## 5 Thermal Analysis  
### Thermal Model-Lunar Environment  
Since the utilisation of regolith extends beyond radiation protection to structural support of the habitat and regolith makes up the majority of the building material in this study, its thermal insulation property is important to be assessed as the optimal thickness for radiation protection is sought. The 14-day night can be below \(-180\)\({}^{\circ}\)C and the 14-day day can be over 100 \({}^{\circ}\)C at the lunar surface, whereas the inside temperature of the habitat should be maintained around 20 \({}^{\circ}\)C. The thermal model is attempting to calculate the thermal profile in a regolith wall covering the habitat, during several shifts between the lunar nights and days. The one-dimensional time-dependent temperature profile is given by the Fourier equation:  
\[\frac{\partial T}{\partial t}=\frac{\lambda(T)}{\rho c_{p}}\frac{\partial^{2} T}{\partial x^{2}}, \tag{1}\]  
where \(T\) is the temperature of the wall as a function of time \(t\) and the position \(x\) in the wall, along the normal direction, \(\lambda\) is the thermal conductivity which is temperature dependent, \(\rho\) is the volumetric mass density and \(c_{p}\) is the specific heat capacity. The thermal conductivity \(\lambda\) of the regolith is estimated from different sources. The Lunar Reconnaissance Orbiter (LRO) observations give values between 0.0001 and 0.03 W/mK [49], and the measurements for the JSC-1A simulant exhibit values between 0.0019 and 0.0026 W/mK [50]. In [51], the thermal conductivity of native highlands regolith is 0.01 W/mK. Here, it is considered as dependant on density and temperature as in [52] using [53] model:  
\[\lambda=K+BT^{3} \tag{2}\]  
where \(K\) is the solid phonon conductivity and fixed at 0.015 W/mK, \(B\) is the radiative conductivity factor and \(\sim\)10\({}^{-11}\) W/mK\({}^{4}\)[52] and \(T\) is the temperature. Even in compressed regolith the morphology of the grains will remain close to that of native highlands regolith with the same shape and size of the grains embedded in a vacuum environment. In comparison, sintered or melted regolith building blocks will have massive and continuous morphologies with no vacuum inside which will lead to much higher thermal conductivity with an increase by one or two orders of magnitude as measured for processed simulants in [54]. Figure 6: Illustration of EMU’s effectiveness against radiation; the doses exhibited correspond to a 6-h period, which is the typical duration of an EVA; the image includes European Space Agency’s (ESA) Moon base model. The specific heat capacity of regolith depends on temperature, and some attempts have been made to define an empirical relationship, see expression in [49]. From the Apollo missions, temperature-dependent tables are available for the specific heat capacity. In the present calculations, the heat capacity was kept at a constant 840 J/kgK [51]. These coefficients, along with the density, affect largely the solution to the transient state of the conductive heat transfer equation. Finally, the regolith diffusivity is taken as \(\lambda(T)/\rho c_{p}\), which for \(T\) = 0 gives 1.12 \(\times\) 10\({}^{-8}\) and 4.5 \(\times\) 10\({}^{-9}\) m\({}^{2}\)/s for non-compressed (1.6 g/cm\({}^{3}\)) and compressed (4.0 g/cm\({}^{3}\)) regolith respectively. The initial conditions also affect the evolution of the temperature profile, however their effects fade out after few full cycles, i.e., lunar day and lunar night several times over, as will be shown in Section 5.2. Five complete continuous thermal cycles are simulated with the following conditions:  
* Initial Condition, \(t\) = 0: \(-\)70 \({}^{\circ}\)C for all \(x\). To start the calculation, it is assumed that the wall has homogeneous temperature, which is the approximate medium temperature between the outside extremes of the night and the day. * Boundary Conditions: \(\circ\): Convection in the habitat is taken into consideration in this model as \(dT/dx=-h(T(x=0)-20)\). Ref.