# Design and Optimization of Modular Solid Rocket Grain Matching Multi-Thrust Performance Curve
## 2 Design Methods
### Chamber Pressure Calculation  
Considering \(i\)th grain segment (star + slot), the mass flow rate (kg/s) is as follows:  
\[Q_{i,j}=s_{\text{b,st}}^{(2)}\big{(}r_{\text{st},i,j}t+w_{\text{st},i,j}\big{)} L_{\text{st},i,j}r_{\text{st},i,j}\rho_{\text{st},i}+s_{\text{b,sl}}^{(2)}\big{(}r_{ \text{st},i,j}t+w_{\text{st},i,j}\big{)}L_{\text{st},i,j}r_{\text{st},i,j}\rho _{\text{st},j} \tag{14}\]  
where the first term is the contribution from the star grain and the second term is the contribution from the slot grain. The value \(t\) is time starting from the burning of the \(j\)th segment to present. The value \(r_{\text{st},i,j}\) means the line burning rate (m/s) of star grain at the \(j\)th segment during \(i\)th thrust phase. The value \(r_{\text{st},i,j}\) represents the line burning rate of slot grain at the \(j\)th segment during the \(i\)th thrust phase. The values \(w_{\text{st},i,j}\) and \(w_{\text{st},i,j}\) are the  
Figure 5: The approximation of burning perimeter. initial burned web thickness of star grain and slot grain at \(t=0\), respectively. The value \(L_{\text{st},j}\) and \(L_{\text{st},j}\) are the length of the star grain and slot grain, respectively. In this work, erosive burning is not considered, so the line burning rate can be calculated by the empirical APN formula  
\[r_{\text{st},j}=\left(\frac{p_{\text{c},i}}{p_{\text{Ref}}}\right)^{n_{p, \text{st},j}}r_{\text{Ref, st},j} \tag{15}\]  
where \(p_{\text{c},i}\) is the chamber pressure during the \(i\)th phase; \(r_{\text{Ref, st},j}\) is the reference burning rate of the star grain at the \(j\)th segment under reference pressure \(p_{\text{Ref}}\), which means different segments may have different reference burning rates; \(n_{p,\text{st},j}\) is the pressure index of the star grain at the \(j\)th segment. Similarly, for the slot grain at the \(j\)th segment, there is  
\[r_{\text{st},j,i}=\left(\frac{p_{\text{c},i}}{p_{\text{Ref}}}\right)^{n_{p, \text{st},j}}r_{\text{Ref, st},j} \tag{16}\]  
where \(r_{\text{Ref, st},j}\) is the reference burning rate of the slot grain at the \(j\)th segment under reference pressure \(p_{\text{Ref}}\); \(n_{p,\text{st},j}\) is the pressure index of the slot grain at the \(j\)th segment. Initial burned web thickness can be calculated as follows:  
\[w_{\text{st},i,j}=\sum\limits_{k=1}^{i-1}r_{\text{st},k,j}t_{\text{b},k};w_{ \text{st},i,j}=\sum\limits_{k=1}^{i-1}r_{\text{st},k,j}t_{\text{b},k} \tag{17}\]  
Then, substitute Equations (7) and (13) into Equation (14). After simplification, this can be organized into a quadratic function about time, as follows:  
\[Q_{i,j}=a_{i,j}+b_{i,j}t+c_{i,j}t^{2} \tag{18}\]  
where  
\[\left\{ \begin{array}{l}a_{i,j}=\left(a_{\text{st},\ j}+w_{\text{st},\ i,j}b_{ \text{st},\ j}+c_{\text{st},\ j}w_{\text{st},\ i,\ j}{}^{2}\right)L_{\text{st}, \ j}r_{\text{st},\ i,j}\rho_{\text{st},j}+\left(a_{\text{st},\ j}+w_{\text{st}, \ i,\ j}b_{\text{st},\ j}+c_{\text{st},\ j}w_{\text{st},\ i,\ j}{}^{2}\right)L_{ \text{st},\ j}r_{\text{st},\ i,j}\rho_{\text{st},j}\\ b_{i,j}=\left(b_{\text{st},\ j}+2c_{\text{st},\ j}w_{\text{st},\ i,j}\right)L_{ \text{st},\ j}r_{\text{st},\ i,j}{}^{2}\rho_{\text{st},j}+\left(b_{\text{st}, \ j}+2c_{\text{st},\ j}w_{\text{st},\ i,\ j}\right)L_{\text{st},\ j}r_{\text{st}, \ i,j}{}^{2}\rho_{\text{st},j}\\ c_{i,j}=c_{\text{st},\ j}L_{\text{st},\ j}r_{\text{st},\ i,j}{}^{3}\rho_{ \text{st},j}+c_{\text{st},\ j}L_{\text{st},\ j}r_{\text{st},\ i,j}{}^{3}\rho_{ \text{st},j}\end{array} \right. \tag{19}\]  
It is worth noting that at any moment during the \(i\)th thrust phase, the grain segments having indices smaller than \(i\) have burned out. These segments have no contribution to the total mass flow rate. Therefore, the total mass flow rate during the \(i\)th thrust stage can be calculated as follows:  
\[Q_{i}=\sum\limits_{j=i}^{N}Q_{i,j} \tag{20}\]  
The total mass flow rate can also be organized into a quadratic function  
\[Q_{i}=a_{i}+b_{i}t+c_{i}t^{2},\ \text{where}\ \left\{