QPROP Formulation

Mark Drela, MIT Aero & AstroJune 2006

This document gives the theoretical aerodynamic formulation of QPROP, which is an analysisprogram for predicting the performance of propeller-motor combinations. The same formulationapplies to the companion design program QMIL, which generates propeller geometries for theMinimum Induced Loss (MIL) condition, or windmill geometries for the MIL or Maximum TotalPower (MTP) conditions.

QPROP and QMIL use an extension of the classical blade-element/vortex formulation, devel-oped originally by Betz [1], Goldstein [2], and Theodorsen [3], and reformulated somewhat byLarrabee [4]. The extensions include

• Radially-varying self-induction velocity which gives consistency with the heavily-loaded ac-tuator disk limit

• Perfect consistency of the analysis and design formulations.

• Solution of the overall system by a global Newton method, which includes the self-inductioneffects and powerplant model.

• Formulation and implementation of the Maximum Total Power (MTP) design condition forwindmills

Nomenclaturer radial coordinateR tip radiusT thrustQ torqueP shaft powerη overall efficiencyβ(r) local geometric blade pitch angleφ(r) local flow angle (= arctan(Wa/Wt))α(r) local angle of attack (= β − φ)Γ(r) local blade circulationT ′(r) local thrust/radiusQ′(r) local torque/radius( )a, ( )t axial, tangential velocity components

B number of bladesV freestream velocityΩ rotation rateu(r) local externally-induced velocity at diskv(r) local rotor-induced velocity at diskW (r) local total velocity relative to bladecℓ(r) local blade lift coefficientcd(r) local blade profile drag coefficientλ advance ratio (= V/ΩR)λw(r) local wake advance ratio (= (r/R)Wa/Wt)ρ fluid densityµ fluid viscositya fluid speed of sound

1 Flowfield velocities

1.1 Velocity decomposition

Figure 1 shows the velocity triangle seen by the blade at some radius r. The axial and tangentialcomponents of the total relative velocity W are decomposed as follows.

Wa = V + ua + va (1)

Wt = Ωr − ut − vt (2)

W =√

W 2a +W 2

t (3)

1

W

Ωr

V

v

v

t

av

φ

t

a

uu

u

φφ

W

Wa

Wt

Figure 1: Decompositions of total blade-relative velocity W at radial location r.

All velocities shown in Figure 1 are taken positive as shown. Typically, va and vt will bepositive for a propeller with positive thrust and torque, and negative for a windmill which willhave negative thrust and torque. The externally-induced ut will be negative if it comes from anupstream counter-rotating propeller, and zero if there is no upstream torque-producing device.

1.2 Circulation/swirl relations

The tangential velocity vt, or “swirl”, is associated with the torque imparted by the rotor on thefluid. It can also be related to the circulation on the rotor blades via Helmholtz’s Theorem. Thetotal circulation on all the blades at radius r is BΓ(r), and this must be completely shed on theblade portions inboard of r. Hence, this is also the half the circulation about a circumferentialcircuit in the rotor plane at radius r

2πr vt =1

2BΓ (4)

vt =BΓ

4πr(5)

where vt is the circumferentially-averaged tangential velocity. The factor of 1/2 in equation (4) isdue to the circuit seeing semi-infinite rather than infinite trailing vortices, as shown in Figure 2.

The circumferential-averaged tangential velocity vt and the velocity on the blade vt are assumedto be related by

vt = vt F√

1 + (4λwR/πBr)2 (6)

The square root term becomes significant near the axis, and the modified Prandtl’s factor F becomessignificant near the tips and accounts for “tip losses”.

F =2

πarccos

(

e−f)

(7)

f =B

2

(

1−r

R

)

1

λw(8)

λw =r

Rtanφ =

r

R

Wa

Wt(9)

In this modified F , the usual overall advance ratio λ = V/ΩR of the rotor has been replaced by thelocal wake advance ratio λw. This is more realistic for heavy disk loadings, where λ and λw differconsiderably. The fact that λw varies with radius somewhat does not cause any difficulties in theformulation.

