Torque-Power Approximations

According to the BET, the incremental power coefficient on the rotor can be written as

dCP = dCQ = І (фСі + Cd) r3 dr. (3.27)

Using the result that X — фг and expanding out Eq. 3.27 gives

dCp — —фСіг3 dr + — C^r3 dr

= —Cikr" dr H—Cdr3 dr 2 2

= dCP. +dCPo, (3.28)

Torque-Power Approximations Подпись: (3.29)

where dCpj is the induced power and dCPo is the profile power. Recall from Eq. 3.15 that the incremental thrust coefficient can be written as dCp — oCir2dr so that dCpt — XdCp. Therefore,

and the total power coefficient is

Подпись: (3.30)pr= /*1 і

CP = I XdCp + I – crCdr3dr. Jr=о Jо 2

By assuming uniform inflow and Cd = Cdo = constant, then after integration we obtain

Подпись: (3.31)Cp = XCp + – ffQ0.


But in hover with uniform inflow then we know that X = VCY/2, and so

C3/2 1

Cp = -2=- + – aCdo. (3.32)

Under these assumptions it is apparent that the first term in this latter equation reduces to the simple momentum theory result given previously in Chapter 2, as it should. The second term is the extra power predicted by the BET that is required to overcome profile drag of the rotor blades, which was a concept first introduced in Eq. 2.42.