Research ArticlePropeller Force-Constant Modeling for Multirotor UAVs fromExperimental Estimation of Inflow Velocity

Gaurang Gupta and Shaaban Abdallah

Department of Aerospace and Engineering Mechanics, University of Cincinnati, Cincinnati, OH 45221, USA

Correspondence should be addressed to Gaurang Gupta; guptagg@mail.uc.edu

Received 11 July 2017; Revised 6 November 2017; Accepted 6 December 2017; Published 10 April 2018

Academic Editor: Angel Velazquez

Copyright © 2018 Gaurang Gupta and Shaaban Abdallah. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Design and simulation of an unmanned aerial vehicle (UAV) highly depends on the thrust produced by a motor-propellercombination. The aim of this paper is to model a generalized mathematical relationship between the motor RPM and thecorresponding thrust generated for the preliminary design process of low Reynold’s number applications. A method isdeveloped to determine a generalized mathematical model which relates inflow velocity to coefficient of thrust usingexperimental data from 291 motor-propeller data points, comprising of input RPM and corresponding output thrust. Using thisrelationship, the Force Constant is calculated, which defines each Thrust-RPM mathematical model. In the first part, expressionof the inflow ratio obtained from Blade Element and Momentum Theory (BEMT) is approximated to a simplified form. In thelater part, the proposed mathematical model is validated against two new sets of pairs of motor-propeller combinations. Aspecial note in the Appendix talks about the application of this mathematical model. The computed results are found to be ingood agreement with the experimental data.

1. Introduction

The last few decades have shown an increase in the usageand development of UAVs. Most of today’s UAVs usedfor reconnaissance, surveillance, and disaster managementmissions are powered using an electrical propulsion systemwhich has shown significant improved propulsion effi-ciency and noise reduction over conventional combustionengine systems. UAVs have proven themselves very usefulduring the time of disaster management, where they hoverin one place to securely drop medical supplies or get visualsof places [1, 2].

Energy density is defined as the amount of energy that asubstance or component can store or transform per unitmass of itself. Fuel, which can be burnt, has an energy densityof the order of magnitude three to four times more than thatof solid-state fuel cell storage like a lithium polymer (LiPo)battery [3, 4]. But the downside of using an engine is that itis highly inefficient and creates a lot of pollution. As studiessuggest that the brushless direct-current (BLDC) motorsare highly efficient, their usage has become very common in

electric UAVs. As the propulsion system of an electric UAVconsists of batteries, electronic speed controllers (ESC),motors, propellers, and so forth, a general survey shows thatthe weight of the propulsion system can account for approx-imately 50% of the total weight of the entire system as shownin [5–7]. Examples can also be taken from [8–10]. Thus, theoptimization of the propulsion system of a UAV becomes avery crucial aspect of UAV design. There can be many othercomponents in the propulsion system but the battery, motor,and propeller have a far more significant impact on the over-all system. The other important aspect is to stabilize theUAV. Most of the general control methods are based onthrust force and the angular velocity of the motor assuminga simple parabolic relation. The performance and character-istics of the vehicle depend on the strong interaction betweenthem. The purpose of this paper is not to present an optimi-zation method or an optimal study but rather to study theinteraction between these components and generalize them.The results of this paper have a significant application scopein the preliminary design process of any UAV (small ormedium sized). This will help save time and drive the costs

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 9632942, 10 pageshttps://doi.org/10.1155/2018/9632942

of the project down by eliminating the necessity of numerousamounts of initial bench tests using different propellers andmotors to determine the best combination of propulsion sys-tem for the vehicle.

The method structured in this paper starts with estimat-ing the inflow velocity which is simplified using approxima-tions. There exists an alternative approach to estimate theinflow velocity. Commonly known as the 3/4 thumb rule,the geometric characteristics of the propeller are consideredat 3/4 radius of the propeller from the center. The majorparameters considered are the twist angle, the width of thepropeller, and the airfoil section at that point. All theseparameters will have to be calculated or measured from theactual propeller as these specifications are not available inoff-the-shelf specifications. The advantage of the method dis-cussed in this paper is that it uses only the parameters whichexist in off-the-shelf available specifications like propellerdiameter and pitch.

