## 4 Heat Transfer through the Gas Gap
Denote by \(E_{g1}\) and \(E_{g2}\) the energies of gas flows moving from plate 1 to plate 2 and back, respectively. \(T_{g1}\) and \(T_{g2}\) are the temperatures of these flows. Let \(E_{1}\) and \(E_{2}\) be the energies of similar gas flows with temperatures \(T_{1}\) and \(T_{2}\). The heat flux \(q_{g}\) carried by the flows between the plates is proportional to the energy difference \(E_{g1}-E_{g2}\). The discontinuities of the temperature at the surfaces of the plates are proportional to this heat flux. Let \(T_{1}>T_{2}\). Then, the boundary conditions at the plates can be represented by Equations (11) [62]. \[\sigma_{1}(E_{1}-E_{g2}) =(E_{g1}-E_{g2}), \tag{11a}\] \[\sigma_{2}\big{(}E_{g1}-E_{2}\big{)} =(E_{g1}-E_{g2}), \tag{11b}\]  
where \(\sigma_{1}\) and \(\sigma_{2}\) are the thermal accommodation coefficients of gas molecules on the surfaces of plates 1 and 2. Therefore, \((E_{g1}-E_{g2})=\frac{\sigma_{1}\sigma_{2}}{(\sigma_{1}+\sigma_{2}-\sigma_{1} \sigma_{2})}(E_{1}-E_{2})\), as follows from the boundary conditions (11). Usually, the coefficients \(\sigma_{1}\) and \(\sigma_{2}\) are difficult to determine separately for thermal contact surfaces. However, the thermal contact can be characterized by the combined coefficient \(\sigma=\frac{2\sigma_{1}\sigma_{2}}{(\sigma_{1}+\sigma_{2})}\). Then, we obtain \(\frac{\sigma_{1}\sigma_{2}}{(\sigma_{1}+\sigma_{2}-\sigma_{1}\sigma_{2})}= \frac{\sigma}{(2-\sigma)}\). The thermal conductance \(G_{g}=\Delta q_{g}/(T_{1}-T_{2})\) between plates 1 and 2 can be obtained from Equation (10) by multiplying by the coefficient \(\frac{\sigma}{(2-\sigma)}\); see Equation (12). \[G_{g}=(c_{v}+\frac{k_{B}}{2})\frac{\sigma}{(2-\sigma)}\frac{p}{\sqrt{2\pi mk_{ B}T}}. \tag{12}\]  
The accommodation coefficient \(\sigma\) is about 0.6 for nitrogen gas in contact with a clean surface [11; 63]. Thus, \(G_{g}\) is about \(5.1\cdot 10^{4}\) W/m\({}^{2}\)K at \(p=10^{5}\) Pa and \(T=300\) K for nitrogen gas with \(\sigma=0.6\), \(c_{v}=5k_{B}/2\), and \(m=4.65\cdot 10^{-26}\) kg [58]. In the case of a thick gas gap \(d\), the coefficient \(G_{g}\) can be represented by Equation (13)  
\[G_{g}(d)=\frac{\varkappa_{g}}{d+\delta_{1}+\delta_{2}}, \tag{13}\]where \(\varkappa_{g}\) is the gas thermal conductivity [11]. The distances \(\delta_{i}\) associated with the temperature jumps on the first or second surfaces are represented by Equation (14) [11]. \[\delta_{i}=\frac{(2-\sigma_{i})}{\sigma_{i}}\frac{2\gamma}{(1+\gamma)}\frac{l_{ mfp}}{\text{Pr}}, \tag{14}\]  
where \(\gamma=C_{p}/C_{\text{v}}\) is the ratio of the specific heats at constant pressure and constant volume. Considering that \(l_{mfp}=\
u\sqrt{\pi m/2k_{B}T}\), \(\text{Pr}=\
u\rho C_{p}/\varkappa_{g}\), and \(\rho\) is the gas density [61; 62], we obtain Equation (15). \[\delta_{i}=\frac{(2-\sigma_{i})}{\sigma_{i}}\frac{\varkappa_{g}}{2c_{v}+k_{B}} \frac{\sqrt{2\pi mk_{B}T}}{p}, \tag{15}\]  
Thus, as follows from Equation (15), Equation (13) transforms into Equation (12) for \(d\to 0\), and \(\sigma_{1}=\sigma_{2}=\sigma\). \(G_{g}(d)\) increases with decreasing gas gap distance \(d\) according to Equation (13) and reaches the maximum possible value \(G_{g}^{max}\) at \(d\to 0\), where \(G_{g}^{max}\) is determined by Equation (12). This theory is well demonstrated in the following experiment, in which the ITC between two flat specimens of C45 steel was measured in real time while cycling the contact pressure \(p_{c}\) at \(T\) about 350 K in air at atmospheric pressure [64]. An increase in ITC from \(0.8\cdot 10^{4}\) to \(3.2\cdot 10^{4}\) W/m\({}^{2}\)K was observed when the contact pressure \(p_{c}\) changed from 25.8 to 120 MPa. Then, measurements were conducted with a cyclic change in the contact pressure \(p_{c}\) in the range of 50-120 MPa. The ITC varied within \(3\cdot 10^{4}\)-\(3.2\cdot 10^{4}\) W/m\({}^{2}\)K with such a cyclic change in pressure. The initial surface roughness of the samples was \(R_{a}=6.5\) um.