2

BΓ

Γ

12

Figure 2: Circulation circuits for obtaining circulation/swirl relation.

Using the empirical relation (6) in (5) gives the final relation between the local circulation andlocal tangential induced velocity relative to the blade.

vt =BΓ

4πr

1

F√

1 + (4λwR/πBr)2(10)

The axial induced velocity then follows with the assumption that v is perpendicular to W

va = vtWt

Wa(11)

which is correct for the case of a non-contracting helical wake which has the same pitch at all radii.This assumption is strictly correct only for a lightly-loaded rotor having the Goldstein circulationdistribution, but is expected to be reasonably good for general rotors.

It’s useful to note that relations (10) and (11) are purely local. In effect, the circulation argumentshown in Figure 2, together with the approximate Prandtl tip factor F , have been used in lieu of aBiot-Savart integration over the entire wake. The latter would have related the local vt and va tothe overall Γ distribution, and thus introduced a considerable complication.

2 Blade geometry and analysis solution

2.1 Local lift and drag coefficients

The propeller geometry and velocity triangle are shown in Figure 3.

The local angle of attack seen by the blade section is

α(r) = β − φ (12)

= β − arctanWa

Wt

(13)

which then produces some local blade lift and profile drag coefficients.

cℓ = cℓ(α,Re,Ma) (14)

cd = cd(α,Re,Ma) (15)

3

φφ

W

Wa

Wt

βφφ

W

Wa

Wt

φ

φ

cl

cdcα

Figure 3: Blade geometry and velocity triangle at one r location.

This lift coefficient also determines the corresponding local blade circulation.

Γ =1

2W c cℓ (16)

2.2 Local analysis solution

Given some blade geometry c(r), β(r), blade airfoil properties cℓ(α,Re,Ma), cd(α,Re,Ma) for each radius,and operating variables V and Ω, the radial circulation distribution Γ(r) can be calculated for eachradius independently. This is performed by solving the preceding nonlinear governing equationsvia the Newton method. Rather than iterating on Γ directly, it is beneficial to instead iterate onthe dummy variable ψ, shown in Figure 4.

W

v

ψ

12

vt

va

tW

Wa

a

t

U

U

a

t

U

U

V

au

tu

rΩ

u

U

U

U

Figure 4: Velocity parameterization by the angle ψ.

This ψ parameterizes all the other variables as follows. The convenient intermediate velocity

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components Ua, Ut, are the overall velocities imposed on the rotor, excluding the rotor’s owninduced velocites va and vt.

Ua = V + ua (17)

Ut = Ωr − ut (18)

U =√

U2a + U2

t (19)

Wa(ψ) = 12Ua + 1

2U sinψ (20)

Wt(ψ) = 12Ut + 1

2U cosψ (21)

va(ψ) = Wa − Ua (22)

vt(ψ) = Ut − Wt (23)

α(ψ) = β − arctan(Wa/Wt) (24)

W (ψ) =√

W 2a +W 2

t (25)

Re(ψ) = ρWc/µ (26)

Ma(ψ) = W/a (27)

It is useful to note that with the above expressions, v and W are inscribed in the circle in Figure4, and hence are always perpendicular regardless of the value of ψ.

The circulation is also related to tangential induced velocity via the Helmoltz relation (10).Again parameterizing everything with ψ, we have

λw(ψ) =r

R

Wa

Wt(28)

f (ψ) =B

2

(

1−r

R

)

1

λw(29)

F (ψ) =2

πarccos

(

e−f)

(30)

Γ(ψ) = vt4πr

BF

√

1 + (4λwR/πBr)2 (31)

Finally, the Newton residual is the cℓ–Γ relation (16) recast as follows.

R(ψ) = Γ −1

2W c cℓ(α,Re,Ma) (32)

A Newton update of ψ

δψ = −R

dR/dψ(33)

ψ ← ψ + δψ (34)

will then decrease |R| in the next Newton iteration. The convergence is quadratic, and only a fewiterations are typically required to drive R to machine zero.

