2017 Compressible Flow Midterm Exam  
Thursday November 9th 2017
16:30 — 18:30

Question #1
You are working for a satellite company. Your first design project consists of optimizing the rocket nozzle used for altitude control on board the satellite. Your idea consists of replacing the converging-diverging “De Laval” nozzle by a simple converging nozzle. Determine how much thrust would be lost in doing so, and on what flow and geometric parameters this depends (find the ratio between the thrust of the De Laval nozzle and the thrust of the converging-only nozzle and simplify the expression as much as possible). For a fair comparison, assume that the stagnation properties and the mass flow rate are the same for both configurations. Also assume that the flow exiting the De Laval nozzle reaches terminal velocities.
Question #2
A flow is referred to as “hypersonic” when the Mach number is 4 or above. In order to achieve hypersonic level Mach numbers, a shock tunnel is usually used, as illustrated below. Initially, the pressure and temperature are 0.05 atm and 293 K respectively, and are uniform throughout the whole assembly. A normal shock ($M_1=4.5$) travels towards the nozzle section. If $A_1 \gg A_2$, the nozzle section is effectively like a solid wall, from which the incident shock reflects. The idea of the shock tunnel is to use the high pressure behind the reflected shock as a reservoir to drive the hypersonic nozzle. After shock reflection, the gas is effectively stationary.
hypersonictunnel.png  ./download/file.php?id=3902&sid=0d8c1cd4fed27551d54c6217c0472484  ./download/file.php?id=3902&t=1&sid=0d8c1cd4fed27551d54c6217c0472484
(a)  Calculate the pressure behind the reflected shock assuming normal reflection. What are the resulting stagnation pressure and temperature?
(b)  Find the area ratio $A_3/A_2$ required to achieve parallel isentropic flow throughout (i.e. no shocks or underexpanded jets at exit). What is the calculated flow velocity in the test section? A quick estimate from part (a) yields about 8 atm and about 3000 K for the stagnation pressure and temperature, respectively.
(c)  By inserting a razor blade parallel to the flow in the test section, the angle of the Mach wave (with respect to the flow direction) is measured to be 18$^\circ$. What is the efficiency of the hypersonic nozzle? Recall the expression we derived previously for the nozzle efficiency: $$ \eta_{\rm nozzle}=\left( \frac{2}{(\gamma-1)M_{\rm e}^2}+1 \right)^{-1} \left(1-\left(\frac{P_{\rm e}}{P_\circ} \right)^\frac{\gamma-1}{\gamma} \right)^{-1} $$
Question #3
Consider the following ramjet engine:
Q3.png  ./download/file.php?id=3897&sid=0d8c1cd4fed27551d54c6217c0472484  ./download/file.php?id=3897&t=1&sid=0d8c1cd4fed27551d54c6217c0472484
Knowing that the mass flow rate of the fuel injected is of 8 kg/s, the gas constant in the diffuser is 287 J/kgK, the gas constant in the nozzle is of 400 J/kgK, that the specific heat ratio in the diffuser is $\gamma_D=1.4$, and the specific heat ratio in the nozzle is $\gamma_N=1.2$ and that

$\rm A_1=1~m^2$$\rm A_3=1~m^2$$\rm A_4=1~m^2$
$P_1=10$ kPa$P_4=600$ kPa$P_6=10$ kPa
$T_1=240$ K$T_4=2000$ K$M_1=3$
Do the following:
(a)  Find $M_3$, $P_3$, and $T_3$.
(b)  Find $M_4$.
(c)  find $M_6$, $T_6$, $A_6$.
Question #4
Consider a supersonic wind tunnel. When the tunnel is operating in normal conditions, a shock sits at the throat of the diffuser as follows:
Q4.png  ./download/file.php?id=3898&sid=0d8c1cd4fed27551d54c6217c0472484  ./download/file.php?id=3898&t=1&sid=0d8c1cd4fed27551d54c6217c0472484
Knowing that the pressure ratio across this shock is of 1.8, and knowing that the throat area of the diffuser needs to be increased by a factor of 1.5 when starting the tunnel (i.e. $(A_t)_{\rm startup}=1.5 (A_t)_{\rm normal}$), find the Mach number in the test section under normal operating conditions.
1.  $\gamma/\sqrt{(\gamma+1)(\gamma-1)}$.
2.  2859 K, 7.43 atm, 3.98, 10.53, 3.236, 0.89.
3.  0.14, 362 kPa, 670 K, 0.195, 3.14, 1011 K, 2.64 m$^2$.
4.  2.21.
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