Blade Element Momentum Theory

Transcription

1 Blade Element Theory has a number of assumptions. The biggest (and worst) assumption is that the inflow is uniform. In reality, the inflow is non-uniform. It may be shown that uniform inflow yields the lowest induced power consumption. Slide 1

2 Model y dy Annulus At a distance y Width dy: area da=2πydy Mass flow rate Slide 2

3 Thrust and Power The thrust for the annulus is : Or in the non-dimensional form: The induced power coefficient is: Slide 3

4 Thrust and Power For the hover case λ C =0. The thrust coefficient is then: The power coefficient is: Valid for any form of λ Slide 4

5 Thrust Coefficient Let s assume that λ(r)= λ tip r n for n 0. We can then relate λ tip with C T Slide 5

6 Power Coefficient Calculating the power coefficient Substituting in the λ tip equation : Since we know that Slide 6

7 Induced Inflow In the previous equation when n=0 then κ=1, uniform inflow which has the lowest induced power. For n>0 then κ>1. With increasing n the inflow is more heavily biased towards the blade tip. Therefore what we want to achieved is, through the blade properties (pitch angle and chord), the first situation Slide 7

8 From BET we have found Then we have Which can be expressed: Slide 8

9 Or in another way: This quadratic equation has the solution: Slide 9

10 Ideal Twist In hover (λ C =0) it simplifies to Since the objective is uniform inflow then we need θr=const.= θ tip. The ideal twist is given by: Slide 10

11 Thrust coefficient in Ideal Twist We can then calculate the thrust coefficient with this ideal twist θr= θ tip Recalling that Slide 11

12 Blade Pitch angle In hover We can then obtain θ tip Slide 12

13 Final notes Recall that: Therefore the thrust varies linearly with r. We have therefore a linear (triangular) variation of lift over the blade. Slide 13

14 Ideal Rotor vs. Optimum Rotor Ideal rotor has a non-linear twist:θ=θ tip /r This rotor will, according to the BEM theory, have a uniform inflow, and the lowest induced power possible. The rotor blade will have very high local pitch angles θ near the root, which may cause the rotor to stall. Ideally Twisted rotor is also hard to manufacture. For these reasons, helicopter designers strive for optimum rotors that minimize total power, and maximize figure of merit. This is done by a combination of twist, and taper, and the use of low drag airfoil sections. Slide 14

15 Optimum Hovering Rotor Minimum P i requires λ=const. (uniform inflow) Minimum P 0 requires α= α(min C d /C l )= α 1 Then for minimum induce power θ= θ tip /r and each blade element must operate at α 1 With (BEMT) dc T =4λ 2 rdr then: Slide 15

16 Optimum Hovering Rotor We have seen that the minimum induced power requires a uniform inflow. Therefore the previous equation is constant over the disk. Let s assume that α 1 is the same for all airfoils along the blade and is independent of Re and M From the equation since α 1 =const and we now that λ=const then σr must be constant too. Slide 16

17 Optimum Hovering Rotor The previous situation is achieved when Slide 17

18 Optimum Hovering Rotor Let s now study the blade twist for uniform inflow and constant AOA: Expressing now in terms of the blade pitch Slide 18

19 Optimum Hovering Rotor The total thrust can be calculated The local solidity of the optimum rotor can be written: And the blade twist Slide 19

20 Optimum Hovering Rotor Slide 20

21 Optimum Hovering Rotor Slide 21

22 Power estimates In a real rotor the non uniformity of λ means that we must calculate numerically: We also saw that we could then calculate the induce power factor κ : Slide 22

23 Power estimates Let s now calculate the rotor profile power. Remember that (BET): We can use for C d the expression: Slide 23

24 Power estimates Slide 24

25 Power estimates The profile power is then: Slide 25

26 Power estimates For a ideally twisted blade λ=const and θr= θ tip : Since Slide 26

27 Power estimates Slide 27

28 Figure of Merit Since we now have the expressions for all power coefficients we can calculate the rotor FM: Slide 28

29 Prandtl s Tip-loss Function The idea of tip losses was already been introduced with the factor B Prandtl provided a solution to the problem of loss of lift near the blade tip. In this solution the curved helical vortex sheets of the rotor wake were replaced by a series of 2D sheets, meaning that the blade tip radius of curvature is large Slide 29

30 Prandtl s Tip-loss Function Prandtl s final result can be expressed in terms of an induced velocity correction factor F: Where f: Remember that Slide 30

31 Prandtl s Tip-loss Function Non dimensional radial position r Slide 31

32 Prandtl s Tip-loss Function Note that when N b (actuator disk) then F 1 The function can be incorporated in the BEMT: Since from BET We can write Slide 32

33 Prandtl s Tip-loss Function With the solution Since F is a function of λ the solution must be found numerically Slide 33

34 Compressibility Corrections So far we have considered that all aerodynamic characteristics independent of Mach number. To introduce a correction to take into account the influence of M, let s used Glauert s rule: Slide 34

35 Compressibility Corrections The local blade M is: Which gives a lift-curve-slope correction of: We had obtained: Slide 35

36 Compressibility Corrections Assuming the ideal twist (uniform inflow): Calculating the total thrust coefficient: Slide 36

37 Compressibility Corrections In the previous expression: If M 0 then K 1 and we obtain the incompressible result Slide 37

38 Weighted Solidity We have seen that for a optimum rotor c varies with r. In these cases where c varies along the blade span the rotor solidity is different from the local blade solidity: The objective of weighted solidity is to help compare performance of different rotors with different blade planforms. Slide 38

39 Trust Weighted Solidity The thrust coefficient is: Assuming a constant C l : With the equivalent chord Slide 39

40 Power Weighted Solidity Extending the study to the power coefficient Therefore Slide 40