## 2 Rocket Engine Heat Transfer  
Assuming the flow in the thrust chamber is isentropic, and considering that the thrust chamber profile is a discretized domain in the axial direction, the flow area ratio equation, Equation (1), could be numerically solved at each discretized point to obtain the Mach number throughout the thrust chamber [28]:  
\[\frac{A}{A_{t}}=\frac{1}{M}\left[\frac{1+\frac{\gamma-1}{2}M^{2}}{1+\frac{ \gamma-1}{2}}\right]^{\frac{\gamma+1}{2\gamma-2}} \tag{1}\]  
where \(M\) is the Mach number, \(A\) is the corresponding area, \(A_{t}\) is the throat area, and \(\gamma\) is the specific heat ratio of the gas. Gas stagnation temperature can be found by Equation (2) using the gas temperature at the throat, \(T_{t}\), with \(M=1\). At any discretized point, the temperature of the flowing gas could be found using the stagnation temperature, \(T_{o}\) and the respective Mach number, \(M\), which is found using Equation (2). \[\frac{T_{o}}{T}=1+\frac{\gamma-1}{2}M^{2} \tag{2}\]  
Between the hot gas and the coolant, there are three thermal resistances: the hot gas film, the thrust chamber wall, and the coolant film, as depicted in Figure 2. In a high Mach number hot gas flow, instead of the free stream gas temperature, it is assumed that the adiabatic wall temperature, \(T_{aw}\), is the driving temperature for convection, which is given as follows [29]:  
\[T_{aw}=T_{o}\frac{1+\sqrt[3]{\Pr}\frac{\gamma-1}{2}M^{2}}{1+\frac{\gamma-1}{2 }M^{2}} \tag{3}\]  
where \(\Pr\) is the Prandtl number at the particular discretized point. On the other hand, the convection coefficient of the combustion gas film is found by Bartz correlation [30]:  
\[u_{g}=\frac{0.026}{d_{t}^{0.2}}\Bigg{(}\frac{\mu_{g}^{0.2}c_{p,g}}{Pr_{g}^{0.6}} \Bigg{)}_{o}\left(\frac{P_{cc}}{c^{*}}\right)^{0.8}\Bigg{(}\frac{A_{t}}{A} \Bigg{)}^{0.9}\sigma \tag{4}\]  
where \(t\), \(cc\), and \(g\) subscripts denote the throat, combustion chamber, and hot gas, respectively. In this equation, \(c^{*}\) is the characteristic velocity, \(d_{t}\) is the throat diameter, \(c_{p,g}\) is the specific heat of the gas, and \(\mu_{g}\) is the dynamic viscosity of the gas. Here, the correction factor, \(\sigma\), is given by the following expression:  
\[\sigma=\left[0.5\frac{T_{w}}{T_{o}}\Bigg{(}1+\frac{\gamma-1}{2}M^{2}\Bigg{)}+ 0.5\right]^{-0.68}\Bigg{(}1+\frac{\gamma-1}{2}M^{2}\Bigg{)}^{-0.12} \tag{5}\]  
where \(T_{w}\) denotes the wall temperature on the hot gas side. Therefore, the heat flux, \(q\), from the hot combustion gases to the inner wall of the thrust chamber would be the following:  
\[q=u_{g}(T_{aw}-T_{w}) \tag{6}\]  
This heat flux would be conducted radially outwards through the engine wall, as shown in Figure 2. Since the engine wall thickness is generally too small compared to the diameter, the radial heat transfer could be approximated as planar. Therefore, the heat flux given by Equation (6) also equals the following:  
\[q=\frac{k_{w}}{t_{w}}(T_{w}-T_{w,\rho}) \tag{7}\]  
where \(k_{w}\) is the wall thermal conductivity, \(t_{w}\) is the wall thickness, and \(T_{w,\rho}\) is the wall temperature at the base of the cooling channel. The coolant flow is assumed to be turbulent and fully developed. The convection coefficient for the coolant flow, \(u_{c}\), is found using the Dittus-Boelter correlation given by [31]:  
\[u_{c}=\frac{0.023\,\text{Re}^{0.8}\text{Pr}^{0.4}k_{c}}{d_{h}} \tag{8}\]  
where Re is the Reynolds number of the coolant flow inside the cooling channel, \(k_{c}\) is the thermal conductivity of the coolant, and the hydraulic diameter is given by \(d_{h}=4A_{c}/S_{c}\), where \(A_{c}\) is the area, and \(S_{c}\) is the perimeter of the cooling channel. In ODREC, the cooling channel cross section is assumed to be a rectangle as depicted in \[\frac{T_{o}}{T}=1+\frac{\gamma-1}{2}M^{2} \tag{2}\]0  
\[\frac{T_{o}}{T}=1+\frac{\gamma-1}{2}M^{2} \tag{2}\]1 Display of the thermal equilibrium at a section of thrust chamber and the thermal resistances.