Propeller theories. Blade element theory

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2 Propeller theories Blade element theory The blade elements are assumed to be made up of airfoil shapes of known lift, C l and drag, C d characteristics. In practice a large number of different airfoils are used to make up one propeller blade. Each of these elements shall have its own lift, C l and drag, C d coefficient characteristics. 2

3 The thrust, dt created by an element of elemental radial length dr is created with contributions from the airfoil with lift, dl and drag, dd 3

4 Using the blade elemental lift and drag characteristics the working capacity of the blade element may be found as : Thrust produced, dt = dl.cos φ dd.sin φ = ½.ρ.V R 2.c.dr. (C l cos φ C d sin φ) Torque to be supplied, dq = (dl.sin φ + dd.cos φ). r = ½.ρ.V R 2.c.dr.(C l.sin φ + C d.cos φ) 4

5 Substituting for Resultant inflow velocity Incident and aligned to the blade element, V R = V /Sin φ, and for Incoming flow Dynamic head based on forward velocity of the element q = ½ ρ V 2 5

6 The elemental thrust is : q.. c dr dt = (C 2 l cos φ - C d sin φ) sin φ and The elemental torque is : q... c r dr dq = C 2 l sin φ + C d cos φ sin φ ( ) 6

7 Propeller thrust and torque are now computed by integrating from the root to the tip of the blade and for number of blades, B R c. dr =.. (C cos - C sin ) T qb φ φ 2 l d sin φ 0 R c.. r dr =. (C 2 l sin + C d cos ) sin Q qb φ φ φ 0 7

8 Thus, the net thrust and the torque are seen to be directly proportional to the number of blades, B and the chord, c. This is not quite true in practice, as more is the number of blades and wider the blade chord - it shall result in more surface area, more flow blockage and higher consequent aerodynamic losses. The optimum number of blades need to be found separately and not from the blade element theory. 8

9 The blade element efficiency, η el = Thrust power produced Torque power supplied In terms of elemental airfoil characteristics C l and C d, blade efficiency is : η el v. dt V C cos φ - C sin φ C cos φ - C sin φ = = = 2 πn. dq 2πnr C sin φ + C cos φ C sin φ + C cos φ l d l d..tan l d l d φ 9

10 Applying maxima condition it can be shown that maximum efficiency, η el-max occurs at φ = π Cd 4 2. C for a blade element airfoil characterized by its C d & C l The estimations from blade element theory is within 10% of the actually obtained results. l 10

11 If the elemental performances are plotted in the form of dc T /dx and dc Q /dx variation in X, the span-wise direction of a blade (root to tip) 11

12 Low speed aircraft propeller Characteristics Lect 30 Cruise Point Courtesy: Theodorsen,

13 High speed aircraft propeller characteristics Lect 30 Cruise Point Courtesy: Dommasch et al.,

14 Efficiency of Propeller Variable Pitch Propeller Cruise Point selection Thrust Co-efficient of Propeller Courtesy: NACA, USA 14

15 Power Cruise Point selection Typical Propeller Thrust-Power Characteristics using NACA airfoils Variable Pitch Propeller Courtesy: NACA, USA 15

16 C s, the speed power coefficient, defined by, Cs = (ρ.v 5 /P.n 2 ) 1/5 Is often used for design / selection of propeller If coeff of power, Cp as a function of J, is known, Cs can be obtained from Cs = J/Cp 1/5 The usefulness of C s is in the process of defining it -- diameter was eliminated. Thus the propeller design or selection related flow parameters may be estimated even before the propeller size is fixed. 16

17 Propeller Cs Characteristics 17

18 Transonic Swept bladed propellers (a) Tractor ; (b) counter-rotating Pusher 18

19 In an aircraft application: Propeller Power, P prop = P Engine.η shaft.η prop Propeller Torque, Q prop = Q engine Typically, at Take off, Q prop is low, β is low, P E is High, rpm is high at Cruise, Q prop is high, β is high, P E is low, rpm is low 19

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