1. MAE 4261: AIR-BREATHING ENGINES Velocity Triangles Example April 12, 2012 Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. EXAMPLE: SEE SECTION 8.2 FROM H&P Draw velocity triangles assuming that  r = 2 times the axial velocity w (w = constant) a b c

3. VELOCITY TRIANGLES AT b w Start by drawing the axial velocity to some scale (10 units here)

4. VELOCITY TRIANGLES AT b w VbVb  b =75º Draw the absolute velocity vector

5. VELOCITY TRIANGLES AT b w VbVb  b =75º vbvb Draw v  in direction of rotation from the axis to absolute velocity vector

6. VELOCITY TRIANGLES AT b rr w VbVb  b =75º vbvb Add the rotational velocity (  r) and remember Vabs=Vrel+Vcs

7. VELOCITY TRIANGLES AT b rr vbvb w VbVb  b =75º Draw in the velocity to the rotor as seen from the rotating frame

8. VELOCITY TRIANGLES AT b rr vbvb w VbVb  b =75º Stationary frame inlet velocity to rotor relative frame inlet velocity to rotor

9. VELOCITY TRIANGLES AT c rr Either start with the fixed axial velocity or fixed rotational speed

10. VELOCITY TRIANGLES AT c rr  c ’=55º w Add the velocity from the rotor blades in the relative frame

11. VELOCITY TRIANGLES AT c rr  c ’=55º w vcvc Add the velocity exiting the rotor in the absolute frame

12. VELOCITY TRIANGLES AT c rr  c ’=55º w vcvc relative frame exit velocity of rotor stationary frame exit velocity of rotor Again, draw v  in the direction of rotation to the absolute velocity vector

13. COMPOSITE TRIANGLE rr vbvb w  b =75º  c ’=55º vcvc To draw the composite velocity triangle, overlay the rotational velocity w Fixed or ‘metal’ blade angles

14. QUESTIONS Is this a compressor or a turbine? How can you tell? On which blade row(s) is there a torque applied? Why? Describe in words the energy exchange process in each of the two blade rows

15. QUESTIONS Is this a compressor or a turbine? –This is a turbine. The stationary frame tangential velocity (v  ) in the direction of rotor motion is reduced across the moving blade row On which blade row(s) is there a torque applied? Why? –Torque is applied to both blade rows since there is a change in angular momentum across each of them. However, power is extracted only from the moving blades. Describe in words the energy exchange process in each of the two blade rows –In the first blade row, fluid internal energy is converted to swirling kinetic energy by accelerating the flow through a nozzle. No additional energy is added or removed from the flow. –In the second blade row, swirling kinetic energy is extracted from the flow reducing the overall level of energy in the flow and transferring it to the spinning rotor blades.

16. ADDITIONAL QUESTION rr vbvb w VbVb  b =75º Stationary frame inlet velocity to rotor relative frame inlet velocity to rotor So far, we have looked at trailing edge angles of the blades (  b and  c ’) Why do we care about exit velocities from stator in the relative frame? Why do we even draw this on velocity triangles? Why draw this?

17. ADDITIONAL QUESTION Information about how to shape leading edge of rotor blade Doesn’t come into ideal Euler equation but obviously important for aerodynamic Purposes (rotor relative inflow angle)