## 3 Modelling
### Thermal Insulation  
The investigation and evaluation of the reference and the alternative concept is based on steady state conditions, whereby the solid medium is assumed to be homogenously heated by the electrical heating system at the maximum temperature (\(T_{PH}\)). Due to the integrated bypass path inside the alternative construction, a two-dimensional formulation of the thermal insulation with its material-specific thermal diffusivity \(a_{Ins}\) is necessary, which enables the calculation of the axial (\(z\)) and radial (\(r\)) temperature profiles (\(T_{Ins,j}\)) as described in Equation (6) [32].  
\[\frac{\partial T_{Ins,j}}{\partial t}=0=a_{Ins}\bigg{(}\frac{\partial^{2}T_{ Ins,j}}{\partial z^{2}}+\frac{\partial^{2}T_{Ins,j}}{\partial r^{2}}+\frac{1}{r} \frac{\partial T_{Ins,j}}{\partial r}\bigg{)} \tag{6}\]  
A one-dimensional model (Equation (7)) is used to determine the cooling flow air temperatures (\(T_{F,c}\)) at a given velocity \(w_{c}\).  
\[\frac{\partial T_{F,c}}{\partial t}=0=\frac{a_{c}}{\rho_{F}c_{F}}\bigg{[}a_{V _{c},1}\bigg{(}\frac{\partial T_{Ins}}{\partial r}\bigg{|}_{r=D/2+\Delta s_{1 }}-T_{F,c}\bigg{)}+a_{V_{c},2}\bigg{(}\frac{\partial T_{Ins}}{\partial r} \bigg{|}_{r=D/2+\Delta s_{1}+\Delta s_{F}}-T_{F,c}\bigg{)}\bigg{]}-w_{c}\frac {\partial T_{F,c}}{\partial z} \tag{7}\]  
Here, heat is transferred from the inner (\(j=1\)) and outer (\(j=2\)) thermal insulation (\(T_{Ins,j}\)) to the cooling air at the gap specific surfaces (\(a_{V,c,j}\)) through a convective heat transport coefficient (\(a_{c}\)) according to [33].  
These heat balance equations form the basis for the conducted simulation studies in Section 4, whereby only Equation (6) is required for the reference concept and Equations (6) and (7) for alternative concept. To solve the resulting systems of equations, boundary conditions are required as summarized in Tables 1 and 2 depending on the thermal insulation system.  
Adiabatic conditions are defined for the axial boundaries (Equations (8), (11) and (14)) and convective heat losses by \(\alpha_{W}\)[34] as well as temperature conditions by \(T_{PH}\) for the radial boundaries at the outer (Equations (10) and (16)) and at the inner insulation surface (Equations (9) and (12)) respectively. Additionally, radial boundary conditions for the alternative concept between the inner and outer thermal insulation surfaces are required (Equations (13) and (15)), which include heat transport via radiation by C\({}_{r}\) and convection by \(\alpha_{c}\).  
This radiation parameter C\({}_{r}\) is based on a geometry-dependent formulation of the parameters \(\varphi_{12}\) and \(\varphi_{21}\) for the inner (\(j=1\)) and outer (\(j=2\)) thermal insulation dimension according to [35], the emission coefficient (\(\beta_{j}\)) and Stefan-Boltzmann constant (\(\sigma\)) as described in Equations (18) and (19):  
\[\mathrm{C}_{r,12}=\sigma\frac{\beta_{1}\beta_{2}\varphi_{12}}{1-(1-\beta_{1}) (1-\beta_{2})\varphi_{12}\varphi_{21}} \tag{18}\]  
\[\mathrm{C}_{r,21}=\sigma\frac{\beta_{1}\beta_{2}\varphi_{21}}{1-(1-\beta_{1}) (1-\beta_{2})\varphi_{12}\varphi_{21}} \tag{19}\]  
Here, a value of 0.7 is being used for the emission coefficients leading to radiation parameters C\({}_{r}\) for the conducted simulation studies of about \(1\times 10^{-8}\) W/m\({}^{2}\)K\({}^{4}\) to \(3\times 10^{-8}\) W/m\({}^{2}\)K\({}^{4}\).  
With the temperature-based boundary condition for the cooling flow (Equation (17)) and after spatial discretization, the set of nonlinear equations is solved by using the Newton-Raphson method in a commercial software (Matlab). Mesh studies were performed in order to allow a deviation of less than \(10^{-4}\) relating to the insulation surface temperature leading to an axial discretization of at least 15 nodes.