# Propeller Selection by Means of Pareto-Optimal Sets Applied to Flight Performance

^{*}

## Abstract

**:**

_{s}is also used. Use of Pareto sets leads to considerable performance increase for the set of contradictory requirements. Therefore, high performance for a low price for the given aircraft can be achieved. The described method can be used for propeller optimization in similar cases.

## 1. Introduction

_{s}is described in [6,7,8]. This procedure is relatively simple and determines optimal diameter and blade angle for the group of propellers having the same blade shape and number of blades. This works only for a single operating point.

## 2. Problem Formulation

## 3. Methods

#### 3.1. Pareto Sets of Flight Performance for Selection of Optimal Propellers

_{2}(X) and maximum 2 for u

_{1}(X)), the extreme is achievable only for the first or second function. The change on Pareto boundary when dropping under the extreme of the first function approximates in the optimal way the solution to the extreme of the second function (for each variable X, between 1 and 2, both functions reach their maximum values). All variables X on Pareto boundary between these two extremes (Pareto-optimal front) are optimal (the most appropriate) because it is not possible to decide which combination of both objective functions is better. None of these solutions is worse or better—the solutions are mutually non-dominant.

_{T}(λ), c

_{P}(λ)) that together with an engine power curve enables to set the available isolated thrust curve of the power unit.

- Assessment of free flight aerodynamic characteristics of the airplane (lift curve c
_{L}(α), lift-drag polar c_{L}(c_{D}) without the propeller influence on the airplane drag in the relevant flight configuration of the flight performance. - Determination of propeller aerodynamic characteristics—thrust and power coefficients c
_{T}(λ), c_{P}(λ). - Calculation of the isolated thrust curve T
_{is}(V) corresponding to the flight condition with the given engine regime, T_{is}(V) correction to the true T(V) and effective thrust T_{ef}(V). - Calculation of all flight performance with the free airplane aerodynamic characteristics corrected for the respective ground effect of each flight condition.
- Set up of the contradictory pairs of the flight conditions and creation of Pareto set graphs.
- Evaluation of optimal Pareto fronts on Pareto sets graphs as the optimization selection.

#### 3.2. Aerodynamic Characteristics of the Aeroplane

^{2}trapezoidal wing of the aspect ratio equals to 7.2. The wing is equipped with a combined high-lift device on the leading and trailing edge—a slotted leading edge (slat) and Fowler flap. The whole wing leading edge is equipped with slat; Fowler flap is installed on 60% of the wing trailing edge. Wing airfoil was developed from GA(W)-1 in order to increase lift coefficient. The airplane is drawn up with the standard side-by-side seat arrangement and a fixed taildragger landing gear with fairing.

- Take-off lift configuration—slat + Fowler flap 15°.
- Cruise configuration—retracted lift devices.

#### 3.3. Powerplant

#### 3.4. Propellers

#### 3.4.1. Propeller Family

#### 3.4.2. Aerodynamic Characteristics of Propellers

_{T}(λ) and c

_{P}(λ). The vortex blade theory of an isolated propeller is used. Free helix vortex surfaces with a constant pitch leaving the boundary vortex of each blade generate a field of the induced velocities that adjust the magnitude and direction of free flow along the propeller blade. The aerodynamic forces acting along the blade can be considered as two-dimensional airfoil characteristics with the incoming free flow corrected to the induced angle of attack. The calculations of the propeller aerodynamic characteristics were performed using the numerical model [18,19]. Input geometric data involve, except diameter D and number of blades N, also the distribution of the chord length, twist and thickness of the airfoils used along the blade. The calculation was done for every blade pitch setting and number of blades. More detailed description of the method can be found in Appendix A.

#### 3.5. Thrust Curves of the Propeller

#### 3.6. Flight Performance for Pareto Sets

^{−3}and air temperature 288.15 K (15 °C). Pareto sets require the pairing of such requirements, which are contradictory. The respective pairs of flight conditions required to achieve their extreme values (minimum, maximum) act on each other so that increasing one flight condition towards the extreme decreases the second flight condition from its desired extreme.

- Maximum horizontal flight speed (continuous engine regime)–Take-off distance (take-off engine regime).
- Maximum horizontal flight speed (cruise engine regime)–Take-off distance (take-off engine regime).
- Maximum horizontal flight speed (continuous engine regime)–Maximum rate of climb (take-off engine regime).
- Maximum horizontal flight speed (cruise engine regime)–Maximum rate of climb (take-off engine regime).
- Take-off distance (take-off engine regime)–Maximum rate of climb (take-off engine regime).
- Maximum horizontal flight speed (continuous engine regime)–maximum horizontal flight speed (cruise regime).

