Question

Air expands isentropically from \(2.2 \mathrm{MPa}\) and \(77^{\circ} \mathrm{C}\) to 0.4 MPa. Calculate the ratio of the initial to the final speed of sound.

Short answer

Answer: The ratio of the initial to the final speed of sound for the isentropic expansion is approximately 1.255.

Step-by-step solution

Calculate the initial and final temperatures using isentropic expansion

Since the expansion is isentropic, we can use the following relation between the initial and final pressures and temperatures: \( \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{(\gamma - 1)/\gamma}\) First, convert the given temperatures to Kelvin: \(T_1 = 77^{\circ} \mathrm{C} + 273.15 \mathrm{K} = 350.15 \mathrm{K}\) For air, \(\gamma = 1.4\) and \(R = 287 \mathrm{J/kg \cdot K}\). Now, we can calculate the final temperature (\(T_2\)) using the given pressures and the isentropic relationship: \(T_2 = T_1 \left(\frac{P_2}{P_1}\right)^{(\gamma - 1)/\gamma} = 350.15\left(\frac{0.4 \times 10^6 \mathrm{Pa}}{2.2 \times 10^6 \mathrm{Pa}}\right)^{(1.4 - 1)/1.4} \mathrm{K}\)

Calculate the speed of sound using the temperature for both cases

Use the speed of sound formula, \(a = \sqrt{\gamma RT}\), for both initial and final temperatures to find the initial and final speed of sound: \(a_1 = \sqrt{(1.4)(287 \mathrm{J/kg \cdot K})(350.15 \mathrm{K})}\) \(a_2 = \sqrt{(1.4)(287 \mathrm{J/kg \cdot K})(T_2)}\)

Find the ratio of the initial and final speed of sound

Finally, calculate the ratio of the initial and final speed of sound: Ratio = \(\frac{a_1}{a_2} = \frac{\sqrt{(1.4)(287)(350.15)}}{\sqrt{(1.4)(287)(T_2)}}\) Now, substitute the value of \(T_2\) that we calculated in Step 1 and compute the ratio.