## 1 Introduction
### Thermal Conduction in Materials with Nanopores  
As the porosity of the material increases, the heat conduction through the solid skeleton decreases, because the large amount of pores limits the propagation of the phonons in the aerogel backbone. The contribution of the gas phase is due to elastic collisions between the gas molecules as it is presented in Figure 1. The thermal conductivity of the gas phase depends on the average free path length and the pore size of the gas molecules in the pores. For the aerogel to be an effective thermal insulator, its thermal conductivity must be lower than the conductivity of free atmospheric air (0.025 W/mK at 300 K and 1 atm). This is achieved if the pore size is smaller than the average free path length of the air molecules (70 nm, 300 K and 1 atm), thereby inhibiting the thermal diffusion of the gas (often referred to as the Knudsen effect). This is possible by pressure reduction. The ratio of the average free path of the gas molecules to the pore diameter is known as the Knudsen number. Here, the gas molecules enclosed in the pore collide with the pore wall and not with each other. For materials where the pore diameter is between 10 and 100 nm, such as an aerogel, heat is propagated by random movement through adjacent oscillators. In these non-crystalline materials, the thermal conductivity can be three orders of magnitude lower than that of the crystalline materials.  
It is also available in a composition optimized for different temperature ranges, in a fire-retardant design and in different thicknesses (2-20 mm). The contact surfaces between its particles are small. The cavities of the nanopores are smaller (5-50 nm) than the length of the path required for the free collision of the gas molecules within them. The gas molecules collide with the walls of the cavities, so convective heat flow is minimal. Nanopores are smaller in size than the wavelength of infrared heat rays, so their surfaces reflect and scatter most of the heat rays.  
Increasing the absorption coefficient of materials at a given temperature (by adding a light-shielding or reflective additive) is an effective way to suppress the thermal radiation conductivity of \(\lambda_{\rm r}\), such as a graphite additive for polystyrene, with which the effective (total) thermal conductivity can be further reduced. At the macroscopic level, the effective or total thermal conductivity mentioned above depends mostly on the type, density (\(\rho\)), temperature (T), moisture content (\(\omega\)) and age of the insulation material (aging or deterioration in fibrous materials) but, of course, it also depends on the method of manufacturing. At the microscopic level, it depends on the size and arrangement of the cells and fibers, where gas filling typically plays a role. Their effects can be taken into account with correction factors, according to the MSZ EN ISO 10456: 2008 standard [24]. This standard also provides a calculation method for determining the declared (manufacturer's) value of thermal conductivity factors.  
Accordingly, the effective thermal conductivity (\(\lambda_{\rm eff}\)) can be given as  
\[\lambda_{\rm eff}={\rm funct}(\rho_{i}\omega_{i}{\rm T}{\rm age}) \tag{4}\]  
Figure 1: Different forms of heat propagation in the fiber-reinforced aerogel (nanoporous material). Red is the solid conduction, yellow is the radiative part and blue is the conduction of gas molecules. Black is the glass fiber.