## 4. Results

_{s}is used for comparison (see c

_{s}opt points in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11). Performance with original constant-speed propeller is also shown for comparison. Results are further discussed in Section 5. Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6 contain coordinates of the points on the Pareto fronts for corresponding Pareto graphs. Pareto fronts are formed only by two-blade propellers; propeller diameter and blade angle are mentioned for every point. Performance with selected optimal propeller is also plotted in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11).

_{s}is on the Pareto-optimal set. Figure 7 shows similar situation for the cruise speed. In this case, Pareto-optimal set contains also uniquely two-blade propellers, but the solution obtained by the means of the speed power coefficient is no more on the Pareto-optimal set.

## 5. Selection of Optimal Propeller

_{0.75}= 20°. This point is part of the Pareto fronts in Figure 6, Figure 8 and Figure 11. It is also close to the Pareto front in other cases. Aircraft performance with the optimal propeller is described in Table 7.

## 6. Conclusions

- Maximum horizontal flight speed–Take-off distance.
- Maximum horizontal flight speed–Maximum rate of climb.
- Take-off distance–Maximum rate of climb.
- Maximum horizontal flight speed–maximum horizontal flight speed (cruise regime).

- Only two-blade propellers belong to all Pareto-optimal fronts.
- One cut-off point of Pareto-optimal fronts for maximum horizontal speed corresponds to D = 1.95 m, pitch blade angle ϕ
_{0.75}= 22.5° for the continuous engine regime and to D = 2 m, ϕ_{0.75}= 20° for cruise regime. - The opposite cut-off points of Pareto-optimal front for both maximum rate of climb and minimal take-off distance corresponds one propeller: D = 2.1 m, pitch blade angle ϕ
_{0.75}= 15°. - The constant speed propeller achieves better performance than the best performance of the fixed pitch and ground adjustable propellers for all four optimization criteria.

_{0.75}= 22.5° to 12.5°. Optimal propeller is then selected from this group.

_{0.75}= 20°. The speed power coefficient methods are limited by single-regime design defined by given flight speed and engine regime, and the design is not clearly connected to flight performance. Two-blade propeller of this family considered as a constant speed propeller with diameter obtained by the speed power coefficient presents an increase in flight performance for the fixed pitch propeller.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

D | Propeller diameter, Drag |

H | Altitude |

L | Lift |

N | Number of propeller blades |

P | Engine power |

P_{max} | Maximum engine power |

Q | Tangential force |

S | Wing area |

S_{P} | Propeller disc area |

S_{wet} | Aircraft area wetted by the propeller flow |

T | True propeller thrust |

T_{eff} | Effective propeller thrust, T_{eff} = (T − ΔD) |

T_{is} | Isolated propeller thrust |

T_{req} | Required thrust |

U_{0} | Rotational flow speed on the propeller (freestream) |

V | Flight speed |

V_{0} | Axial flow speed on the propeller (freestream) |

V_{Hmax} | Maximum horizontal flight speed–continuous engine regime |

V_{HC} | Maximum horizontal flight speed–cruise engine regime |

V_{min} | Stalling speed |

V_{1} | Safe lift-off speed |

V_{1P} | Flow velocity through the propeller disc (actuator disc) |

V_{2} | Safe take-off climb speed (Safe speed of transition) |

W | Total flow speed on the propeller |

ΔD | Airplane drag induced by the propeller, ΔD = ∆D_{fr} + ∆D_{pr} |

∆D_{fr} | Friction component of airplane-propeller drag |

∆D_{pr} | Pressure component of airplane-propeller drag |

R | Propeller radius |

c | Blade chord |

c_{P} | Power coefficient of the isolated propeller, c_{P} = P/(ρ n_{s}^{3} D^{5}) |

c_{T} | Thrust coefficient of the isolated propeller, c_{T} = T_{is}/(ρ n_{s}^{2} D^{4}) |

c_{s} | Speed power coefficient of the isolated propeller, c_{s} = [ρ V^{5}/(P n_{s}^{2})]^{1/5} |

