# Estimating the Thermal Conductivity of Unsaturated Sand
## 4 Numerical Models with the MPTC Model
### Discussion of Volumetric Heat Capacity  
The volumetric heat capacity (\(\rho c\)) of soil is another of the most important parameters in a proper simulation of the temperature field of the ground surface and subsurface formation, which describes how well the soil stores heat. However, the volumetric heat capacity of sand could not be obtained from the literature because not enough specific details were given. The value of the volumetric heat capacity of sand selected for numerical simulation is discussed in the following paragraphs.  
The volumetric heat capacity of soil was calculated from the sum of the heat capacities of the soil constituents through many trials and theoretical derivations [44; 45; 46; 47]. Thus,  
\[\rho c=\rho_{\rm s}\ c_{\rm s}\ \phi_{\rm s}+\rho_{\rm w}\ c_{\rm w}\ \phi_{\rm w}+\rho_{\rm a}\ c_{\rm a}\ \phi_{\rm a} \tag{10}\]  
where \(\rho\) is the density (kg m\({}^{-3}\)), \(c\) is the specific heat (kJ kg\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\)), and \(\phi\) is the volume fraction of the component; the subscripts s, w, and a indicate the soil, water, and air constituents, respectively, where \(\phi_{\rm i}=\frac{V_{\rm i}}{V}\), and Equation (10) is given by:  
\[\rho c=\frac{V_{\rm s}}{V}\rho_{\rm s}\ c_{\rm s}+\frac{V_{\rm w}}{V}\ \rho_{\rm w}c_{\rm w}+\frac{V_{\rm a}}{V}\ \rho_{\rm a}c_{\rm a} \tag{11}\]  
Figure 11: Temperature curves of discrete circular temperature boundary.  
Figure 10: The discrete quarter circular temperature boundary.  
since the porosity \(n\) and degree of saturation \(S_{\rm r}\) are given by \(n=\frac{V_{\rm r}}{V}\), and \(S_{\rm r}=\frac{V_{\rm r}}{V_{\rm e}}\), respectively. Equation (11) can be rewritten in the form:  
\[\rho c=(1-n)\;\rho_{\rm s}c_{\rm s}+n\;S_{\rm r}\;\rho_{\rm w}c_{\rm w}+n\;(1-S_ {\rm r})\;\rho_{\rm a}c_{\rm a} \tag{12}\]  
The values of the specific heat of the water and air were taken as 4.18 kJ kg\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\) and 1.00 kJ kg\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\), respectively. Generally, the value of \(c_{\rm s}\) for clay was 1.10 kJ kg\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\) and 0.90 kJ kg\({}^{-1}\) for sand particles [48, 49]. For that reason, five typical values of specific heat were selected (0.70, 0.80, 0.90, 1.00, and 1.10 kJ kg\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\)) to discuss the influence of volumetric heat capacity on the temperature field of sand particles. The other parameters remained constant, including porosity \(n\) (0.45), \(\rho_{\rm s}\) (2.65 kg/m\({}^{3}\)), and \(\rho\) (1.45 kg/m\({}^{3}\)). The numerical analyses of thermal conductivities of sand used the results of Section 4.1 and employed the model presented in Section 4.2. Among them, point C was taken as an example for the following analysis, and the temperature curves of point B were employed.  
Figure 12 illustrates that the value adopted for the specific heat of sand particles had little influence on the variation in the temperature of the numerical analysis. A small amount of the variance in the results across the studies occurred when \(S_{\rm r}=0\), and the differences decreased with increasing \(S_{\rm r}\). The reason was because water has a greater specific heat than sand particles and air, and water had a dominant effect on the volumetric heat capacity of the sand because the sand had large pores. Thus, the value of \(c_{\rm s}\) for sand particles was selected to be 0.90 kJ kg\({}^{-1}\), and it was feasible in the numerical analysis.