## 2 Methodology
### The Modified Layer-by-Layer Model  
The layer-by-layer (LBL) model is a classical method for the thermal insulation calculation of MLI proposed by [PERSON] [19]. He summarized the development of the MLI heat transfer theory over the past decades and proposed the new heat transfer model, which is a layer-by-layer model. Empirical coefficients in traditional models are replaced by quantified physical parameters. The heat flux leaking across MLI can be determined in three different heat transfer mechanisms separately, i.e., radiation, solid conduction, and residual gas conduction. The solid conduction heat flux across a spacer can be determined by the following equation,  
\[q_{\rm s}=\frac{c}{D}(T_{i}-T_{i-1}) \tag{1}\]  
where \(c\) represents the thermal conductivity of the spacer, W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\); \(D\) represents the thickness of the spacer between two adjacent reflectors, m. \(T_{i}\) and \(T_{i-1}\) (\(i\) is the current position of reflector) are the temperatures of the adjacent reflectors, K.  
The radiation heat flux can be calculated as follows,  
\[q_{\rm r}=\sigma\epsilon\Big{(}T_{i}^{4}-T_{i-1}^{4}\Big{)} \tag{2}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline \(T_{\rm h}({\rm K})\) & 293 \\ \(T_{\rm c}({\rm K})\) & 20 \\ \(\varepsilon\) & \(7.39\times 10^{-4}\times T^{2/3}\) \\ \(c\) (W/m\({}^{*}\)K) & \(8.823\times 10^{-6}+1.04\times 10^{-7}\times T\) \\ Residual gas & Helium \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The basic parameters and properties of the MLI. Figure 1: The schematic diagram of the MLI. where \(\varepsilon=1/(1/\varepsilon_{1}+1/\varepsilon_{2}-1)\), \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are the emissivities of the adjacent reflectors; \(\sigma\) represents the Stephan-Boltzmann constant (5.675 \(\times\) 10\({}^{-8}\) W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-4}\)). The residual gas conduction heat flux can be determined by the following equation,  
\[q_{\rm g}=\frac{\gamma+1}{\gamma-1}\sqrt{\frac{R}{8\pi mT}}p_{i}(x)\alpha(T_{i }-T_{i-1}) \tag{3}\]  
where \(R\) represents the gas constant, 8.314 kJ\(\cdot\)mol\({}^{-1}\cdot\)K\({}^{-1}\), \(\gamma\) represents the specific heat ratio, \(m\) represents the molecular weight of the residual gas, kg\(\cdot\)mol\({}^{-1}\); \(\alpha\) represents the accommodation coefficient. \(p_{i}(x)\) represents the pressure on the current layer, Pa. The total heat flux density is calculated by the following equation,  
\[q_{\rm t}=q_{\rm s}+q_{\rm g}+q_{\rm r} \tag{4}\]  
In the classical LBL model, the interlayer pressure of MLI is regarded as an ideal vacuum (e.g., 10\({}^{-3}\) Pa). If the pressure of each layer in the MLI could be determined experimentally or theoretically, the variation of the residual gas conduction heat flux across the MLI could be elaborated. It can be understood that the interlayer pressure varies in the heat leaking direction, and the distribution of the reflector temperatures across the MLI is impacted by the interlayer pressure correspondingly. The relationship between the interlayer pressure and the reflector positions can be given as a function,  
\[p_{i}=k_{1}i^{m}+k_{2}i^{m-1}+\ldots+k_{n-1}i+k_{n} \tag{5}\]  
where \(i\) represents the current layer position; \(n\) is the highest order of the polynomial function; \(k_{i}\), (\(i\) = 1,2,3 \(\ldots\)\(z\)) represents the polynomial coefficients. Using a gradient descent method and giving the pressure points at certain interlayer locations, the polynomial coefficients can be fitted. It is worth noting that the highest order of the polynomial function should be assumed before the calculation. Once the polynomial coefficients are determined, the interlayer pressure can be interpolated to any layer in the MLI by Equation (5). For the ideal vacuum of a MLI structure, set \(k_{i}=0\) (\(i\) = 1, 2, 3, \(\ldots\), \(k-1\)), \(k_{\rm n}=p_{\rm c}\). \(p_{\rm c}\) is the vacuum chamber pressure.