Heat Transfer from Animals to a Substrate

8.5 Heat Transfer from Animals to a Substrate

Equation (8.3) can be solved with a different set of initial and boundary conditions to obtain another result of interest to environmental biophysi- cists. The practical problem is that of estimating conduction heat loss or heat gain when an animal with body temperature comes in contact with soil or another substrate with initial temperature

The mathematical problem which approximates this is to find temperature as a function of depth and time for a

K , and initial temperature when the surface is instantaneously raised to a tempera- ture at time zero. The solution can be found in standard texts on heat transfer. It is:

medium of

where is the error function, a function which is tabulated in standard mathematical tables. To find the heat flow through the surface of the soil, differentiate Eq. (8.21) with respect to depth to get the temperature gradient, multiply the gradient by the thermal conductivity, and set depth to zero. The result is:

The numerator of the term multiplying the temperature difference is the thermal admittance of the soil and the denominator is the square root of multiplied by the length of time since the surface temperature was changed. As time increases the rate of heat flow into the soil decreases.

To make Eq. (8.22) into a form that can be used with the conductances the past two chapters the average heat input to the soil could be found over the total time of animal contact with the soil and then cal- culate and average conductance for that time period. The average heat input is obtained by integrating Eq. (8.22) over time and dividing by the time. The result is that the average heat flux density is exactly twice the instantaneous heat flux at time t given by Eq. (8.22). Now, using Eq. (6.8) as a definition of conductance, an equivalent soil conductance can be obtained:

where is the molar specific heat of air. The conductance is directly proportional to the soil admittance and inversely related to the square root of time. Values are plotted in Fig. 8.6 for mineral and organic soils. The admittance for concrete is about the same as for wet soil, and the admittance of straw or leaves is similar to that of the organic soil, so

Heat Transfer

Animals to a Substrate

wet soil

dry soil

organic

Time (hrs)

F IGURE 8.6. Thermal conductance of three soil materials averaged for the times shown.

Fig. 8.6 can be used to estimate heat loss or gain for most substrates. The average heat loss decreases by a factor of about five in going from contact periods of a few minutes to contact periods of a day. There is also roughly a factor of five difference between wet soil and dry soil of dry soil and organic material. These numbers should not be completely foreign to your experience. Just compare how you feel about sitting on

C to how you feel about sitting on a bale of straw. Before leaving this subject we need to indicate some cautions and limitations. First, Eqs. (8.21) through (8.23) are for one-dimensional heat flow. For a large animal lying on a substrate for a relatively short time a one-dimensional analysis is probably adequate, but the smaller the animal and the longer the time, the worse the one-dimensional analysis fits the problem. A second point to mention is that the soil conductance is in series with the coat and tissue conductances of the animal. The boundary conditions we used to solve the differential equation are therefore not strictly correct. They should, however, provide a good approximation. The third point is that energy budgets, and therefore conductances, are generally for the whole animal, while these calculations are just for the part of the animal in contact with the solid substrate. The conductance for

a concrete bench when the temperature is

the whole is obtained by multiplying the conductance in Eq. (8.23) by the ratio of area in contact with the substrate to total surface area of the animal.

128 Heat Flow in the Soil

References

Campbell; G. S., J. D. Jungbauer, Jr., W. R.

D. Hungerford. (1994) Predicting the effect of temperature on soil thermal conductivity. Soil Sci.

de Vries, D.A. (1963) Thermal properties of soil. In Physics of plant environment. W.R. Van Wick (ed.). North Holland Pub. Co., Amsterdam, pp.

Van Wijk, W.R. (ed.), (1963) Physics of the Plant Environment, New York: Wiley.

Problems

8.1. A common saying among farmers is "a wet soil is a cold soil." Is this true? At what water content would you expect a mineral soil to warm fastest (or have the largest damping depth)? What factors other than damping depth and thermal admittance might affect the rate of warming of a soil in the spring?

8.2. Compute the thermal conductivity of a dry sandy soil at

C with

a bulk density of 1.5 (quartz content equals zero).

8.3. Rattlesnakes often seek out rocky locations for their dens where they can retreat several meters underground in winter. If the daily near-surface temperature of the rocks is 30" C in summer and -5" C in winter, what is the lowest temperature the snakes would experi- ence during the year if they choose their depth to maximize their

temperature?

8.4. Compare the conductive heat losses for a deer on frozen, saturated soil (at

C) and a deer on a thick bed of leaves at the same temperature. Assume a body temperature of 37" C, and that 30 percent of the deer's surface is in contact with the substrate for a period of eight hours. Would the conductive losses be about the same, less, or more if the

deer were on a frozen, dry soil?

8.5. If the temperature of your bare foot is

C and the temperature of the floor is 20" C, calculate the interface temperature between your foot and the floor for a tile floor that has a thermal admittance three times that of your foot, and for a carpet floor that has a thermal

admittance

that of your foot.

The transport equation that we need to consider is Darcy's law (Eq. (6.4)). This law describes the transport of water in porous materials such as soils. Darcy's law describes most of the water flow that takes place in soils. Since water plays such an important role in the energy balance of soils, plants, and animals, an understanding of at least some simple applications of Darcy's law is important to environmental biophysicists. The processes that are important in determining the water budget of a soil are infiltration of applied water, redistribution of water in the soil profile, evaporation of water from the soil surface, and transpiration of water by plants.

We are mainly interested in applying Darcy's law to problems of one-dimensional water flow, with flow occurring vertically upward or downward. The components of the water potential (Ch. 4) responsible for flow are the matric and gravitational potentials. We can therefore substitute the matric and gravitational potentials for in Eq. (6.4) to obtain:

Two aspects of this equation make it more complicated mathematically than the equations for

and heat conduction. One is that the hy- draulic conductivity has a strong dependence on the dependent variable (matric potential). The other is the flow caused by the gravitational po- tential gradient. We do not try a frontal attack on Eq. (9. l), but do look for some simple cases for which we can get approximate solutions.

The Hydraulic Conductivity

The most important factor determining the behavior of Eq. (9.1) is the hydraulic conductivity function. When the soil is saturated with water (all pores filled) the hydraulic conductivity has a value called the saturated conductivity. As the pores drain, the conductivity falls rapidly. With half

130 Water Flow in Soil

the pore space drained (roughly field capacity) the conductivity has de- creased, typically, by a factor of a thousand. When three-fourths of the pore space has drained (roughly

wilting point) the conductivity is only one-millionth of its value at saturation.

A simple equation that gives a good approximation of the hydraulic conductivity is (Campbell, 1985):

where

is the saturated conductivity of the soil. The parameter b is the exponent of the mois- ture release equation, similar to Eq. (4.4):

is called the air entry water potential and

where 8 is the volumetric water content, and 8, is the saturation water content. The values of b,

and depend on soil physical character- istics such as texture. Saturated conductivity is large for coarse textured soils and small for textured, while the inverse is true for Since and both depend on the size of the largest pores in the soil, they are not independent, and, in fact are related by the equation:

= a 9.1 gives values of

b and other hydraulic properties for a range of soil texture. This table, with Eqs. (9.2) and

allows one to estimate the

T A BLE 9.1. Hydraulic properties of soils as a function of soil texture (recomputed from Rawls et al. 1992).

Texture

Clay

sand loamy sand sandy loam loam silt loam sandy clay loam clay loam silty clay loam sandy clay silty clay clay

Infiltration of Water into Soil 131

hydraulic conductivity and matric potential for various soils at any water content.