(a) In case of isentropic flow of a compressible fluid through a variable duct, show that, $$ \frac{A}{A_{c}}=\frac{1}{M}\left[\frac{1+\frac{1}{2}(\gamma-1) M^{2}}{\frac{1}{2}(\gamma+1)}\right]^{\frac{\gamma+1}{2(\gamma-1)}} \\ $$ where $\gamma$ is the ratio of specific heats, $M$ is the Mach number at a section whose area is $A$ and $A_{c}$ is the critical area of flow. (b) A supersonic nozzle is to be designed for air flow with Mach number 3 at the exit section which is $200 \mathrm{~mm}$ in diameter. The pressure and temperature of air at the nozzle exit are to be $7.85 \mathrm{kN} / \mathrm{m}^{2}$ and $200 \mathrm{~K}$ respectively. Determine the reservoir pressure, temperature and the throat area. Take : $\gamma=1.4$.

UPSC Exam

Solution:

Mach number, $ \quad M=3 \\ $

Area at the exit section, $ A=\pi / 4 \times 0.2^{2}=0.0314 \mathrm{~m}^{2} \\ $

Pressure of air at the nozzle, $ \quad(p)_{\text {nozzle }}=7.85 \mathrm{kN} / \mathrm{m}^{2} \\ $

Temperature of air at the nozzle, $ (T)_{\text {nozzle }}=200 \mathrm{~K} \\ $ …