Multi-Point Design of Optimal Propellers for Remotely Piloted Aircraft Systems

Abstract

This paper proposes a solution for the design of high-performance propellers optimized for various flight conditions. Considering both propulsion and electric motor efficiencies, a new design optimization methodology is proposed. The optimization of the electric propulsive system is directly achieved by simultaneously analyzing the aerodynamic performance of the propeller and the motor. This study is focused on small, low-speed Remotely Piloted Aircraft Systems, addressing the design of fixed pitch propellers that operate efficiently over the entire speed range. The aerodynamic methodology uses combined blade element and momentum theory, which is adequate for a preliminary design phase with low computational time. For the aerodynamic coefficients of the airfoils used in these applications, at low Reynolds numbers, a new database was developed that incorporates airfoil experimental data and analytical methods to cover a wide range of angles of attack, beyond stall. For the modelling of the motor behavior, an idealization of the circuit was carried out, which considers its basic electric parameters. The results show significant improvements with respect to the information available for a current commercial propeller.

1. Introduction

The current strong interest in RPASs (Remotely Piloted Aircraft Systems) has provided a new impetus to the optimization of the different parts of these aircraft, and particularly their propulsive system. Focusing on propeller propulsion, many authors are working on aerodynamic design improvements or new experimental test methods, such as in the study by Susi et al. [1]. Extensive compilations of the classical BEMT (Blade Element Momentum Theory) can be found in the work of Okulov et al. [2]. Several recent works have focused on improving the capabilities of the available models. The study by Epps [3] models the wake using combinations of the lifting line theory, while other authors focus on the practical selection of the most suitable propeller for a given performance, as demonstrated by Slavik et al. [4]. In the case of Brandt et al. [5] and Deters et al. [6], the authors have focused on completing experimental studies in wind tunnels, especially with low Reynolds numbers.

The large number of new and demanding missions requires propeller design to be the result of a complex optimization procedure. In the case of smaller RPASs, the usual solution uses FPPs (Fixed Pitch Propellers), whose process of adaptation to the different flight phases is much more difficult to achieve efficiently compared with a variable pitch propeller.

There are many works in the literature that only address aerodynamic optimization and consider a single flight condition. Adkins and Liebeck et al. [7] conducted the simplest studies using BEMT and, in 1994, extended the initial studies, limited to the case of propellers with light loads, to the more general case of propellers with an arbitrary load. There are also intermediate models using the VLM (Vortex Lattice Method) with the lifting line theory, such as the study by Gandia-Aguera et al. [8]. Using the most comprehensive lifting surface theory, Cho et al. [9] demonstrated improved results, while Burger et al. [10] and Chattopadhyay et al. [11] successfully optimized the propeller for multiple design points, the former using a genetic algorithm and the latter using a nonlinear programming technique.

For the more advanced aerodynamic design phases, the greater capacity of Computational Fluid Dynamics (CFD) is used to detail the design. The study by Sodja et al. [12] demonstrated good agreement between the BEMT, CFD, and experimental results. Kwon et al. [13] and Toman et al. [14] developed a multilevel design combining the high computational efficiency of the BEMT and the accuracy of CFD.

None of these methodologies guarantees adequate performance out of the chosen design point, which is precisely the main objective of the current work. In addition, the more complex and more precise models are not appropriate for the initial stages of the design process because of the high computational time.

To date, in the case of multidisciplinary aerodynamic and aeroacoustics design, some authors, such as in the study by Gur et al. [15], have solved the acoustic optimization problem, but only for a single flight condition. Succi [16] and Patrick et al. [17] carried out a first aerodynamic optimization and then an acoustic one, so that the first solution may be penalized. In the case of Marinus [18], the optimization only affects a single flight condition, e.g., cruise flight, and is then limited to verifying that there are no special problems in the rest of the conditions.

In the present work, a new model was implemented that is able to perform a multi-point optimization, considering several flight conditions. In addition, the recent interest in achieving RPASs with minimal acoustic footprint, such as air taxis operating in an urban environment and ensuring passenger comfort, makes it necessary to include acoustic criteria in the design process. For this reason, in the proposed model, it was necessary to introduce, at least, a simple restriction to the generated acoustic impact, so it can be considered as a first basic approach to multidisciplinary optimization. This restriction consists of a limit to both the helical tip Mach number and the maximum value of the distribution of angles of attack at which the entire blade operates in the different flight conditions. This qualitative method is always necessary, although not sufficient, and valid for the initial stages of acoustic design.

The aerodynamic model uses the classic physical models of the BEMT extended for arbitrary load levels [7], which accurately describe the motion around the blades. At least in a preliminary design phase, the results are sufficiently satisfactory with acceptable computational times, without the need to move to more expensive models, as demonstrated in the studies by Kotwicz Herniczek et al. [19] in 2017 and Yu et al. [20] in 2020. Moreover, in the case of the study of smaller and slower RPASs, the aerodynamic coefficients of the airfoils are dominated by Reynolds number effects; therefore, a comprehensive experimental database was added to allow these effects to be considered in the different flight conditions and along the blade.

In the field that studies the effect of the electrical system on RPAS performance, Lee and Yee’s study [21] provides detailed models of all elements of the system. Focusing on lighter RPAS systems, the typical system includes a BLDC motor (Brushless DC), the ESC (Electronic Speed Controller), and the Li-PO batteries. For a first study of motorpropulsive efficiency, it was decided to analyze only the effect that the motor has on the system. The model allows for estimating the minimum energy required for a given flight mission, which is the starting point for sizing the batteries, as shown by Rodriguez and Sanchez [22]. The model consists of the study of a basic equivalent electrical circuit that includes both the losses in the copper and in the core with sufficient accuracy, as demonstrated by Stepaniak et al. [23] and Drela [24].

The aim of this paper is to show a new design process for the definition of the geometry of an FPP so that once the thrust required at various flight conditions (speeds) is defined, the total motor-propeller power consumption is minimized. Therefore, a multidisciplinary and multi-point-type optimization algorithm is created, unlike classic models, such as those developed by Adkins and Liebeck and others, which are limited to a single speed.

The structure of this article is as follows: Section 2 presents the aerodynamic and electrical models used in contrast to existing ones. Section 3 shows the general methodology used to achieve the objectives and the details of the optimization process. Section 4 presents the results obtained from three optimization methods with different criteria. Section 5 discusses the capacity and limitations of the results obtained. Finally, conclusions are drawn in Section 5.

2. Related Work

This section presents the physical models used with special emphasis on the conventional design of propellers. First, the Adkins and Liebeck model is introduced. Like many other traditional aerodynamic methods for propeller design, this model only supports one flight condition per design. An experimental database of the aerodynamic coefficients of the airfoil is introduced later. Finally, the model for motor study is presented.