The first part of the paper discusses the developmentmethod and techniques utilized to simplify the conventionalrelation between the thrust generated by a propeller and itsgiven RPM which is derived from very famous theories,namely the Blade Element and Momentum Theory. The nextsection discusses the assumptions undertaken to simply theestimation of inflow velocity which is the most importantcomponent in estimating the thrust generated by a propeller,using the existing relation. Secondly, the method used in thepaper ensures that the output thrust is a function of only thoseparameters or a propeller that are available in any off-the-shelf propeller and not the geometric parameters which areused to design a propeller. The later sections demonstratethe validation of the proposed model and the estimates oferror induced due to simplification of the original coefficientof thrust versus RPM relation.

2. Propeller Static Thrust—RPM Modeling

Most of the propeller designs are based on the work of Betzmentioned in [11, 12]. The design principle is based on opti-mizing the propeller’s geometry for a certain specific operat-ing condition such that the power required for that operationis minimized or can also be understood as maximizing thethrust generated for the given power. The thrust estimationmodel for a propeller in this study is based on the veryfamous Blade Element and Momentum Theory (BEMT).This theory helps in estimating the aerodynamic loads devel-oped on the propeller, which can be used to estimate thethrust generated at a given RPM.

2.1. Thrust Model. First, we estimate CT using the axialmomentum theory [13, 14]. The major assumptions are asfollows: (i) no rotational motion is imparted to the flow bythe propeller disk; (ii) the Mach number is small, so the fluidcan be assumed to be incompressible; and (iii) the flow issteady as the propeller is assumed to be a thin disk (ofcross-section area A), through which air passes, and theinduced velocity is assumed to be constant at all points whichlie on the same radius.

A simplified model of a propeller stream tube is shown inFigure 1. The disk is assumed to be uniformly loaded, and thevelocity of air across the rotor disk is V + v1, which isassumed to be uniform across the disk and has the samemag-nitude before and after the disk [15]. On the rotor disk(Figure 2), at radius r from the center, consider a ring withinfinitesimal thickness dr. It is assumed that this elementalarea of the disk is uniformly loaded. Thus, the elementalthrust coefficient, dCT , that is, the nondimensional form ofthrust for this elemental area, is

dCT = 4 λc + λi λirdr, 1

where inflow ratio λi is defined as the ratio of inflowvelocity v1 to tip velocity ΩR; similarly, λc is defined asthe ratio of freestream velocity V to tip velocity ΩR andr as r/R. Detailed derivation to obtain dCT is explainedin Appendix A.

The modern propellers have a complex geometry, wherethe chord, pitch angle, and airfoil geometry vary along theradius of the blade to improve propeller efficiency. As theaerodynamic forces on a fixed wing (propeller blade) are

V V + v1 V + v2T

𝛺

Figure 1: Propeller slipstream with axial velocity representation.

dr

r

R

Figure 2: The propeller as a rotor disk.

2 International Journal of Aerospace Engineering

governed by the rotating velocity, the effects of this can bedescribed using a more fundamental Blade Element Theory[16–18], which takes into consideration the geometry ofthe propeller and the operating conditions. This theoryassumes that each section of the wing acts as a 2D airfoilto produce aerodynamic forces (lift and drag as shown inFigure 3) which are then resolved to give elemental thrustper unit span.

Considering Nb as the number of propeller blades, totalelemental thrust coefficient at radius r (Figure 2) can bedefined as

dCT = NbcR

θr2 − λc + λi r dr, 2

where θ = tan−1 p/πd . Detailed derivation to obtain dCT ,using Blade Element Theory, is explained in Appendix A.

On relating (1) and (2), derived elemental thrust coeffi-cients from two different theories, we get

4 λc + λi λirdr =NbcR

θr2 − λc + λi r dr, 3

λi2 + λc + k λi + kλc − kθr = 0, 4

where k =Nbc/4R or k =Nbc/2d. Solving (4) for λi and con-sidering only a positive value, we get

λi = −λc + k2 + 1

2 λc − k 2 + 4kθr, 5

for hover, λc = 0; (2) and (5) can be written as

dCT = 4k θr2 − λir dr, 6

λi = −k2 + 1

2 k2 + 4kθr 7

2.2. Inflow Velocity Estimation. From geometric data col-lected using nine propellers (refer to Appendix B) [19, 20],at the root, θ ranges from θroot,max ≈ 0 48rad, for high-pitch-type propellers, and θroot,min ≈ 0 11rad, for low-pitch-typepropellers, to θtip,avg ≈ 0 07rad towards the tip. The variationof the twist angle with the radius can be a high-order polyno-mial but is assumed to be linear here. Figure 4 shows the var-iation of the pitch along the radius from root to tip, betweenhigh-pitch-type and low-pitch-type propellers. The geomet-ric data collected from the nine propellers shows that theaverage effective radius of the blade that produces lift isapproximately 86% of the total propeller radius, whichincludes hub losses.

