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1 Turbomachinery in Propulsion Engines -- Exercises -- Rensselaer Hartford Campus Dr. William T. Cousins United Technologies Research Center Text: S. Larry Dixon and C. Hall, Fluid Mechanics and Thermodynamics, 7 th ed, Elsevier Inc., 2014 ISBN-10: 0124159540 ISBN-13: 978-0124159549

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2 Problem 1 Air in a jet engine enters a nozzle at 1800 R, 30 psia and 90 ft/s and exits the nozzle at 1500 R, 13 psia. Assuming that there is no heat loss, A)What is the exit velocity of the nozzle? B)What type of nozzle is it? (converging, diverging, or converging/diverging) Solution:

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3 Problem 2 Air enters a diffuser at 14.7 psia, 540 R, with a velocity of 600 ft/s. The inlet cross-sectional area of the diffuser is 0.2 in 2. A At the exit, the area is 1.75 in 2 and the exit velocity is 60 ft/s. Determine the exit pressure and temperature of the air. Solution:

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4 Problem 3 Air at 60 ft/s, 480 R, 11 lb f /in 2 with 10 lb m /s flows into a turbojet engine and it flows out at 1500 ft/s, 1440 R, 11 psia. What is the change (power) in flow of kinetic energy? Solution:

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5 Problem 4 In a jet engine a flow of air at 1000 K, 200 kPa and 40 m/s enters a nozzle where the air exits at 500 m/s, 90 kPa. What is the exit temperature assuming no heat losses and a frictionless flow? Solution:

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6 Problem 5 The front of an engine acts as a diffuser receiving air at 900 km/h, -5C, 50 kPa, bringing it to 80 m/s relative to the engine before entering the fan. If the flow area is reduced to 80% of the inlet area, find the temperature and pressure at the fan inlet. Solution:

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7 Problem 6

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8 Problem 7 Shown schematically in Part (a) of the figure is the blading of a singlestage axial turbomachine. What kind of machine is represented by cases 1 and 2? What would happen in case 3? Would it be desirable to build a compressor stage as shown in Part (b) of the figure? Part (a) Part (b) Hint: Draw the velocity diagrams for cases one, two, and three and examine them. Look at things like the U, C , work done, etc. Also draw the velocity diagram for Part (b) and examine it.

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9 Problem 8 At the mean radius (r m = 30 cm), the blade configuration is as shown in Part (b). For simplicity, assume the air angles and blade angles are identical (i.e., no deviation angle, etc.). The overall adiabatic efficiency of the stage is 90%. The hub-tip radius ratio is 0.8, which is high enough so that the conditions at the mean radius are a good average of the conditions from the hub to the tip. The axial velocity component at design flow rate is uniformly 125 m/s and the inlet air is at 1 atm and 20 C. a) What would be the power required to drive the compressor and the shaft speed under these conditions? b) What would be the stage total pressure rise? Hint: Draw the velocity triangles and use your geometric knowledge to calculate the power. Use your knowledge of the compressor map relationship (efficiency, pressure ratio, temperature ratio) to find the pressure ratio.

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10 Problem 9 At a certain operating condition the mid-radius velocity triangles for an axial compressor stage are as shown in the figure. As usual, the subscripts 1 and 2 denote the entrance to the rotor and the stator, respectively. The stagnation temperature and pressure at the entrance to the rotor are 340 K and 185 kPa. Neglecting frictional effects, determine: a)The specific work in kJ/kg; b)The total and static temperatures between the rotor and the stator; c)The total and static pressures between the rotor and stator; and d)The pressure coefficient at mid-radius, C P = (P 2 P 1 ) / 1 W 1 2. X Hint: In part a, use your knowledge of calculating work from geometry. Then, in part b, use your knowledge of the 1 st Law and stagnation enthalpy. There are several ways to do part c, but the easiest is with your knowledge of the isentropic relationships. Remember, you are working with air (an ideal gas).

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11 Information for Problems 10, 11, 12 and 13 If needed, use: For the Turbine:Cp = 1.157 kJ/kg K, R = 0.287 kJ/kg K, = 1.33 For the Compressor:Cp = 1.005 kJ/kg K, R = 0.287 kJ/kg K, = 1.4 Engine Inlet conditions are standard day inlet conditions. The single stage turbine in the J-30 was designed to drive the 10-stage axial compressor. The design point specifications for the compressor are m dot = 30 lb m /sec and P 02 /P 01 = 3.8, at an operating speed of 17,000 rpm, with an efficiency of 0.90. The average mean radius of the compressor is 0.1835 m. The measured hub radius, tip radius, the mean radius values of angles 2, 2 and 3 of the turbine, the mean radius spacing s m, and the axial chord b Z of the blades, was measured as shown in the two following figures, and the values obtained. r h = 0.152 m r t = 0.216 m 2 = 60 2 = 30 3 = -57 s m = 0.02 m b Z = 0.025 m

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12 Problem 10 Estimate (be sure to list any assumptions you might make) the density of the combustion products in the turbine. Assume a 5% loss of total pressure in the burner, and a combustion temperature of 1100 K.

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13 Problem 11 (page 1) Using the data provided, calculate C X /U m for the turbine. Assume that mdot fuel = 3% of the mass flow rate of the air. Also, while you have the measured turbine hub and tip radius, assume there is a 5% area blockage (reduction) due to the fact that there are turbine blades in the flow path.

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14 Problem 12 Using the turbine design spreadsheet presented in class, find h 0 /U m 2 and 3. Find the turbine efficiency tt from the Smith chart. Sketch the velocity triangles.

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15 Problem 13 Using Z = 0.8, predict the blade spacing s, and compare the prediction with the measurements.