The aim of this section is to set the basis for Section 3, which describes the multidisciplinary multi-point optimization problem that is the subject of this study.

2.1. Adkins and Liebeck Design Method

The model presented by Adkins and Liebeck [7] provides a procedure for the design of optimal propellers (with minimum energy losses) for a given flight condition, based on the BEMT. It is, in fact, an extension of Larrabee’s previous work for moderate and heavy-loaded propellers that eliminates small-angle approximations and includes viscous terms in the induced velocities while maintaining Betz’s optimal condition on the far wake velocities (Larrabee [25], Betz and Prandtl [26]). For each blade element, the momentum losses due to radial flow are modeled through the Prandtl momentum loss factor, F, and, when necessary, the Prandtl–Glauert compressibility correction is applied.

This model solves the nonlinear system of equations that governs the aerodynamics of the problem by means of an algorithm that iterates over the ratio between the wake displacement speed, w′, and the speed of the flow upstream of the rotor, V_∞, being ζ = w′/V_∞. The design problem begins with the definition of a series of geometry parameters and propeller operating conditions, such as the number of blades, B; the selected airfoils (and all their coefficients as a function of the angle of attack); the design thrust, T, or input power, P; the lift coefficient at the design condition, C_l; the altitude, h; the flight speed, V_∞; and the propeller rotational speed at the design condition, Ω.

Starting with a first guess of ζ (which can be ζ = 0) and the propeller speed ratio, λ = V_∞/(ΩR), with R being the propeller radius, it is possible to calculate the flow angle at the tip of the blades, which is used to obtain the radial distributions of the flow angle, ϕ.

The initial estimate of the design lift coefficient, C_l, is used together with the aerodynamic coefficient curves of the airfoils to obtain the local values of the angle of attack, α, and the drag coefficient, C_d. To minimize the local value of ϵ = C_d/C_l, each blade element was chosen to work with a C_l equal to its optimal value. Then, the model determines the local speed at each section of the blade and defines the propeller geometry regarding its chord and pitch angle radial distributions.

To verify the convergence of the algorithm, it should be noted that this model assumes thrust and power to be quadratic functions of the wake displacement speed. Based on those quadratic functions, a new value of ζ is obtained after each iteration. If the stated value does not satisfy the convergence criterion imposed, the initial value of ζ is replaced by this new result, and the process starts again.

The final design (radial chord and pitch distributions) obtained from this procedure satisfies the initial conditions imposed, ensuring that it is an optimal propeller with minimum energy losses.

2.2. Airfoil Aerodynamic Database

The correct estimation of the aerodynamic coefficients of the blade sections has a great impact on the physical validity of the results. Therefore, it is important to have a database that the algorithm can check at any time, as well as interpolation and extrapolation functions capable of adapting the coefficients to any operating regimes.

With this purpose, a database of experimental airfoil coefficients was generated at low Reynolds numbers. In addition, in applications such as aircraft propellers, information beyond the stall is also required, as there might be cases where certain regions of the blades are stalled, either during operation or in an iterative process when solving the equations of the model. However, experimental information beyond stall is not usually available; in this case, extrapolation models proposed by Viterna et al. [27] and Montgomerie [28] were implemented. The experimental data was obtained from several references (mainly Selig et al. [29], Jacobs and Sherman et al. [30], and Miley et al. [31]).

Although the database created has several different airfoils, the results obtained in this study correspond to designs that only use NACA 4415, as it is the airfoil that has a maximum lift-to-drag ratio of the same order as the airfoils used for this type of application and is one of the best characterized for a large range of Reynolds numbers. Its coefficients at high angles of attack were extrapolated using the Viterna method.

All this airfoil information is stored in a file that the code can access to interpolate the lift coefficient as a function of a specific angle of attack and Reynolds number, Re, and then use the Mach number to apply a compressibility correction to it. The associated drag coefficient is obtained from the airfoil polar using the corrected lift coefficient.

2.3. Electrical Model of the Motor

To obtain the electrical efficiency of the motor, η_m, the simplified procedure described by Drela [24] is followed. The speed constant of the motor, K_v, its electrical resistance modeling copper losses, R_m; the no-load current as a simple modeling of iron or core losses, I₀; and operating conditions through its rotational speed, Ω, and shaft power, P_out, are assumed to be known. According to the equivalent idealized circuit shown in Figure 1, it is assumed that if the motor is powered externally by a voltage, V_m, and a current, I_m, finally, the useful voltage that serves to move the motor is V_net, and the current is I_net.

All these variables are related using the following expression:

$$V_m = I_m{R}_m + V_{\mathrm{net}}$$

(1)

$${ I}_m=I_{\mathrm{net}}+{ I}_0$$

(2)

Since the out-shaft power, P_out, is V_net I_net and the input power to the motor, P_in, is V_m I_m, the efficiency of the motor is determined using the following expression:

$${\eta }_m = {\displaystyle \frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}}$$

(3)

In each step of the optimization process, these expressions allow for sequentially obtaining the voltage V_net required to maintain the revolutions through the relationship Ω = K_vV_net, the current I_net from P_out, and, finally, with them, the voltage V_m and current I_m necessary to assess η_m.

3. Methodology

The main objective of this work is to achieve a method for the design of propellers with reduced electrical power consumption throughout the entire mission of an aircraft with multiple flight phases, in contrast to the conventional design models, which consider a single condition. The motor-propeller obtained is expected to provide a minimum thrust (the necessary thrust to fly according to the drag polar, as previously calculated) in each flight condition, and, working at low Mach numbers and angles of attack, which are necessary to avoid penalizing aerodynamics and noise emissions.

It is important to highlight that the model is intended to be applied to the specific case of small RPASs, such as micro and mini categories, that require the installation of an FPP. In this scenario, the selection of motor-propeller design parameters that maximize propeller performance throughout the flight envelope is particularly important and justifies the development of the present model.

To achieve this, the mission chosen for the optimization of the propeller is divided into N flight phases. Each of these phases, i, is defined by its altitude, h_i, the required flight speed, V_i, the rotor speed, Ω_i, the required thrust, T_i (to match the drag), the propeller input power, P_pi, and its propulsive efficiency, η_i. It is also necessary to consider those associated with the geometry of the propeller (regarding the radial distributions of chord, c, and pitch, β), i.e., those used for its design: flight speed, V_des, blade rotational speed, Ω_des, radius, R_des, and thrust, T_des. The result of the optimization is the design speed, V_des, and the propeller geometry associated with it, for which the energy consumed in the different phases, i, is minimum.

The Adkins and Liebeck analytical model was used to generate geometries, while the performance estimations were obtained using a model derived from that of Drela [32].

