## 3 Interfacial Thermal Resistance at the Membrane/Gas Interface
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Gas** & \begin{tabular}{c} **Thermal Accom-Modation** \\ **Coefficient** \\ **\(\sigma^{\prime}\)** \\ **Dimensionalless** \\ \end{tabular} & \begin{tabular}{c} **Mean Free Path of** \\ **Gas Molecules** \\ \(\mu_{f\prime}\) \\ \end{tabular} & \begin{tabular}{c} **Temperature Jump** \\ **Distance** \\ \end{tabular} & \begin{tabular}{c} **Thermal** \\ **Conductivity** \\ \(\delta\) \\ \end{tabular} & \begin{tabular}{c} **Maximum Possible** \\ **Parameter** \\ \end{tabular} & \
\begin{tabular}{c} **Maximum Interfacial** \\ **Thermal Conductance** \\ \end{tabular} &
\begin{tabular}{c} **Maximum Cooling** \\ **Rate** \\ \(\delta\) \\ \end{tabular} \\
**Dimensionless** & \begin{tabular}{c} **\(\mu\)m** \\ \end{tabular} & \begin{tabular}{c} **\(\mu\)m** \\ \(\mu\)m \\ \end{tabular} & \begin{tabular}{c} **W/Wm-K** \\ \end{tabular} & \begin{tabular}{c} **M/Wm-K** \\ \end{tabular} & \begin{tabular}{c} **MW/Wm-K** \\ \end{tabular} & \
\begin{tabular}{c} **MW/Wm-K** \\ \end{tabular} &
\begin{tabular}{c} **K/s** \\ \(\mu_{f\prime}\) \\ \end{tabular} \\ \hline H\({}_{2}\) & 0.7 & 0.126 & 0.40 & 0.192 & 0.48 & 0.48 & \(2.1\cdot 10^{8}\) \\ \hline H\({}_{2}\) & 0.5 & 0.126 & 0.64 & 0.192 & 0.30 & 0.30 & \(1.3\cdot 10^{8}\) \\ \hline N\({}_{2}\) & 0.95 & 0.068 & 0.12 & 0.026 & 0.21 & 0.22 & \(0.9\cdot 10^{8}\) \\ \hline He & 0.5 & 0.200 & 1.13 & 0.156 & 0.14 & 0.14 & \(0.6\cdot 10^{8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermal and contact parameters of gases for a bright membrane at 300 K and \(10^{5}\) Pa.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Gas** & \begin{tabular}{c} **Mean Free Path** \\ \(l_{inf}\) \\ **\(\mu\)m** \\ \end{tabular}  & \begin{tabular}{c} **Thermal Diffusivity** \\ \(\mu\)m \\ \end{tabular}  & \begin{tabular}{c} **Thermal Diffusion Time** \\ \(\tau_{0}=\sqrt{\text{R}_{2}T/\text{min}}\) \\ **\(\mu\)m** \\ \end{tabular}  & \ \begin{tabular}{c} **Average Velocity of Gas Molecules** \\ \(V_{m}=\sqrt{\text{R}_{2}T/\text{min}}\) \\ **\(\mu_{f\prime}/V_{m}\)** \\ \end{tabular}  & \begin{tabular}{c} **Average Time between Collisions** \\ \(\tau_{0}=l_{inf}/V_{m}\) \\ **\(\mu\)m** \\ \end{tabular}  \\ \hline H\({}_{2}\) & 0.126 & \(1.6\cdot 10^{-4}\) & \(6.2\cdot 10^{-9}\) & 1.78 & \(0.7\cdot 10^{-10}\) \\ \hline He & 0.200 & \(1.9\cdot 10^{-4}\) & \(5.3\cdot 10^{-9}\) & 1.26 & \(1.6\cdot 10^{-10}\) \\ \hline N\({}_{2}\) & 0.068 & \(2.2\cdot 10^{-5}\) & \(45\cdot 1 Figure 2 shows the dependence of the rate \(R\) on \(r_{0}\) for different gases, obtained by Equation (8). \[R=\frac{2\lambda(T-T_{t})/r_{0}}{\rho_{m}c_{m}d_{m}}. \tag{8}\]  
\(R\) increases as \(1/r_{0}\) up to the maximum possible rate corresponding to the maximum possible heat flux \(G_{max}(T-T_{t})\), see Equation (6). For the estimation of \(R_{max}\), the thermal parameters of the gas and membrane are considered independent of temperature. In fact, the heat capacity of the membrane changes insignificantly at temperatures above 300 K [33]. The increase in gas thermal conductivity \(\lambda\) with increasing temperature [33] is compensated by an increase of the temperature jump distance \(\delta\). Thus, we get approximately the same \(R_{max}\) (with an accuracy of about 30%) for thermal parameters corresponding to 600 K; these parameters are available in [33]. The uncertainty in \(\sigma\) (which depends on the surface quality) is more significant. Thus, we obtain an estimate of the maximum possible interfacial thermal conductance (\(G_{max}\)) and the minimum radius (\(r_{min}\geq\delta\)) of the hot zone of the membrane. In fact, it makes no sense to reduce the radius \(r_{0}\) to less than 1 \(\upmu\)m, at least for the most heat-conducting hydrogen and helium gases, see Table 2 and Figure 2b. Thus, a cooling rate \(10^{8}\) K/s is possible at \(r_{0}\) of about 1 \(\upmu\)m. However, the maximum possible cooling rate \(R_{max}\) is even higher due to the additional heat transfer from the hot zone of the membrane to the periphery of the membrane.