## 2 Methodology
### LV Model
#### 2.1.1 Propellant Mass and Tank Calculations  
A forward calculation is used in the optimization techniques, which requires Thrust \(F\), \(p_{c}\) and burn time ( \(\Delta t\)) to be specified. Therefore, initial values are given; however, the final values are obtained as an optimized output. Using the output values calculated in Section 2.1 and standard equations as defined in [30], specific impulse, \(I_{sp}\), propellant mass flow rate, \(\dot{m}_{p}\), and overall propellant mass, \(M_{prop}\), are calculated using the following equations:  
\[I_{sp}=c^{*}\times\frac{C_{F}}{g_{0}} \tag{4}\]  
\[\dot{m}_{p}=\frac{F}{I_{sp}\times g_{0}} \tag{5}\]  
\[M_{prop}=\dot{m}_{p}\times\Delta t \tag{6}\]  
where \(g_{0}\) is the standard value of Earth's gravity defined as 9.81 m/s\({}^{2}\). The multiplication by \(g_{0}\) is for the purpose of unit conversion only as \(I_{sp}\) is defined as total impulse per unit weight of propellant. Using the optimum MR calculated in the previous section, the mass of fuel, \(M_{f}\), is calculated via the following.  
\[M_{f}=\frac{M_{prop}}{1+MR} \tag{7}\]  
As fuel has been predefined, its density is already known; therefore, the volume of fuel and, hence, volume of fuel tank can be calculated as per Equation (9). An additional 10% volume is added to account for voltage volume.  
\[v_{f}=\frac{M_{f}}{\rho_{f}} \tag{8}\]  
\[v_{fTank}=1.1v_{f} \tag{9}\]  
Assuming the tank is cylindrical in shape, with a Length:Diameter (L/D) of 7, rearranging the standard equation for volume of a cylinder, the radius and length are calculated in the following equations.  
\[r_{fTank}=\sqrt[3]{\frac{v_{fTank}}{14\pi}} \tag{10}\]  
\[l_{fTank}=\frac{v_{fTank}}{r_{fTank}^{2}\pi} \tag{11}\]The selected propellant tank material for both propellants is Stainless Steel 316 [30]. Safety factors of 1.25 and 1.5 are, respectively, applied to the yield strength and ultimate strength to calculate the maximum allowable operating stress, \(\sigma_{tank}\).  
The total operating pressure is calculated as given below where the pressure drops due to the injector head assumed to be 10 bar, and an additional 10% is added due to losses in the fittings.  
\[p_{tot}=p_{c}+p_{injDrop}+0.1p_{c} \tag{12}\]  
As the tank material and allowable operating stress is known, the equation for Hoop Stress as defined in [30] is rearranged to obtain the tank's thickness via the following equation.  
\[t_{fTank}=\frac{p_{tot}\times r_{fTank}}{\sigma_{tank}} \tag{13}\]  
Finally, the propellant fuel tank mass is determined using Equation (14):  
\[M_{fTank}=2\pi\times r_{fTank}\times l_{fTank}\times t_{fTank}\times\rho_{tank} \tag{14}\]  
where \(\rho_{tank}\) is already known as the tank material is pre-defined. Equations are then repeated to calculate the oxidizer mass, \(M_{ox}\), and oxidizer tank mass, \(M_{oxTank}\). Instead of Equation (7), the following equation is used to calculate the oxidizer mass.  
\[M_{ox}=M_{prop}-M_{f} \tag{15}\]  
In this framework, it is assumed both tank radii are the same. Therefore, once the radii of both tanks are calculated, the maximum value is taken as the radii for both, and the tank lengths and thicknesses are recalculated accordingly.