Drela’s formulation uses a parameterization of the local velocities that differs from classical models. Together with the replacement of the momentum equations by a semi-empirical relation between the tangential speed induced in the propeller plane and the circulation in the far wake, it enables description of the behavior of flow around the blades and estimation of the propeller performance through an iterative process with very high convergence rates.

The variables Ω_i, V_des, Ω_des, R_des, and T_des (that is, the rotational speed of the propeller in each phase of the mission, and the variables that determine its geometry) are the optimization variables in this problem. On the other hand, the flight speed and minimum thrust in each phase, V_i and T_i,min, have fixed values.

In summary, the optimization of the propeller geometry and its rotational speed in the different phases of the mission are addressed, given the flight speed and required thrust in each of them, so that the average power consumed by the combination of the propeller and the motor throughout the mission is minimum (or, equivalently, the average efficiency is maximum). This results in the same formulation of restricted multi-point and multidisciplinary optimization, and both are solved at the same time.

3.1. Optimization Process

The optimization algorithm starts by focusing on the propeller and then analyzes the motor. It is first necessary to provide several constants that characterize the geometry and performance in each phase of flight:

• Phase i: Flight speed, V_i, minimum thrust, T_i,min, and electrical power weighting factor for phase i over the entire mission, ω_i (it must be satisfied that ω₁ + ω₂ + … + ω_N = 1).
• Constraint: Minimum and maximum values allowed for all the optimization problem variables, Ω_i, V_des, Ω_des, R_des, and T_des, as well as the maximum helical tip Mach number permitted during operation in each phase of the mission, M_max, and the range of angles of attack where all blade elements must operate in all defined phases. These constraints on the Mach number and the angles of attack over the blade are intended to reduce the aerodynamic noise emissions during the propeller operation. The helical tip airspeed, necessary to perform the Mach number, is obtained as the module of the vectorial addition of the flight speed and rotational speed.

The algorithm evaluates two functions for each candidate solution: one related to the objective function and another to constraint satisfaction. The two functions are described as follows:

• Objective function: Based on the mission parameters initially fixed and the characteristics of a given solution (or combination of the optimization variables Ω_i, V_des, Ω_des, R_des, and T_des), the objective function runs a design function to obtain the propeller geometry associated with the values of V_des, Ω_des, R_des, and T_des, and then analyzes its performance in the different phases of the mission, using both the initial parameters for each one (which are fixed) and the rotational speeds provided by each specific solution, Ω_i. Once the propeller power at each phase, P_pi, is obtained, the motor model determines the electrical input power to the motor at that phase, P_i. The result of the objective function is an average electrical power required by the propeller over the whole mission, $\overline{P}$, which is determined using the following expression:

$$\stackrel{-}{P}={\omega }_1{P}_1+{\omega }_2{P}_2+\dots +{\omega }_N{P}_N,$$

(4)

with ω_i being the weight factors of each flight phase. Their values are initially set to satisfy the specific mission of the aircraft to be designed. It should be noted that the choice of the average power as the objective function is not arbitrary: after initial tests with different objective functions such as weighted power or propeller efficiency (which, for fixed speed and thrust, are equivalent), it was observed that for this particular problem, the weighted power objective function provided a more satisfactory performance regarding convergence of the optimization algorithm.

• Constraint function: The constraint function is identical to the objective function regarding input data (the optimization variables) and the calculations performed (a first geometry design and a subsequent estimation of performance in all flight phases). However, this function is only responsible for checking that a solution meets all the constraints imposed on the problem. If any constraint is violated, the case is rejected. The constraints applied to this problem are as follows:

T_i,min ≤ T_i ≡ T (V_i, Ω_i, V_des, Ω_des, R_des, T_des)

(5)

M (V_i, Ω_i, r = R_des) ≤ M_max

(6)

α _min (r) ≤ α (V_i, Ω_i, V_des, Ω_des, R_des, T_des) ≤ α _max (r)

(7)

The optimization algorithm evaluates the constraint function first and, only if all the constraints imposed on the problem are satisfied, executes the objective function to consider only the valid results, saving significant computational time.

The optimizer contains the following two routines:

• The geometry design starts with values of V_des, Ω_des, R_des, and T_des, using the Adkins and Liebeck optimal design method.
• An analysis of the performance of the designed geometry in all flight phases is conducted, with some operating conditions fixed, and with the propeller rotational speed in each stage being the optimization variable: Ω_i. For these calculations, an adaptation of the Drela model [32] was used.

Figure 2 shows a diagram of the optimization process described in this section, where the blocks corresponding to the fixed input data, the variables used by the optimizer, and the final variables obtained are separated.

3.2. Study of the Convergence of the Results with the Mesh

To ensure the quality of this study, an analysis of the effect that the number of blade elements, N_e, and their distribution have on the thrust and power coefficients, C_T and C_P, and the propeller efficiency, η, was carried out.

The number of blade elements ranged between 10 and 110 and was distributed in different ways. As expected, the results show that the variables improve asymptotically with increasing Ne. On the other hand, a higher density of elements in the blade tip area can model, with better resolution, the effects of functions such as the Prandtl loss function, F, and the shapes of the C_T and C_P distributions.

Figure 3 presents, from left to right, the relative error in the thrust and power coefficients, and propeller efficiency (all of them with respect to their theoretical value, which was considered the one obtained with 500 blade elements), committed by the use of a number of blade elements between 10 and 110, as well as the difference between using a distribution of equidistant or concentrated blade elements at the tip of the blades. The radial coordinates of the blade elements for these two distributions are determined using the following expressions:

$$r_{k equidis }= R_{\mathit{hub}}+{\displaystyle \frac{k - 1}{N_e - 1}}\left(R {- R}_{\mathrm{hub}}\right) ; r_{k conc } =R_{\mathit{hub}} + \left(R - R_{\mathit{hub}}\right)\mathit{cos}\left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{N_e - k}{N_e - 1}}\right)$$

(8)

The errors in C_T, C_P, and η are, respectively, defined as follows:

$$\mathit{Error} \mathit{in} C_T={\displaystyle \frac{C_T\left(N_e\right)}{C_T\left(500\right)}} - 1 ; \mathit{Error} \mathit{in} C_P={\displaystyle \frac{C_P\left(N_e\right)}{C_P\left(500\right)}} - 1 ; \mathit{Error} \mathit{in} \eta = {\displaystyle \frac{\eta \left(N_e\right)}{\eta \left(500\right)}} - 1$$

(9)

Significant differences are observed depending on the distribution method applied to the blade elements along the span, with the results obtained being less accurate with an equidistant distribution because of its lower capacity to catch effects and gradients in areas such as the blade tips: double or triple the number of blade elements is required to obtain errors of the same order as those of the concentrated blade element distribution. Using this second distribution, only 20 blade elements are required to ensure results with errors of less than 0.01% of the theoretical value under the conditions of this study, thus guaranteeing quality results without increasing the computational time required excessively. This is why the distribution of 20 blade elements concentrated at the blade tips was used to validate the model and obtain the results presented in this report.

