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ME-882 Compressible Flow Homework Set 1 (Chapters 1 and 2) 1. (Problem 1.4) The pressure and temperature ratios across a given portion of the shock wave in air are p2/p1 = 4.5 and T2/T1 =1.687 where 1 and 2 denote conditions ahead of and behind the shock wave, respectively. Calculate the change in entropy in units of ft. lbf/ slug R and J/ kg K. 2. Air is expanded in an insulated cylinder equipped with a frictionless piston. The initial temperature of the air is 1400 K. The original volume is 10% of the final volume. Calculate the change in temperature, the work removed from the gas and pressure ratio. 3. (Problem 1.5 modified) Assume that the flow of air through a given duct is isentropic. At one point in the duct, the pressure and temperature are 1800 lb/ft2 and 500 R, respectively. At a second point, the temperature is 400 R. Calculate the pressure and density at the second point. Estimate the velocity at the second point. (Hint: Use momentum equation) Homework Set 2 (Chapter 3) 1. Air flows through a constant area, insulated passage. Entering conditions are 520 R, 50 psia, M1 = 0.45. At a point downstream the Mach number is found to be unity. Solve for downstream temperature and pressure. What is the entropy change between these two sections? Determine the wall frictional force if the duct is 1 ft. in diameter. 2. (Problem 3.5) Consider a Pitot static tube mounted on the nose of an experimental airplane. Calculate the free-stream Mach number at which the airplane is flying for the following conditions: i) Pitot pressure = 1.22 bar; static pressure = 1.01 bar ii) Pitot pressure = 7222 lb/ft2; static pressure = 2116 lb/ft2 3. (Problems 3.9 and 3.10) Air enters the combustor of a jet engine at 10 atm, 1000 R and Mach number of 0.2. Fuel is injected and burned, with a fuel-air ratio (by mass) of 0.06. The heat released during the combustion is 4.5 X 108 ft-lb per slug of fuel. Assuming one-dimensional frictionless flow with =1.4 for the fuel-air mixture, calculate the Mach number, pressure and temperature at the exit of the combustor. What would be the maximum fuel-air ratio beyond which the flow will be choked at the exit? 4. At some point in a flow system of air M1 = 3.0, p1 = 35 psia and Tt1 = 800 R. At a section farther along in the duct, the Mach number has been reduced to M2 = 1.5 by heat transfer. Find the static and stagnation temperatures and pressures at the downstream section. Determine the direction and amount of heat transfer. 5. At one section in a constant area duct the stagnation pressure is 66.8 psia and the Mach number is 0.80. At another section the pressure is 60 psia and the temperature is 120 F. Compute the temperature at the first section and the Mach number at the second section if the fluid is air. Which way is the air flowing? What is the friction length (f x/D) of the duct? 6. A 12 in.-diameter duct has a friction factor of 0.02 and no heat transfer from section 1 to section 2 which is 10.6 ft long. There is negligible friction from section 2 to section 3 in which sufficient heat is added to just choke the air flow at the exit. Draw the T-s diagram for the system showing the complete Fanno and Rayleigh lines involved. Determine the Mach number and stagnation conditions at section 2. Determine the static and stagnation conditions at section 3. How much heat was added to the flow? Homework Set 3 (Chapter 4) 1. (Problem 4.3) Calculate the maximum surface pressure that could be achieved on the forward face of a wedge flying at Mach 3 at standard sea level conditions (p1 = 1.01 bar) with an attached shock wave. 2. (Problem 4.4) In the flow past a compression corner, the upstream Mach number and pressure are 3.5 and 1 atm, respectively. Downstream of the corner, the pressure is 5.48 atm. Calculate the deflection angle of the corner. 3. (Problem 4.7) An incident shock wave with wave angle of 300 impinges on a straight wall. If the upstream flow properties are Mach 2.8, 1 atm and 300 K, calculate the pressure, temperature, Mach number and total pressure downstream of the reflected wave. 4. (Problem 4.9) Consider the intersection of two shocks of opposite families (see Fig. 4.23). For M1 =3, p1= 1 atm, 2 =200 and 3 =150, calculate the pressure in regions 4 and 4’ and the flow direction , behind the refracted shocks. 5. (Problem 4.11) For a given Prandtl-Meyer expansion, the upstream Mach number is 3.0 and the pressure ratio across the wave is 0.4. Calculate the angles of the forward and rearward Mach lines of the expansion fan relative to the free-stream direction. 6. (Problem 4.12) Consider a supersonic flow with an upstream Mach number of 4 and a pressure of 1 atm. This flow is first expanded around an expansion corner with 150, and then compressed through a compression corner with equal angle of 150, so that it is returned to its original upstream direction. Calculate the Mach number and pressure downstream of the compression corner. 7. (Problem 4.14) Consider a supersonic flow past a compression corner with an angle of 200. The upstream properties are Mach 3.0 and 2116 lb/ft2. A Pitot tube is inserted in the flow downstream of the corner. Calculate the value of pressure measured by the Pitot tube. 8. (Problem 4.17) Consider a flat plate with a chord length of 1 m. The free-stream properties are Mach 3.0, pressure of 1 atm and temperature of 270 K. Using shock-expansion theory, tabulate and plot the following properties as functions of angle of attack from 00 to 300 in increments of 50. Pressure on the top surface, Pressure on the bottom surface, Temperature on the top surface, Temperature on the bottom surface, Lift per unit span, Drag per unit span and Lift-to-Drag ratio Homework Set 4 (Chapter 5) 1. (Problem 5.5) Consider a subsonic flow through a divergent duct with exit-to-inlet area ratio of 1.7. If the inlet conditions are 300 K, 250 m/s and the exit pressure is 1 atm, calculate the inlet pressure and exit velocity. 