Power Approximations

From Equation 3.20 the differential power coefficient dCP (= dCQ) may be written as:

dCp — dCQ

= s(fCL + CD)x3 dx

— sCLf x3 dx + sCDx3 dx (3.46)

— sCL1x2 dx + sCDx3 dx

— dCpi + dCp0

Подпись: Thus: Подпись: (3.47) (3.48)

where dCPi is the differential power coefficient associated with induced flow and dCP0 is that associated with blade section profile drag. The first term, using Equation 3.17, is simply:

Подпись: CP Подпись: 1 sCDx3dx 0 Подпись: (3.49)

whence:

Assuming uniform inflow and a constant profile drag coefficient CD0, we have the approximation:

Cp = l • Ct + -^ (3-50)

In the hover, where:

l = 1 УСТ (3.51)

this becomes:

Cp = 1 (Ct)3/2 + SC4D0 (3.52)

The first term of Equations 3.50 or 3.52 agrees with the result from simple momentum theory (Equation 2.12). The present l, defined by Equation 3.14, includes the inflow from climbing speed VC (if any), so the power coefficient term includes the climb power:

Pclimb = Vc • T (3.53)

The total induced power in hover or climbing flight is generally two or three times as large as the profile power. The chief deficiency of the formula in Equation 3.50 in practice arises from the assumption of uniform inflow. Bramwell (p. 94ff.) shows that for a linear variation of inflow the induced power is increased by approximately 13%. This and other smaller correction factors such as tip loss (Section 3.7) are commonly allowed for by applying an empirical factor kj to the first term of Equation 3.50, so that as a practical formula:

Cp = ki • l • Ct +(3-54)

is used, in which a suggested value of kj is 1.15. The combination of Equations 3.54 and 3.27 provides adequate accuracy for many performance problems.

Подпись: CP Подпись: (3.55)

For the hover, we have:

Подпись: M = CPIDEAL CP ACTUAL Подпись: (3.56)

The figure of merit M may be written:

which demonstrates that for a given thrust coefficient a high figure of merit requires a low value of the product sCD0. Using a low solidity seems an obvious way to this end but it must be tempered because the lower the solidity, the lower the blade area, which means the higher the blade angles of incidence required to produce the thrust and the profile drag may then be increased significantly from either Mach number effects or the approach of stall. A low solidity, subject to retaining a good margin of incidence below the stall, would appear to be the formula for producing an efficient design.

For accurate performance work the basic relationships in Equations 3.18 and 3.21 are integrated numerically along the span. Appropriate aerofoil section data can then be used, including both compressibility effects and stalling characteristics. Further reference to numerical methods is made in Chapter 6.

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