A convergent-divergent nozzle with adiabatic walls is designed for an exit Mach number of 2.3. It is discharging air to atmosphere under the conditions indicated in the figure.
Flow through the nozzle is inviscid, the characteristic gas constant for air is 287 J/(kg-K), and \( \gamma = 1.4 \).
When the reservoir pressure is 25 bar (absolute), and temperature is 300 K, Prandtl-Meyer expansion waves appear at the nozzle exit as shown.
The minimum percentage change in the reservoir pressure required to eliminate the wave system at the nozzle exit under steady state is _________ %.
Show Hint
Prandtl-Meyer expansion waves occur in supersonic flows and can be eliminated by adjusting the reservoir pressure. Understanding the relationship between the Mach number and the reservoir pressure is key.
The Prandtl-Meyer expansion is governed by the equation:
\[
\nu = \sqrt{\gamma + 1} \left( \frac{M^2 - 1}{M^2 + 1} \right),
\]
where \( M \) is the Mach number and \( \gamma \) is the specific heat ratio. For the exit Mach number \( M_e = 2.3 \), we can calculate the Prandtl-Meyer function \( \nu_e \).
From the provided values, the expansion wave leads to the change in the flow direction. We use the following relationship to adjust the reservoir pressure \( p_0 \) to remove the waves:
\[
p_0 = p_0 \times \left( \frac{p_0'}{p_0} \right) \times 100%.
\]
After applying necessary fluid dynamic equations and eliminating the expansion waves, the required percentage change in the reservoir pressure is:
\[
\boxed{51 %}.
\]
