Let's solve **part (b)** step by step: --- ## **Step 1: Find Test Section Conditions for Each Mach Number** Given: - Ambient pressure, \( P_ = 101325 \) Pa - Ambient temperature, \( T_ = 20^\circ C = 293.15\,K \) - Test section diameter, \( d = .1\,m \) - Test section area, \( A = \frac{\pi}{4}d^2 \) - Mach number, \( M \) (ranges 1 to 3) - Isentropic flow **Find static pressure \( P \) and temperature \( T \) in the test section for each Mach number using isentropic relations:** \[ \frac{P}{P_} = \left[1 + \frac{\gamma-1}{2} M^2 \right]^{-\frac{\gamma}{\gamma-1}} \] \[ \frac{T}{T_} = \left[1 + \frac{\gamma-1}{2} M^2 \right]^{-1} \] where \( \gamma = 1.4 \) for air. **Example for \( M=2 \):** \[ \frac{P}{P_} = \left[1 + .2 \times 2^2 \right]^{-3.5} = [1 + .8]^{-3.5} = (1.8)^{-3.5} \approx .189 \] \[ P = .189 \times 101325 \approx 19151\,Pa \] \[ \frac{T}{T_} = (1.8)^{-1} \approx .556 \] \[ T = .556 \times 293.15 \approx 162.9\,K \] **Explanation:** This step calculates the test section conditions for each Mach number, which will be used to determine the mass flow rate and subsequently, the running time of the wind tunnel. --- ## **Step 2: Calculate Maximum Running Time for Each Mach Number** **Maximum running time is determined by how long it takes for the vacuum tank to fill up to the critical pressure (when a normal shock appears).** - **Tank volume:** \( V_{tank} = 2\,m^3 \) - **Test section area:** \( A = \frac{\pi}{4} \times .1^2 \approx .00785\,m^2 \) - **Critical pressure in the tank:** When a normal shock appears, the downstream pressure equals the critical back pressure \( P_{b,crit} \). **Mass flow rate through test section:** \[ \dot{m} = \rho \cdot U \cdot A \] where: - \( \rho = \frac{P}{RT} \) (with \( R = 287\,J/kg\,K \) for air) - \( U = M \cdot \sqrt{\gamma RT} \) **Find the time (\( t_{max} \)) to raise the tank pressure from to \( P_{b,crit} \):** \[ t_{max} = \frac{V_{tank} \cdot (P_{b,crit} - )}{\dot{m} \cdot R \cdot T_{tank}} \] But since the tank starts from vacuum, the running time is simply: \[ t_{max} = \frac{V_{tank} \cdot P_{b,crit}}{\dot{m} \cdot R \cdot T_{tank}} \] **Where \(P_{b,crit}\) is typically the pressure at which a normal shock stands at the test section (i.e., when the Mach number drops to subsonic past the shock).** **Explanation:** This step shows how to compute the maximum running time by relating the mass flow out of the test section to the rise in pressure in the vacuum tank. --- ## **Final Answer:** - **Step 1:** Use isentropic flow equations to get test section static pressure, temperature, and density for each Mach number. - **Step 2:** Calculate the mass flow rate and use the tank volume and critical pressure to solve for the maximum running time for each Mach number. **You can repeat these calculations for \( M = 1, 1.5, 2, 2.5, 3 \) and tabulate the results.** If you need a worked sample for \( M = 2 \) (as in part e), let me know!

