# Effect of Cavity Vacuum Pressure Diminution on Thermal Performance of Triple Vacuum Glazing
## 2 Materials and Methods
The position of any point in the element is given by Equation (5). \[x=x^{0}(\mathcal{O}_{1},\ \mathcal{O}_{2})+s_{3}n(\mathcal{O}_{1},\ \mathcal{O}_{2}) \tag{5}\]  
where \(x^{0}\) is the locus and \(n\) is the unit normal of a point in the reference surface. The interpolation of the temperature is written as in Equation (6):  
\[T=N^{N}(\mathcal{O}_{1},\ \mathcal{O}_{2})M^{p}(s_{3})[\dot{\mathbb{T}}]^{NP} \tag{6}\]  
where \(M^{p}(s_{3})\) is a piecewise parabolic interpolation, \(N^{N}(\mathcal{O}_{1},\ \mathcal{O}_{2})\) is the interpolator in the reference surface, and \([\dot{\mathbb{T}}]^{NP}\) are nodal temperature values (at node \(N\), point \(P\) via the breadth). The thermal energy balance Equation (2) is now rewritten as Equation (7), which accounts for the heat conduction of each element with the approximate Jacobian matrix for the Newton method. \[\int_{V}\rho\delta T\frac{d\dot{E}}{dT_{t+[\Delta]t}}\ dTdV+\ \int_{V}\frac{\partial\delta T}{ \partial x}\ k.\frac{\partial dT}{\partial x}dV\ -\ \int_{A}\delta T\ \frac{\partial q}{dT_{t+[\Delta]t}}dT\ dA \tag{7}\]  
where \(dT\) is the correction to the temperature solution at time \(t+[\Delta]t\). The form of these terms for the elemental heat conduction, e.g., through the edge seal and support pillars, is obtained by introducing the interpolator and neglecting the change in area with respect to \(s_{3}\) of surfaces parallel to the reference surface. The piecewise quadratic interpolation through the breadth of finite elements is represented by Equation (8). \[\bigg{\langle}\frac{\partial T}{\partial s_{1}}\bigg{|}\frac{\partial T}{ \partial s_{2}}\bigg{|}\frac{\partial T}{\partial s_{3}}\bigg{\rangle}=\left[ \gamma^{P}\right]\bigg{\langle}T^{P}\bigg{|}\frac{\partial T^{P}}{\partial s _{1}}\bigg{|}\frac{\partial T^{P}}{\partial s_{2}}\bigg{\rangle} \tag{8}\]  
where \(\gamma^{P}\) is a one-step operator for stabilising the heat-transfer processes and the elemental heat transfer through convection modelled with a nonsymmetric Jacobian matrix, thus both transient and steady-state capabilities are incorporated. The transient capability introduces a limit on the time increment; the time increment is adjusted to satisfy this limit. Equation (9) is matched to the work by [PERSON] and [PERSON] [23,24]. \[\int\delta T\bigg{[}\rho c\bigg{\{}\frac{\partial T}{\partial t}+v.\frac{ \partial T}{\partial x}\bigg{\}}-\frac{\partial}{\partial x}\bigg{(}k.\frac{ \partial T}{\partial x}\bigg{)}-q\bigg{]}dV+\int_{Aq}\delta T\bigg{[}n.k.\frac{ \partial T}{\partial x}-q_{s}\bigg{]}\ dA=0 \tag{9}\]where \(T(x,t)\) is the temperature distribution; \(\delta T(x,t)\) is the arbitrary variation field; \(\rho(T)\) is the fluid density; \(c(T)\) is the specific heat of the air molecules, which can then be reduced for the perfect vacuum; \(k(T)\) is the conductivity of the air molecules at a particular pressure; \(q\) is the auxiliary heat distribution in the volume; \(q_{s}\) is the surface heat distribution per volume where temperature is not prescribed \((A_{q})\); \(n\) is the outward normal to the surface; \(x\) is the spatial position; and \(t\) is time. Low-pressure air in the cavity will have isotropic conductivity, so that \(k=kI\) (where \(k(T)\) is a scalar and \(I\) is the unit matrix). The boundary conditions are such that \(T(x)\) is coupled to \(A_{q}\) and heat flux inflowing to the domain across the rest of the surface, \(q_{s}(x)\), is associated by convection and/or radiation conditions and is expressed in Equation (10). \[q_{s}=-\text{n.k.}\ \frac{\partial T}{\partial x} \tag{10}\]  
where \(q_{s}\) is the heat flux linked to the conduction through the surface only, and any convection of energy across the surface is not included in \(q_{s}\). It makes negligible variance if the surface is part of a solid body (where \(q_{s}\) is defined by heat transfer into the adjacent body), since then, the normal velocity (v.n.) into that body is equal to 0, due to no continuous fluid crossing the surface, and here is assumed as static conditions in a TVG.