## 2 Modeling of Hybrid-Propellant Launch Vehicle
### Propulsion Model  
Playing a critical role in launch vehicle flight simulation, the propulsion model is a key determinant of the launch vehicle's thrust-generating capability and flight performance. This model encompasses the characteristics of the launch vehicle's motor, thrust, propellant consumption, and other parameters related to propulsion. In the simulation of a launch vehicle's flight, the propulsion model is utilized to calculate dynamic parameters such as the launch vehicle's velocity, altitude, and trajectory. Employed in optimizing the design of the launch vehicle, configuring the launch vehicle for a specific mission, and predicting the flight performance of the launch vehicle, this model holds significant importance. The accurate creation and integration of the propulsion model are crucial for ensuring the reliable and effective achievement of the launch vehicle's objectives. In this study, the effects and performance of fundamental characteristics of the propulsion system, such as thrust, total impulse, and specific impulse, were examined within the context of flight simulation. Effects related to combustion, internal ballistics, and similar factors within propulsion were not considered in this study. These effects have been addressed separately in dedicated studies outside the scope of flight simulation [44; 45]. In the design and optimization problem, the hybrid propulsion equations, along with main \(T\) definitions and fundamentals, are utilized to calculate the propulsion characteristics of the modified last stage. The HTPB is chosen as the propellant due to its low cost, easy processing, and non-self-deflagration properties. The LOX, on the other hand, offers high safety and excellent performance at a low cost. The LOX/HTPB propellant combination is selected as it provides a non-toxic and relatively smoke-free exhaust while exhibiting similar chemical properties and performance to a LOX/kerosene bi-propellant system [46]. To determine the theoretical characteristic exhaust velocity (\(c^{*}\)) and specific heat ratio (\(k\)) for the reaction gases of the LOX/HTPB combination, a function of the mixture ratio is employed [46]. In the optimization study, the solution for the optimization problems incorporates a circular \(F\) grain port, as depicted in Figure 2. This choice is motivated by the ease of the production process and computational convenience. In the optimization process, the outside diameter of the modified vehicle is derived from the non-modified launch vehicle to prevent conflicts with the lower stages. The calculation of \(T\) coefficients is based on propulsion fundamentals [46]. Figure 2: Circular fuel grain port. The equation representing the combustion port radius (\(R\)) as a function of time (\(t\)) is provided as Equation (2) [46]:  
\[R(t)=\left\{a(2n+1)\bigg{(}\frac{\dot{m_{Q}}}{\pi N}\bigg{)}^{n}t+R_{i}^{(2n+1)} \right\}^{1/(2n+1)} \tag{2}\]  
where \(a\), and \(n\) represent regression rate coefficients, \(\dot{m_{O}}\) represents the mass flow rate of the oxidizer, and \(N\) represents the port number. For the chosen fuel combination of HTPB/LOX, empirical coefficients \(a\) and \(n\), derived experimentally for specific propellant formulations, are 0.104 and 0.681, respectively. Additionally, the fuel grain port is set at 7. Once \(R_{i}\) is determined, the \(F\) burn distance, oxidizer mass (\(m_{O}\)), initial \(F\) regression rate, and grain length are obtained using hybrid propulsion equations [46]. The total fuel mass, \(m_{F}\), can be calculated using Equation (3):  
\[m_{F}=\pi N\rho_{F}L\left[\left\{a(2n+1)\bigg{(}\frac{\dot{m_{O}}}{\pi N} \bigg{)}^{n}t+R_{i}^{2n+1}\right\}^{\frac{2}{2n+1}}-R_{i}^{2}\right] \tag{3}\]  
where \(L\) represents the motor length and \(\rho_{F}\) represents fuel density.