## 2 Materials and Methods
### Data Post Processing  
At the end of each test session, the regression rate law \(a\) value, and the combustion efficiency (\(\eta_{c^{*}}\)) were estimated. \(a\) is the first constant in the [PERSON] law Equation (4) that models the grain port regression. This number is calculated by assessing the radial thickness of the fuel consumed throughout the burning period. \(\eta_{c^{*}}\) is defined by:  
\[\eta_{c^{*}}=\frac{c_{exp}^{*}}{c_{th}^{*}} \tag{6}\]  
where \(c_{th}^{*}\) is retrieved from the NASA CEA code and \(c_{exp}^{*}\) is computed using the following equation:  
\[c^{*}=\frac{P_{cc}A_{t}}{\dot{m}_{ox}(1+\frac{1}{\mathcal{O}/F})} \tag{7}\]  
The experimental \(c^{*}\) value is calculated using both the data collected during the tests and the internal component measurements. To do so, the lab-scale hybrid motor is disassembled at the end of each test to assess the engine consumables, namely, fuel grain, thermal protections and nozzle insert. Figure 9 illustrates the parts after the disassembly. Then, the mass of each component is evaluated to estimate the amount of material involved in the combustion. The sum of the burned mass contributions is called the fuel mass (\(m_{f}\)). Also, at the conclusion of each test, both the fuel grain and the nozzle insert are measured by means of the vernier caliper. By measuring the throat diameter of the graphite insert, the erosion value of the throat area at the end of the test may be computed (\(A_{th-fin}\)). The grain web thickness burnt during the test is computed using the measurement of the fuel final port diameter (\(\Phi_{port-fin}\)). However, the degradation of the grain port is not constant along the longitudinal axis. Furthermore, also the rear face is consumed, especially in the cases where the post chamber is present. So, along the grain port, the diameter is measured in three different points; moreover, in the case of rear face consumption, its diameter and depth are taken. It is important to emphasize that the average of the diameters along the grain port and the quantity of fuel burned less the mass calculated from the face consumption are utilized during post processing to obtain the regression rate constant \(a\). Consequently, rear face consumption is seen as a continuous fuel injection during the burnt that is independent of port diameter erosion. Figure 9: Disassembled parts of the lab-scale hybrid motor after the firing test. Once the sensor data have been collected, the estimation of the ignition and burning time can be computed by analyzing the combustion chamber pressure plot. The ignition time is considered as the time spanned between the first change in the outlet catalyst bed pressure plot (after the _V_test_ opening electric input) and the achievement of 90% of the combustion chamber pressure. The burning time (\(t_{b}\)) spans from the end of the ignition time and the _V_test_ closing electric input. These two time intervals are highlighted in Figure 11. Average engine performance parameters are evaluated referring to the burning time just defined. Once the burning time is retrieved, it is possible to compute the average \(O/F\) as:  
\[O/F=\frac{\dot{m}_{ox}}{\dot{m}_{fuel}} \tag{8}\]  
where \(\dot{m}_{fuel}=m_{fuel}/t_{b}\). An iterative approach is used to perform the regression rate analysis and simulate the Marxman law over the burning time span. The initial conditions are (1) the initial \(G_{ox}\), calculated from the measured \(\Phi_{port-in}\) and the collected oxidizer mass flow; and (2) the first guess value of \(a\). Then, the code will calculate the grain port regression during the burning  
Figure 11: Ignition and burning times identified on the pressure plot of a F5 15-second test. The logged pressure signal has been filtered. Figure 10: The fuel grain geometrical measurements. Grain port non-uniform longitudinal regression and rear face consumption. time and the fuel mass burned.