# Effect of Cavity Vacuum Pressure Diminution on Thermal Performance of Triple Vacuum Glazing
## 2 Materials and Methods
A graded mesh with a higher number of elements in the pillars was employed to achieve adequate representation of the heat transfer [6; 17]. An example of the finite element mesh employed is shown in \[\int_{V}\rho\dot{E}\delta TdV+~{}\int_{V}\frac{\partial\delta T}{\partial x}.k.\frac{\partial T}{\partial x}dV~{}=~{}\int_{V}\delta TdV+~{}\int_{A}\delta TqdA \tag{2}\]5 A series of 15 convergence tests were performed to ensure that the density of the elements used was sufficient to predict the thermal performance with an accuracy of more than 97%. It is vital in the 3D finite element model to perform the simulation with repetitive results and confirm with the literature. In the convergence tests, the error found was less than 0.4% and found to be precisely repeatable. The material properties and parameters of the simulated triple vacuum glazing are listed in \[\int_{V}\rho\dot{E}\delta TdV+~{}\int_{V}\frac{\partial\delta T}{\partial x}.k.\frac{\partial T}{\partial x}dV~{}=~{}\int_{V}\delta TdV+~{}\int_{A}\delta TqdA \tag{2}\]6 \begin{table}
\begin{tabular}{c c c} \hline
**Type** & **Boundary** & **Parametric Condition** \\ \hline \multirow{3}{*}{TVG dimensions} & Pane (K glass) top & 0.3 m \(\times\) 0.3 m \\  & Pane (K glass) middle & 0.3 m \(\times\) 0.3 m \\  & Pane (K glass) bottom & 0.3 m \(\times\) 0.3 m \\ \hline \multicolumn{3}{c}{Pane thickness} & 4 mm \\ \hline Pilkington K glass pane & Heat conductivity & 1 Wm\({}^{-1}\) K\({}^{-1}\)[29] \\ \hline \multirow{3}{*}{Emittance} & Low-emissivity coating (each pane) & 0.15/SnO\({}_{2}\) \\ \hline \multirow{3}{*}{Edge seal} & Alloy & Cerasolzer CS186 \\  & Width & 10 mm \\  & Heat conductivity & 46.49 Wm\({}^{-1}\) K\({}^{-1}\) * \\ \hline \multirow{3}{*}{Support pillar} & Alloy & Stainless steel 304 \\  & Diameter & 0.3 mm \\ \cline{1-1}  & Height & 0.15 mm \\ \cline{1-1}  & Pillar separation & 24 mm \\ \cline{1-1}  & Heat conductivity & 16.2 Wm\({}^{-1}\) K\({}^{-1}\)[16] \\ \hline \end{tabular}  
* Measurement of the thermal conductivity of Cerasolzer CS-186 reported by [PERSON] [17].  
\end{table} 
Table 1: Constructional boundary elements of the 10-mm wide edge-sealed TVG (Triple Vacuum Glazing). Figure 1: A quarter portion (size of 150 mm \(\times\) 150 mm) of the 10-mm wide edge-sealed triple vacuum glazing showing the developed finite element mesh. A thermal contact between glass interfaces with a cavity gap between them is specified by incorporating the gap conductance coefficient between two closely adjacent surfaces with small clearance (similar to glass surfaces separated by support pillars). Thermal gap conductance between two surfaces is represented by nodes 1 and 2 with temperatures \(T_{1}\) and \(T_{2}\) (\(T_{1}>T_{2}\)), and the clearance \(d\) between them is shown in Figure 2.  
The gap heat transfer is defined by the summation of gaseous conductance heat transfer and radiation heat transfer between two adjacent surfaces, i.e., Equation (15). \[q_{gap}=q_{c}+q_{r} \tag{15}\]  
where \(q_{gap}\) is the gap heat transfer, \(q_{c}\) is the gas conductance heat transfer, and \(q_{r}\) is the radiation heat transfer. The gas conductance heat transfer over a fixed gap clearance \(d\) is given by Equation (16). \[q_{c}=k_{g}(T_{2}-T_{1}) \tag{16}\]  
where \(k_{g}\) is the conductance coefficient of the gas and \(T_{1}\) and \(T_{2}\) are the temperatures at nodes 1 and 2, respectively. Subroutine GAPCON can help to define the dependency of \(k_{g}\) on different variables, including temperature, distance, and pressure between the bodies. In this study, it is used to define the \(k_{g}\) as a function of pressure between the glass surfaces [22]. The radiation heat transfer (\(q_{r}\)) between two nodes in contact is given by Equation (17). \[q_{r}=\sigma(T_{2}-T_{1}) \tag{17}\]  
where \(\sigma\) is the Stefan-Boltzmann constant and \(T_{1}\) and \(T_{2}\) are the temperatures at nodes 1 and 2, respectively. The theoretical model of the thermal gap conductance is implemented into Abaqus using the user-defined GAPCON subroutine.