Regarding the precision of the routines called by the optimizer to design a propeller and study its performance at particular flight conditions, they were validated comparing the results to experimental studies available in the literature (Biermann and Hartman [33] and the APC Sport 11×8 propeller [34]) and state-of-the-art open-source software for the design and analysis of the propellers XROTOR 7.55 [35] and OpenProp [36]). Figure 4 presents an extract of the validation process performed with the experimental results obtained by Biermann and Hartman for a 6-bladed Hamilton propeller. The results show that the relative error committed in the central range of advance ratios was about 3% for the power coefficient. The error obtained in other parameters, such as the thrust coefficient, was of the same order.

It is important to highlight that although the propeller selected to validate the model rotates only at 500 rpm, Etewa et al. [37] showed that this kind of model can be applied up to the range of 10,000 rpm, maintaining errors below 10%. Therefore, the model is considered validated for its application in RPASs.

3.3. Optimization Algorithm

A gradient-based algorithm was selected to obtain the results presented in Section 4 to verify the validity of the model proposed in the foregoing sections. This choice was made because it is a standard algorithm that is currently widely used, both in industry and other academic studies in the field of aerodynamics, as indicated by Yu et al. [38] and Yao et al. [39].

Gradient-based algorithms are local optimization algorithms that use the gradient of the objective function as the search direction for finding the minimum of the function in the design space. They start from an initial point, usually provided by the user, and proceed in the direction of the gradient: at each iteration the gradient at the point under study is calculated and its direction followed, until the value of the objective function is no longer reduced (or the change between one iteration and another is small enough) or the maximum number of iterations set is reached.

In the present study, the gradient-based optimizer available in MATLAB R2022a, fmincon [40], provided satisfactory results in an advantageously reduced computational time for the optimization parameters presented in

Table 1. Values set for the parameters of the gradient-based algorithm.

Parameter	Value
Optimality tolerance	10−4
Constraint tolerance	10−4
Function tolerance	10−4
Step tolerance	10−4
Maximum iterations allowed	500
Maximum function evaluations allowed	300

. The active-set algorithm was selected for the following reasons:

• This algorithm does not penalize solutions close to the boundaries of the design space, which are defined by the user through the constraints established in the problem. This does not occur in other algorithms, such as interior-point. As a disadvantage, compared with interior-point, active-set uses intermediate calculations. Therefore, the computational burden for large problems (a high number of design variables) is considerably higher.
• The algorithm can use long steps between iterations, potentially reducing computational time. The step size is the measure of how much the solution changes between two iterations and is calculated from the gradient of the function at each point. Long steps allow rapid progress in solving the problem but run the risk of overfitting and skipping the optimal point; therefore, each algorithm applies specific procedures to find a balance.

4. Results

The method developed in this study was used to design a propeller for the BIXLER 3 model aircraft. This aircraft has been studied in other projects of the Department of Aircraft and Spacecraft of the Universidad Politécnica de Madrid [41]; thus, all the necessary data was available to characterize both the operation of the aircraft (using its aerodynamic characteristics) and its physical characteristics (geometric and mass properties). In addition, the propeller of the studied BIXLER 3 was the APC 8×4E, which allowed the use of the data provided by the manufacturer [35] to compare it with the propeller designed with the design method presented here. The APC 8×4E propeller was selected because it is the one recommended by the manufacturer of the BIXLER 3 model (although the specific conditions of its design are unknown). This propeller can also be considered representative of other models with a weight similar to that of the BIXLER 3, which is 1.1 kg.

In the three optimizations performed, the ratio between the hub radius and the propeller radius was kept constant, setting its value at 15% of the total propeller radius, which is the proportion observed in the APC 8×4E.

Additionally, for an electric motor suitable for the Bixler 3, the following electrical constants were applied: K_v = 700 rpm/V, R_m = 0.505 Ω, and I₀ = 0.385 A.

4.1. First Optimization: Propeller with a Fixed Radius Considering Only One Flight Speed

To make an accurate comparison of the new method with the APC 8×4E reference propeller, a first optimization was carried out in which the radius of the propeller, R, was kept constant and equal to that of the APC 8×4E (0.102 m), while also seeking maximum efficiency at a single flight speed to be able to compare, later on, the advantages of selecting a greater number of speeds. This first optimization is identified as Single-Point Fixed R. The design speed selected is 15 m/s, which is an intermediate speed in the range of speeds of the aircraft under study. As previously mentioned, the design speed of the APC 8×4E propeller is not really known; however, the speed selected may be a representative value for comparing the performance obtained.

To provide a detailed comparison between the designed propeller and the APC 8×4E, it was initially thought to use the performance data supplied by the manufacturer, APC; however, two difficulties arose.

First, the calculation method used by APC to estimate the performance of their propellers is very similar to the one used in our work, except that APC does not consider the effects of the low Reynolds number, as indicated by the manufacturer itself on its web page [42]. As is well known, these effects significantly reduce the maximum efficiency of the airfoils used.

Second, there is no certainty about the airfoil used by APC in this propeller (APC provides the performance, chord distribution, and pitch angles for each of its propellers, but not the actual airfoil used). Considering these two difficulties, the performance of the APC 8×4E propeller (with the same radius, chord distributions, and pitch angles as the commercial propeller, but with the NACA 4415 airfoil used in the optimization process) was obtained using the same performance routine used in the optimization process. This ensures a more accurate comparison with the results obtained in this study.

The developed model has the capacity to study several profiles along the blade; however, to simplify comparison of the results, NACA 4415 was used as the only profile in the three optimizations carried out.

All optimizations were tested at sea level, which allows a better comparison with the APC 8×4E data also provided at that altitude.

As parameters of the optimization, the gradient-based algorithm requires the initial values of the design variables to be set to begin the optimization process, and, as presented in Section 3.3, several initial points (and therefore several optimizations) were used so that the search for the optimal propeller would cover a wider design space. Within the ranges defined by each design variable, six equidistant points are initially chosen so that, taking the points of each variable correlatively, six different optimizations are defined:

• V_des = 15 m/s, Ω_des: [1000, 26,000] rpm, R_des = 0.102 m, and T_des = 1.9 N.

The ranges used to produce these initial values for the design variable, Ω_des, correspond to the minimum and maximum limits set for these variables (see Section 4.2). The rotational speeds were limited to between 1000 and 26,000 rpm (these are the values for which information on the operation of the APC 8×4E is available, and it is considered a sufficiently wide range, as can be deduced from observing the results). The design thrust, T_des, is the thrust required for the design speed, according to the estimated drag polar of the aircraft.