c_{D} | Drag coefficient |

c_{Dmin} | Minimum drag coefficient |

c_{L} | Lift coefficient |

c_{Lmax} | Maximum lift coefficient |

c_{L}^{α} | Lift-curve slope |

f | Rolling friction coefficient |

h_{arc} | Height of the transition arc |

h_{W} | Distance of the wing under the ground |

i | Gear ratio of the engine speed reducer |

m | Aircraft weight |

n, n_{max} | Engine speed, Maximum engine speed |

n_{s} | Propeller speed per second |

n_{y} | Lift load factor |

r | Radial coordinate |

u | Induced tangential speed on the propeller |

v | Induced axial speed on the propeller |

v_{y} | Rate of climb |

η | Propeller efficiency, η = c_{T} λ/c_{P} |

λ | Propeller advance ratio, λ = V/(n_{s} D) |

λ_{W} | Wing aspect ratio |

ρ | Air density |

φ_{0.75} | Propeller pitch blade angle at 75% of the radial distance |

Ω | Propeller angular velocity |

Abbreviations | |

ISA | International Standard Atmosphere |

STOL | Short Take-off and Landing |

## Appendix A. Computation of Propeller Characteristics

_{T}

_{P}is defined in a following way

_{1}and V

_{1}are defined by

_{L}is coupled with circulation by

**Figure A3.**Comparison of computed values of thrust coefficient c

_{T}(solid lines) with experimental results from [8] (discrete points) for the three-blade propeller 5868-R6.

**Figure A4.**Comparison of computed values of power coefficient c

_{P}(solid lines) with experimental results from [8] (discrete points) for the three-blade propeller 5868-R6.

## Appendix B. Computation of Flight Performance

#### Appendix B.1. Thrust Curve of the Propeller Power Unit

#### Appendix B.1.1. Thrust of Isolated Fixed-Pitch and Ground-Adjustable Propellers

_{P}(λ) for a given flight speed V on the other side. The equilibrium speed is solved for a number of speeds from the full range of the flight speed. The pair of the flight speed and the equilibrium rotational speed determinates the advance ratio λ and by help of the thrust coefficient c

_{T}(λ) the isolated thrust can be evaluated:

_{max}, it is assumed that the pilot will maintain this limit rotational speed by the throttle. The equilibrium rotational speed in such overload regime is thus the maximum speed n

_{max}.

#### Appendix B.1.2. Thrust of Isolated Constant-Speed Propellers

_{T}(λ) determined for the constant-speed propeller working in the required power regime.

#### Appendix B.1.3. Thrust Curve with Influence of Height

_{N}according to [24]

#### Appendix B.1.4. True Thrust of Installed Propellers

_{i}is estimated as the ratio of the actuator disc thrust of the propeller with a mean flow speed through the disc and thrust of the isolated actuator disc. The mean flow speed corresponds to the deceleration downstream due to a body behind the propeller. The mean value of the decelerated flow through the disc is determined from the equality of the flow momentum by the actuator disc with a constant and variable velocity.

_{1P}/V

_{1P}) in the plane of the propeller disc in the area of the central part of the propeller. The central part is defined by the cross-section of the engine nacelle S

_{n}. The velocity drop ratio depends on the cross-sectional area S

_{n}and propeller disc area S

_{P}and a compensatory analytical form of the graph has the form:

_{1P}through the propeller disc can be expressed depending on the thrust at flight speed V according to the theory of the ideal actuator disc:

_{n}is replied by the cross-section area of the engine part of the fuselage.

#### Appendix B.1.5. Effective Thrust

_{fr}is depended on the thrust and a wetted area of an airplane influenced by the propeller flow. Its mean value is presented in References [20,24]:

_{pr}is taken from Reference [19] for the engine nacelle case:

#### Appendix B.2. Curves of Available and Required Thrust and Power

#### Appendix B.2.1. Available Thrust Curve

_{eff}; see (A20). The available thrust curve is the dependence of the effective thrust on the flight speed V.

#### Appendix B.2.2. Required Thrust Curve

- Step 1: To determine the needed lift coefficient from the balance of lift and weight for the chosen flight speed V:

- Step 2: To state the corresponding drag coefficient from the aerodynamic polar of the respective flight configuration (without additional aerodynamic drag caused by propeller flow) for the corresponding lift coefficient. Drag coefficient c
_{D}is determined from aerodynamic polar (Figure 2). - Step 3: The required thrust is equal to the aerodynamic drag:

_{min}(minimum horizontal flight speed at the given configuration–stalling speed) to the maximum flight speed V

_{HC}(or V

_{Hmax}) corresponding to the intersection of the required thrust curve T

_{req}(V) with the available thrust curve T

_{eff}(V) for a given altitude. An example of the curve of available and required thrust of a small sport aircraft at a cruise regime is depicted in Figure A5.

