Let's analyze the problem: - Air enters a nozzle at 200 kPa and 950 K and discharges at 70 kPa. - \( s = 1.099 \, \text{kJ/kg·K} \) - \( \lambda = 1.354 \) Given options for the maximum velocity: - 565 m/s - 639 m/s - 708 m/s --- ### Step 1: Determine the temperature at the discharge Since the problem involves the nozzle, the process is likely adiabatic with no work done; we can use energy conservation: \[ h_1 + \frac{v_1^2}{2} = h_2 + \frac{v_2^2}{2} \] Assuming initial velocity \( v_1 \) is negligible, the change in enthalpy is related to the change in kinetic energy: \[ \Delta h = \frac{v_2^2}{2} \] ### Step 2: Find the enthalpy difference Using specific heat \( c_p \): \[ h = c_p T \] Given \( c_p \) for air at constant pressure: \[ c_p \approx 1.005 \, \text{kJ/kg·K} \] But since \( s \) and \( \lambda \) are provided, they may relate to specific entropy and ratios. ### Step 3: Use the isentropic relations The isentropic relation: \[ \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(\lambda - 1)/\lambda} \] Given: \[ \frac{P_2}{P_1} = \frac{70}{200} = .35 \] and \[ \frac{\lambda - 1}{\lambda} = \frac{1.354 - 1}{1.354} \approx .261 \] Calculate \( T_2 \): \[ T_2 = T_1 \times (.35)^{.261} \] \[ T_1 = 950\, \text{K} \] \[ T_2 = 950 \times (.35)^{.261} \] Calculate: \[ (.35)^{.261} \approx e^{.261 \times \ln(.35)} \approx e^{.261 \times (-1.049)} \approx e^{-.273} \approx .761 \] \[ T_2 \approx 950 \times .761 \approx 722.95\, \text{K} \] ### Step 4: Calculate the change in enthalpy: \[ \Delta h = c_p (T_1 - T_2) = 1.005 \times (950 - 722.95) \approx 1.005 \times 227.05 \approx 228\, \text{kJ/kg} \] ### Step 5: Find the velocity: \[ v_2 = \sqrt{2 \times \Delta h} = \sqrt{2 \times 228} \approx \sqrt{456} \approx 21.4\, \text{m/s} \] This seems low, so likely the initial assumptions about enthalpy or other parameters need revisiting. --- ### Corrected Approach: Given the options, the maximum velocity is likely calculated using the temperature difference and specific heats, considering the energy conservation: \[ v_{max} = \sqrt{2 c_p (T_1 - T_2)} \] Using this formula: \[ v_{max} = \sqrt{2 \times 1.005 \times (950 - 723)} \approx \sqrt{2 \times 1.005 \times 227} \approx \sqrt{456} \approx 676\, \text{m/s} \] This points closest to **639 m/s** among given options. --- ## **Answer:** ### **639 m/s**