Besides the bounds imposed on the design variables, constraints were imposed on the helical tip Mach number and the distribution of angles of attack over the blade, as shown in Equations (3) and (4). For the Mach number at the tip of the blade, a maximum of 0.85 was selected, which is a reference value in the aerodynamics of propellers since it is close to the critical Mach number of commonly used airfoils. The maximum and minimum limits imposed on the distribution of angles of attack along the blade are different for each defined flight condition and depend on the calculated angles of attack at which the maximum and minimum lift coefficients of each blade section are obtained (particularly, their value is 90% of those), having used the Reynolds number in each section and the experimental aerodynamic database of the airfoil.

In all optimizations performed, the minimum required thrust constraint is always active. It is logical that this restriction should be active since the method searches for a propeller with minimum power consumption (i.e., maximum efficiency, η_i = T_iV_i/P_i) for both flight phases.

Once the propeller is optimized for the design speed of 15 m/s, its performance is also tested for the two different speeds and an additional off-design speed to be defined for the optimizations, as shown in Section 4.2 and Section 4.3. In this way, it is possible to compare all the optimizations performed with each other, for the same reference speeds.

Table 2 shows the geometrical differences between the APC 8×4E and the Single-Point Fixed R, while Table 3, Table 4, Table 5 and Table 6 show a comparison regarding performance. Table 6 presents the flight conditions defined during the design, whereas Table 3, Table 4 and Table 5 show the results for the other reference speeds.

The tables present the rotational speed, Ω, and power, P, required by each propeller to generate the demanded thrust at a particular flight speed, as well as the resulting helical tip Mach number, M_R, which is closely related to the noise produced by the propeller at each flight condition.

Table 2. Radius, pitch, and chord values of the propeller designed with a fixed radius and one flight speed, V_des = 15 m/s, in comparison with the values of the commercial propeller, APC 8×4E.

	β3/4R (◦)	Δβ3/4R	c3/4R (m)	Δc3/4R
APC 8×4E	18.1	-	0.017	-
Designed prop. (Single-Point Fixed R)	25.4	+40.3%	0.009	−47%

Table 2. Radius, pitch, and chord values of the propeller designed with a fixed radius and one flight speed, V_des = 15 m/s, in comparison with the values of the commercial propeller, APC 8×4E.

	β3/4R (◦)	Δβ3/4R	c3/4R (m)	Δc3/4R
APC 8×4E	18.1	-	0.017	-
Designed prop. (Single-Point Fixed R)	25.4	+40.3%	0.009	−47%

Table 3. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius and one flight speed, V_des = 15 m/s, compared with the APC 8×4E propeller, in the low-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	4767	-	18.3	-	0.150	-
Designed prop. (Single-Point Fixed R)	5362	+12.5%	22.7	+24%	0.169	+12.4%

Table 3. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius and one flight speed, V_des = 15 m/s, compared with the APC 8×4E propeller, in the low-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	4767	-	18.3	-	0.150	-
Designed prop. (Single-Point Fixed R)	5362	+12.5%	22.7	+24%	0.169	+12.4%

• Low Speed: V = 5.0 m/s, T = 1.4 N

Table 4. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius and one flight speed, V_des = 15 m/s, compared with the APC 8×4E propeller, in high-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	9961	-	116.6	-	0.318	-
Designed prop. (Single-Point Fixed R)	8393	−16%	111	−4.8%	0.270	−15.2%

Table 4. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius and one flight speed, V_des = 15 m/s, compared with the APC 8×4E propeller, in high-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	9961	-	116.6	-	0.318	-
Designed prop. (Single-Point Fixed R)	8393	−16%	111	−4.8%	0.270	−15.2%

• High Speed: V = 20 m/s, T = 3.2 N

Table 5. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius and one flight speed, V_des = 15 m/s, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 4 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	5291	-	25	-	0.167	-
Designed prop. (Single-Point Fixed R)	6671	26.1%	40.4	+61.6%	0.210	26%

Table 5. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius and one flight speed, V_des = 15 m/s, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 4 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	5291	-	25	-	0.167	-
Designed prop. (Single-Point Fixed R)	6671	26.1%	40.4	+61.6%	0.210	26%

• Off-Design: V = 4 m/s, T = 1.9 N

Table 6. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius and one flight speed, V_des = 15 m/s, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 15 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	7620	-	53.2	-	0.243	-
Designed prop. (Single-Point Fixed R)	6414	−16%	49.5	−7%	0.206	−15.3%

Table 6. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius and one flight speed, V_des = 15 m/s, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 15 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	7620	-	53.2	-	0.243	-
Designed prop. (Single-Point Fixed R)	6414	−16%	49.5	−7%	0.206	−15.3%

• Design: V = 15 m/s, T = 1.9 N

Figure 5 illustrates the geometry of the resulting propeller, comparing it to the geometry of the APC 8×4E.

The optimal propeller geometry found for the design speed of 15 m/s differs appreciably from the APC 8×4E, with a 40.3% larger pitch angle and a 45% smaller chord at 3/4R.

At the design speed, the power consumption is 7% lower and the revolutions are 16% lower, which translates into a 15.3% reduction in the M_R value. The weighted power consumption for speeds of 5 and 20 m/s is only 1% lower than that of the APC 8×4E. These results seem to indicate good behavior of the proposed optimization process.

As a negative effect of this first single-speed optimization, although the power also decreases at the speed of 20 m/s, as shown in Table 4, its increase is too large at low speeds, as presented in Table 3 and Table 5.

4.2. Second Optimization: Propeller with a Fixed Radius Considering Different Flight Speeds

In this second design, again keeping the radius of the APC 8×4E propeller constant, the objective is to evaluate whether it is possible to reduce the total power consumption while optimizing for different flight speeds. Therefore, this second design is identified as Multi-Point Fixed R.

First, this section presents the flight phases selected for the multi-point design of a propeller using the gradient-based algorithm. For the sake of simplicity, only two flight phases were considered, and since the operating characteristics of the aircraft were known, they were chosen to present markedly different speeds and thrust requirements:

• Phase 1: Low speed. For the low-speed condition of the BIXLER 3, a flight speed of 5 m/s was considered, which, according to the estimated drag polar of the aircraft, requires a thrust production of 1.4 N.
• Phase 2: High speed. The flight speed chosen for the second phase was 20 m/s, which entails a demand of 3.2 N of thrust.

As parameters of the optimization, in this case, the defined ranges to produce the initial values for each design variable are as follows:

• V_des: [4, 34] m/s, Ω_des: [1000, 26,000] rpm, R_des = 0.102 m, T_des: [1.4, 3.2] N.