If we consider infinite propellers in a given regime ofdiameter varying from 4 inches to 12 inches, the probabilityof occurrence of each point (where each point refers to a pro-peller with θroot in between θroot,max and θroot,min) in triangleCDE is equal. Thus, we try to estimate the average effect ofθroot over infinite possibilities such that in a broad domain,every propeller in this domain will have the same value ofproduct of twist with varying radii shown as a red-shadedrectangle area in Figure 4. Here, it can safely be assumedthat that infinite data set will have a normal distribution;that is, there will exist an equal number of propellers for

all given values of θroot. Assuming a linear twist and normaldistribution (considering equal probability) of data setsbetween θroot,max and θroot,min to θtip,avg, we calculateθr mean as below and represent it as a uniform pitch propel-ler (as shown by the red-shaded rectangular area)

θr mean =Area ABCE + Area ABCD

2 × r1

0 148

Solving for (8), we get θr mean ≈ 0 25, giving 4 θr mean =1 and θmean ≈ 0 29rad. Using the above approximation, weassume that the inflow velocity across the entire rotor diskis uniform. From (7) λi can thus be defined as

λi = −k2 + 1

2 k2 + 4k θr mean 9

From (8) and (9), we have a relation defined for λi forhover defined as

λi = −k2 + k − k

2 10

Considering that about 80% to 92% of the propellerblade diameter effectively produces lift, the analytical

D

L T

𝛺r

Vr V + v1

𝛼

𝛽

𝜃

Figure 3: The blade cross-section showing simplified velocitytriangle and aerodynamic forces.

0.5

0.4

0.3

0.2

0.29

�et

a (ra

d)

r/R

0.48 D

High-pitch propeller pitch change

0.1

0

0 0.2 0.4 0.6 0.8 1

1.0 B0.14 A

0.11 E 0.07 CLow-pitch propeller pitch change

Figure 4: Twist variation along the blade radius.

3International Journal of Aerospace Engineering

solution, considering diameter effectiveness ed , for (6) isas follows:

CT =1

1−ed4k θr2 − λir dr 11

From (10) and (11), we have

CT = 43 kθ 1 − 1 − ed

3 − k k 1 + k − k 1 − 1 − ed2 ,

12

where θ = tan−1 p/πd and k and ed are selected fromTables 1 and 2. Thrust is defined as

T = CTρπR2 ΩR 2 13

The final Thrust-RPM mathematical model and ForceConstant can be defined as

T = kfΩ2, kf =16 ρπR

4e4dCT , 14

where CT can be calculated from (12) and ed is selectedfrom Table 1.

It is well known that the entire propeller is not capableof producing lift as there always exists hub and tip losses.Thus, a range of values of effective diameters has to beapproximated using the available geometric data of ninepropellers (refer to Appendix B).

3. Verification of the Mathematical Model

Verification of the above mathematical model is done using291 available experimental data points obtained fromhttp://www.hobbyking.com, which consists of various motorpropeller combinations. Each data point is described as thethrust generated at a given RPM for a set of a motor andpropeller. 36 propellers are used ranging from 4-inch diame-ter to 16-inch diameter, and 18 different motors are usedranging from 360 kV to 3100 kV. The air density used forthe estimation of all data is 1.18 kg/m3, and the typical edvalues (from Table 1) were used to calculate (estimate) thethrust. A small MATLAB (R2016a version) script is used tocalculate the thrust and generate all graphs.

The graphs in Figure 5 show the thrust estimated at dif-ferent given RPM for few combinations (out of the 291 datapoints) of the motor and propeller in comparison to experi-mental data. These graphs are used to build and verify theproposed mathematical model. On all graphs, “o” pointsare the experimental thrust values at a given RPM and “Δ”points are the estimated thrust values from the proposedmodel for the respective given RPM. This gives us a betterunderstanding of the comparison between the experimen-tally obtained thrust and the estimated thrust. It can beobserved from the results that for typical ed values (fromTable 1) used, the estimated thrust lies within the rangeof ±10% of the experimental data. The four chosen propellershave pitch-to-diameter ratios varying from 1.125 (4× 4.5prop) to 0.3375 (16× 5.4 prop). This ensures that the

proposed mathematical model is valid for a wide rangeof propellers available and the validity of the proposed edas a function of p/d.

