## 4 Numerical Model
### Boundary Conditions  
The test case considered to demonstrate the advantages of adopting iterative submodelling approaches is a cooling channel of a regeneratively cooled thrust chamber of the final demonstrator of the Hyprob Project. The size, heat fluxes, and pressures considered are those encountered during the \"hot phase\", which represents the stage where the combustion of the hot gases causes a significant increase in the chamber pressures and temperatures during a typical fire ground test. The thrust chamber section considered is that illustrated in Figure 1 and is located in the cylindrical part of the chamber: here, heat fluxes are not as high as in the throat section, but the coolant is characterized by higher temperatures and the channel section is considerably larger than the throat one; consequently, it is worthwhile analyzing, from a thermal and mechanical point of view, the behavior of the cooling channel in that portion of the chamber. A simplified rectangular representation of the cooling channel, instead of trapezoidal geometry, has been chosen for this study, in order to minimize the geometrical and discretization operations. Such approximation is considered acceptable as the aim of the current work is to demonstrate the usefulness and the advantages of the implemented methodology rather than accurately reproduce the real component functioning. Finally, displacements on the top of the channel have been restrained in the vertical direction (y axis) to avoid a structural liability of the system. In order to save computation time, a half cooling channel, for both the thermal and the structural analyses, has been modelled taking advantage of the symmetry conditions (see Figure 5). The geometric parameters of the cooling channel are summarized in Table 1:  
Plane strain conditions have been applied for the two-dimensional analysis. The thermal and structural loads are respectively represented by the body temperature distribution in the hot phase and the maximum pressure in the cooling channels. Further details on pressure and heat flux laws during a fire test can be found in [5]. The results of the Computational Fluid Dynamic (CFD) analyses provide the convective boundary conditions to be applied on the internal surface of the combustion chamber and inside the cooling channel. The convective heat transfer coefficients and the adiabatic wall temperatures considered in the thermal analyses are summarized respectively in Tables 2 and 3: \(\mathbf{h^{C}}_{hot}\) and \(\mathbf{h^{C}}_{cold}\) are respectively the convective coefficients for the combustion gases and the coolant, while \(\mathbf{T^{C}}_{hot}\) and \(\mathbf{T^{C}}_{cold}\) are the combustion gases and coolant bulk temperatures (the subscripts \"\(hot\)\" and \"\(cold\)\" refer respectively to the hot gases and cooling channel sides). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**a (mm)** & **b (mm)** & **s (mm)** & **t\({}_{1}\) (mm)** & **t\({}_{2}\) (mm)** \\ \hline
2 & 0.62 & 0.9 & 0.5 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of the cooling channel. Figure 5: Cooling channel geometry and thermo–mechanical boundary conditions. Steady state thermal analyses and static structural analyses have been performed. The chamber pressure is 5.5 MPa, while the coolant pressure is 7.2 MPa. An iterative submodelling approach has been applied to both two-dimensional and three-dimensional models. The three-dimensional model has been obtained by a 20 mm extrusion of the cooling channel (see Figure 6) in the axial z direction of the thrust chamber. Axial displacements for the nodes of the faces at z = 0 and z = 20 mm have been restrained (\(\mathbf{u}_{z}=0\)). Figure 7 shows the cooling channel geometry and the positions of the interface between the global and local models considered. \begin{table}
\begin{tabular}{c c} \hline \(\mathbf{h^{C}}_{hot}\big{(}\frac{W}{m^{2}\,K}\big{)}\) & \(\mathbf{h^{C}}_{cold}\big{(}\frac{W}{m^{2}\,K}\big{)}\) \\ \hline
5600 & 280.000 \\ \hline \end{tabular}
\end{table}