The ranges used for the design thrust are composed of the thrust requested in both flight phases, as it was understood that the final value would be an intermediate solution.

Considering an RPAS mission profile in which the cruise phase of flight is the most extensive, for this optimization a weight, ω₁, of 10% was assigned to the low-speed phase and ω₂ equal to 90% to the high-speed phase, based on the understanding that this is an arbitrary choice that can be changed to analyze the influence of these weights on the final characteristics. As mentioned above, the choice of the power weight factors is determined by the specific mission of the RPAS under study and is mainly related to the fraction of time the RPAS operates at a given design speed with respect to total endurance.

To delimit the design space of the propellers, lower and upper limits were established to constrain the allowed variation in the variables. These limits were set to reduce the time required to design each propeller while maintaining sufficiently wide margins, also ensuring that the values corresponding to the APC 8×4E were within the intervals. Bounds on the velocities, [4, 35] m/s, and design thrusts, [0.5, 10] N, were set.

The optimization found that for a design speed, V_des, of 10.8 m/s, a blade geometry is defined with the active minimum thrust constraints for the low- and high-speed phases, and the constraints on the maximum values of the helical tip Mach number and the distribution of angles of attack on the blade, which requires minimum power.

As a result, Table 7 and Figure 6 show the geometric differences between the APC 8×4E and the designed propeller with a fixed radius, while Table 8, Table 9, Table 10 and Table 11 depict a comparison regarding performance.

Table 7. Radius, pitch, and chord values of the propeller designed with a fixed radius for two flight speeds in comparison with the values of the commercial propeller APC 8×4E.

	β3/4R (◦)	Δβ3/4R	c3/4R (m)	Δc3/4R
APC 8×4E	18	-	0.017	-
Designed prop. (Multi-Point Fixed R)	19.9	+11%	0.016	−6%

Table 7. Radius, pitch, and chord values of the propeller designed with a fixed radius for two flight speeds in comparison with the values of the commercial propeller APC 8×4E.

	β3/4R (◦)	Δβ3/4R	c3/4R (m)	Δc3/4R
APC 8×4E	18	-	0.017	-
Designed prop. (Multi-Point Fixed R)	19.9	+11%	0.016	−6%

Table 8. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius for two flight speeds, compared with the APC 8×4E propeller, in the low-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	4767	-	18.3	-	0.150	-
Designed prop. (Multi-Point Fixed R)	6870	44.1%	18.2	−1%	0.216	43.8%

Table 8. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius for two flight speeds, compared with the APC 8×4E propeller, in the low-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	4767	-	18.3	-	0.150	-
Designed prop. (Multi-Point Fixed R)	6870	44.1%	18.2	−1%	0.216	43.8%

• Low Speed: V = 5 m/s, T = 1.4 N

Table 9. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius for two flight speeds, compared with the APC 8×4E propeller, in high-speed flight conditions.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	9961	-	117	-	0.32	-
Designed prop. (Multi-Point Fixed R)	13,060	31.1%	105	−10%	0.41	29%

Table 9. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius for two flight speeds, compared with the APC 8×4E propeller, in high-speed flight conditions.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	9961	-	117	-	0.32	-
Designed prop. (Multi-Point Fixed R)	13,060	31.1%	105	−10%	0.41	29%

• High Speed: V = 20 m/s, T = 3.2 N

Table 10. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius for two flight speeds, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 4 m/s.

	Ω (rpm)	ΔΩ	P [W]	ΔP	MR	ΔMR
APC 8×4E	5291	-	25	-	0.167	-
Designed prop. (Multi-Point Fixed R)	7841	48%	25.3	+1%	0.250	50%

Table 10. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius for two flight speeds, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 4 m/s.

	Ω (rpm)	ΔΩ	P [W]	ΔP	MR	ΔMR
APC 8×4E	5291	-	25	-	0.167	-
Designed prop. (Multi-Point Fixed R)	7841	48%	25.3	+1%	0.250	50%

• Off-Design: V = 4 m/s, T = 1.9 N

Table 11. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius for two flight speeds, compared with the APC 8×4E propeller, to generate 1.085 N of thrust at a flight speed of 15 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	7620	-	53.2	-	0.243	-
Designed prop. (Multi-Point Fixed R)	9977	31%	48.1	−10%	0.320	30%

Table 11. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller with a fixed radius for two flight speeds, compared with the APC 8×4E propeller, to generate 1.085 N of thrust at a flight speed of 15 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	7620	-	53.2	-	0.243	-
Designed prop. (Multi-Point Fixed R)	9977	31%	48.1	−10%	0.320	30%

• Off-Design: V = 15 m/s, T = 1.9 N

In this case, the designed propeller has a 11% larger pitch angle and a 6% smaller chord at 3/4R and improves the power consumption in the low-speed and high-speed phases, resulting in a reduction of 10% of its value weighted over the entire aircraft mission, which, considering the results at 15 m/s, shows that the proposed design method provides more potential than just a single-speed design.

In addition, to qualitatively compare the results obtained from the first two optimization algorithms with the APC reference, Figure 7 shows motor-propulsive efficiency, ${\eta }_{mp}$, as a function of flight speed. This efficiency is defined as the product of the electrical efficiency of the motor, ${\eta }_m$, obtained using Equation (3), and the propulsive efficiency of the propeller, which is determined using the following expression:

$${\eta }_p = {\displaystyle \frac{\mathrm{TV}}{P_s}}$$

where P_s is the power consumed at the propeller shaft.

The two optimizations improve the APC in the high-speed range, and the multi-point matches the APC at low speeds.

4.3. Third Optimization: Variable-Radius Propeller with Two Flight Speeds

In Section 4.2, the results show that the design philosophy Multi-Point Fixed-R is better than the Single-Point Fixed-R, with a lower weighted power consumption of the two design speeds than the APC 8×4E.

In this section, a third and last, more general, design is carried out in which the radius of the optimized propeller can also vary within certain limits, with the aim of demonstrating that the proposed optimization also correctly captures the effect of this important design variable. This new design is identified as Multi-Point Var R.

As parameters of the optimization, the defined ranges to produce the initial values for each design variable are the same as in Section 4.2, but now the range of R is as follows:

• R_des: [0.8 × 0.102, 1.15 × 0.102] m.

Therefore, the range of variation in the initial values of the radius varies between 80% and 115% of the APC 8×4E propeller radius, where the upper limit was fixed to avoid collisions between the propeller and the tail boom of the studied aircraft. Structural or manufacturing constraints are examples of limits that could be reached before the radius limit, which is a topic of study envisioned for future work.

