3.2.4. Soil

(a) Energy loss of heat leakage in soil *Qs,l* includes the energy loss to the soil *Qs,l,s* and to the air *Qs,l,a*. The leakage of heat from the soil to the air is conducted through an insulating layer covering the ground and the heat leakage to the surrounding soil is abstracted as heat transfer between two layers of cylindrical surfaces of a hollow cylinder:

$$\begin{cases} Q\_{s,l} = Q\_{s,l,s} + Q\_{s,l,a} \\ Q\_{s,l,s} = \frac{2\pi\lambda\_s L\_a (t\_s - t\_{s,c})}{\ln\frac{r\_{s,c}}{r\_s}} \times \frac{1}{C\_{\rm NG}} \times 10^{-3} \\\ Q\_{s,l,a} = \frac{\lambda\_l A\_{il} (t\_s - t\_0)}{\Delta l} \times \frac{1}{C\_{\rm NG}} \times 10^{-3} \end{cases} \tag{23}$$

(b) Energy that the soil eventually use to heat up *Qs,a*:

$$Q\_{\mathbb{S},\mathfrak{a}} = Q\_{\mathfrak{s},\mathfrak{m}} - Q\_{\mathfrak{s},\mathfrak{l}} \tag{24}$$
