Compressible Flow Assignment 1 — Quasi-One-Dimensional Flow I  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
An aircraft is flying at an altitude of $12000$ feet at a Mach number of $0.50$. The cross-sectional area at the mouth of the inlet diffuser before the compressor stage is 0.6 m$^2$. Determine the mass of air entering the compressor per second, the speed of the aircraft, and the stagnation pressure and temperature of the air at the diffuser entry.
Question #2
The pressure, velocity, and temperature of air at the entry of a nozzle are $2$ bar, $145$ m/s, and 330 K; the pressure at the exit is $1.5$ bar. What is the shape of the nozzle? Determine the entry and exit Mach numbers and the mass flow rate per unit area at the entry and exit of the nozzle. Then, find the maximum possible mass flow rate per unit area that could be attained by the flow.
Question #3
A gas stored in a reservoir expands from $P=10$ bar, $T=525^\circ$C through a nozzle to a pressure of $7.6$ bar. If the mass flux is 1.5 kg/s, determine the pressure, temperature, and velocity at the nozzle throat and exit, the maximum possible velocity attainable by the gas within this nozzle and the type of nozzle and its throat area. Take $\gamma=1.3$ and $R=0.464$ kJ/kgK.
Question #4
Stagnant air at 25 bar and 1050 K is expanded through a supersonic nozzle to an exit pressure of 4.35 bar. The exit area of the nozzle is 100 $\rm cm^2$. Determine the throat area, the pressure and temperature at the throat, exit temperature, mass flux, and exit velocity as a fraction of the terminal velocity.
Question #5
Starting from the conservation of mass principle, Newton's law, and the first law of thermo: $$\dot{m}_{\rm out}=\dot{m}_{\rm in}$$ $$\vec{F}=m {\rm d}\vec{v}/{\rm d} t$$ $${\rm d}(me)+P {\rm d}V=\delta Q-\delta W$$ show that the equations of motion for one-dimensional flow in a duct with a varying cross-sectional area correspond to: $${\rm d} (\rho v A)=0$$ $$\rho v {\rm d}v + {\rm d} P=0$$ $${\rm d}\left(h+ \frac{1}{2}v^2\right)=0$$ Outline all assumptions. Note: this question is worth double the points awarded to the other questions.
1.  163 m/s, 83 kg/s, 0.75 atm, 278 K.
2.  0.399, 0.77, 489 kg/sm$^2$, 463 kg/sm$^2$, convergent.
3.  convergent, 7.6 bar, 476$~^\circ$C, 444 m/s, 15.4 cm$^2$, 647 m/s.
4.  637 K, 21.71 kg/s, 69.5 cm$^2$, 13.2 bar, 875 K, 0.627.
Due on Thursday September 20th at 16:30. Do Questions 1 and 2.
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