The optimal propeller geometry and its performance were compared with that of the APC8×4E at the same four speeds as in the previous optimizations.

The algorithm has designed a propeller capable of providing the necessary thrust in both low-speed and high-speed phases. The final values of the design variables provided by the optimization algorithm for minimizing the average consumed power (with ω₁ = 0.1 and ω₂ = 0.9) were V_des = 10.3 m/s, Ω_des = 7491 rpm, R_des = 0.117 m, T_des = 4.56 N, Ω₁ = 4000 rpm, and Ω₂ = 8238 rpm. These values correspond to a converged optimization where the optimum was reached with the following active constraints:

• The minimum thrust requirement for both low-speed and high-speed phases;
• The limit to the highest admissible values of the helical tip Mach number and the distribution of angles of attack over the blade;
• The radius upper bound, which is necessary due to the geometric constraints of the studied aircraft.

Figure 8 and Table 12 illustrate the geometry of the resulting propeller, comparing it to the geometry of the APC 8×4E. Since the propeller radius was allowed to vary as a design variable, and considering its influence on efficiency, the optimal propeller obtained has a larger radius. The negative effect of weight gain is discussed in the next section.

On the other hand, the geometry obtained has larger pitch angles and a significant reduction in the ¾ chords. The combination of all these geometrical differences results in a lower rotational speed required to obtain the same thrust as the APC 8×4E for all flight conditions.

Table 12. Radius, pitch, and chord values of the propeller designed in the third optimization in comparison with the values of the commercial propeller APC 8×4E.

	R (m)	ΔR	β3/4R (◦)	Δβ3/4R	c3/4R (m)	Δc3/4R
APC 8×4E	0.102	-	18.1	-	0.017	-
Desig. prop. (Multi-Point Var R)	0.117	+14.7%	18.6	2.8%	0.012	−29.4%

Table 12. Radius, pitch, and chord values of the propeller designed in the third optimization in comparison with the values of the commercial propeller APC 8×4E.

	R (m)	ΔR	β3/4R (◦)	Δβ3/4R	c3/4R (m)	Δc3/4R
APC 8×4E	0.102	-	18.1	-	0.017	-
Desig. prop. (Multi-Point Var R)	0.117	+14.7%	18.6	2.8%	0.012	−29.4%

Table 13, Table 14, Table 15 and Table 16 depict a comparison regarding performance at the four reference speeds previously defined.

Table 13. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller in the third optimization, compared with the APC 8×4E propeller, in the low-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	4767	-	18.3	-	0.150	-
Desig. Prop. (Multi-Point Var R)	4076	−14.5%	17.1	−7%	0.148	−1.9%

Table 13. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller in the third optimization, compared with the APC 8×4E propeller, in the low-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	4767	-	18.3	-	0.150	-
Desig. Prop. (Multi-Point Var R)	4076	−14.5%	17.1	−7%	0.148	−1.9%

• Low Speed: V = 5 m/s, T = 1.4 N

The designed propeller operates at less rotational speed and Mach number than the APC 8×4E, both in the low-speed condition, as shown in Table 13, to generate 1.4 N of thrust at 5 m/s, and in the high-speed phase, Table 14, to generate 3.2 N of thrust at 20 m/s.

Table 14. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller in the third optimization, compared with the APC 8×4E propeller, in the high-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	9961	-	116.6	-	0.318	-
Desig. Prop. (Multi-Point Var R)	8175	−18%	112.3	−4%	0.300	−5.7%

Table 14. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller in the third optimization, compared with the APC 8×4E propeller, in the high-speed flight condition.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	9961	-	116.6	-	0.318	-
Desig. Prop. (Multi-Point Var R)	8175	−18%	112.3	−4%	0.300	−5.7%

• High Speed: V = 20 m/s, T = 3.2 N

It can also be observed that the designed propeller reduces the required power in the high-speed condition by 4%, while it requires about 7% less power for the low-speed one. Overall, the weighted power is reduced by the designed optimal propeller, which is 4.3% smaller than the power required by the APC 8×4E (having obtained a result of ω₁ = 0.1 and ω₂ = 0.9). Moreover, operating at 4 m/s, the designed propeller reduces the required power by 8% with respect to the APC 8×4E (Table 15), and it reduces the power by 4% at 15 m/s. Regarding the Mach number at the tip, the optimal propeller reduces its value in comparison with the APC 8×4E in the four studied flight conditions, which can be beneficial for noise reduction.

Therefore, the propeller designed in this study represents a promising propulsion alternative to the APC 8×4E.

Table 15. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller in the third optimization, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 4 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	5495	-	25	-	0.167	-
Desig. Prop. (Multi-Point Var R)	4409	−16.7%	23	−8%	0.159	−4.4%

Table 15. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller in the third optimization, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 4 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	5495	-	25	-	0.167	-
Desig. Prop. (Multi-Point Var R)	4409	−16.7%	23	−8%	0.159	−4.4%

• Off-Design: V = 4 m/s, T = 1.9 N

Table 16. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller in the third optimization, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 15 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	7620	-	53.2	-	0.243	-
Desig. Prop. (Multi-Point, Var R)	6271	−17.7%	51.1	−4%	0.230	−5.4%

Table 16. Values of rotational speed, Ω, electrical input power, P, and Mach number at blade tip, M_R, for the designed propeller in the third optimization, compared with the APC 8×4E propeller, to generate 1.9 N of thrust at a flight speed of 15 m/s.

	Ω (rpm)	ΔΩ	P (W)	ΔP	MR	ΔMR
APC 8×4E	7620	-	53.2	-	0.243	-
Desig. Prop. (Multi-Point, Var R)	6271	−17.7%	51.1	−4%	0.230	−5.4%

• Off-Design: V = 15 m/s, T = 1.9 N

The condition imposed on the angle of attack distribution with the same purpose is also perfectly fulfilled in the three optimizations. As an example for the case of the third one, Figure 9 shows (with a solid line) the radial distributions of angles of attack that the airfoils of the propeller are exposed to in the design conditions: low speed (left) and high speed (right). Additionally, the maximum (dash-dotted line) and minimum allowed angles of attack (dashed line) are shown, making it easier to verify compliance with the imposed constraint on the angles of attack related to the criterion of reducing aerodynamic noise emissions.

It can be observed that radial distributions of angles of attack are maintained between their minimum and maximum values in both design phases. Multi-point optimization has generated designs where the angles of attack at low speed are as close as possible to their maximum admissible value, so that in phases of flight at speeds lower than 5 m/s, the angles of attack would be greater than those allowed. However, a much larger margin is observed in the angles of attack at high-speed conditions.

