Let's break down the problem and solve it step by step: --- ## **Given Data** - **Ambient temperature:** \( T_0 = 390 \) R - **Specific heat ratio:** \( \gamma = 1.4 \) - **Specific heat at constant pressure:** \( c_p = 0.24 \) Btu/(lbm·R) - **Heating value of fuel:** \( h_{PR} = 18400 \) Btu/lbm - **Turbine inlet temperature:** \( T_{t4} = 3000 \) R - **Afterburner exit temperature:** \( T_{t7} = 4000 \) R --- ## **1. Temperature Ratios** Define ratios for temperature at different stages: - **Stagnation (total) temperature at inlet:** \[ T_{t0} = T_0 \left[1 + \frac{\gamma - 1}{2} M_0^2\right] \] - **Compressor exit temperature:** \[ T_{t2} = T_{t0} \cdot \pi_c^{\frac{\gamma - 1}{\gamma}} \] - **Turbine exit temperature:** \( T_{t4} \) (given) - **Afterburner exit temperature:** \( T_{t7} \) (given) --- ## **2. Fuel-to-Air Ratio Calculation** ### **a. Main Combustor (\( f_1 \)):** \[ f_1 = \frac{c_p (T_{t4} - T_{t2})}{h_{PR} - c_p T_{t4}} \] ### **b. Afterburner (\( f_2 \)):** \[ f_2 = \frac{c_p (T_{t7} - T_{t4})}{h_{PR} - c_p T_{t7}} \] ### **c. Total Fuel-to-Air Ratio (\( f \)):** \[ f = f_1 + f_2 \] --- ## **3. Specific Thrust (\( F/m_a \))** The specific thrust (per unit mass flow rate of air) is: \[ \text{Specific Thrust} = V_9 - V_0 \] Where: - \( V_0 = M_0 \sqrt{\gamma R T_0} \) (flight speed) - \( V_9 = \sqrt{2 c_p (T_{t7} - T_0)} \) (ideal expansion to ambient) --- ## **4. Specific Fuel Consumption (SFC)** \[ \text{SFC} = \frac{f}{V_9 - V_0} \] --- ## **5. Efficiencies** ### **a. Thermal Efficiency:** \[ \eta_{thermal} = \frac{(\text{Thrust Power})}{(\text{Fuel Energy Input})} = \frac{(V_9 - V_0) \cdot V_0}{f \cdot h_{PR}} \] ### **b. Propulsive Efficiency:** \[ \eta_{propulsive} = \frac{2 V_0}{V_9 + V_0} \] ### **c. Overall Efficiency:** \[ \eta_{overall} = \eta_{thermal} \cdot \eta_{propulsive} \] --- ## **6. Step-by-Step Solution (Symbolic Example)** Let's take nominal values: - \( M_0 = 0.8 \) - \( \pi_c = 10 \) ### **Step 1: Calculate \( T_{t0} \)** \[ T_{t0} = 390 \left[1 + \frac{1.4-1}{2} \times (0.8)^2 \right] = 390 \left[1 + 0.2 \times 0.64\right] = 390 \left[1 + 0.128\right] = 390 \times 1.128 = 440 \text{ R} \] ### **Step 2: Calculate \( T_{t2} \)** \[ T_{t2} = 440 \times 10^{(0.4/1.4)} \approx 440 \times 10^{0.286} \approx 440 \times 1.93 = 849.2 \text{ R} \] ### **Step 3: Calculate \( f_1 \)** \[ f_1 = \frac{0.24 \times (3000 - 849.2)}{18400 - 0.24 \times 3000} \] \[ = \frac{0.24 \times 2150.8}{18400 - 720} \] \[ = \frac{516.19}{17680} \approx 0.0292 \] ### **Step 4: Calculate \( f_2 \)** \[ f_2 = \frac{0.24 \times (4000 - 3000)}{18400 - 0.24 \times 4000} \] \[ = \frac{0.24 \times 1000}{18400 - 960} \] \[ = \frac{240}{17440} \approx 0.0138 \] ### **Step 5: Total Fuel-to-Air Ratio** \[ f = 0.0292 + 0.0138 = 0.043 \] ### **Step 6: Flight and Jet Velocities** - \( R = 1716 \) ft·lb/(lbm·R) (for air) - \( V_0 = M_0 \sqrt{\gamma R T_0} \) \[ V_0 = 0.8 \times \sqrt{1.4 \times 1716 \times 390} \] \[ = 0.8 \times \sqrt{1.4 \times 1716 \times 390} \] \[ = 0.8 \times \sqrt{1.4 \times 1716 \times 390} \] \[ = 0.8 \times \sqrt{1.4 \times 1716 \times 390} = 0.8 \times \sqrt{937,104} \approx 0.8 \times 968 = 774.4 \text{ ft/s} \] - \( V_9 = \sqrt{2 \cdot c_p \cdot (T_{t7} - T_0) \cdot 778.16} \) (convert Btu to ft-lbf) \[ = \sqrt{2 \times 0.24 \times (4000 - 390) \times 778.16} \] \[ = \sqrt{0.48 \times 3610 \times 778.16} \] \[ = \sqrt{0.48 \times 2,808,158} = \sqrt{1,347,916} \approx 1,161 \text{ ft/s} \] ### **Step 7: Specific Thrust** \[ \text{Specific Thrust} = 1,161 - 774.4 = 386.6 \text{ ft/s} \] ### **Step 8: SFC** \[ \text{SFC} = \frac{f}{\text{Specific Thrust}} = \frac{0.043}{386.6} = 1.11 \times 10^{-4} \text{ (lb fuel)/lb·s} \] --- ## **Summary Table** | Parameter | Value | |----------------------|----------------| | \( T_{t0} \) | 440 R | | \( T_{t2} \) | 849.2 R | | \( f_1 \) | 0.0292 | | \( f_2 \) | 0.0138 | | \( f \) | 0.043 | | \( V_0 \) | 774.4 ft/s | | \( V_9 \) | 1,161 ft/s | | Specific Thrust | 386.6 ft/s | | SFC | \(1.11 \times 10^{-4}\) | | \(\eta_{propulsive}\)| \(2 \times 774.4/(774.4+1161) \approx 0.81\) | --- ## **How to Generate Performance Curves** 1. **Create a grid** for \( M_0 \) (e.g., 0 to 2) and \( \pi_c \) (e.g., 5 to 40). 2. **For each point**, repeat the above calculations. 3. **Plot**: - Specific Thrust vs \( M_0 \) and \( \pi_c \) - SFC vs \( M_0 \) and \( \pi_c \) - \( f \) vs \( M_0 \) and \( \pi_c \) - Efficiencies vs \( M_0 \) and \( \pi_c \) You can implement these steps in Excel or MATLAB for a full set of curves. --- ### **Final Answers (for sample values):** - **Total fuel-to-air ratio (\(f\)):** 0.043 - **Specific Thrust:** 386.6 ft/s - **SFC:** \(1.11 \times 10^{-4}\) (lb fuel)/lb·s - **Propulsive Efficiency:** ~0.81 --- **If you need MATLAB code or Excel formulas for plotting, let me know!**