**Figure A5.**Curves of available and required thrust for a small sport aircraft m = 473 kg at a cruise configuration with three-blade propeller: D = 1.75 m, φ

_{0.75}= 22°, maximum cruise-regime power 47.9 kW.

#### Appendix B.2.3. Required Thrust Curve with Influence of Height

_{W}of the wing above the ground, the aspect ratio of the wing λ

_{W}and its area S:

#### Appendix B.2.4. Available Power Curve

#### Appendix B.2.5. Required Power Curve

#### Appendix B.3. Maximum Horizontal Speed, Maximum Rate of Climb

_{vymax}:

#### Appendix B.4. Take-Off Distance

#### Appendix B.4.1. Take-Off Ground Run

_{1}.

_{1}is within the range (110–115)% of the stall speed with the ground effect. Stalling speed with the ground run effect corresponds to the relationship (A26):

_{ground run}represents an accelerating force during the ground run:

#### Appendix B.4.2. Take-Off Ground Flight

_{1}up to the safe take-off climb speed V

_{2}.

_{2}is required at least 120% of the stall speed in the ground flight regime. Stalling speed with the ground flight effect corresponds to the relationship (A26):

_{ground flight}represents an accelerating force during the ground flight:

#### Appendix B.4.3. Take-Off Transition

_{2}.

_{y}caused by the curvilinear flight:

_{2}is

_{Lmax ground flight}and leads to the shortest length of the take–off transition phase, but it is on the edge of the sharp flow separation. A smaller value n

_{y}should be considered, e.g., 80% n

_{ymax}.

_{W}of the wing above the ground is applied. The mean value can be estimated, e.g., as a mean value between the height of arc h

_{arc}calculated with the ground effect of the take-off ground flight and without the ground effect (free flight).

#### Appendix B.4.4. Climb Out

_{obs}(usually 15.25 m = 50 ft)) is reached:

_{arc}and take-off output angle θ are determined for h

_{W_transition}.

#### Appendix B.4.5. Total Take-Off Distance

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**Figure 1.**Scheme of Pareto set (redrawn according to [16]).

**Figure 2.**Polar graph of the aircraft for cruise (i.e., flaps retracted) and take-off configuration (i.e., flaps 15°).

**Figure 6.**Pareto frontier for maximum horizontal speed (continuous engine regime) and take-off distance.

**Figure 7.**Pareto frontier for maximum horizontal flight speed (cruise engine regime)–take-off distance.

**Figure 8.**Pareto frontier for maximum horizontal flight speed (continuous engine regime) and maximum rate of climb (take-off engine regime).

**Figure 9.**Pareto front for maximum horizontal flight speed, (cruise engine regime) and maximum rate of climb (take-off engine regime).

**Figure 10.**Pareto frontier for take-off distance (take-off engine regime) and maximum rate of climb (take-off engine regime).

**Figure 11.**Pareto frontier for maximum horizontal flight speed (continuous engine regime) and maximum horizontal flight speed (cruise regime).

**Table 1.**Points on the Pareto front for maximum horizontal speed (continuous engine regime) and take-off distance (see Figure 6).

φ_{0.75} [°] | D [m] | Maximum Level Speed [km/h] | Take-Off Length [m] |
---|---|---|---|

22.5 | 1.95 | 214.54 | 125.25 |

20 | 2.05 | 212.22 | 122.61 |

20 | 2.0 | 204.68 | 119.63 |

17.5 | 2.15 | 204.37 | 119.03 |

17.5 | 2.1 | 197.7 | 115.99 |

15 | 2.25 | 192.88 | 115.05 |

17.5 | 2.05 | 190.97 | 114.55 |

15 | 2.2 | 187.06 | 112.34 |

15 | 2.15 | 181.18 | 111.1 |

15 | 2.1 | 175.25 | 110.63 |

**Table 2.**Points on the Pareto front for maximum horizontal speed (continuous engine regime) and take-off length (see Figure 7).

