Abstract
The simultaneous diffusion of heat and moisture through soil is described by two coupled partial differential equations in which the diffusion coefficients are highly non-linear functions of the dependent variables. The system has been regarded as analytically intractable for any generality of coupled flow. However, for an asymptotically steady state, the equations show a marked periodic stability. Computer simulation indicates that the behaviour quickly becomes entrained to input boundary periodicity for any initial state, regardless of the detailed functional form of the diffusion coefficients. This property allows an harmonic series solution to be assembled. Factors such as amplitude decay, phase shift and wave form evolution may be evaluated. The solution is adapted to boundary conditions pertaining to arid soils and the results validated against the 1968 field data of Rose and the 1973 experiment by Jackson.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- ∇:
-
gradient operator
- ∇:
-
divergence operator
- θ A :
-
amplitude of surface moisture content variation
- θ l :
-
volumetric liquid content, m3/m3
- θ c :
-
value for moisture content, at which vapour diffusivity decays to zero
- θ M :
-
mean of surface moisture content variation
- θ s :
-
saturation value of moisture content
- α :
-
tortuosity factor, m/m
- η i :
-
eigenvalues of λ0
- λ :
-
hypothetical thermal conductivity, J/m/sec/K
- ρ 0 :
-
density of saturated water vapour, kg/m3
- ρ l :
-
density of liquid water, kg/m3
- ρ v :
-
density of water vapour, kg/m3
- σ :
-
surface tension, kg/sec2
- ψ :
-
matric potential, m
- C :
-
volumetric heat capacity, J/m3/K
- D * :
-
molecular diffusivity of water vapour in the porous medium, m2/sec
- D atm :
-
molecular diffusivity of water vapour in air, m2/sec
- D TV :
-
thermally induced vapour diffusivity, m2/sec/K
- D Tl :
-
thermally induced liquid diffusivity, m2/sec/K
- D Θv :
-
isothermal vapour diffusivity, m2/sec
- D Θl :
-
isothermal liquid diffusivity, m2/sec
- L :
-
latent heat of vaporisation, J/kg
- P :
-
atmospheric pressure at soil surface,Pa
- R :
-
gas constant of water vapour, J/kg/K
- T :
-
temperature,K
- T M :
-
mean temperature at surface, K
- T A :
-
temperature amplitude at surface, K
- g :
-
acceleration due to gravity, m/sec2
- h :
-
relative humidity, dimensionless
- p :
-
partial pressure of water vapour,Pa
- q v :
-
water vapour flux, kg/m2/sec
- t :
-
time, sec
- z :
-
depth, (measured downwards), m
References
Brust K. J., Van Bravel, C. H. M. and Stirk, G. B., 1968: Hydraulic properties of a clay loam soil,Soil Sci. Am. J. 32, 322–326.
Buckingham, 1915: Studies in the movement of soil moisture, U.S. Dept. Agr. Bur. Soils, Bull. 38.
De Vries, D. A., 1958: Simultaneous transfer of heat and moisture in porous media,Amer. Geophys. Union. 32(5).
Dorsey, W. E., 1940:Properties of Ordinary Water Substance, Reinhold Publ. Corp.
Edlefsen, W. E. and Anderson, B. C. 1943: The thermodynamics of soil moisture,Hilgardia,15(2)
Gardner, W. R., 1953: Some steady state solutions of the unsaturated moisture flow equation,Soil Sci.,85, 228–232.
Jackson, R. D., 1973: Diurnal changes in soil water content during drying. Spec. Publ. No.5,Soil Sci. Soc. Amer.
Kimball, B. A.,et al. (1976):Soil Sci. Soc. Am. J.,40, 18–24.
Krischer, O. and Rohnalter, H., 1940: Warmleitung und Dampf Diffusion im Feuchten Gutern,Verein. Deut. Ing. Forschungs. Left. 402.
Lykov A. V., and Mikhaylov Yu. A., Teoriya Teplo-i-Massoperenosa, 1963:Gosenergoizdat.
Philip, J. R., 1957: The theory of infiltration, 1.Soil Sci. 83, 345–357.
Philip, J. R., 1957: The theory of infiltration, 2.Soil Sci. 83, 435–448.
Philip, J. R., 1957: The theory of infiltration, 3.Soil Sci. 84, 251–264.
Philip, J. R., 1958: The theory of infiltration, 4.Soil Sci. 84, 421–438.
Philip, J. R. and De Vries, D. A., 1957: Moisture movement in porous materials under a temperature gradient,Trans. Am. Geophys. Un. 38, 222–232.
Marshall, T. J., and Holmes, J. W., 1988:Soil Physics, Cambridge University Press.
Rollinset al., 1953: Movement of soil moisture under a thermal gradient,Highway Res. Board Proc. 33.
Rose, C. W., 1968: Water transport in soil with a daily temperature wave, 1,Aust. J. Soil Res. 6, 31–44.
Rose, C. W., 1968: Water Transport in soil with a daily temperature wave, 2,Aust. J. Soil Res. 6.
Shepherd, R. and Wiltshire, R. J., 1994: A periodic solution to a non-linear diffusion equation,Transport in Porous Media,15, 175–182.
Van Bavel, C. H. M., 1952: Gaseous diffusion and porosity in porous media,Soil Sci. 73, 91–104.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Shepherd, R., Wiltshire, R.J. An analytical approach to coupled heat and moisture transport in soil. Transp Porous Med 20, 281–304 (1995). https://doi.org/10.1007/BF01073177
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01073177