It is clear from [21–24] that there are far too manyassumptions made in deriving the Blade Element andMomentum Theory for axial flow across a propeller. To gen-eralize the thrust-RPM relation, there were more assump-tions made with respect to the geometry of the propellerand the flow across it. Thus, it becomes necessary to quan-tify the magnitude of error induced in the results in compar-ison to the experimental data. The results shown in Figure 6are the individual estimated thrust values at a given RPM fora set of motor and propeller combinations. The graph con-sists of all the 291 studied data points, which shows a com-parison between the collected experimental thrust data fromdifferent motor-propeller combinations with the estimatedthrust data which is calculated using the proposed mathe-matical model. The graph shows a normal distribution ofthe error in estimation of the thrust. The error measured isgiven as

Error = Test − TexpTexp

∗ 100, 15

where Test is the value of thrust estimated using the proposedmathematical model against Texp, the experimental value ofthe thrust.

It can be calculated from the graph that more than 70% ofthe estimated values (shaded area in the graph) lie within arange of ±10% of the experimental data. Few estimatedvalues, which are off by more than ±10%, can be due to thelack of knowledge of the experimental setup data, experimen-tal conditions (example air density), exact propeller geome-try, airfoil characteristics, and so forth.

Table 2: Estimated c/d ratio for k calculation.

d inch c/d

4 0.09

5-6 0.1

7–9 0.11

10–12 0.12

13-14 0.13

15-16 0.14

Table 1: Estimated effective propeller diameter.

p/d ed range Typical edp/d < 0.4 0.90–0.92 0.91

0.4≤ p/d< 0.8 0.87–0.89 0.88

0.8≤ p/d< 0.9 0.82–0.86 0.86

p/d≥ 0.9 0.80-0.81 0.80

4 International Journal of Aerospace Engineering

4. Validation of the Mathematical Model

As the verification of the proposed Thrust-RPM mathemat-ical model was done using open source data available onthe website https://hobbyking.com, it will be good practiceto validate the proposed mathematical model using datapublished by different authors who have conducted suchexperiments. The proposed model is validated using twodata sets published by two authors. In the first set(Figure 7(a)) taken from [25], we validate a model whichuses a 10× 4.7 propeller. The blade effectiveness factor, ed ,used is 0.89, and the air density considered is 1.225 kg/m3.It is observed that the estimated data at higher RPM isundercalculated. This implies that the assumed hub losses

for the propeller are greater than what actually occurs inreality and the propeller is more efficient than estimated bythe model.

In the second set (Figure 7(b)) taken from [10], we vali-date a model which uses a 16× 4 propeller. The blade effec-tiveness factor of 0.92 is used, and the density of airconsidered is 1.225 kg/m3. It is observed that the estimateddata is undercalculated but pretty much matches with theexperimental data as the estimated data lies within the −10%limit of the experimental data. From these results, we cansay that the proposed mathematical model of the propellercan estimate promising data.

The above estimations are also affected by the increasein Reynolds number at a higher RPM which alters the

1.4

Experimental dataEstimated data

1.2

0.8

0.6

0.4

0.2

00 2000 4000 6000 8000 10000

N (rpm)

ed (typ) = 0.8

Thru

st (N

)

12000 14000 16000 18000

1

Prop: 4 × 4.5Motor: 1306−3100 kV

(a)

10

Experimental dataEstimated data

9

8

00 0.5 1 1.5 2 2.5 3

×104N (rpm)

ed (typ) = 0.8

Thru

st (N

)

7

6

5

4

3

2

1

Prop: 5 × 4.5Motor: 2205−2750 kV

(b)

7

Experimental dataEstimated data

6

5

4

3

2

1

00 1000 2000 3000 4000 5000 6000 7000 8000 9000

N (rpm)

ed (typ) = 0.88

Thru

st (N

)

Prop: 9.4 × 4.3Motor: 2312−960 kV

(c)

15

Experimental dataEstimated data

10

5

00 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

N (rpm)

ed (typ) = 0.91

Thru

st (N

)Prop: 16 × 5.4Motor: 3510−360 kV

(d)

Figure 5: Verification for the proposed model using experimental data.

5International Journal of Aerospace Engineering

airfoil characteristics (Cl and Cd); that is, the assumptionof having a constant Cl for the entire blade over theentire range of RPM is not always good for high RPM.But we will see in the later part of the paper that thisundercalculation can be compensated for by adjustingfor ed.

