UNSAT-H, Applications
Heat Flow Test for UNSAT-H
Introduction
Campbell (1977) reported an analytic solution to a heat conduction problem in which the temperature T at the soil surface varies by
T(z,t) = Tm + A(0)sin(wt)
where z = depth, cm
t = time, h
Tm = mean surface temperature, K
A(0) = amplitude of surface temperature, K
w = angular frequency of the temperature oscillation, 1/h.
Assuming the soil is uniform and infinitely deep, the solution (Campbell 1977) is
Equation 1
T(z,t) = T + A(0)e -z/D sin(wtd - z/D)
where
w = PI*2/24
and
D = [2 kh / (w Ch)]1/2
For this case, the angular frequency yields a complete surface temperature cycle in 24 h. To specify that the peak temperature occurs at noon, as is done for this example, the td value in Eq. (1) is modified by subtracting 6 h. The parameter, D, in Eq. (1) is the damping depth, which is the depth at which the temperature fluctuation has been reduced to 37% (i.e., 1/e) of its surface value.
Problem Description
A 1-m-deep soil profile is subjected to a temperature variation of 10 K from a mean surface temperature of 288 K occurring at noon. A total of 101 nodes, evenly spaced 1 cm apart, are used to discretize the soil profile. The soil type is a loamy sand known as lysimeter sand or L-soil (Rockhold et al. 1988). The hydraulic properties are described using the Brooks-Corey functions, with ThetaS = 0.4326, ThetaR = 0.0381, he = 9.4 cm, b = 1.2846, and Ks = 35.3 cm/h. The thermal properties were assigned values associated with the lysimeter sand at 22.5°C listed in Table 2 of Cass et al. (1981). The initial temperature at all nodes is 288 K; the initial suction at all nodes is 100 cm (ThetaI = 0.1007).
A horizontal profile is simulated; vapor flow is neglected so that water contents and thermal conductivities remain constant during the simulation. For the analytic solution, kh = 27.448 J cm-1 mole-1 K-1, Ch = 1.1927 J cm-3 K-1, and D = 13.26 cm.
Results
Figure 1 contains soil temperature results from UNSAT-H for several depths over a 24-h period. The analytic solution is included in Figure1 for each depth. Figure 2 shows a similar comparison, but for all depths at selected times. The closeness of the match between the analytic solution and the simulated temperatures shown in Figures 1 and 2 indicates that UNSAT-H correctly solves the heat conduction equation.
References
Campbell GS, 1977. An introduction to environmental biophysics. Springer-Verlag, New York.
Cass A, GS Campbell, and TL Jones, 1981. "Hydraulic and thermal properties of soil samples from the Buried Waste Test Facility," PNL-4015, Pacific Northwest Laboratory, Richland,Washington.
Rockhold ML, MJ Fayer, and GW Gee, 1988. "Characterization of unsaturated hydraulic conductivity at the Hanford Site," PNL-6488, Pacific Northwest Laboratory, Richland, Washington.