According to the above, the design obtained guarantees, for a given geometry, radius, and rotor speed, that the airfoils of the blade elements are far from the aerodynamic stall in the two flight conditions used for the design. Therefore, it can be stated that the noise emission in both phases could be reduced because the helical tip Mach number is always kept at low values and the angles of attack are far from the airfoil boundary layer detachment. In the future, further analysis of these phenomena and quantification of the noise emission generated by the propeller in each phase is planned.

Finally, to qualitatively compare the results obtained in this third Multi-Point Var R optimization with the Single-Point Fixed R design and the APC reference, Figure 10 shows the motorpropulsive efficiency as a function of flight speed. It can be observed that the Multi-Point Var R optimization has a higher motorpropulsive efficiency (lower electrical power input to the motor) than the APC over the whole speed range.

5. Discussion

In general, the results obtained in the optimization proposed for the electric moto-propeller system appear to be satisfactory, complying with the conditions imposed at the same time in various phases of flight and with different real criteria on the radius, R. The limitations in the distribution of angles of attack and Mach at the tip in all flight conditions are also considered to be satisfied.

The results obtained at the two design speeds seem to indicate that the model accurately reflects the physical effects of the choice of weighting, the variation in propeller diameter, and the electrical characteristics of the motor. This provides confidence that in the future, the use of more daring strategies, such as larger diameter increases (without forgetting the weight penalty), will allow for even greater reduction in power consumption. The model enables this objective to be achieved for any RPAS similar to the one studied.

Furthermore, in the third design, Multi-Point Var R, the importance of diameter is highlighted, reinforcing the idea that it should be a design variable, as it contributes to the overall reduction in power. However, it is necessary to consider that increasing the diameter has the negative impact of increasing the weight of the propeller and, ultimately, the electrical power consumed during flight.

For this reason, the effect of the increase in propeller weight was included in the optimization. Although in general the penalty can be significant, in the case studied, by limiting the increase in radius to only 15% of the initial value, the increase in propeller weight relative to the relative weight of the aircraft is very small (+0.5%); therefore, the increase in the weighted electrical power is negligible.

It is necessary to assess the effect of the choice of power weighting factors, ω_i. As previously mentioned, the choice of these factors is determined by the specific mission of the aircraft under study and is related to the fraction of time the aircraft operates at a given design speed with respect to total endurance. To evaluate the influence of the choice of these factors, a new design was performed, using the same values of the design variables and constraints as in the second optimization, the Multi-Point design, with constant radius and ω₁ = 0.1 (Section 4.2), but now with a different value of ω₁ = 0.9.

In this case, compared with the reference (ω₁ = 0.1), the results show that regarding electrical power, the propeller designed for ω₁ = 0.9 at 20 m/s requires 7.3% more power, while at 5 m/s it requires 2% less power. This makes sense, since by increasing the weight factor without changing any other conditions, the optimal design shifts to speed values closer to 4 m/s, although the changes are minimal since the required thrust is maintained at both speeds.

Regarding geometry, large variations appear: the propeller designed for ω₁ = 0.9 requires a pitch angle (0.75R) 8% smaller and a chord (0.75R) 38% smaller than the reference propeller. These significant changes are attributed to the fact that the revolutions required by the new propeller are appreciably higher. Finally, as expected, the percentage reduction in the weighted power is significant, with the propeller designed for ω₁ = 0.9 requiring 71% less as it flies longer at lower speeds.

Regarding the computation time and the design space sweep, the algorithm ensures reduced computation times at the cost of a moderate exploration of the design space, where each candidate propeller is relatively like the previous one studied. In the problem studied, the duration of time required to obtain a new propeller design was about 40 min using an HP Pavilion laptop with 12 GB of installed RAM and an Intel Core i5-8250U processor.

As previously mentioned, an experimental campaign is planned in the future to verify the results presented in this study. Prior to this campaign, to leverage the multidisciplinary optimization capabilities of the proposed model, it is considered necessary to add a structural module that can include limits to the blade geometry and revolutions if critical values of the structural stresses are reached during the optimization process.

Furthermore, this study was conducted under the assumption that the final geometry obtained by means of the optimization process is manufacturable. Current techniques of 3D printing allow users to print almost any geometry with an acceptable level of surface roughness. These techniques could be used for printing small propellers. Thus, the hypothesis of neglecting manufacturing constraints seems to be reasonable. However, if it is necessary to introduce any constraint related to the manufacturing process, the methodology presented could be easily extended to consider it, as demonstrated in studies by Malim et al. [43] and by Chen et al. [44].

Finally, as a general comment on the three designs, it should be noted that the propeller rotational speed varies appreciably with respect to that of the APC 8×4E. It is well known that a reduction in propeller speed is usually beneficial to improve propulsive efficiency, but the efficiency of the motor may be reduced, and thus, the efficiency of the complete system (motor and propeller) may not improve. As a result, a future study will incorporate an advanced optimization, including the complete electrical system (battery, ESC, and motor) into the basic combination of the motor–propulsive optimization performed here, to ensure improvement of the complete system.

6. Conclusions

This study presents a new method for preliminary design of fixed pitch propellers that extends conventional procedures and allows the development of the electrical motor-propeller system optimized for more than one flight condition at the same time. The proposed method is adapted to the needs of today’s increasingly demanding propellers with novel and specialized missions.

For flight conditions fixed via the required thrust and flight speed, the present design finds an optimal motor-propeller system with minimum power required for the selected combination of flight phases (i.e., minimum electrical input power consumption over the defined mission). Additional aerodynamic constraints were incorporated for noise reduction purposes, which are currently a constant in aeronautical design. As a result, the optimal propeller designed satisfies the flight conditions as well as the constraints imposed: distribution of angles of attack contained between maxima and minima, and helical tip Mach number lower than the defined limit.

The results of the presented method have been compared with a commercial propeller (APC 8×4E) used in a real model aircraft, showing consistent improvements.

The method combines optimization techniques with an experimental aerodynamic database and procedures for propeller design and performance study based on the state of the art. The resulting tool achieves short execution times with accurate results, demonstrating that it is highly suitable for preliminary design and improving the performance of real commercial solutions.

The work performed extends knowledge in the design of a small RPAS motorpropeller system for multiple flight conditions with various constraints. For future stages of development, the versatility of the presented multi-point and multidisciplinary procedure allows for the addition of different design objectives and constraints, such as propeller minimum weight or structural and manufacturing concerns, that would further restrict the diameter increase observed in the results. Likewise, the method may be adapted to different optimization techniques and performance levels, as well as noise analysis models of higher complexity.

Due to the abovementioned difficulties in properly validating the optimization procedure with the available data, the appropriate tunnel tests leading to detailed verification of this model are planned for the near future. Moreover, further validation and improvements using Computational Fluid Dynamics tools are also envisioned.