# Studies on Influence of Chromium Layer on Inner Surface of Steel Tube on Heat Transfer
## 2 Material and Numerical Methods  
### Heat Transfer Coefficient and Gas Temperature  
The heat transfer coefficient is calculated in the cross-section \(P_{1}\) to \(P_{30}\) of the tube from an expression [38]:  
\[h=\frac{6.1}{D^{0.2}}\left(\rho(t)w(t)\right)^{0.8} \tag{1}\]  
where:  
\(D\)--the inner diameter of the tube,  
\(\rho\)--density of gas,  
\(w\)--velocity of gas.  
Velocity \(w(t)\), density \(\rho(t)\) and gas temperature \(T_{g}(t)\) are calculated by solution of the internal ballistics model [39; 40; 41]:  
* when the piston is inside the tube: \[\frac{d\left(RT_{g}\right)}{dt}=\frac{\left(f-RT_{g}\right)m_{p}\frac{dz_{p}}{ dt}-\left(\gamma-1\right)\varphi mv\frac{dv}{dt}}{m_{p}z_{p}}\] (2) where: \(z_{p}\)--fraction of mass burned of propulsion material,  
\(v\)--velocity of piston,  
\[\rho=\frac{m_{p}z_{p}}{V_{0}+sl-\frac{m_{p}}{\rho_{p}}\left(1-z_{p}\right)- \gamma m_{p}z_{p}} \tag{3}\]  
assuming a linear variation of the velocity of the gas in the tube  
\[w=\frac{l_{i}}{l_{0}+l}\cdot v \tag{4}\]  
where: \(l_{i}\)-- distance from the bottom of the tube to the cross-section \(P_{i}\),  
\(l_{0}\)-- distance from the bottom of the tube to the first zone boundary (\(PS_{1}\) = 216 mm),  
\(l\)--travel of piston in tube,
* when the tube is open (piston leaves the tube): \[\frac{d\left(RT_{g}\right)}{dt}=\frac{\left(f-RT_{g}\right)\frac{dz_{p}}{dt}- \left(\gamma-1\right)RT_{g}\frac{d\zeta}{dt}}{z_{p}-\zeta}\] (5) \[\rho=\frac{m_{p}\left(z_{p}-\zeta\right)}{V_{0}+sl_{m}-\frac{m_{p}}{\rho_{p} }\left(1-z_{p}\right)-\eta m_{p}\left(z_{p}-\zeta\right)}\] (6) where: \(\zeta\)--fraction of the mass of gas that flowed out from the tube.  
Assuming that the gas flowing out of the tube reaches critical parameters  
\[w=\frac{l_{i}}{l_{0}+l_{m}}\cdot w_{cr} \tag{7}\]  
where \(w_{cr}=\sqrt{\gamma RT_{cr}}=\sqrt{\frac{2\gamma}{\gamma+1}RT_{g}}\); \(w_{cr}\), \(T_{cr}\)-- critical velocity and critical temperature of gas at the tube outlet.  
Calculations were carried out until the gas pressure dropped to 0.18 MPa [42]. The main parameters for the calculations [25] are shown in Table 2.  
An example of the calculation results of the heat transfer coefficient \(h_{i}(t)\) in the 6 cross-sections, \(P_{4}\), \(P_{10}\), \(P_{16}\), \(P_{21}\), \(P_{25}\) and \(P_{30}\), and the gas temperature \(T_{S}(t)\) are shown in Figure 3.  
For \(t\) = 4.54 ms, the heat transfer coefficient \(h_{i}(t)\) sharply drops in each of the 6 chosen cross-sections. This is the moment when the piston leaves the tube.  
The envelopes of the maxima \(h_{i}\) in all zones are shown in Figure 4. The highest value is achieved by the heat transfer coefficient in the S7, S8 and S9 zones. In the following zones, the values of \(h_{i}\) begin to slowly decrease.  
\begin{table}
\begin{tabular}{c c c} \hline \hline mass of piston & \(m\), kg & 0.380 \\ \hline mass of propulsion material & \(m_{p}\), kg & 0.376 \\ \hline cross-sectional area of tube’s hole & \(s\), m\({}^{2}\) & \(9.98\times 10^{-4}\) \\ \hline volume of propulsion material chamber & \(V_{0}\), m\({}^{3}\) & \(373\times 10^{-6}\) \\ \hline total travel of piston in tube & \(l_{m}\), m & 2.9342 \\ \hline coefficient of secondary works of gas & \(\varphi\) & 1.37 \\ \hline “force” of the propulsion material & \(f\), J\(\cdot\)kg\({}^{-1}\) & \(1.071\times 10^{6}\) \\ \hline covolume of gas & \(\eta\), m\({}^{3}\cdot\)kg\({}^{-1}\) & \(1.064\times 10^{-3}\) \\ \hline ratio of specific heats of gas & \(\gamma\) & 1.2 \\ \hline gas constant & \(R\), J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\) & 340 \\ \hline density of propulsion material & \(\rho_{p}\), kg\(\cdot\)m\({}^{-3}\) & 1600 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Main parameters for the calculations.  
Figure 4: Envelopes of the maxima \(h_{i}\) in all zones and the gas temperature \(T_{S}\) at the moment of maximum \(h_{i}\).  
Figure 3: Heat transfer coefficient \(h_{i}\) and the gas temperature \(T_{S}\) in the 6 chosen zones.