2. (Problem 5.10) Consider a supersonic nozzle with a Pitot tube mounted at the exit. The reservoir pressure and temperature are 10 atm and 500 K, respectively. The pressure measured by the Pitot tube is 0.6172 atm. The thorat area is 0.3 m2. Calculate the exit Mach number, exit area, exit pressure, exit temperature and mass flow through the nozzle. 3. (Problem 5.11) Consider a covergent-divergent nozzle duct with exit and throat areas of 0.5 m2 and 0.25 m2, respectively. The inlet reservoir pressure is 1 atm and the exit static pressure is 0.6 atm. For this pressure ratio, the flow will be supersonic in a portion of the nozzle, terminating with a normal shock inside the nozzle. Calculate the local area ratio at which the shock is located inside the nozzle. 4. (Problem 5.14) In a supersonic nozzle flow, the exit-to-throat area ratio is 10. The reservoir pressure is 10 atm and the back pressure is 0.04 atm. Calculate the angle through which the flow is deflected immediately after leaving the lip of the nozzle exit. 5. (Problem 5.15) Consider an oblique shock wave with M1 = 4.0 and = 500. This shock wave is incident on a constant pressure boundary (see Fig. 5.26). For the flow downstream of the reflected expansion wave, calculate the Mach number M3 and the flow direction relative to the flow upstream of the shock. 6. (Problem 5.17) We wish to design a Mach 3 supersonic wind tunnel, with a static pressure and temperature in the test section of 0.1 atm and 400 R, respectively. Calculate the exit-to-throat area ratio of the nozzle, the ratio of diffuser thorat area to the nozzle throat area, reservoir pressure and reservoir temperature. HOT PROBLEMS FOR PRACTICE (Set 5: Chapters 3 and 5) 1. A converging-diverging nozzle receives air from a tank at 100 psia and 600 R. The pressure is 28.0 psia immediately preceding a plane shock that is located in the diverging section. The Mach number at the exit is 0.5 and the flow rate is 10 lbm/s. Determine the throat area, the area at which the shock is located, the outlet pressure required to operate the nozzle in this manner, the outlet area and the design Mach number (exit Mach number if there was no shock). 2. A converging-diverging nozzle has an exit-to-throat area ratio of 3.0. The stagnation conditions of the inlet are 150 psia and 550 R. A constant area duct with a length of 12 diameters is attached to the nozzle outlet. The friction factor in the duct is 0.025. Compute the receiver pressure that would place a shock i) at the nozzle exit and ii) at the duct exit. 3. Air enters a duct with M1 = 2.0, p1 = 0.7 bar and T1 = 170 K. Heat transfer takes place while the flow proceeds down the duct. A converging section (A2/A3 = 1.45) is attached to the duct outlet and the exit Mach number, M3 is unity. Assume that the inlet conditions and exit Mach number remain fixed. Find the amount and direction of heat transfer i) if there are no shocks in the system and ii) if there is a normal shock someplace in the duct. Does it make any difference where the shock occurs? Homework Set 6 (Chapter 7) 1. (Problem 7.2 and 7.3) Consider a normal shock wave moving with a velocity of 680 m/s into still air at standard atmospheric conditions ( 1 atm, 288 K). Determine the temperature, pressure and velocity that exist after passage of the shock wave. What is the entropy change experienced by the air? 2. (Problem 7.7) Consider a blunt-nosed aerodynamic model mounted inside the driven section of a shock tube. The axis of the model is aligned parallel to the axis of the shock tube, and the nose of the model faces towards the on-coming incident shock wave. The driven gas is air initially at a temperature and pressure of 300 K and 1 atm, respectively. After the diaphragm is broken, an incident shock wave with a pressure ratio of p2/p1 = 40.4 propagates into the driven section. Calculate the pressure and temperature at the nose of the model shortly after the incident shock sweeps by the model. Calculate the pressure and temperature at the nose of the model after the reflected shock sweeps by the model. Homework Set 7 (Chapter 9) 1. (Problem 9.3) In the low-speed flow, the pressure coefficient at a point on an airfoil is -0.9. Calculate the value of Cp at the same point for M =0.6 by means of a) Prandtl-Glauert rule b) Laitone’s correction and c) Karman-Tsien rule. 2. (Problem 9.9) Repeat problem 4.17 using linearized theory. Compare the results using this (approximate analysis) and the shock-expansion theory (exact analysis). Homework Set 8 (Chapter 10) 1. (Problems 10.1 and 10.2) Consider a 150 half-angle cone at 00 angle of attack in a free stream at standard sea level conditions with M = 2.0. Obtain a) the shock wave angle b) pressure, temperature, density and Mach number immediately behind the shock wave and c) pressure, temperature, density and Mach number on the cone surface. For this cone, below what value of M will the shock wave be detached? Compare this the analogous value for the wedge. COURSE PROJECT (Chapter 11) Compute and graph the contour of a two-dimensional minimum-length nozzle for the expansion of air to a design exit Mach number of 2.0. Write the code such a way that we can change the design exit Mach number and still be able to compute and graph the contour for it. The report must be formal that includes the methodology and the results with appropriate discussion. Include references and any code developed. Presentation must be clear.