1. Chalmers University of Technology Elementary axial turbine theory –Velocity triangles –Degree of reaction –Blade loading coefficient, flow coefficient Problem 7.1 Civil jet aircraft performance Lecture 7 – Axial flow turbines

2. Chalmers University of Technology Axial flow turbines Working fluid is accelerated by the stator and decelerated by the rotor –Expansion occurs in stator and in relative frame of rotor Boundary layer growth and separation does not limit stage loading as in axial compressor

3. Chalmers University of Technology Elementary theory Energy equation for control volumes (again): Adiabatic expansion process (work extracted from system - sign convention for added work = +w) –Rotor => -w = c p (T 03 -T 02 ) w = c p (T 02 -T 03 ) –Stator => 0 = c p (T 02 -T 01 ) => T 02 = T 01

4. Chalmers University of Technology How is the temperature drop related to the blade angles ? We study change of angular momentum at mid of blade (as approximation)

5. Chalmers University of Technology Governing equations and assumptions Relative and absolute refererence frames are related by: We only study designs where: –C a2 =C a3 –C 1 =C 3 We repeat the derivation of theoretical work used for radial and axial compressors:

6. Chalmers University of Technology Principle of angular momentum Stage work output w: C a constant:

7. Chalmers University of Technology (1) (2) (1)+(2) =>

8. Chalmers University of Technology Combine derived equations => Exercise: derive the correct expression when  3 is small enough to allow  3 to be pointing in the direction of rotation. Energy equation Energy equation: We have a relation between temperature drop and blade angles!!! :

9. Chalmers University of Technology Dimensionless parameters Blade loading coefficient Degree of reaction Exercise: show that this expression is equal to => when C 3 = C 1

10. Chalmers University of Technology  can be related to the blade angles! C 3 = C 1 => Relative to the rotor the flow does no work (in the relative frame the blade is fixed). Thus  T 0,relative is constant => Exercise: Verify this by using the definition of the relative total temperature:

11. Chalmers University of Technology  can be related to the blade angles! Plugging in results in definition of  => The parameter  quantifies relative amount of ”expansion” in rotor. Thus, equation 7.7 relates blade angles to the relative amount of expansion. Aircraft turbine designs are typically 50% degree of reaction designs.

12. Chalmers University of Technology Dimensionless parameters Finally, the flow coefficient: Current aircraft practice (according to C.R.S): Aircraft practice => relatively high values on flow and stage loading coefficients limit efficiencies

13. Chalmers University of Technology Dimensionless parameters Using the flow coefficient in 7.6 and 7.7 we obtain: The above equations and 7.1 can be used to obtain the gas and blade angles as a function of the dimensionless parameters

14. Chalmers University of Technology Exercise: show that the velocity triangles become symmetric for  = 0.5. Hint combine 7.1 and 7.9 Exercise: use the “current aircraft practice” rules to derive bounds for what would be considered conventional aircraft turbine designs. What will be the range for  3 ? Assume  = 0.5. Two suggested exercises

15. Chalmers University of Technology Turbine loss coefficients: Nozzle (stator) loss coefficients: Nozzle (rotor) loss coefficients:

16. Chalmers University of Technology Problem 7.1

17. Chalmers University of Technology Civil jet aircraft performance

18. Chalmers University of Technology Four forces of flight Resulting force perpendicular to the flight path Net thrust from the engines resulting force parallell to the flight path α angle of attack V velocity Newton’s second law

19. Chalmers University of Technology Aerodynamic equations L=Lift = q·S·C L [N] D=Drag = q·S·C D [N] q = dynamic pressure [N/m²] S = reference wing area [m²] C L = coefficient of lift C L = f(α,Re,M) C D = coefficient of drag C D = f(α,Re,M)

20. Chalmers University of Technology Reference wing area The area is considered to extend without interruption through the fuselage and is usually denoted S.

21. Chalmers University of Technology Lift versus angle of attack

22. Chalmers University of Technology

23. The ISA Atmosphere From lecture 5

24. Chalmers University of Technology Equations

25. Chalmers University of Technology Lift equation

26. Chalmers University of Technology Drag equation

27. Chalmers University of Technology Drag polar

28. Chalmers University of Technology High speed drag polar

29. Chalmers University of Technology A flight consists of: Taxi Take off Climb Cruise Descent Approach and landing Diversion to alternate airport?

30. Chalmers University of Technology Cruise For an airplane to be in level, unaccelerated flight, thrust and drag must be equal and opposite, and the lift and weight must be equal and opposite according to the laws of motion, i.e. Lift = Weight = mg Thrust = Drag

31. Chalmers University of Technology Range

32. Chalmers University of Technology Range

33. Chalmers University of Technology Breguet range equation For a preliminary performance analysis is the range equation usually simplified. If we assume flight at constant altitude, M, SFC and L/D the range equation becomes which is frequently called the Breguet range equation

34. Chalmers University of Technology Breguet range equation The Breuget range equation is written directly in terms of SFC. Clearly maximum range for a jetaircraft is not dictated by maximum L/D, but rather the maximum value of the product M(L/D) or V(L/D).

35. Chalmers University of Technology Breuget range equation From the simplified range equation, maximum range is obtained from Flight at maximum Low SFC High altitude, low ρ Carrying a lot of fuel

36. Chalmers University of Technology Range

37. Chalmers University of Technology Endurance Endurance is the amout of time that an aircraft can stay in the air on one given load of fuel

38. Chalmers University of Technology Endurance

39. Chalmers University of Technology Learning goals Understand how turbine blade angles relate to work output Be familiar with the three non-dimensional parameters: –Blade loading coefficient, degree of reaction, flow coefficient Understand why civil aircraft are –operated at high altitude –at flight Mach numbers less than 1.0 –How engine performance influences aircraft range