## 1 Introduction
Study results demonstrate that a double layer of tuff with a massive layer in the internal side and a resistive layer outside is the most suitable solution in the Mediterranean climate to maintain a comfortable operative temperature [13; 14; 15]. [PERSON] et al. analyzed the position of insulating layer, focusing on relative humidity and the way it influences heat transfer mechanisms. Their results show that insulation on the internal side of the wall is the worst under an internal relative humidity condition of 70%; therefore, insulation on the external side of the wall is the most suitable solution in the climate characterized by a hot summer and cold winter [16]. [PERSON] et al. showed that the thermal insulation layer on the outside represents the best solution to reduce heating and cooling energy demand [17]. The interest for the issue of thermal inertia in buildings is still remarkable; [PERSON] et al. [18] highlighted the impacts of building thermal inertia on thermal comfort and on heating or cooling energy demand. The capacity of attenuating heat fluctuations and time shifting are related to the thermal inertia of the building. It is widely recognized that a suitable solution in building design for most climates and intended uses is represented by a wall with high thermal mass located at the inner side of the thermal insulation. However, thermal inertia is a complex phenomenon and should be evaluated considering several factors, i.e., building location, intended use, and the thermal insulation of the building envelope. The majority of the buildings of urban fabric in old Italian towns has a huge thermal mass wall, which do not respect the current normative limits imposed for the thermal resistance (R) and, therefore, for the thermal transmittance (U). In order to improve thermal efficiency and guarantee living indoor comfort, identifying the best insulation techniques valid for both winter and summer periods is of utmost importance. Several authors, including [PERSON] et al., conducted studies aimed at identifying the best energy-efficient retrofitting methods; thus, analyzing how the position of insulating layer influences the behavior of building characterized by high thermal mass walls, especially, in a Mediterranean climate. Results show how passive strategies, such as natural ventilation, can lower the overheating of an internal environment in hyper-insulated buildings. Furthermore, the experimental study on several building retrofitting methods highlight that the best solution is represented by a massive external and well-insulated wall coupled with ventilated cavities [19]. [PERSON] et al. proved the importance of high thermal inertia in the building envelope, focusing on the problem of reduction of the cooling energy demand. This study also demonstrates that defining reference values for superficial mass (M\({}_{\text{s}}\)) and periodic thermal transmittance (Y\({}_{\text{mn}}\)) is not exhaustive to investigate the effects of thermal inertia on the summer performances of walls in Southern Europe. For this reason, other parameters are studied in a high thermal capacity wall, i.e., internal areal heat capacity (k\({}_{1}\)) and decrement factor (f). Moreover, in order to determine energy consumption, weather conditions are of utmost importance: Given the same Y\({}_{\text{mn}}\) and k\({}_{1}\) values, the research shows different energy consumptions in several Italian geographic areas (Milan and Catania). Furthermore, the mass of external wall represents another fundamental parameter because the best results were found in heavyweight walls [20; 21; 22]. Moreover, [PERSON] et al. highlighted the limits of the use of a lightweight building and hyper-insulated wall, focusing their analysis on the study of two coupled factors, namely the internal areal heat capacity (k\({}_{1}\)) and decrement factor (f). Their results show that the optimal combination in a hot summer climate is represented by high internal areal heat capacity equal to 30 k/m\({}^{2}\)K and decrement factor bigger than 0.07 [23].