Optimum performance of an actuator disk by a compact momentum theory including swirl

A new compact form of momentum theory is introduced for actuator disks including swirl. The new form unifies both the axial and angular momentum balances into a single momentum equation, applicable over the entire range of thrust and power coefficients. While completely consistent with earlier momentum theories, such as that of Glauert in Ref. 1, the compact form allows analytic expressions for the parameters of an optimum actuator disk and reveals additional insight into the limiting efficiency of rotors, propellers, and wind turbines. Closed-form results presented here include the optimum values of: induced flow, inflow angle, thrust, induced power, and efficiency. Closed-form expressions are also given for optimum twist, chord distribution, and solidity in the presence of profile drag (along with the resulting over-all efficiencies). For the limiting case of the optimum rotor in hover, the compact form leads to closed-form expressions for both contraction ratio and pressure distribution in the far wake. This report also gives a formal proof that the Betz inflow distribution results in the maximum figure of merit, and it further demonstrates that some approximations used in earlier actuator-disk momentum theories have been inconsistent.