Prof. Daniel J. Bodony (AE 312/ME 410) Spring 2020 HW #7 Due April 27, 2020 by 11:59 p.m. on compass2g

Problem 1 A tank of air at 15 MPa pressure and 20 ◦C has a valve knocked off to form a convergent nozzle with a throat diameter of 0.01 m. The ambient pressure is 100 kPa. Neglect friction and heat transfer. (a) Calculate the pressure and velocity at the end of the nozzle. (b) Calculate the mass flow in kg/s. (c) Calculate the thrust of this nozzle in N. (d) Sketch what the flow looks like immediately downstream of the nozzle. (Use your pressure value from part (a) to assess any flow features that must be present.)

Problem 2 Consider the isentropic flow through a quasi-1-D nozzle.

(a) Show that the mass flow rate,m ˙ , through a quasi-1-D nozzle connected to a reservoir at conditions (p0, T0), with gas properties (γ, R) can be expressed as r Ap γ M m˙ = √ 0 .  − (γ+1)/[2(γ−1)] T0 R γ 1 2 1 + 2 M √ (b) Using your result from part (a), plotm ˙ T0/(p0A) versus the M for air.

(c) Observe that the plot in part√ (b) attains a maximum value at a particular value of the Mach number. Find both the maximum value ofm ˙ T0/(p0A) and the Mach number at which it occurs.

Problem 3 Air flows isentropically at a rate of 1 kg/s through a nozzle. The is 310 K and the is 810 kPa. If the exit pressure is 101.3 kPa, determine (a) The throat area. (b) The exit area. (c) The exit velocity.

Problem 4 What is the percentage increase in net thrust of the rocket motor shown below if a divergent portion of area ratio A2/At = 1.5 is added to the sonic nozzle? Assume isentropic flow throughout. The working fluid is not air and has cp = 1.2 kJ/(kg K) and γ = 1.3. Problem 5 The gas turbine engines for fighter aircraft are often equipped with an afterburner to boost their thrust. In the afterburner the hot exhaust gas from the turbine is mixed with new fuel and releases additional heat. This heated gas then flows through the converging-diverging nozzle and exits into the atmosphere at supersonic speed. The following data are available

• mass flow rate through the engine :m ˙ = 75 kg/s

• pressure at turbine exit : p1 = 545 kPa

• temperature at turbine exit : T1 = 830 K

• Mach number at turbine exit : M1 = 0.3. • back pressure is 101 kPa Answer the following questions

(a) What is the design Mach number M4 of the nozzle when operating without the afterburner?

(b) Estimate the flow area at the turbine exit, A1, the throat area At, and the nozzle exit area A4. Assume the flow is fully isentropic.

(c) If the afterburner is turned on and the stagnation temperature of the exhaust gas is raised to T03 = 1530 K (a) can the same mass flow rate be maintained without changing the throat area? (b) find the desired throat area with afterburner turned on, and the desired exit area for best thrust. Note. You will need to use your Rayleigh flow results.

Problem 6 You need to design a new nozzle for a rocket that will expel a gas with ratio of specific heats γ = 1.2, stagnation pressure 1 MPa and stagnation temperature 3000 K. The important design parameter is the exit-to-throat area ratio Ae/At for a given back pressure, pb.

(a) Assuming pb = 101.3 kPa, what is the optimum Ae/At to yield maximum thrust? What kind of nozzle is this and what is special about this operating condition? Note that you will need to determine whether a normal shock exists in the nozzle for each Ae/At you try. Note that your answer will be in terms of thrust per unit area.

(b) Using your nozzle design from part (a), determine the nozzle’s thrust as a function of pb ranging from 101.3 kPa (corresponding to on-the-ground conditions) to 10 kPa. Sketch what happens to the flow outside the nozzle as the back pressure is reduced.