III Blade Element Theory for Rotor in Hover and Climb In the previous section we looked at momentum theory This theory gave us four useful pieces of information 1 Induced velocity far downstream in the rotor wake called downwash is twice that at the rotor disk called inflow CT 3 2 2 The ideal power coefficient Cp in hover equals 3 The induced power is minimized for a given thrust coefficient if the induced 2 velocity in the far wake is uniform 4 The induced velocity at the rotor disk is related to the thrust coefficient in hover by i CT v R 2 1 Momentum theory can not help us analyze specific rotor blades or distinguish between the number of blades and their other physical characteristics such as twist taper camber etc In order to do these we turn to a theory called blade element theory Blade element theory is similar to the strip theory in fixed wing aerodynamics The blade is assumed to be made of several infinitesimal strips of width dr The lift and drag are estimated at the strip using 2 D airfoil characteristics of the airfoil at that strip and what we know about the local flow magnitude such as the angular velocity climb speed V and inflow v The lift L and drag D multiplied by the in plane velocity of the rotor are integrated with respect to r from root to tip to obtain the thrust T and the power P consumed by a single rotor blade For multi bladed rotors this integrated expression is multiplied by the number of blades b Note Wayne Johnson uses the symbol N for number of blades r dr V v arctan r Line of Zero Lift V v r effective Consider a typical element or strip shown below The blade sees an in plane velocity UT that is tangential to the plane of rotation The magnitude of U T is of course r where r is the radial position of the strip This element has a pitch angle equal to That is the angle between the plane of rotation and the line of zero lift is If there were no climb velocity V or induced inflow v this would be the section angle of attack These two components of velocity V and v change the flow direction by amounts as shown in the figure above Here V v arctan r 2 Thus the effective angle of attack is The airfoil lift and drag coefficients Cl and Cd at this effective angle of attack may be looked up from a table of airfoil characteristics The lift and drag forces will be perpendicular to and along the apparent stream direction These forces are given by 1 L U T2 U P2 cC l 2 1 D U T2 U P2 cC d 2 3 The L and D have units of force per unit span They must be rotated in directions normal to and tangential to the rotor disk respectively and multiplied by the strip width dr to get the thrust and drag components as shown below dT L cos D sin dr 1 2 dFx D cos L sin dr U T2 U P2 c C l cos C d sin dr 1 2 U T2 U P2 c C d cos C l sin dr 4 dP U T dFx rdFX Finally the thrust and power T and P may be found by integrating dT and dP above from root to tip r 0 to r R and multiplying the results by the total number of blades b The above integration can in general be only numerically done since the chord c the sectional lift and drag coefficients may vary along the span Finally the inflow velocity v depends on T Thus an iterative process will be needed to find the quantity v Approximate expressions for thrust and power may however be found if we are willing to make a number of approximations a The chord c is constant b The inflow velocity v and climb velocity V are small Thus 1 and i 1 We can approximate cos I by unity and approximate sin i by i c The lift coefficient is a linear function of the effective angle of attack that is Thus C l a 5 where a is the lift curve slope For low speeds a may be set equal to 5 7 per radian d Cd is small So Cd sin may be neglected e The in plane velocity UT is much larger than the normal component UP over must of the rotor except near the hub With these assumptions thrust T may be expressed as r R 1 V v 2 T cba 2 r dr 2 r r r 0 r R 1 V v V v P cba 3 Cd r 3dr 2 r r r r r 0 6 To perform the integration we need to know how the pitch angle varies with r Many rotor blades are twisted and it is not reasonable to assume that the pitch angle is constant Two choices are common Linearly Twisted Blade Here we assume that the pitch angle varies as E Fr 7 where E and F are constants Using this definition and performing the integration check we get b 1 3 75 V v 3 b 2 T 2 ca E FR R ca R R 2 2 R 2 4 2 3 3 abc 75 a 75 CT 2 3 2 2 R 3 2 where solidity BladeArea DiskArea bc R V v Inflow Ratio R Notice that the thrust coefficient is linearly proportional to the pitch angle at the 75 Radius This is why the pitch angle is usually defined at the 75 R in industry The expression for power may be integrated in a similar manner if the drag coefficient Cd is assumed to be a constant equal to C d0 The final expression is check C P C T C d 0 8 8 The above expressions are true only for a linearly twisted rotor Ideally Twisted Rotor Here the twist angle is inversely proportional to the radial location r Such rotors are hard to manufacture but turn out to have the lowest power consumption t R r 9 Here t is the pitch angle at the blade tip Using this in the expression for thrust given in equation 6 we get T 1 2 ab c 2 r r 0 t R V v 2 1 r d r abc R 3 t r r 4 10 Or CT a 4 t 11 The expression for the coefficient for power for an ideally twisted rotor turns out to be identical to that for a linearly twisted rotor In summary according to the blade element theory the following expressions are obtained For a linearly twisted rotor in hover or climb CT a 75 2 3 2 For an ideally twisted rotor in hover or climb CT a 4 t For both types of twist the power coefficient is given by C P C T C d 0 8 The first term in the power …