5. Conclusion

As observed from all the above results, even though therewill always be some discrepancy between the experimentaldata and the estimated data, the proposed mathemati-cal model

T = 116 ρπR

4e4dCTΩ2, 16

with CT defined as

CT = 43 kθ 1 − 1 − ed

3

− k k 1 + k − k 1 − 1 − ed2 ,

17

does give promising results. The simplification of theinflow velocity estimation is likely to introduce ±10% errorin the magnitude of thrust generated in the majority ofcases. It is interesting to note how the effects of pitch var-iation and rotor diameter effectiveness is captured in themathematical expression developed for CT . Thus, this pro-cess is different from curve fitting as we cannot predict thechange in thrust that can occur due to change in anyother parameter in (16) by curve-fitted data. Figure 6shows the discrepancies between the estimated data andthe experimental data. It is important to note that this

discrepancy occurs at high RPM where it becomes difficultto generalize the aerodynamic phenomenon over such ahuge range of propeller sizes and RPM in this inefficientregime of operation. It was only possible to validate theabove model using propellers ranging from 4-inch diame-ter to 16-inch diameter, the size range which is generallypreferred for small-size UAVs. Appendix C discusses anapplication note of this mathematical model in a real-world scenario.

Appendix

A. Blade Element and Momentum Theory

General momentum theory is a magnificently huge theorywhich deals with the flow characteristics across the propel-ler disk at a much broader level than Blade Element Theoryas discussed in [26] with all its development and assump-tions around it. We first estimate CT using momentum the-ory [13, 14]. A simplified model of a propeller stream tubeis shown in Figure 1. A uniformly loaded disk is consideredand as it has been proved in [27, 28] that even for a uni-formly loaded disk the axial velocity at the propeller diskis not the same throughout, we assume it to be the samefor simplicity of the derivation, and we also consider thatthe axial velocity of the air across the rotor disk, V + v1,has the same magnitude before and after the disk as consid-ered in [15]. There are many more assumptions made inthis theory which are talked about in detail in [21–23]. Asmass is conserved, the mass flow rate across the disk isgiven as

m = ρA V + v1 A 1

From Newton’s Second Law, the rate of change ofmomentum can be defined as the resultant force applied onthe airflow, that is, thrust, which can be defined as the prod-uct of the mass flow rate and the change in velocity

T = ρA V + v1 V + v2 −V = ρA V + v1 v2 A 2

Assuming an ideal system, work done by thrust force perunit time equals the rate of change of translational kineticenergy of the airflow which can be defined as

T V + v1 = 12m V + v2

2 −V2 A 3

Solving (A.3) for v2, using the above derived relations, weget a relation between the far-field velocity in the slipstreamand the inflow velocity at the disk as v2 = 2v1. This relationis an alternative form of the well-known Froude’s law whichstates that the axial velocity at the disk is the arithmetic meanbetween the one at the upstream and downstream (in thewake) infinity [29].

Considering a rotor disk, (Figure 2), at radius r from thecenter, a ring with infinitesimal thickness dr, and consideringthe assumptions mentioned above about uniform disk

Good estimateHigh error estimateed (typical)

−200

10

20

30

40

50

Num

ber o

f sam

ples

Percent error

60

70

80

90

−15 −10 −5 0 5 10 15 20

Figure 6: Comparison between the experimental thrust andestimated thrust for 291 samples.

6 International Journal of Aerospace Engineering

loading, the elemental thrust, dT, generated by this elementalarea is described as

dT = 2ρ V + v1 v1dA = 2ρ V + v1 v12πrdr A 4

Thus, the nondimensional form of the elemental thrustcan be defined as

dCT = dT

ρπR2 ΩR 2 A 5

Effective radius

Lift distributionalong propeller blade

Figure 8: Reference propellers.

8

Experimental dataEstimated data

7

6

5

4

3

2

1

00 1000 2000 3000 4000 5000 6000 7000

N (rpm)

ed = 0.89Thru

st (N

)

Prop: 10 × 4.7

(a)

9

Experimental dataEstimated data

8

7

6

5

4

3

2

1

00 500 1000 1500 2000 2500 3000 3500 4000 4500

N (rpm)

ed = 0.92

Thru

st (N

)

Prop: 16 × 4

(b)

Figure 7: Validation of the Thrust-RPM mathematical model.

