## 3 Data Processing Method  
The initial internal diameter of the grain is measured by vernier calipers before the test and the internal diameter of the grain after the test is measured by a CT scan. The calculation of Equations (1)-(6) refers to [23]. The average regression rate \(\tilde{\tilde{r}}\) is obtained as:  
\[\tilde{\tilde{r}}=\frac{r_{e}-r_{i}}{t} \tag{1}\]  
where \(r_{e}\) is the initial inner radius of the grain, \(r_{i}\) is the final inner radius of the grain, and \(t\) is the working time of the motor. Then the oxidizer mass flux \(G_{o}\) can be calculated as follows:  
\[G_{o}=\frac{\dot{m}_{o}}{\pi r^{2}} \tag{2}\]  
where \(\dot{m}_{o}\) is the oxidizer mass flow rate, and \(r\) is the radius of the grain. The combustion efficiency of the motor \(\eta_{c^{*}}\) is acquired by:  
\[\eta_{c^{*}}=\frac{c^{*}}{{c_{th}}^{*}} \tag{3}\]  
where \({c_{th}}^{*}\) is the theoretical characteristic velocity, which is calculated by RPA. \(c^{*}\) is obtained by:  
\[c^{*}=\frac{\int_{t_{0}}^{t_{uo}}p_{\rm c}\cdot A_{\rm t}dt}{\int_{t_{0}}^{t_{ uv}}(\dot{m}_{f}+\dot{m}_{o})dt} \tag{4}\]  
where \(p_{\rm c}\) is the chamber pressure, \(A_{\rm t}\) is the throat area of the nozzle, \(\dot{m}_{f}\) is the mass flow rate of the fuel, \(t_{w}\) is the moment when the motor operation ends, and \(t_{0}\) is the moment when the motor operation starts. Figure 2: Theoretical vacuum specific impulse and theoretical vacuum density-specific impulse. (**a**) Theoretical vacuum specific impulse. (**b**) Theoretical vacuum density-specific impulse. The specific impulse \(I_{\text{sp}}\) is acquired by:  
\[I_{\text{sp}}=\frac{\int_{t_{0}}^{t_{w}}Fdt}{\int_{t_{0}}^{t_{w}}\left(\dot{m}_{ f}+\dot{m}_{o}\right)dt} \tag{5}\]  
where \(F\) is the thrust of the motor. The ablation rate of nozzle \(r_{s}\) is obtained by:  
\[r_{s}=\frac{d_{2}-d_{1}}{2t} \tag{6}\]  
where \(d_{1}\) is the initial nozzle throat diameter, and \(d_{2}\) is the final nozzle throat diameter. For motors operating for long periods of time, the size of the grain passage varies significantly, and the oxidizer flux and regression rate change continuously during operation. In this paper, a method is developed to analyze the instantaneous regression rate of a hybrid rocket motor. By means of one or several tests, the change curve of the regression rate under different oxidizer flow rates can be obtained, and the regression rate equation can be fitted. Assuming that the pressure of the combustion chamber measured in the test is \(p_{c,\text{exp}}\), and the combustion efficiency of the motor is \(\eta_{c*}\), then by the equation for conservation of mass in the combustion chamber, the following equation can be given:  
\[\dot{m}_{f}=\frac{p_{c,\text{exp}}A_{t}}{c^{*}\eta_{c*}}-\dot{m}_{o} \tag{7}\]  
During the operation of a hybrid rocket motor, the combustion chamber pressure and the oxygen-fuel ratio are continuously varied, and the theoretical characteristic velocity is a function of them. The value of \(c^{*}\) is obtained by a thermodynamic calculation:  
\[c^{*}=f(p_{c},r_{of}) \tag{8}\]  
Moreover, the oxygen-to-fuel ratio \(r_{of}\) can be calculated as:  
\[r_{of}=\frac{\dot{m}_{o}}{\dot{m}_{f}}=\frac{\dot{m}_{o}}{\frac{p_{c,\text{exp }}A_{t}}{f(p_{c}r_{of})\eta_{c*}}-\dot{m}_{o}} \tag{9}\]  
For a given test chamber pressure \(p_{c,\text{exp}}\), nozzle throat area \(A_{t}\), combustion efficiency \(\eta_{c*}\), and oxidizer flow rate \(\dot{m}_{o}\), the oxygen-to-fuel ratio and fuel flow rate can be calculated iteratively from the above equation until the given accuracy is met:  
\[G_{o}=\frac{\dot{m}_{o}}{\pi r^{2}} \tag{2}\]0  
Based on the fuel mass flow rate obtained after iterative convergence, the oxidizer mass flux \(G_{o}\) and the instantaneous regression rate \(\dot{r}\) can be calculated by combining the cross section of grain. After integration, parameters of the next moment, such as channel area and flesh thickness, can be obtained.