A Compact , Closed - Form Solution for the Optimum , Ideal Wind Turbine

The classical momentum solution for the optimum induced-flow distribution of a wind turbine in the presence of wake swirl can be found in many textbooks. This standard derivation consists of two momentum balances (one for axial momentum and one for angular momentum) which are combined into a formula for power coefficient in terms of induction factors. Numerical procedures then give the proper induction factors for the optimum inflow distribution at any radial station; and this, in turn, gives the best possible power coefficient for an ideal wind turbine. The present development offers a more straightforward derivation of the optimum turbine. The final formulas give the identical conditions for the ideal wind turbine as do the classical solutions— but with several important differences in the derivation and in the form of the results. First, only one momentum balance is required (the other being redundant). Second, the solution is provided in a compact, closed form for both the induction factors and the minimum power—rather than in terms of a numerical process. Third, the solution eliminates the singularities that are present in current published solutions. Fourth, this new approach also makes possible a closed-form solution for the optimum chord distribution in the presence of wake rotation.