7International Journal of Aerospace Engineering

Solving (A.5) for dT , we get (1).Using Blade Element Theory [11, 12], elemental aerody-

namic forces, lift and drag per unit span are respectivelygiven as

dL = 12 ρV

2r cCldr,

dD = 12 ρV

2r cCddr,

A 6

which are resolved (Figure 3) to give the elemental thrust as

dT = 12 ρV

2r c Cl cos β − Cd sin β dr, A 7

where Vr and c are the relative velocity of air experiencedby the blade and the chord of the blade at radius r, respec-tively [19] (Figure 3). Cl and Cd are lift and drag coeffi-cients, respectively, for the propeller airfoil, considered tobe the same, radially, for each cross-section throughoutthe blade. β is the relative inflow angle defined as β = tan−1V + v1 /ΩR . Assuming angle β has very small magnitude,

cos β ≈ 1, sin β ≈ β, and β ≈ V + v1 /ΩR. Substituting thesevalues in (A.8) we get

dT = 12 ρV

2r c Cl − Cdβ dr A 8

Further, considering thin airfoil theory [14] and airfoildata to be origin centered, the product, Cdβ, is assumed tobe very small and the lift curve slope can be approximated

6

Typical estimated data

5

4

3

Thru

st (N

)2

1

00 1000 2000 3000 4000

N (rpm)5000

ed (typ) = 0.91

6000 7000 8000

Prop: 10 × 3.3

(a)

Experimental dataTypical estimated data

7

6

5

4

3Thru

st (N

)

2

1

00 1000 2000 3000 4000 5000 6000 7000 8000

N (rpm)

ed (typ) = 0.91

Prop: 10 × 3.3Motor: 2213−920 kV

(b)

Experimental data

7

6

5

4

3�ru

st (N

)

2

1

00 1000 2000 3000 4000 5000 6000 7000 8000

N (rpm)

ed (adj)=0.95

ed (typ)=0.91

Typical estimated dataAdjusted estimated data

Prop: 10 × 3.3Motor: 2213−920 kV

(c)

Figure 9: Fine tuning of the model based on the experimental data.

8 International Journal of Aerospace Engineering

as Clα ≈ 2π, where α = θ − β and θ is the blade twist. Withthese assumptions, (A.9) can be simplified to

dT = 12 ρV

2r c Clαα

dr = 12 ρV

2r c 2π θ − β dr A 9

Next, considering V + v12 ≪ ΩR 2 gives V2

r ≈ Ωr 2.Using these approximations, we can now define the elemen-tal thrust as

dT = ρπ Ωr 2c θ −V + v1Ωr

dr A 10

From (A.5) and (A.11), the elemental thrust coefficient atradius r can be given as (2).

B. Propeller Geometrical Data Approximation

Nine propellers are shown in Figure 8, which range from 4-inch to 12-inch diameters, and are used to get geometric datalike twist angle, c/d ratio, and effective diameter of the pro-pellers. An approximate effective diameter is measured byeliminating the propeller hub and thin tip section. The rela-tion θ = tan−1 p/πd suggests that θmax will occur for themaximum value of the p/d ratio and vice versa is true forθmin. θroot,max and θroot,min are thus calculated using a 4× 4.5(leftmost) propeller and 12× 4 (rightmost) propeller, respec-tively. The values calculated for θroot,max and θroot,min are 0.34rad and 0.11 rad, respectively.

If a propeller is carefully observed, the blade sectionnear the hub and the tip is tapered to the extent where itbecomes incapable of generating any significant amountof lift. Based on these observations and measurementsmade (of diameter with significant lift-generating area), afew typical effective diameter, ed , values are proposed.There may exist extreme cases where the value of ed fallsout of the proposed range, and one such case is discussedin Appendix C.

C. Application Note

Let us consider a design case where we have concludedthat the propeller suitable for a UAV is a 10× 3.3-inchpropeller to be used with some low-kV motor (typicallyless than 1000 kV). Using the proposed Thrust-RPMmathematical model, we have the following thrust versusRPM characteristic curve as shown in Figure 9(a), plottedusing a typical value of ed. From the validation, we can saythere is a high probability of the estimated data to bewithin ±10% of the actual values. But before the finaldesign, the experiments conducted give the following dataset at discrete points (Figure 9(b)) which can be used tofine-tune the parameters in the proposed model as shownin Figure 9(c).

If we look at the mathematical model carefully, we canunderstand the parameters which vary from system to sys-tem. The value of the thrust is directly influenced by CTand ρ. CT is a function of k, θ, and ed . Further, k is a functionof the ratio c/d. The parameters which have been highlyapproximated in the proposed model are k and ed . Is it

suggested that the tuning of the model should be done inthe following way:

(1) Start with using the corrected value of air density.

