Esercizi Gasdinamica

Let's go through the provided problems step-by-step to ensure a thorough understanding and correct solutions. ### Problem 1: Compressible Flow in a Converging Diverging Nozzle **Given:** - \( M_1 = 2 \) - \( d_1 = 0.8 \, \text{m} \) - \( L = 0.5 \, \text{m} \) **Required:** - Determine the exit Mach number \( M_2 \) and the pressure ratio \( P_2/P_1 \). **Solution:** 1. **Mach Number at Exit (\( M_2 \)):** - For a converging-diverging nozzle, if \( d_1 > 0 \), it is a subsonic flow. - Using the area-Mach number relation for subsonic flow: \[ \frac{d}{d_1} = \left( \frac{2}{\gamma + 1} \right)^{\frac{\gamma + 1}{2(\gamma - 1)}} M^2 \] - Given \( d > d_1 \), the flow is subsonic, and we can use: \[ M_2 = \sqrt{\left( \frac{2}{\gamma + 1} \right)^{\frac{\gamma + 1}{2(\gamma - 1)}} \left( \frac{d}{d_1} \right)^2} \] - For air, \( \gamma = 1.4 \): \[ M_2 = \sqrt{\left( \frac{2}{1.4 + 1} \right)^{\frac{1.4 + 1}{2(1.4 - 1)}} \left( \frac{0.8}{d_1} \right)^2} \] - Assuming \( d_1 = 1 \, \text{m} \): \[ M_2 = \sqrt{\left( \frac{2}{2.4} \right)^{\frac{2.4}{0.8}} \left( 0.8 \right)^2} \] - Simplifying: \[ M_2 \approx 1 \] 2. **Pressure Ratio (\( P_2/P_1 \)):** - For subsonic flow in a converging-diverging nozzle, the pressure ratio can be found using the isentropic relations: \[ \frac{P_2}{P_1} = \left( 1 + \frac{\gamma - 1}{2} M_1^2 \right)^{-\frac{\gamma}{\gamma - 1}} \] - Substituting \( M_1 = 2 \) and \( \gamma = 1.4 \): \[ \frac{P_2}{P_1} = \left( 1 + \frac{1.4 - 1}{2} \cdot 2^2 \right)^{-\frac{1.4}{0.4}} = \left( 1 + 1.8 \right)^{-3.5} = (2.8)^{-3.5} \] - Calculating: \[ P_2/P_1 \approx 0.16 \] **Final Answers:** - \( M_2 \approx 1 \) - \( P_2/P_1 \approx 0.16 \) ### Problem 2: Heat Transfer in a Converging Diverging Nozzle **Given:** - \( d = 2 \, \text{m} \) - \( L = 0.5 \, \text{m} \) - \( M_1 = 2 \) - \( Q_m = 8000 \, \text{W/m}^2 \) **Required:** - Determine the exit Mach number and the pressure ratio. **Solution:** 1. **Mach Number at Exit (\( M_2 \)):** - Similar to Problem 1, for \( d > d_1