# Numerical Simulation and Design of a High-Temperature, High-Pressure Fluid Transport Pipe
## 2 Methodology and Implementation
### Theoretical Approach  
#### 2.2.1 Heat Transfer and Structure Analysis  
Figure 3 shows a schematic of the heat transfer in a hand caliber, showing the inner and outer radii, \(\rm{R_{i}}\) and \(\rm{R_{o}}\), respectively. The radius of the interface between the two metal layers is \(\rm{R_{L}}\). In the heat transfer, the governing equation, Equation (1), is the heat diffusion equation--[PERSON]'s conduction equation. This is combined with the energy equation [16]. Here, the subscript \(\rm{i}\) of temperature \(\rm{T_{i}}\) and the thermal diffusivity, \(\rm{\alpha_{i}}\), for time \(\rm{t}\) refers to the region of barrel (\(\rm{i}=1\)) and the region of the barrel housing (\(\rm{i}=2\)). The friction heating between the projectile driving band and the barrel is neglected [17]. The effect of gravity on the convective heat transfer is also neglected because the heat transfer in the barrel is caused by forced convection, in which the flow is generated by external forces rather than gravity. As the heat transferred to the internal barrel surface via radiation is very low, it is ignored. As expressed in Equation (2), the initial uniform temperature (\(\rm{t}=0\)) is \(\rm{T_{0}}\). Regarding time \(\rm{t}>0\), the boundary conditions are given by Equations (3)-(6). \[\frac{\partial\rm{T_{i}}}{\partial\rm{t}}=\rm{\alpha_{i}}V\cdot(V\rm{T_{i}}), \;i=1,\;2. \tag{1}\]  
\[\rm{T_{i}}(r,0)=\rm{T_{0}},\;i=1,\;2\;R_{i}\leq r\leq R_{o}\;for\;t=0 \tag{2}\]  
\[-\rm{k_{1}}\frac{\partial\rm{T_{1}}(R_{i},\;t)}{\partial r}=\rm{h_{g}}\Big{(} T_{g}-\rm{T_{1}}(R_{i},\;t)\Big{)}=\rm{q_{in}}(t)\;at\;r=R_{i} \tag{3}\]  
\[\rm{k_{1}}\frac{\partial\rm{T_{1}}(R_{b},t)}{\partial r}=\rm{k_{2}}\frac{ \partial\rm{T_{2}}(R_{b},t)}{\partial r}\;at\;r=R_{b} \tag{4}\]  
\[\rm{T_{1}}(R_{b},t)-\rm{T_{2}}(R_{b},t)=-\rm{R_{k}}\frac{\partial\rm{T_{1}}( R_{b},t)}{\partial r}\;at\;r=R_{b} \tag{5}\]  
\[-\rm{k_{2}}\frac{\partial\rm{T_{2}}(R_{o},t)}{\partial r}=\rm{h_{oo}}(T_{2}( R_{o},t)-\rm{T_{oo}})+\rm{e}\sigma\Big{(}T_{2}(R_{o},t)^{4}-\rm{T_{oo}}^{4} \Big{)}=\rm{q_{out}}(t)\;at\;r=R_{o} \tag{6}\]  
Note that \(\rm{q(t)}\) is the heat flux, \(\rm{k}\) is the thermal conductivity, and \(\rm{R}\) is the interlayer thermal contact resistance. \(\rm{h_{g}}\), \(\rm{T_{g}}\), \(\rm{h_{oo}}\), and \(\rm{T_{oo}}\) are the convective heat transfer coefficient between the internal surface of the barrel and the hot propellant gases, the film temperature, the convective heat transfer coefficient between the external barrel housing surface and the surrounding atmosphere, and the temperature of outer surrounding atmosphere, respectively. \(\rm{e}\) and \(\rm{\sigma}\) are the emissivity of the barrel housing and the Stefan-Boltzmann constant, respectively. The convective heat-transfer coefficient for a fully developed turbulent pipe flow can be calculated using the Nusselt number (\(\rm{N_{u}}\)) as a function of the Reynolds (\(\rm{Re}\)) and Prandtl numbers (\(\rm{P_{r}}\)), using the  
Figure 3: Schematic of heat transfer in the barrel and barrel housing. Dittus-Boelter equation [18]. Notably, the hydrodynamic entrance region should be considered because the flow is not fully developed in the breech. Therefore, the convective heat transfer coefficient applied to the internal barrel surface should be calculated by incorporating the entrance factor, \(\mathrm{Ke}\), in the Dittus-Boelter equation, using Equations (7)-(9) [19]. \[\mathrm{h_{g}=f(N_{u})=\frac{k_{f}\times N_{u}}{D}} \tag{7}\]  
\[\mathrm{N_{u}=0.023\times Re^{0.8}P_{r}^{0.3}\times Ke} \tag{8}\]  
\[\mathrm{Re=\frac{D\times V\ avg}{\gamma}} \tag{9}\]  
where \(k_{f}\), \(D\), \(Ke\), \(V\ avg\), and \(\gamma\) are the fluid thermal conductivity, the hydraulic diameter, average propellant velocity, and the kinematic viscosity, respectively. \(Ke\) is a function of the axial position considering that the flow is not fully developed in the breech. Its value is approximately 1.75 at the breech, and it decays to 1 at a distance of the hand caliber, where the flow is thought to be fully developed.