# Effects of the airfoil section, the chord and pitch distributions on the aerodynamic performance of the propeller

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## Abstract

The main objectives of this study are to investigate parametrically the possible use of alternative airfoils (Joukowski and Göttingen) for propellers and to assess the effects of varying the chord and pitch angle distributions as well as the use of multiple airfoils along the blade on the performance parameters of the propeller. In this study, a validated home-built FORTRAN code based on the BEM method with incorporated tip and compressibility losses is used. The detailed investigation of the blade geometry is done to help in selecting a configuration that is efficient and easy to manufacture. The linear pitch distribution is found to reduce the coefficients of thrust and power as well as higher blade loading at the intermediate region and lower loading at the tip region in comparison with the Göttingen 796-based propeller. The results show that the power coefficient and efficiency of the generalized Joukowski-based propeller are greater than the respective coefficients of Göttingen 796-based propeller for advanced ratio *J* = 0.85 and higher. The predicted results indicate that the use of the elliptical chord distribution provokes reduction in the blade loading at the tip region and increases at the intermediate region of the blade. It is found also that it reduces the coefficient of thrust, torque and power in comparison with the blade having the reference chord distribution.

## Keywords

Small propeller Momentum theory Blade element theory Panel method Blade aerodynamics Airfoil section## List of symbols

- \(a\)
Inflow factor

*a*_{0}Lift curve slope at zero Mach number (i.e., in incompressible flow) (radians

^{−1})*a*_{M}Lift curve slope at zero Mach number (radians

^{−1})- \(b\)
Swirl factor

- \(B\)
Number of blades of the propeller

- \(c\)
Local blade chord (m)

- \(C_{\text{d}}\)
Two-dimensional drag coefficient of the local blade chord

- \(C_{\text{l}}\)
Two-dimensional lift coefficient of the local blade chord

- \(D\)
Diameter of the propeller (m)

- \(f_{\text{tip}}\)
Tip loss correction used to calculate Prandtl loss factor

*F*- \(f_{\text{hub}}\)
Hub loss correction used to calculate Prandtl loss factor

*F*- \(F\)
Prandtl loss factor for combined tip and hub losses which arise due to the finite number of the propeller blades

- \(J\)
Advance ratio of the propeller \(J = {V \mathord{\left/ {\vphantom {V {(n{\kern 1pt} D)}}} \right. \kern-0pt} {(n{\kern 1pt} D)}}\)

- \(k_{\text{P}}\)
Power coefficient of the propeller \(k_{\text{P}} = {P \mathord{\left/ {\vphantom {P {(\rho {\kern 1pt} n^{3} D^{5} )}}} \right. \kern-0pt} {(\rho {\kern 1pt} n^{3} D^{5} )}}\)

- \(k_{\text{Q}}\)
Torque coefficient of the propeller \(k_{\text{Q}} = {Q \mathord{\left/ {\vphantom {Q {(\rho {\kern 1pt} n^{2} D^{5} )}}} \right. \kern-0pt} {(\rho {\kern 1pt} n^{2} D^{5} )}}\)

- \(k_{\text{T}}\)
Thrust coefficient of the propeller \(k_{\text{T}} = {T \mathord{\left/ {\vphantom {T {(\rho {\kern 1pt} n^{2} D^{4} )}}} \right. \kern-0pt} {(\rho {\kern 1pt} n^{2} D^{4} )}}\)

*M*Local Mach number of the relative flow

- \(n\)
Rotational speed of the propeller (rps)

- \(N\)
Rotational speed of the propeller (rpm)

- \(p\)
Geometric pitch of the blade section (m)

- \(P\)
Power supplied at the propeller axis (Nm/s)

- \(Q\)
Torque applied on the propeller (Nm)

- \(r\)
Radius of the transversal section of the blade of the propeller (m)

- \(R\)
Radius of the blade tip of the propeller (m)

- \(Re_{75}\)
Reynolds number of the propeller based on the local chord and resultant velocity at a radial distance of 0.75 of the tip radius

- \(T\)
Thrust force of the propeller (N)

- \(V\)
Advance velocity of the propeller (m/s)

- \(V_{0}\)
Axial component of the flow velocity relative to the blade (m/s)

- \(V_{\text{R}}\)
Resultant flow velocity relative to the blade (m/s)

- \(V_{\text{S}}\)
Axial component of the flow velocity relative to the propeller at exit of the slipstream (m/s)

- \(V_{\text{w}}\)
Rotational component of the flow velocity relative to blade (m/s)

- \(\alpha\)
Angle of attack is the angle between the resultant velocity vector \(V_{\text{R}}\) and the zero lift line of the blade airfoil (radians)

- \(\alpha_{\text{c}}\)
Angle between the resultant velocity vector \(V_{\text{R}}\) and the chord line of the blade airfoil (radians)

- \(\delta {\kern 1pt} k_{\text{Q}}\)
Torque loading coefficient of the blade element

- \(\delta {\kern 1pt} k_{\text{T}}\)
Thrust loading coefficient of the blade element

- \(\eta\)
Efficiency of the propeller

- \(\theta_{\text{c}}\)
Pitch angle of the blade section (radians)

- \(\lambda\)
Taper ratio of the propeller blade

- \(\rho\)
Specific mass of the fluid (air) (kg/m

^{3})- \(\sigma\)
Solidity of the rotor

- \(\phi\)
Angle of the resultant velocity \(V_{\text{R}}\) with the plane of rotation of the propeller (radians)

## Notes

### Acknowledgements

The first author wishes to thank the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) for the PQ Research Grant.

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