## 4 Discussion  
This work had several goals: (1) demonstrate that vortex injection can be used successfully with liquefying fuels; (2) investigate how low swirl numbers (\(SN_{g}<1\)) affect hybrid motor performance; (3) exhibit the potential of playing with two design variables (fuel composition and swirl number) to perform regression rate tailoring; and (4) identify the optimal set of parameters to fulfill the PHAEDRA project requirements. The first objective, the effective use of vortex injection together with liquefying fuels, has been confirmed. In fact, the first and third test campaigns have demonstrated:  
Figure 14: The values of the \(a_{exp}\) constant are plotted against the corresponding \(SN_{g}\). The black dots correspond to the experimental points. The blue dashed line is the least squares line and the relation is reported in the graph. * Rapid fuel ignition, with the burning time almost corresponding to the oxidizer valve opening time;
* Stable combustion for the majority of cases, with no anomalies from the graphs of the two pressure sensors in the motor. * An approximate straight line correlation between swirl number and regression rate, proving the possibility of tailoring the grain geometry by selecting the proper fuel-\(SN_{g}\) combination. This has been verified for two very different paraffin formulations and four different swirls numbers (i.e., 0.75, 1.00, 2.00, 3.33). It is important to underline that the relation between the regression rate and the swirl number can be approximated with a straight line that intersects the \(Y\)-axis with a positive bias, and therefore the behavior of the regression rate with the swirl number is sub-linear, i.e., doubling the swirl number provides a much lower increase in fuel consumption. Moreover, the real behavior of the regression rate with the swirl number should be more accurately approximated by a relation like \((1+SN_{g}^{2})^{m}\)[10], so the fits shown in this paper should be used with caution, particularly outside the ranges from which they were inferred. For the high regressing F8 paraffin with a \(SN_{g}>2\), the regression rates achieved are up to 5 times higher than usual for paraffin, and even 15 times higher than classical fuels with axial injection. These impressive results could pave the way for the effective utilization of single port hybrid rocket motors at larger scales (rule of thumb > 100 kN) and/or shorter burning times than usually considered. Unfortunately, the lab scale involved in the tests is on the opposite spectrum of size with respect to these favorable cases, producing very short, stubby fuel grains (\(L/D<1\)) and small combustion chamber volumes. The short residence time probably negatively affected the vaporization of the entrained droplets. The fact that the efficiency with vortex injection is already above 90% level at this small scale is a promising indication that very high levels should be possible to be achieved at the targeted larger scales where the residence time is longer and the grain geometry is more appropriate, allowing better vaporization and mixing of the fuel in the oxidizing stream. An exception to the general trend occurred in the first test of the campaign, F8_001, where the ignition occurred later than expected, due to the very unfavourable \(O/F\) value and the initial high \(G_{ox}\). Generally, the oxidizer-to-fuel ratio has a strong influence on the combustion reactions, both in the transient phase of the flame ignition and in the nominal operating phase. Efficiency is also dependent on the \(O/F\) ratio; however, in the performed experiments, higher mixture ratios were also related to shorter fuel grains and smaller port volumes, thus further contributing to the final results. Regarding the second objective of the test campaign, i.e., the possible use of low swirl numbers down to 0.5, the results are encouraging. The tests with HDPE, with a longer grain and no droplet entrainment, showed very high efficiency at small scale.