Aerospace Propulsion Assignment 7 — Nozzles with Friction


B. Parent • AME425 Aerospace Propulsion

Aerospace Propulsion Assignment 7 — Nozzles with Friction

Instructions

Write your solutions in single column format, with one statement following another vertically. Write your solutions neatly so that they are easy to read and verify. Don't write one line with two equal signs. Highlight your answers using a box. Solve the problems starting from the Aerospace Propulsion Tables. Do not use any other document to solve the problems.

05.04.14

Question #1

(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 at the nozzle entrance, $M_{\rm e}$ 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 stagnation pressure at the nozzle entrance is 1 atm. Determine the nozzle efficiency and the Mach number at the exit. Take $\gamma=1.4$ for air.

04.28.23

Question #2

A preliminary design based on simple 1-D theory is made of a converging-diverging axisymmetric nozzle intended to be used in a supersonic turbofan engine. The nozzle is insulated and a mixture of air and combustion products enters the nozzle with a stagnation temperature of $1000^\circ$C and a stagnation pressure of $\rm 500~kN/m^2$. The throat to exit area ratio for the nozzle is 0.5444. The nozzle discharges into the atmosphere where the pressure is 48 $\rm kN/m^2$. Knowing that 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 #3

Consider that you are faced with designing a nozzle for a supersonic turbojet which is to have exit Mach number of $M=2$ and an exit circular cross-section of approximately 10 cm in diameter.

(a)

As a first try, you decide to start out by machining a simple converging-diverging conical nozzle to get some feel for what's required:

nozzle1.png ./download/file.php?id=14380&sid=f30f2185f7deac43bec659df3100ee03 ./download/file.php?id=14380&t=1&sid=f30f2185f7deac43bec659df3100ee03

The nozzle is equipped with a static pressure hole in the exit plane. The flow entering the nozzle has a stagnation pressure of 10 bar and a stagnation temperature of $2000^\circ$C. The supersonic nozzle exhausts into the atmosphere at a pressure of 0.1 bar. For the above conical nozzle under the specified operating conditions, pitot tube-static pressure measurements in the test section near the exit plane indicate pressures of $$ {P_{\rm pitot~tube}}=5.6677~{\rm bar}$$ $$ {P_{\rm static~hole}}=1.1023~{\rm bar}$$ Determine the exit Mach number and the nozzle efficiency.

(b)

Based on these preliminary tests, you are now in a position to machine out the conical nozzle to a final contoured converging-diverging supersonic de Laval nozzle, as shown here:

nozzle2.png ./download/file.php?id=14382&sid=f30f2185f7deac43bec659df3100ee03 ./download/file.php?id=14382&t=1&sid=f30f2185f7deac43bec659df3100ee03

For the same nozzle inflow and back pressure conditions, you now find that the pitot-static pressure measurements in the exit plane give a pressure ratio of: $$ {P_{\rm pitot~tube}}=7.2084~{\rm bar}$$ $$ {P_{\rm static~hole}}=1.2780~{\rm bar}$$ Determine the exit Mach number and the nozzle efficiency.

(c)

The final contoured de Laval nozzle, described in (b), is now put through its paces by testing it over a whole range of intake and exhaust conditions. In one such test, the stagnation pressure and temperature of the flow entering the nozzle is of 15 bar and 2000$^\circ$C, respectively. A pitot tube measurement in the exit plane gives a pressure of 11.65 bar. Determine the exit Mach number, the exit pressure and sketch out roughly the Mach number distribution along the length of the nozzle from the throat to the exit.

Answers

1.90, 89.7%, 2.00, 100%, 0.52, 9.69 bar.

10.25.23

Due on Tuesday November 5th at 23:59. Do Questions #2 and #3 only.

10.28.24

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