United Technologies Research Center 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”, 7th ed, Elsevier Inc., 2014 ISBN-10: 0124159540 ISBN-13: 978-0124159549

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, What is the exit velocity of the nozzle? What type of nozzle is it? (converging, diverging, or converging/diverging) Solution:

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 in2. A At the exit, the area is 1.75 in2 and the exit velocity is 60 ft/s. Determine the exit pressure and temperature of the air. Solution:

Problem 3 Air at 60 ft/s, 480 R, 11 lbf/in2 with 10 lbm/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:

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:

Problem 5 The front of an engine acts as a diffuser receiving air at 900 km/h, -5°C, 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:

Problem 6 In an aircraft engine at takeoff, the combustion products expand adiabatically in an exhaust nozzle. At the entrance to the nozzle, the pressure is 0.180 MPa and the temperature is 1200 K. The kinetic energy entering the nozzle is very much smaller than the kinetic energy of the gas leaving the nozzle. The specific heat of the exhaust gas varies with temperature as follows: CP = 0.959 + 1.16x10-4T + 3.65x10-8T2 in which the units of CP are kJ/kg K and T is the temperature in degrees Kelvin. The molecular weight of the exhaust gas is 30. Show how the exhaust gas velocity and pressure depend upon the temperature for 900 K, 1000 K, and 1100 K. At each temperature, determine also the speed of sound γ𝑅𝑇 . Determine the required nozzle types. The Universal Gas Constant is equal to: 8.31441 kJ/kmol-K. Solution:

Problem 7 Shown schematically in Part (a) of the figure is the blading of a single–stage 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, ΔCθ, work done, etc. Also draw the velocity diagram for Part (b) and examine it.

Problem 8 At the mean radius (rm = 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.

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: The specific work in kJ/kg; The total and static temperatures between the rotor and the stator; The total and static pressures between the rotor and stator; and The pressure coefficient at mid-radius, CP = (P2 – P1) / ½ρ1W12. X Hint: In part a, use your knowledge of calculating work from geometry. Then, in part b, use your knowledge of the 1st 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).

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, γ = 1.33 For the Compressor: Cp = 1.005 kJ/kg K, R = 0.287 kJ/kg K, γ = 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 mdot = 30 lbm/sec and P02/P01 = 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 α2 , β2 and β3 of the turbine, the mean radius spacing sm , and the axial chord bZ of the blades, was measured as shown in the two following figures, and the values obtained. rh = 0.152 m rt = 0.216 m α2 = 60° β2 = 30° β3 = -57° sm = 0.02 m bZ = 0.025 m

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.

Problem 11 (page 1) Using the data provided, calculate CX/Um 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.

Problem 12 Using the turbine design spreadsheet presented in class, find Δh0/Um2 and α3 . Find the turbine efficiency ηtt from the Smith chart. Sketch the velocity triangles.

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