# Effect of Cavity Vacuum Pressure Diminution on Thermal Performance of Triple Vacuum Glazing
## 2 Materials and Methods  
To analyse TVG thermal performance or the prediction of U values, a 3D finite element model was developed using a validated approach based on the boundary conditions. The U value was quantified with the boundary conditions of: Temperature, \(T=~{}T(x,t)\); heat flux (surface) per unit area, \(q=~{}q(x,t)\); heat flux (volumetric) per volume, \(q=r(x,t)\); heat-transfer convection (surface), \(q=h(T-T^{\circ})\), where the film coefficient \(h=h(x,t)\) and temperature (sink) \(T^{\circ}=T^{\circ}(x,t)\); and heat-transfer radiation, \(q=[\epsilon]\left[(T-T^{\circ})^{4}-~{}(T^{\circ}-T^{\circ})^{4}\right]\), where \([\epsilon]\) is the radiation constant and the \(T^{\circ}\) value is absolute zero on the temperature scale used. The heat conduction occurred as a solid body in which the conductivity is subjected to the temperature, internal energy with effects of latent heat, and convection and radiation equations. According to [PERSON] and [PERSON] [21], the energy balance approach used is stated in Equation (1), and with the Fourier law Equation (2), is acquired by applying the standard [PERSON] spatial discretisation method. \[\int_{V}\rho\dot{E}dV=~{}\int_{A}qdA+~{}\int_{V}r\,dV \tag{1}\]  
\[\int_{V}\rho\dot{E}\delta TdV+~{}\int_{V}\frac{\partial\delta T}{\partial x}.k.\frac{\partial T}{\partial x}dV~{}=~{}\int_{V}\delta TdV+~{}\int_{A}\delta TqdA \tag{2}\]  
where \(V\) is the volume of the glazing components having: \(A\), area of the surface; \(r\), the heat supplied externally per volume; \(\rho\), the density of the materials; \(E\), the material time rate of the internal energy; and \(q\), the heat flux per unit area. It is an adopted assumption that iterative solutions are disjoined in such a way that only \(\dot{\mathrm{E}}=\dot{\mathrm{E}}(T)\), where \(T\) is the temperature of the material. \(\delta T\) is the arbitrary differential field satiating the essential boundary conditions. The body is estimated geometrically with finite elements, so the temperature is interpolated as in Equation (3):  
\[T=N^{N}~{}(x)T^{N},\;\mathrm{N}~{}=~{}1,~{}2, \tag{3}\]  
where \(T^{N}\) are nodal temperatures. The [PERSON] approach presumes \(\delta T\), that the variation field is interpolated by the same functions as \(\delta T=N^{N}~{}\delta T^{N}\), and that \(\delta T^{N}\) is arbitrarily chosen. First- and second-order polynomials consist of 1D, 2D, and 3D for \(N^{N}\). With these interpolations, the variation of Equation (2) is rewritten as:  
\[\delta T^{N}\bigg{\{}\int_{V}N^{N}\rho\dot{E}dV+\ \int_{V}\frac{\partial N^{N}}{ \partial x}.k.\frac{\partial T}{\partial x}dV\ =\ \int_{V}N^{N}rdV+\ \int_{Aq}N^{N}qdA\bigg{\}} \tag{4}\]  
The 8-node brick in a second-order heat-transfer element method [22] used a numerical Gaussian integration rule, because by using this for the TVG, a smooth solution can be achieved. The heat transfer mechanisms through conduction elements are proposed and are based on the combination of piecewise quadratic interpolations of \(T^{N}\) via the breadth of the element. The isoparametric interpolation functions for the element associated with the heat-conduction reference surface are utilised. \(T^{N}\) are coupled at a set of points via the breadth at each node of the element (nodes \(N\)). For computational numerical integration of the finite element equations, a 2 \(\times\) 2 Gauss integration scheme with a 2 \(\times\) 2 nodal integration scheme is used for the internal energy and specific heat term used for the quadrilateral element with a 3 \(\times\) 3 Gauss integration method. Let \((\mathcal{O}_{1},\mathcal{O}_{2})\) be individual TVG elements directing to the reference surface point of the element related with heat conduction, and \(s_{3}\) measures the position through the thickness of the heat conduction object, such as the edge seal, glass panes, and support pillars, so that \(-\frac{h}{2}-z_{0}\leq s_{3}\leq\frac{h}{2}-z_{0}\), where \(h\) is the thickness of the element and \(z_{0}\) is the offset of the reference surface from the mid-surface.