maximum occurs at about the same time as the minimum at the surface.

To the heat flux density at the soil surface differentiate Eq. substitute from Eq.

and set to zero. Doing this gives

Equation (8.9) shows that the maximum heat flux density occurs cycle

before the maximum temperature (Eq. (8.6)). This flux can

be integrated over a half-cycle to the total heat input to the soil. From the integration

is obtained, which is the same as the heat storage that would occur in a layer of soil of thickness

which changed temperature by

can be thought of as an effective depth for thermal exchange with the soil. Yet another relationship can be obtained from Eq.

Therefore,

or from the expression for total heat input. Using Eq. (8.8) the following can be written:

Heat Flow in the Soil

Temperature (C)

Depth (m) Time of Day

F I GURE 8.1. Graph of Eq. (8.6) showing how the surface temperature wave is attenuated with depth and shifted in time.

where This square root of the product of thermal conduc- tivity and volumetric heat capacity is called the thermal admittance, It can be seen that this relates directly to the ability of the soil to store heat,

since both the rate of heat storage (Eq. (8.9)) and the total amount of heat stored in a half-cycle are proportional to the thermal admittance. Soils with a high thermal admittance store heat more readily than those with low admittance. When the admittance is high much of the heat available

at the surface goes to heating the soil, while when it is low, most of the heat goes to the atmosphere.

The thermal admittance can be used to help understand how radiant energy that is absorbed at a dry surface might be partitioned between the atmosphere (convection) and the soil (conduction). Since the soil surface is dry, it can be assumed that latent heat loss is near zero, so radiant energy

is approximately equal to G +

H. This is partitioned as:

Some approximate values of

H are given in Table 8.1 for a dry bare soil and a dry mulch. This analysis is only qualitative because Eq. (8.10) assumes is con- stant with height, and the equations derived in ch. 7 show that increases with height. However, it can be seen that a higher atmospheric

Thermal Properties of Soils: Volumetric Heat Capacity

T ABLE 8.1. Dependence of soil and sensible heat flux on atmospheric and soil factors for two dry surfaces.

GIH Medium

Bare, dry soil

Loose straw mulch

Still air 50 20 Calm atmosphere

0.5 0.2 Windy atmosphere

tance (windy atmosphere) or a lower soil admittance (loose straw mulch) decreases the heat going to the soil.

The thermal admittance can also be used to estimate the contact sur- face temperature at the interface between two solid objects, each initially at a different temperature, when they are brought into contact. If one ob- ject with an initial temperature

and thermal admittance is brought into contact with another object with initial temperature

and thermal admittance

then the temperature at the interface, is given by

Clearly the object with the higher thermal admittance will dominate the interface temperature. This is why a tile floor "feels" colder than a carpet. The tile has a much higher admittance than the carpet.