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

## Abstract

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## Abbreviations

$$a$$ :

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

$$c$$ :

$$C_{\text{d}}$$ :

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)

$$R$$ :

$$Re_{75}$$ :

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)

$$\delta {\kern 1pt} k_{\text{Q}}$$ :

$$\delta {\kern 1pt} k_{\text{T}}$$ :

$$\eta$$ :

Efficiency of the propeller

$$\theta_{\text{c}}$$ :

$$\lambda$$ :

Taper ratio of the propeller blade

$$\rho$$ :

Specific mass of the fluid (air) (kg/m3)

$$\sigma$$ :

Solidity of the rotor

$$\phi$$ :

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

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## Acknowledgements

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

## Author information

Authors

### Corresponding author

Correspondence to Kamal A. R. Ismail.