# Effect of Cavity Vacuum Pressure Diminution on Thermal Performance of Triple Vacuum Glazing
## 2 Materials and Methods
The elemental heat transfer through radiation is dependent on the surfaces that are comprised of facets with radiation boundary conditions. In this 3D FEM, a facet is a face of a solid element of TVG and each facet is presumed to be isothermal, having unvarying emissivity. Cavity heat radiation elements, calculated through Abaqus, produce a series of matrices, since they couple the temperature degrees-of-freedom of every node on the cavity surface. Such a method of surface radiation is followed according to [PERSON] [25] and [PERSON] et al. [26]. Radiation flux density \(q_{j}^{c}\) into a cavity facet \(j\) stated in Equation (11) depends on the grey body radiation theory, in which the monochromatic emissivity of the facets is independent of the wavelength of propagation of the radiation and only nondirectional reflection is applied with the approximation of isothermal and isoemissive cavity facets. \[\int_{V}\rho\dot{E}\delta TdV+~{}\int_{V}\frac{\partial\delta T}{\partial x}.k.\frac{\partial T}{\partial x}dV~{}=~{}\int_{V}\delta TdV+~{}\int_{A}\delta TqdA \tag{2}\]0  
where \(\epsilon_{i}\), \(\epsilon_{j}\) are the surface emissivities of facets \(i\), \(j\); \(\sigma\) is the Stefan-Boltzmann constant; \(F_{ij}\) is the geometrical view factor matrix; \(T_{i}\), \(T_{j}\) are the temperatures of facets \(i\), \(j\); \(T^{2}\) is the value of absolute zero on the temperature scale being used; and \(\delta_{ij}\) is the Kronecker delta. Distinctively, in the blackbody radiation, where no reflection takes place, the addition of all emissivities is equal to one and Equation (11) is condensed to Equation (12). \[\int_{V}\rho\dot{E}\delta TdV+~{}\int_{V}\frac{\partial\delta T}{\partial x}.k.\frac{\partial T}{\partial x}dV~{}=~{}\int_{V}\delta TdV+~{}\int_{A}\delta TqdA \tag{2}\]1  
The radiation flux \(q_{i}^{c}\) is made up of finite element facets, which occurs when the heat is exchanged in a way that is dependent on view factors, which is a dimensionless factor \(F_{ij}\) between two elementary areas, \(A_{i}\) and \(A_{j}\), satisfying the relation of Equation (13). \[\int_{V}\rho\dot{E}\delta TdV+~{}\int_{V}\frac{\partial\delta T}{\partial x}.k.\frac{\partial T}{\partial x}dV~{}=~{}\int_{V}\delta TdV+~{}\int_{A}\delta TqdA \tag{2}\]2  
where \(R_{ij}\) is the gap between two areas and \(\mathcal{O}_{i}\), \(\mathcal{O}_{j}\) are the angles between \(R_{ij}\). The dimensionless view factor is a geometrical number, and for the validation of the accuracy of the calculations used in Equation (14), it is a result of the fact that all rays from facet \(i\) collide with other the surface \(j\) in an enclosed cavity. \[\int_{V}\rho\dot{E}\delta TdV+~{}\int_{V}\frac{\partial\delta T}{\partial x}.k.\frac{\partial T}{\partial x}dV~{}=~{}\int_{V}\delta TdV+~{}\int_{A}\delta TqdA \tag{2}\]3To reduce the computational time and due to symmetrical conditions, one quarter (150 mm \(\times\) 150 mm) of the TVG size of 300 mm \(\times\) 300 mm \(\times\) 4 mm was simulated. The model was designed using an eight-node linear brick diffusion element, with a total of 174,100 elements and 204,226 nodes, used to model a quarter of the TVG as shown in \[\int_{V}\rho\dot{E}\delta TdV+~{}\int_{V}\frac{\partial\delta T}{\partial x}.k.\frac{\partial T}{\partial x}dV~{}=~{}\int_{V}\delta TdV+~{}\int_{A}\delta TqdA \tag{2}\]4 The vacuum cavity is modelled as a material with thermal conductivity dynamically dependent on the air molecules and its mean free path as most of the thermal conduction occurs due to molecules adsorbed on the surfaces and/or left in the cavity of the vacuum gap. A 6 \(\upmu\)m layer on the inner surface of the glass panes with the emissivity of tin oxide is the coating modelled for radiation heat-transfer simulations. The stainless-steel support pillars, modelled as a square with the side length of 1.78\(r\), represented the same heat transfer as of the cylindrical type used; this approach was used by [PERSON] et al. [27] and [PERSON] et al. [28].