2018 Compressible Flow Midterm Exam  
Tuesday 6th November 2018
16:30 — 18:30


NO NOTES OR BOOKS; USE COMPRESSIBLE FLOW TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
10.18.18
Question #1
A fixed-geometry, convergent-divergent wind tunnel diffuser is to be designed for Mach number 2. Assuming no friction, compare the minimum possible percent loss in stagnation pressure during operation for the following cases:
(a)  The best possible design is employed
(b)  The design is conservative, with a throat area 5 percent larger than that required for starting, and with the shock located during operation at an area 5 percent greater than the throat area.
(c)  The converging portion is eliminated, and the process comprises a normal shock followed by reversible subsonic compression.
Question #2
Consider the following ramjet engine:
Q7.png ./download/file.php?id=4547&sid=53692e1f694aecbd7add482e56599df6 ./download/file.php?id=4547&t=1&sid=53692e1f694aecbd7add482e56599df6
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 #3
Consider a bottle of compressed nitrogen attached to a converging-diverging nozzle as follows:
Q3.png ./download/file.php?id=4534&sid=53692e1f694aecbd7add482e56599df6 ./download/file.php?id=4534&t=1&sid=53692e1f694aecbd7add482e56599df6
Knowing that $A_4/A_3=1.5$ and that $A_1/A_3=100$, that $P_{\rm back}=0.9P_\circ$, and that $T_\circ=300$ K, and that the mass flow rate of nitrogen escaping the nozzle is of $1~{\rm kg/s}$, do the following:
(a)  Plot the Mach number distribution from the tank till the nozzle exit.
(b)  Determine the force in Newton acting on the structure by the nitrogen gas. Simplify the expression for the force as much as possible and then substitute values.
Question #4
Consider two shockwaves moving towards each other in a duct as follows:
Q4a.png ./download/file.php?id=4529&sid=53692e1f694aecbd7add482e56599df6 ./download/file.php?id=4529&t=1&sid=53692e1f694aecbd7add482e56599df6
where $v_{\rm s1}$ and $v_{\rm s2}$ are the velocities of the left and right shocks with respect to the ground. In zone 1, the gas consists of air at rest with a pressure of 1 atm and a temperature of 300 K. When the shocks reach each other, they reflect as follows:
Q4b.png ./download/file.php?id=4530&sid=53692e1f694aecbd7add482e56599df6 ./download/file.php?id=4530&t=1&sid=53692e1f694aecbd7add482e56599df6
Knowing that $v_{\rm s1}=500$ m/s and $v_{\rm s2}=700$ m/s, do the following:
(a)  Find the flow properties (in the ground frame) after the left incident shock. I.e., find $P_2$, $T_2$, and $v_2$ in the ground frame. Clearly indicate the flow direction in zone 2.
(b)  Find the flow properties (in the ground frame) after the right incident shock. I.e., find $P_3$, $T_3$, and $v_3$ in the ground frame. Clearly indicate the flow direction in zone 3.
(c)  Using the flow properties found in (a) and (b) find the pressure after the reflected shocks $P_4$ and $P_5$. Also, find the velocity of the contact surface in the ground frame, $v_{\rm c}$. Clearly indicate the flow direction in zones 4 and 5.
Hint: a contact surface separates two flow regions of equal pressure and velocity but of different density.
Answers
1.  -16.5%, -22.4%, -27.9%.
2.  1011 K, 2.64 m$^2$.
3.  773 N.
4.  -217 m/s, 8.8 atm.
11.27.18
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