Compressible Flow Assignment 2 — Quasi-One-Dimensional Flow II  
$\xi$ 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
Starting from the conservation of mass $$ {\rm d} (\rho v A)=0 $$ and the conservation of momentum for a quasi-1D flow: $$ \rho v {\rm d} v+ {\rm d} P =0 $$ Show that the force that a fluid exerts on a structure (in the opposite direction of the fluid motion) is: $$ F=\left( \rho v^2 A + PA\right)_2-\left( \rho v^2 A + PA\right)_1 $$
Question #2
A perfect gas $\gamma=1.4$, $R=287.06$ J/kgK is supplied to a converging nozzle at a vanishing velocity and at $750$ kN/m$^2$ and 300 K. The nozzle discharges to atmospheric pressure, 101 $\rm kN/m^2$. Assuming frictionless adiabatic flow, and a mass rate of flow of 1 kg/s calculate (i) the pressure in the exit plane, (ii) the velocity in the exit plane, and (iii) the cross-sectional area of the exit plane.
Question #3
A conical diffuser has entry and exit diameters of 15 cm and 30 cm respectively. The pressure, temperature, and velocity of air at entry are 0.69 bar, 340 K, and 180 m/s. Determine the exit pressure, exit Mach number, exit velocity, and the thrust force exerted on the diffuser walls by the fluid. Take $\gamma=1.4$ and $C_P=1.4$ kJ/kgK.
Question #4
(a)  Defining the nozzle efficiency as: $$ \eta_{\rm nozzle}\equiv \frac{v_{\rm e}^2}{v_{\rm ei}^2}$$ with $v_{\rm e}$ the measured exit velocity, and $v_{\rm ei}$ the exit velocity that would be obtained if the flow is expanded isentopically to the measured exit pressure. Show that for a perfect gas and for adiabatic wall conditions, the nozzle efficiency becomes: $$ \eta_{\rm nozzle}=\left( \frac{2}{(\gamma-1)M_{\rm e}^2}+1 \right)^{-1} \left(1-\left(\frac{P_{\rm e}}{P_\circ} \right)^\frac{\gamma-1}{\gamma} \right)^{-1} $$ with $P_\circ$ the stagnation pressure (in the reservoir driving the nozzle), $M_{\rm e}$ the measured exit Mach number and $P_{\rm e}$ the measured exit pressure.
(b)  From Schlieren photographs of the flow of air at the exit of a converging-diverging insulated nozzle it is observed that the angle of the Mach wave subtended by a disturbing small obstruction is $\alpha=40^\circ$. The measured static pressure at the exit cross section is 0.198 atm, while the pressure in the reservoir which drives the air flow through the nozzle is 1 atm. Determine the nozzle efficiency and the Mach number at the exit. Take $\gamma=1.4$ for air.
Question #5
A preliminary design is made of a converging-diverging axisymmetric nozzle based on simple 1-D theory. The nozzle is insulated and fed from a large reservoir supplied with air at $20^\circ$C and 500 $\rm kN/m^2$. The exit of the nozzle is connected to a discharge chamber whose cross-sectional dimension is much larger than the nozzle exit area. The throat to exit area ratio for the nozzle is 0.5444 and the intention is to use the nozzle to generate supersonic jet flow into the discharge chamber. If the discharge chamber pressure is 48 $\rm kN/m^2$ and the nozzle exit Mach number is measured to be 2.02, explain and quantify as to what is going on in the nozzle. Specifically, plot the Mach number and pressure profiles and compare with those obtained for frictionless (isentropic) flow.
Question #6
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 #7
Consider the following ramjet engine:
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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 $M_6$, $T_6$, $A_6$.
3.  0.81 bar, 42 m/s, 4195 N.
7.  2.62 m$^2$, 1011 K, 3.14.
Due on Thursday October 11th at 16:30. Do problems #4 and #7 only.
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