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Solving propeller optimization problems by using helical vortex and exact penalty methods

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Abstract

An alternative procedure of finding a propeller's optimal circulation distribution and geometry is presented. The method uses a helical vortex model for aerodynamic calculations and an exact penalty method to formulate the constrained optimization problem as an unconstrained nonlinear programming problem. An example case for optimal circulation distribution from the literature was used for comparison. The method showed a good fit with the exact solutions, namely the inviscid and the infinite number of blades cases. For the general case considering the viscosity and a finite number of blades, the model performed better than the benchmarks blade element momentum methods. The geometry was also optimized, and the geometric twist and chord distribution were calculated for the general case, for a given twist or chord distribution and for a given activity factor, increasing in 4% to 6% the efficiency when compared to the geometries available on the references for all cases.

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Abbreviations

\({V}_{\infty }\) :

Freestream velocity

\(\Omega\) :

Angular velocity

\(W\) :

Resultant velocity

\({u}_{\theta }, {u}_{z}\) :

Tangential and axial induced velocity

\(u\) :

Wake velocity

\(v\) :

Slipstream velocity

\(\lambda\) :

Tip–speed ratio

\(d\) :

Propeller diameter

\(R\) :

Propeller radius

\(r\) :

Radial position

\(\mu\) :

Non-dimensional radial position

\(\beta\) :

Blade twist angle

\(\alpha , {\alpha }_{e}\) :

Geometric and effective angle of attack

\(\phi\) :

Wake angle

\(B\) :

Number of blades

\(c\) :

Chord length

\({a}_{0}\) :

\(d{C}_{L}/d\alpha\)

\(\sigma\) :

Blade solidity

\(\mathrm{AF}\) :

Activity factor

\(\rho\) :

Air density@@@

\(\xi\) :

\({C}_{D}/{C}_{L}\)

\(a\) :

Ritz coefficients

\(f\) :

Ritz functions

\(\eta\) :

Propeller efficiency

THP:

Thrust horsepower

BHP:

Brake horsepower

\({C}_{T}\) :

Thrust coefficient

\({C}_{P}\) :

Power coefficient

\({C}_{L}\) :

Lift coefficient

\({C}_{D}\) :

Drag coefficient

\(T\) :

Thrust

\({F}_{\theta }\) :

Tangential force

\(P\) :

Power

\(L\) :

Lift

\(D\) :

Drag

\(Q\) :

Torque

\(\Gamma\) :

Circulation

\(A\) :

Influence matrix

A′:

Kawada coefficients

\(M\) :

Morrison parameter

\(\delta\) :

Discriminant

\(\varepsilon\) :

Auxiliary variable

\({S}_{1}, {S}_{2}\) :

Auxiliary variables

\(K,K\mathrm{^{\prime}}\) :

Auxiliary variables

\(k,k\mathrm{^{\prime}}\) :

Auxiliary variables

\(I(\cdot ), K(\cdot )\) :

Modified Bessel functions

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Authors and Affiliations

  1. Graduate Program in Mechanical Engineering, Federal University of Minas Gerais, Antônio Carlos Avenue, 6627, Pampulha, Belo Horizonte, MG, 31270-901, Brazil

    Danilo Azevedo, Joel L. C. Reis & Ricardo L. U. de F. Pinto

  2. School of Mechanical Engineering, Federal University of Piauí, UFPI, Ininga, Teresina, PI, 64049-550, Brazil

    Danilo Azevedo

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Azevedo, D., Reis, J.L.C. & Pinto, R.L.U.d.F. Solving propeller optimization problems by using helical vortex and exact penalty methods. J Braz. Soc. Mech. Sci. Eng. 45, 374 (2023). https://doi.org/10.1007/s40430-023-04258-y

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