Question

Air enters a nozzle at 30 psia, \(630 \mathrm{R}\), and a velocity of \(450 \mathrm{ft} / \mathrm{s}\). Approximating the flow as isentropic, determine the pressure and temperature of air at a location where the air velocity equals the speed of sound. What is the ratio of the area at this location to the entrance area?

Short answer

To find the pressure, temperature, and area ratios when the air velocity reaches the speed of sound inside a nozzle, we followed these steps: 1. Set the Mach number (M) as 1 since the speed of sound is reached. 2. Find the temperature ratio: \(\frac{T}{T_0} = \left(\frac{2}{k+1}\right)^\frac{k}{k-1}\) 3. Find the pressure ratio: \(\frac{P}{P_0} = \left(\frac{2}{k+1}\right)^\frac{k^2}{(k-1)^2}\) 4. Calculate the actual pressure and temperature at that point: \(P = 30 \mathrm{psia} \cdot \left(\frac{2}{k+1}\right)^\frac{k^2}{(k-1)^2}\) \(T = 630 \mathrm{R} \cdot \left(\frac{2}{k+1}\right)^\frac{k}{k-1}\) 5. Find the area ratio: \(\frac{A}{A^*} = \left(\frac{2}{k+1}\right)^{-\frac{1}{k-1}}\) Using these steps, we can determine the pressure, temperature, and area ratio for air in the nozzle when its velocity reaches the speed of sound.

Step-by-step solution

Find the Mach number when the velocity equals the speed of sound

Since the speed of sound is defined as the point where the Mach number is equal to 1, we can simply set \(M = 1\) in our equations.

Find the temperature ratio

Using the relation between temperature and Mach number, we can find the temperature ratio when \(M = 1\): \(1^2 = \frac{2}{k-1}\left[\left(\frac{T}{T_0}\right)^{-\frac{k-1}{k}} - 1\right]\) Solving this equation for the temperature ratio, we get: \(\frac{T}{T_0} = \left(\frac{2}{k+1}\right)^\frac{k}{k-1}\)

Find the pressure ratio

Using the temperature ratio and the isentropic relations, we can find the pressure ratio: \(\frac{P}{P_0} = \left(\frac{T}{T_0}\right)^\frac{k}{k-1} = \left(\frac{2}{k+1}\right)^\frac{k^2}{(k-1)^2}\)

Find the actual pressure and temperature

Now that we have the pressure and temperature ratios, we can find the actual pressure and temperature at the point where the Mach number is 1: \(P = P_0 \cdot \frac{P}{P_0} = 30 \mathrm{psia} \cdot \left(\frac{2}{k+1}\right)^\frac{k^2}{(k-1)^2}\) \(T = T_0 \cdot \frac{T}{T_0} = 630 \mathrm{R} \cdot \left(\frac{2}{k+1}\right)^\frac{k}{k-1}\)

Find A/A*

Finally, we can find the area ratio: \(A^* = A \frac{\left(1 + \frac{k-1}{2}M^2\right)^\frac{1+k}{2(k-1)}}{M}\) Since \(M = 1\), this simplifies to: \(A^* = A \left(\frac{2}{k+1}\right)^\frac{1}{k-1}\) To find the ratio of the area at this location to the entrance area, we just need to take the inverse of this equation: \(\frac{A}{A^*} = \left(\frac{2}{k+1}\right)^{-\frac{1}{k-1}}\) Now, we have all the required values to answer the given problem.