Analyzing coupled transport of heat and water in soil can be complex (Hartley and Black, 1981; Bittelli et al., 2008). Thermal energy flux in soil is the sum of sensible heat flux, through thermal conduction in soil,qhc and latent heat fluxqhL transported by water vapor.
Total water flux in soil is composed of liquid water flux qlw, temperature driven vapor flux qvT and vapor flux driven by water potential qvP.
Fig. B.1 illustrates the contribution of the different fluxes to the energy and mass budget in wet and dry soil: (a) under wet soil conditions a stable liquid water−water vapor cycle is established, where liquid water flow toward the cable is in equilibrium with vapor flow away from the cable, the latter carrying out a considerable part of thermal energy away from the cable as latent heat, (b) under dry soil conditions the soil hydraulic conductivity is very low and the contribution of the liquid water flow almost disappears, therefore also reducing the dissipation of latent heat away from the cable. Under condition (b), temperature driven vapor flow equilibrates with vapor flow driven by water potential. The net water vapor flow becomes zero, and the thermal energy can no longer dissipate in form of latent heat, instead all heat dissipates by thermal conduction in the soil and the cable temperature is controlled by the low, dry soil thermal conductivity.
Here we provide a short summary of the main equations employed to describe coupled heat, liquid water and vapor flow. For more detailed informations we refer to Bittelli et al.
(2008) were it was tested against experimental data of heat, water and vapor flow dynamics in soils. Kroener et al. (2014a) have used this model to describe heat dissipation from underground electrical power cables. They tested the model with data from a down-scaled
qvP qlw +
qvT +
= 0 qhC
qhL +
qC
=
cable
soil mass balance energy balance
a) wet soil conditions
qvP qlw +
qvT +
= 0 qhC
qhL +
qC
=
cable
soil mass balance energy balance
b) dry soil conditions
Figure B.1: Energy and mass balance for heat dissipation in a) wet soil and b) dry soil. The thickness of the arrows indicates the quantitative contribution of latent heat fluxqhL, sensitive heat flux qhC, heat dissipation from the cable qC, water vapor flux driven by a gradient in water potential qvP, liquid water flux qlw, and water vapor flux driven by a gradient in temperatureqvT to the overall energy and mass budget.
experiment of heat dissipation from electrical underground cables (de Lieto Vollaro et al., 2014).
Richards equation describes liquid water flow in soils. Neglecting gravity, and for a radial geometry of the cable’s environment it is formulated as:
qlw
2πr =−Kdψ
dr (B.1)
where r is the radial coordinate,qlw is flux of water to the cable per unit length of cable (kg s−1 m−1),K is hydraulic conductivity of the soil (kg s m−3) andψis the matric potential of the soil (J kg−1).
Here we neglect the contribution of temperature driven liquid flow which is generally much smaller than liquid flow driven by the water potential gradient.
The hydraulic conductivity can be expressed as:
K = where n is a constant ranging from 2 to 3.5. The saturated conductivityKs and the air entry matric potentialψe, as well asndepend on soil texture and bulk density (Bittelli et al., 2015).
Fick’s first law governs steady vapor transport away from the cable. We can write qv
2πr =−Ddc
dr (B.3)
where c is the vapor concentration (kg m−3) andD is the vapor diffusivity in soil. The vapor diffusivity in soil is (Campbell, 1985):
D=D0βφm (B.4)
where φ is the air filled porosity of the soil and D0 is the binary diffusion coefficient of water vapor in air D0(T, P) (m2 s−1). β = 0.9 and m = 2.3 are constants. The air filled porosity is:
φ=θs−θ=θsh
1−(ψe/ψ)1/bi
(B.5) where θs is the saturation water content of the soil and b a parameter that depends on the soil texture.
Since the soil vapor concentration depends on water potential and temperature, we can expand Eq. (B.3) to express vapor flow as the sum of a water potential driven flux component qvP and a temperature driven flux component qvT:
qv
where h = exp(Mwψ/RTK) is relative humidity, c0v is saturated vapor concentration, Mw is the molecular mass of water, R is the universal gas constant, TK is the soil Kelvin temperature, and sis the slope dc0v/dT of saturation vapor concentration function.
Conductive heat flux is driven by a gradient in temperature and proportional to the soil thermal conductivityλ:
qhC
2πr =−λdT
dr (B.7)
Latent heat flux qhL is proportional to vapor flux and the latent heat of vaporization L:
qhL =Lqv (B.8)
Continuity for mass states that the sum of water and vapor flow from the cable is zero and continuity for energy states that sum of sensible and latent heat flow from the cable are equal to the heat produced by the cable qC. Eqs. B.1, B.6, B.7 and B.8 yield to the coupled system of equations for steady state heat dissipation from the cable:
0 = Kdψ
In the simplified analysis we assume that water movement away from the cable is entirely in the vapor phase and entirely driven by a temperature gradient, and that water flow back toward the cable is entirely in the liquid phase and entirely driven by a matric potential gradient. Condensation of vapor into liquid water leads to a continuous decrease of water and vapor flow with increasing distance from the cable. In this order of magnitude estimation, however, we assume that liquid water qlw and water vapor flow qv do not decrease with increasing distance from the cable.