(2) Next, try to change ed ; the effective radius of a goodpropeller will lie between 0.75 and 0.95.

(3) Lastly, if the expected value of ed lies outside theabove-mentioned range, try to adjust k.

By following the above procedure, we end up with a finelytuned thrust-RPM model of the propeller as shown inFigure 9(c).

Nomenclature

A: Rotor disk areaα: Angle of attackβ: Relative inflow anglec: ChordCT : Thrust coefficientd: Propeller diameterdD: Elemental drag per unit spaned : Diameter effectivenessλi: Inflow ratioλc: Flow ratiodL: Elemental lift per unit spankf : Motor-propeller Force Constantm: Mass flow rate of airN: Rotation per minuteNb: Number of propeller bladesΩ: Propeller angular velocity rad/sp: Propeller pitchR: Rotor radius from hub to tipdr: Incremental radiusρ: Air densityT: Propeller thrustθ: Blade twist angleV: Free-stream velocityVr : Relative velocityv1: Inflow velocityv2: Increment in far-field velocity.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

Acknowledgments

The author would like to thank Shivani Chandak, GargiKadoo, and Kiran Siddappaji (University of Cincinnati) fortheir comments that greatly improved the manuscript. Theauthors would also like to show their gratitude to TedBaldwin and Bryan Brown from the University of Cincinnatifor helping them with reference papers and for providingpropellers for this research, respectively.

9International Journal of Aerospace Engineering

References

[1] S. R. Kesler, Propeller thrust analysis using Prandtl’s lifting linetheory, a comparison between the experimental thrust and thethrust predicted by Prandtl’s lifting line theory [M. S. thesis],Dept. of Mechanical Engineering, The University of Utah, SaltLake City, UT, USA, 2014.

[2] C. Tjhai, Developing stochastic model of thrust and flightdynamics for small UAVs [M. S. Thesis], The University ofMinnesota, Minneapolis, MN, USA, 2013.

[3] P. Lindahl, E. Moog, and S. R. Shaw, “Simulation, design, andvalidation of an UAV SOFC propulsion system,” IEEE Trans-actions on Aerospace and Electronic Systems, vol. 48, no. 3,pp. 2582–2593, 2012.

[4] O. Atlam and M. Kolhe, “Performance evaluation of directlyphotovoltaic powered DC PM (direct current permanent mag-net) motor-propeller thrust system,” Energy, vol. 57, pp. 692–698, 2013.

[5] M. J. Stepaniak, F. van Grass, and M. U. De Haag, “Design ofan electric propulsion system for a quadrotor unmanned aerialvehicle,” Journal of Aircraf, vol. 46, no. 3, pp. 1050–1058, 2009.

[6] P. Ragot, M. Markovic, and Y. Perriard, “Optimization of elec-tric motor for a solar airplane application,” IEEE Transactionson Industry Applications, vol. 42, no. 4, pp. 1053–1061, 2006.

[7] O. Gur and A. Rosen, “Optimizing electric propulsion systemsfor unmanned aerial vehicles,” Journal of Aircraft, vol. 46,no. 4, pp. 1340–1353, 2009.

[8] D. A. Lawrence and K. Mohseni, “Efficiency analysis for long-duration electric MAVs,” in Infotech@Aerospace Conference,pp. 1649–1661, Arlington, Virginia, USA, September 2005.

[9] T. McElory and D. Brian Landrum, “Simulated high-altitudetesting of a COTS electric UAV motor,” in 50th AIAA Aero-space Sciences Meeting including the New Horizons Forumand Aerospace Exposition, Nashville, TN, USA, January 2012.

[10] D. Kaya, A. Tevfik, A. T. Kutay, and O. Tekinalp, “Design andcontrol of a micro UAV,” in AIAA Atmospheric FlightMechanics Conference, AIAA AVIATION Forum, (AIAA2016-3857), Washington, D.C., USA, June 2016.

[11] H. Amini and S. Steen, “Experimental and theoretical analysisof propeller shaft loads in oblique inflow,” Journal of ShipResearch, vol. 55, no. 4, pp. 268–288, 2011.

[12] J. H. Horlock, Actuator Disk Theory, McGraw-Hill, New York,NY, USA, 1978, Chaps 2, 3.

[13] W. F. Phillips, “Propeller momentum theory with slipstreamrotation,” Journal of Aircraft, vol. 39, no. 1, pp. 184–187, 2002.