φ_{0.75} [°] | D [m] | Maximum Level Speed [km/h] | Take-Off Length [m] |
---|---|---|---|

20 | 1.95 | 194.4 | 118.59 |

17.5 | 2.1 | 191.53 | 115.99 |

17.5 | 2.05 | 190.97 | 114.55 |

15 | 2.2 | 187.06 | 112.34 |

15 | 2.15 | 181.18 | 111.11 |

15 | 2.1 | 175.25 | 110.63 |

**Table 3.**Points on the Pareto front for maximum horizontal speed (continuous engine regime) and maximum rate of climb (take-off engine regime) (see Figure 8).

φ_{0.75} [°] | D [m] | Maximum Level Speed [km/h] | Maximum Rate of Climb [m/s] |
---|---|---|---|

22.5 | 1.95 | 214.54 | 6.823 |

20.0 | 2.05 | 212.22 | 6.973 |

22.5 | 1.9 | 207.97 | 7.122 |

20 | 2.0 | 204.68 | 7.315 |

17.5 | 2.1 | 197.70 | 7.531 |

17.5 | 2.05 | 190.97 | 7.723 |

15 | 2.2 | 187.06 | 7.735 |

17.5 | 2.0 | 184.18 | 7.876 |

15 | 2.15 | 181.18 | 7.926 |

15 | 2.1 | 175.25 | 8.049 |

**Table 4.**Points on the Pareto front for maximum horizontal speed (cruise engine regime) and maximum rate of climb (see Figure 9).

φ_{0.75} [°] | D [m] | Maximum Level Speed [km/h] | Maximum Rate of Climb [m] |
---|---|---|---|

20 | 1.95 | 192.04 | 7.505 |

17.5 | 2.1 | 191.53 | 7.531 |

17.5 | 2.05 | 190.97 | 7.723 |

15 | 2.2 | 187.06 | 7.735 |

17.5 | 2.0 | 184.18 | 7.876 |

15 | 2.15 | 181.18 | 7.926 |

15 | 2.1 | 175.25 | 8.049 |

**Table 5.**Point on the Pareto front for take-off distance (take-off engine regime) and maximum rate of climb (take-off engine regime) (see Figure 10).

φ_{0.75} [°] | D [m] | Take-Off Length [m] | Maximum Rate of Climb [m/s] |
---|---|---|---|

15 | 2.1 | 110.63 | 8.049 |

**Table 6.**Points on the Pareto front for maximum horizontal flight speed (continuous engine regime) and maximum horizontal flight speed (cruise regime) (see Figure 11).

φ_{0.75} [°] | D [m] | Maximum Level Speed (Continuous Engine Regime) [km/h] | Maximum Level Speed (Cruise Regime) [km/h] |
---|---|---|---|

22.5 | 1.95 | 214.54 | 185.95 |

20 | 2.05 | 212.22 | 188.17 |

22.5 | 1.9 | 207.97 | 189.18 |

20 | 2.0 | 204.68 | 190.69 |

22.5 | 1.85 | 199.47 | 191.01 |

17.5 | 2.1 | 197.7 | 191.53 |

20 | 1.95 | 197.07 | 192.04 |

**Table 7.**Aircraft performance with chosen optimal propeller (diameter D = 2.05 m and pitch blade angle ϕ

_{0.75}= 20°).

Maximum Level Speed (Continuous Engine Regime) [km/h] | Maximum Level Speed (Cruise Regime) [km/h] | Take-Off Length [m] | Maximum Rate of Climb [m/s] |
---|---|---|---|

212.22 | 188.17 | 122.61 | 6.973 |

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**MDPI and ACS Style**

Slavik, S.; Klesa, J.; Brabec, J. Propeller Selection by Means of Pareto-Optimal Sets Applied to Flight Performance. *Aerospace* **2020**, *7*, 21.
https://doi.org/10.3390/aerospace7030021

**AMA Style**

Slavik S, Klesa J, Brabec J. Propeller Selection by Means of Pareto-Optimal Sets Applied to Flight Performance. *Aerospace*. 2020; 7(3):21.
https://doi.org/10.3390/aerospace7030021

**Chicago/Turabian Style**

Slavik, Svatomir, Jan Klesa, and Jiri Brabec. 2020. "Propeller Selection by Means of Pareto-Optimal Sets Applied to Flight Performance" *Aerospace* 7, no. 3: 21.
https://doi.org/10.3390/aerospace7030021