# A Numerical Model to Explain Experimental Results of Effective Thermal Conductivity Measurements on Unsaturated Soils

• Gianluigi Bovesecchi
• P. Coppa
• M. Potenza
Article

## Abstract

Effective thermal conductivity measurements on unsaturated soils by means of the probe method (thermal conductivity probe, TCP) often present a nonlinear trend of $$\Delta T$$ versus ln ($$\tau$$). Three different slopes are present in the plots, while a homogeneous material should present only one. Being soils composite structures made of different phases (solid earth, liquid water and air), a possible explanation is the presence of phenomena other than pure conduction, such as water evaporation and vapor migration through the soil structure. A numerical model based on finite differences has been developed to simulate these phenomena. The model takes into account several factors including heat conduction, heat storage due to thermal capacity, water evaporation and water diffusion through a porous medium. Results show that two of the three slopes can be successfully simulated by the model, confirming the interpretation of the phenomena. However, the third slope from the experimental data is lower than the model’s slope, likely indicating the presence of other phenomena not yet taken into account, such as capillarity.

## Keywords

Effective thermal conductivity Evaporation kinetic Finite difference method Numerical model Soil Vapor migration

## Latin

a

Extent of reaction

A

Fick equation area [$$\hbox {m}^{2}$$]

$$c_\mathrm{p}$$

Specific heat [$$\hbox {J}\cdot \hbox {kg}^{-1}\cdot \hbox {K}^{-1}$$]

C

Concentration [$$\hbox {kg}\cdot \hbox {m}^{-3}$$]

Fo

Fourier number

I

Electric current [A]

K

Mass transfer coefficient [$$\hbox {kg}\cdot \hbox {m}^{-2}\cdot \hbox {s}^{-1}$$]

L

Probe length [m]

m

Mass [kg]

$${\dot{m}}$$

Mass flow rate [$$\hbox {kg}\cdot \hbox {s}^{-1}$$]

$$\dddot{q}$$

Specific thermal power generated or absorbed [$$\hbox {W}\cdot \hbox {m}^{-3}$$]

$${\dot{Q}}$$

Thermal power [W]

r

R

Heater wire resistance [$$\Omega$$]

$$\Delta R^{*}$$

Dimensionless grid size

s

Thickness [m]

S

External surface of the probe [$$\hbox {m}^{2}$$]

$$\hbox {S}^{{*}}$$

Free water surface [$$\hbox {m}^{2}$$]

T

Temperature [$${^{\circ }}\hbox {C}$$]

x

Volume fraction

$$W_\mathrm{ser}$$

Weight of the series component

## Greek

$$\alpha$$

Thermal diffusivity [$$\hbox {m}^{2}\cdot \hbox {s}^{-1}$$]

$$\delta$$

Stability criteria

$$\lambda$$

Thermal conductivity [$$\hbox {W}\cdot \hbox {m}^{-1}\cdot \hbox {K}^{-1}$$]

$$\rho$$

Density [$$\hbox {kg}\cdot \hbox {m}^{-3}$$]

$$\tau$$

Time [s]

$$\Delta \tau$$

Time step [s]

$$\varphi$$

Relative humidity

$$\chi$$

Humidity ratio [$$\hbox {kg}_{\mathrm {water}} \cdot \hbox {kg}_{\mathrm {dryair}}^{\mathrm {-1}}$$]

$$\psi$$

Degree of saturation

## Subscript

eq

Equivalent

f

Final condition

gas

Gas phase

liq

Liquid phase

p

Relative to probe surface

par

Parallel

pr

Probe

sam

Relative to sample edge

ser

Series

sol

Solid phase

vap

Vapor

0

Initial condition

pr

Probe

## Abbreviations

FD

Finite differences

TCP

Thermal conductivity probe

TGA

Thermo-gravimetric analysis

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## Authors and Affiliations

• Gianluigi Bovesecchi
• 1
• P. Coppa
• 1
• M. Potenza
• 1
1. 1.Department of Industrial EngineeringUniversity of Rome “Tor Vergata”RomeItaly