Derivation of the Analytical Solution of the Thermal
Conduction-Convection Equation under Fourier Series Boundary Conditions
Abstract
The thermal properties of soil play important roles in biogeochemical
cycles. The soil thermal diffusivity can accurately reflect the
transient process of soil heat conduction. In this study, we use
observation data from the 5, 10, 20, 40, and 80 cm layers in Golmud from
October 2012 to July 2013 and comprehensively compare the solution of
soil thermal diffusivity thereafter. A new model is established using
the thermal conduction-convection equation under Fourier boundary
conditions. The results show that (1) the amplitude method and the phase
method are based on a single temperature sine wave, which is used to
describe the general soil, although the accuracy is not high enough; the
logarithmic method and the arctangent method are performed four times a
day, the accuracy of the obtained result is also low; moreover, the
Laplace method does not have a clear soil temperature boundary function
and thus can better address extreme weather effects or nonperiodic
changes in soil temperature. (2) When solving the thermal conduction
equation by a numerical method, format 2 (Crank-Nichalson-Sch format) is
unconditionally stable, the data utilization is higher; in addition, the
obtained soil thermal diffusivity is less discrete, and the result is
more accurate. (3) When the soil temperature is simulated by the Fourier
series, as the order n becomes larger, the result becomes more accurate.
The Fourier series performs well in simulating the soil thermal
properties. This study provides a useful tool for calculating soil
thermal diffusivity, which may help to further characterize
biogeochemical cycles.