# Estimating the Thermal Conductivity of Unsaturated Sand
## 3 Model Development  
### Parallel Theoretical Models (Wiener Model)  
Sand is considered a porous medium that is composed of air, water, and solids; the particles of sand have point contacts with their neighbors. Each sand has a unique mineral composition, size distribution of the solid phase, and geometry of the particle and pore  
Figure 1: General relationship between soil thermal conductivity and degree of saturation.  
structure, and those parameters have a significant influence on the thermal properties of the soil [34; 35; 36]. Furthermore, the water content in the pores of sand plays an important role. Consequently, it is difficult to assess whether a prediction accurately considers all these factors [37]. The proposed PTC model predicts the thermal conductivity of soil based on the volume fraction of the three phases, i.e., solid, liquid, and air, and the models are expressed as follows:  
\[\lambda_{\text{parallel}}=\sum\phi_{i}\lambda_{i}=\phi_{s}\lambda_{s}+\phi_{ w}\lambda_{w}+\phi_{a}\lambda_{a} \tag{1}\]  
where \(\lambda_{\text{parallel}}\) is the thermal conductivities of the PTC model in the unfrozen state; \(\phi_{\text{i}}\) is the volume fraction of component \(i\), and \(\sum\phi_{i}=1\); \(\phi_{s}\), \(\phi_{w}\), and \(\phi_{a}\) and are the volume fractions of solid particles, water, and air, respectively; \(\lambda_{i}\) is the thermal conductivity of component \(i\); \(\lambda_{s}\) is the solid thermal conductivity (2 to 8 Wm\({}^{-1}\)C\({}^{-1}\)); \(\lambda_{w}\) and \(\lambda_{a}\) are the thermal conductivity of water (0.6 Wm\({}^{-1}\)C\({}^{-1}\)) and air (0.02 Wm\({}^{-1}\)C\({}^{-1}\)), respectively.  
The PTC model has the advantage of using clearly defined concepts to estimate the thermal conductivity of soil. In general, the air volume, moisture content, and particle density are represented by selecting the two indexes, porosity n and degree of saturation \(S_{\text{r}}\)[38; 39]. Thus, the PTC model is also expressed by:  
\[\lambda_{\text{parallel}}=\left\{\begin{matrix}(1{-}n)\lambda_{s}+n\lambda_ {a}&S_{r}=0\\ (1{-}n)\lambda_{s}+nS_{r}\lambda_{w}+n(1{-}S_{r})\lambda_{a}&S_{r}\
eq 0\end{matrix}\right. \tag{2}\]  
However, the materials are made up of columns in the PTC model, and the heat conduction is constant at all points and the same in both phases, which depends on the thermal conductivity of each phase. The reason for the upper bound of the thermal conductivity of media was calculated by the PTC model based on the assumption that the components were superposed to form a multiphase mixture, but the form of contact between solid particles was not taken into consideration. Figure 2 indicates the comparison between measured values and calculated values from the PTC model; four examples were derived from the literature [6; 8; 40]. As shown in Figure 2, compared to the measured values, the PTC model gave an upper bound, and the results of the calculation generally overestimated the data. The estimated values of the PTC model showed a linear trend over a wide range of \(S_{\text{r}}\); this tendency of the sand thermal conductivity did not correspond to the actual situation, which showed a nonlinear trend represented by the measured values. The maximum error in the calculated value of the PTC model occurred at \(S_{\text{r}}=0\) and gradually decreased according to the increase in \(S_{\text{r}}\) because the soil was treated as a continuum by the PTC model; broken and scattered particles were not considered.  
Figure 2: Comparison between measured and PTC-model-calculated values.