## 2 Materials and Methods
### Motor Design and Configurations
#### 2.1.2 Combustion Chamber  
The combustion chamber includes the injection insert, the fuel grain, the post chamber and the thermal protections. An exploded view of these consumables is displayed in Figure 4.  
Figure 3: The vortex injector. (**a**) The geometrical parameters characterizing the vortex injector \(SN_{g}\). Reprinted with permission [45]. (**b**) A three-dimensional injector view. Figure 2: The different internal motor configurations used during the experimental test campaign. (**a**) Configuration with a paraffin-based fuel grain and a wider grain port. (**b**) Configuration with a paraffin-based fuel grain and the combustion post chamber. (**c**) HDPE fuel motor configuration. No fuel jacket is needed to accommodate the grain. All the thermal protections components (ochre-coloured) are made of cotton phenolic. The convergent and the upstream flanges protect the motor case metal flanges from the high temperature of the combustion gases. Moreover, the upstream flange holds a graphite injection ring. The insert shields the internal hole of the upstream flange, which otherwise would be excessively eroded by the swirled high oxidizing flow. The cotton phenolic erosion rate is fairly low compared with the fuel grain consumption. In addition, the surfaces exposed to the flame are minor, and thus the contribution to the fuel mass burned is lower than 10%. Moreover, the TP characteristic velocity is only slightly lower than that of the fuel. The fuel grain jacket is not present in motor configurations using the HDPE as the fuel grain material. The tests performed with HDPE do not entirely use the fuel grain. The residual thickness acts as a jacket, preventing the heating of the cylindrical metal case. The post-combustion chamber is used in some tests with paraffin-based fuels. Generally, this element is not needed in a vortex hybrid motor, since a high combustion efficiency is already achieved thanks to the swirled flowfield. Nonetheless, in our investigation, it has been useful for increasing the combustion gases' residence time inside the chamber in particularly compact motor configurations. The post chamber is a hollow cylinder with an internal diameter of 100 mm, while the cylindrical length (\(L_{\mathrm{PC}}\)) varies. The \(L_{\mathrm{PC}}/L_{grain}\) ratio is used to determines the post chamber length. The tested values of this parameter are 23%, 30% and 42%. A simple side-burning grain with a circular port, whose dimension is constant along the motor axis, serves as the fuel. Then, to fully define its shape, three measurements are required: the length, the external diameter and the initial port diameter. Every paraffin-based fuel grain has an exterior diameter of 125 mm, while the HDPE fuels measure 140 mm, due to the absence of the cotton phenolic jacket. For each of the examined grains, a different combination of the other two geometrical factors is used. Regression rate coefficients from earlier experimental data or from the experimental data retrieved in the actual test campaign are used to define both the internal diameter and the length. The initial port diameter ranges from a minimum of 15 mm to a maximum of 65 mm. The minimum value allows to not exceed the threshold of 950 kg/m\({}^{2}\)s for the oxidizer mass flux (\(G_{ox}\)). The maximum allowable value depends on the estimated grain thickness burned during the test. In tests where the port diameter is increased, the objective is to lower the average \(G_{ox}\) value over the burning time, i.e., to reduce the combustion gases' velocity inside the chamber, thus increasing their residence time. The length of the grain is adjusted from test to test to obtain the desired oxidizer-to-fuel ratio (\(O/F\)). In the case of paraffin or HDPE hybrid engines, the optimal \(O/F\) value is between 6.5 and 7.5. Figure 4: The exploded view of the motor cartridge with the consumables named. The influence of the fuel grain material is the other important parameter investigated in this study.