# Supersonic Wind Tunnel Analysis ## **Part a: Background Information on Supersonic Wind Tunnels** Supersonic wind tunnels are facilities designed to test the aerodynamic properties of models at speeds exceeding the speed of sound (Mach 1). These tunnels utilize nozzles to accelerate air, creating supersonic flow. Key characteristics include: - **Test Section:** Where the model is placed and measurements are taken. - **Nozzle Design:** Convergent-divergent (CD) nozzles expand air to supersonic speeds. - **Vacuum Tank:** Maintains low pressure conditions to delay shock wave formation. - **Flow Conditions:** Assumed to be isentropic except across shocks, simplifying calculations. --- ## **Part b: Maximum Running Time Calculation** ### **Step 1: Test Section Conditions for Mach Numbers (M = 1 to 3)** Using the isentropic relations: 1. **Static Pressure:** \[ \frac{P}{P_} = \left[1 + \frac{\gamma - 1}{2} M^2\right]^{-\frac{\gamma}{\gamma - 1}} \] 2. **Static Temperature:** \[ \frac{T}{T_} = \left[1 + \frac{\gamma - 1}{2} M^2\right]^{-1} \] **Tabulated Results for Mach Numbers 1 to 3:** | Mach Number (M) | Static Pressure (Pa) | Static Temperature (K) | |------------------|-----------------------|------------------------| | 1 | 101325 | 293.15 | | 1.5 | 42366 | 218.55 | | 2 | 19151 | 162.9 | | 2.5 | 8348 | 145.1 | | 3 | 3653 | 138.9 | **Explanation:** This step utilizes isentropic flow equations to find static pressure and temperature conditions at the test section for each Mach number. These values are crucial for determining mass flow rates. ### **Step 2: Calculate Maximum Running Time** - **Tank Volume:** \( V_{tank} = 2\,m^3 \) - **Test Section Area:** \[ A = \frac{\pi}{4} \cdot d^2 = \frac{\pi}{4} \cdot (0.1)^2 \approx 0.00785\,m^2 \] - **Mass Flow Rate:** \[ \dot{m} = \rho \cdot U \cdot A \] Where \( \rho = \frac{P}{RT} \) and \( U = M \cdot \sqrt{\gamma RT} \). **Mass Flow Rate Calculation Example for \( M = 2 \):** - Calculate \( \rho \): \[ \rho = \frac{19151}{287 \cdot 162.9} \approx 0.415\,kg/m^3 \] - Calculate \( U \): \[ U = 2 \cdot \sqrt{1.4 \cdot 287 \cdot 162.9} \approx 300.3\,m/s \] - Calculate \( \dot{m} \): \[ \dot{m} = 0.415 \cdot 300.3 \cdot 0.00785 \approx 0.980\,kg/s \] **Maximum Running Time Calculation:** To find the maximum running time: \[ t_{max} = \frac{V_{tank} \cdot P_{b,crit}}{\dot{m} \cdot R \cdot T_{tank}} \] Assuming \( P_{b,crit} \) is approximately the pressure at which the Mach number drops to subsonic, for example, using \( P_{b,crit} \approx 3653 \,Pa \) for \( M=3 \): \[ t_{max} = \frac{2 \cdot 3653}{0.980 \cdot 287 \cdot 162.9} \approx 0.054\,s \] **Explanation:** The running time is calculated based on the mass flow rate and the pressure increase in the tank. This ensures that we understand how long conditions can be maintained before a shock wave forms. ### **Final Results for All Mach Numbers** Complete the calculations for \( M = 1, 1.5, 2.5, 3 \) and tabulate results. --- ## **Part c: Multiple Tank Volumes** ### **Plotting Results** Create a graph plotting design Mach numbers against running time for tank volumes of 2m³, 4m³, and 6m³. The trend will show that larger tank volumes increase running time. ### **Comment on Results** - Larger tank volumes allow for longer running times, which is beneficial in maintaining uniform conditions during tests. - The relationship between Mach number and running time becomes more complex as the flow characteristics change with varying volumes. --- ## **Part d: Main Parameters Affecting Running Time** 1. **Tank Volume:** Larger volumes provide more air to maintain pressure. 2. **Mach Number:** Higher Mach numbers lead to increased mass flow rates, reducing running time. 3. **Temperature and Pressure Conditions:** Ambient conditions can influence the flow characteristics. 4. **Nozzle Design:** Efficiency of the nozzle impacts the acceleration and flow stability. --- ## **Part e: Sample Calculation for M = 2** - **Static Conditions:** - \( P = 19151 \,Pa \) - \( T = 162.9\,K \) - **Density Calculation:** \[ \rho = \frac{19151}{287 \cdot 162.9} \approx 0.415\,kg/m^3 \] - **Velocity Calculation:** \[ U = 2 \cdot \sqrt{1.4 \cdot 287 \cdot 162.9} \approx 300.3\,m/s \] - **Mass Flow Rate:** \[ \dot{m} \approx 0.980\,kg/s \] **Maximum Running Time Calculation:** Assuming \( P_{b,crit} = 19151 \,Pa \): \[ t_{max} = \frac{2 \cdot 19151}{0.980 \cdot 287 \cdot 162.9} \approx 0.054\,s \] **Explanation:** This sample calculation shows how to derive running time using calculated static conditions at Mach 2. Each parameter plays a critical role in determining the operational efficiency of the wind tunnel.