# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 3 Results and Discussions  
### Single-Thrust Case  
The single-thrust motor is a commonly employed propulsion system in space launch vehicles. A low pressure ratio (maximum pressure to mean pressure) design can enable significant reductions in both chamber wall thickness and inert mass, providing an improved motor performance [21]. In this case, we will design a modular grain with a thrust of approximately 100 kN, a duration of 5 s, and a target pressure of 10 MPa. Firstly, it is necessary to determine the appropriate nozzle throat area. Assuming the thrust coefficient \(C_{\rm F}\) is 1.4, the nozzle throat area is as follows:  
\[A_{\rm t}=\frac{F}{p_{\rm c}C_{\rm F}}=0.007143\;\rm m^{2} \tag{32}\]  
Figure 6: Flow diagram of optimization. As the characteristic velocity \(c\)* of the propellant is given as 1400 m/s, the mass flow rate can be calculated by Equation (23). Furthermore, in order to optimize the design, the constraints outlined in Table 2 are considered. These constraints provide guidelines and limitations for the optimization problem. In this particular case, the number of segments \(N\) is 1, indicating that the grain configuration consists of only one single segment. The optimization process for this design problem is performed using the Nelder-Mead algorithm. The optimal results obtained from the optimization process are presented in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & \multicolumn{2}{c}{**Range**} & **Unit** \\ \hline \multirow{4}{*}{Star grains} & \(\varepsilon_{j}\) & [0.6, 0.8] & — \\ \cline{2-4}  & \(\theta_{j}\) & \(\begin{bmatrix}50+4\;\left(n_{\text{st},\;j}\;-3\right)\), \\ 150\end{bmatrix}\) & \(\circ\) \\ \cline{2-4}  & \(r_{j}\) & =0.02 \(R\) & m \\ \cline{2-4}  & \(n_{\text{st},\;j}\) & [6; 8] & — \\ \cline{2-4}  & \(n_{\text{p},\;\text{st},j}\) & =0.3 & — \\ \hline \multirow{4}{*}{Slot grains} & \(h_{i}\) & [0.01 \(R\), 0.1 \(R\)] & m \\ \cline{2-4}  & \(n_{\text{st},\;j}\) & [3; 5] & — \\ \cline{2-4}  & \(n_{\text{p},\;\text{st},j}\) & =0.3 & — \\ \hline \multirow{2}{*}{Common} & \(R\) & [0.05, 0.2] & m \\ \cline{2-4}  & \(J\) & [0, 0.3] & — \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Constrains of the optimization problem. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & \multicolumn{2}{c}{**Type**} & **Unit** \\ \hline \multirow{4}{*}{Star grains} & \(r_{\text{Ref},\;\text{st},j}\) & 0.0188972 & m/s \\ \cline{2-4}  & \(L_{\text{st},j}\) & 1.17073 & m \\ \cline{2-4}  & \(l_{j}\) & 0.0909091 & m \\ \cline{2-4}  & \(\varepsilon_{j}\) & 0.75 & — \\ \cline{2-4}  & \(\theta_{j}\) & 105 & \(\circ\) \\ \cline{2-4}  & \(r_{j}\) & 0.004 & m \\ \cline{2-4}  & \(n_{\text{st},\;j}\) & 8 & — \\ \cline{2-4}  & \(n_{\text{p},\;\text{st},j}\) & 0.3 & — \\ \hline \multirow{4}{*}{Slot grains} & \(r_{\text{Ref},\;\text{st},j}\) & 0.0188853 & m/s \\ \cline{2-4}  & \(L_{\text{st},j}\) & 0.361296 & m \\ \cline{2-4}  & \(h_{i}\) & 0.0124 & m \\ \cline{2-4}  & \(n_{\text{st},\;j}\) & 5 & — \\ \cline{2-4}  & \(n_{\text{p},\;\text{st},j}\) & 0.3 & — \\ \hline Common & \(R\) & 0.2 & m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal result of single-thrust grain. The propellant loading fraction, which refers to the ratio of propellant volume to chamber volume, is determined to be 81.37%. Additionally, the throat-to-port ratio \(J\) is found to be 0.306, slightly larger than the upper limit \(I_{\text{max}}\) due to:  
1. The numerical process of rounding \(n_{\text{st, }j}\) and \(n_{\text{sl, }j}\) from a floating-point number to an integer. 2. The soft-constraint nature of the penalty function in Equation (31). The burning-to-port ratio (\(Po\), also known as the Pobedonostsev number) can be expressed in terms of the throat-to-port ratio (\(J\)):  
\[Po=\frac{JA_{\text{b,0}}}{A_{\text{t}}}=60.2 \tag{33}\]  
where \(A_{\text{b,0}}\) is the initial total burning surface area.