[14] M. Amini, L. Sileo, and S. Steen, “Numerical calculations ofpropeller shaft loads on azimuth propulsors in oblique inflow,”Journal of Marine Science and Technology, vol. 17, no. 4,pp. 403–421, 2012.

[15] K. Siddappaji and M. Turner, “Counter rotating propellerdesign using blade element momentum theory,” in Proceedingsof the 22nd ISABE Conference, Phoenix, AZ, USA, 2015.

[16] R. Martinez-Alvarado, F. J. Ruiz-Sanchez, A. Sanchez-Orta,and O. Garcia-Salazar, “Dynamic response of BLDC-thrusterfor small scale quadrotors under aerodynamic load torque,” in2014 IEEE International AutumnMeeting on Power, Electronicsand Computing (ROPEC), pp. 163–176, Ixtapa, Mexico, 2014.

[17] H. Matthew and J. W. Gregory, “Blade element momentummodeling of low-Re small UAS electric propulsion systems,”in 33rd AIAA Applied Aerodynamics Conference, AIAA AVIA-TION Forum, (AIAA 2015-3296), Dallas, TX, USA, June 2015.

[18] B. W. McCormick, “Blade element theories,” in Aerodynamicsof V/STOL Flight, , pp. 79–93, Dover, NY, USA, 1999.

[19] R. W. Deters and M. S. Selig, “Static testing of micro propel-lers,” in 26th AIAA Applied Aerodynamics Conference, Guid-ance, Navigation, and Control and Co-located Conferences,Honolulu, HI, USA, August 2008.

[20] H. Otsuka and K. Nagatani, “Effect of Reynolds numbers of10,000 to 100,000 on rotor blades of small unmanned aerialvehicles,” in 34th AIAA Applied Aerodynamics Conference,AIAA AVIATION Forum, (AIAA 2016-3423), Washington,D.C., USA, June 2016.

[21] R. Bontempo and M. Manna, “Analysis and evaluation of themomentum theory errors as applied to propellers,”AIAA Jour-nal, vol. 54, no. 12, pp. 3840–3848, 2016.

[22] R. Bontempo and M. Manna, “Highly accurate error estimateof the momentum theory as applied to wind turbines,” WindEnergy, vol. 20, no. 8, pp. 1405–1419, 2017.

[23] R. Bontempo and M. Manna, “Effects of the approximationsembodied in the momentum theory as applied to the NRELPHASE VI wind turbine,” International Journal of Turboma-chinery Propulsion and Power, vol. 2, no. 2, p. 9, 2017.

[24] H. A. Madsen, R. Mikkelsen, S. Øye, C. Bak, and J. Johansen,“A detailed investigation of the Blade Element Momentum(BEM) model based on analytical and numerical results andproposal for modifications of the BEM model,” Journal ofPhysics: Conference Series, vol. 75, 2007.

[25] W. Khan and M. Nahon, “A propeller model for general for-ward flight conditions,” International Journal of IntelligentUnmanned Systems, vol. 3, no. 2/3, pp. 72–92, 2015.

[26] J. N. Sørensen, General Momentum Theory for Horizontal AxisWind Turbines, Springer, Switzerland, 2016.

[27] J. N. Sørensen and R. Mikkelsen, “On the validity of the bladeelement momentum theory,” in Proceedings of the EuropeanWind Energy Conference and Exhibition, pp. 362–366, Copen-hagen, 2001.

[28] G. A. M. van Kuik and L. E. M. Lignarolo, “Potential flow solu-tions for energy extracting actuator disc flows,” Wind Energy,vol. 19, no. 8, pp. 1391–1406, 2016.

[29] R. Von Mises, Theory of Flight, McGraw-Hill, New York, NY,USA, 1945.

10 International Journal of Aerospace Engineering

International Journal of

AerospaceEngineeringHindawiwww.hindawi.com Volume 2018

RoboticsJournal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Shock and Vibration

Hindawiwww.hindawi.com Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwww.hindawi.com

Volume 2018

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013Hindawiwww.hindawi.com

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwww.hindawi.com Volume 2018

International Journal of

RotatingMachinery

Hindawiwww.hindawi.com Volume 2018

Modelling &Simulationin EngineeringHindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwww.hindawi.com Volume 2018

Hindawiwww.hindawi.com Volume 2018

Navigation and Observation

International Journal of

Hindawi

www.hindawi.com Volume 2018

Advances in

Multimedia

Submit your manuscripts atwww.hindawi.com