2.3 Parameter sensitivities

The simplest analysis case of the rotor has a prescribed velocity V , and also a prescribed rotationrate Ω or advance ratio λ. We might also modify all the blade angles by a constant,

β(r) = βo(r) + ∆β (35)

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where βo is the baseline twist distribution, and ∆β is the radially-constant specified pitch anglechange. Each of these parameters can be set a priori, and the analysis proceeds as described above.

However, in many situations it is instead necessary to specify the overall rotor thrust or torque,or perhaps specify some torque/speed relation for a driving motor. In this case, either V , Ω or ∆βwill be treated as an unknown. For design calculations, we wish to find the blade chord c(r) andblade angle β(r) distributions, again to achieve a specified thrust or torque. So c and β must thenbe treated as unknowns.

In the present approach, the analysis and design cases are treated conceptually in the samemanner. The local circulation is treated as a parameter-dependent function of the form Γ(r;V,Ω,β,c),and any of its four parametric derivatives may be required for the analysis or design case.

ΓV (r) ≡∂Γ

∂V

∣

∣

∣

∣

(Ω,β,c)=const

ΓΩ(r) ≡∂Γ

∂Ω

∣

∣

∣

∣

(V,β,c)=const

Γβ(r) ≡∂Γ

∂β

∣

∣

∣

∣

(V,Ω,c)=const

Γc(r) ≡∂Γ

∂c

∣

∣

∣

∣

(V,Ω,β)=const

We first note that the residual R defined by (32) is a function of not just ψ, but also the fourparameters being considered. A physical solution requires that R remain at zero for all cases,

R(ψ;V,Ω,β,c) = 0 (36)

which implicitly defines the ψ(V,Ω,β,c) function. The actual value of ψ is obtained numerically from(36) by the Newton iteration procedure described previously. Its parametric derivatives are thenderived by setting the variation of R to zero, since (36) must hold for any physical perturbation.

δR =∂R

∂ψδψ +

∂R

∂VδV +

∂R

∂ΩδΩ +

∂R

∂βδβ +

∂R

∂cδc = 0 (37)

δψ = −∂R/∂V

∂R/∂ψδV −

∂R/∂Ω

∂R/∂ψδΩ −

∂R/∂β

∂R/∂ψδβ −

∂R/∂c

∂R/∂ψδc (38)

The variation coefficients on the righthand side of (38) are seen to be the derivatives of ψ(V,Ω,β,c).

∂ψ

∂V= −

∂R/∂V

∂R/∂ψ(39)

∂ψ

∂Ω= −

∂R/∂Ω

∂R/∂ψ(40)

∂ψ

∂β= −

∂R/∂β

∂R/∂ψ(41)

∂ψ

∂c= −

∂R/∂c

∂R/∂ψ(42)

With these, the parametric derivatives of any converged quantity can now be computed via thechain rule. For example, the circulation and its parametric derivatives are computed as follows.

Γ(ψ;V,Ω,β,c) = vt4

BF

√

(πr)2 + (4λwR/B)2 (43)

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ΓV =∂Γ

∂V+

∂Γ

∂ψ

∂ψ

∂V(44)

ΓΩ =∂Γ

∂Ω+

∂Γ

∂ψ

∂ψ

∂Ω(45)

Γβ =∂Γ

∂β+

∂Γ

∂ψ

∂ψ

∂β(46)

Γc =∂Γ

∂c+

∂Γ

∂ψ

∂ψ

∂c(47)

Parametric derivatives of other required quantities

Wa(V,Ω,β,c) , Wt(V,Ω,β,c) , cℓ(V,Ω,β,c) , cd(V,Ω,β,c)

are obtained by this same approach. Although the blade angle variables are defined to be βo(r)and ∆β, only the one β-derivative is needed for both, since from (35) we see that ∂/∂β = ∂/∂βo =∂/∂∆β.

3 Thrust and torque relations

After the Newton iteration procedure described previously is performed for each radial station, theoverall circulation distribution Γ(r) is known. This then allows calculation of the overall thrust andtorque of the rotor as follows.

3.1 Local loading and efficiencies

As shown in Figure 5, the blade lift and drag forces are resolved into thrust and torque componentsby using the net flow angle φ.

W

Wa

Wt

φφ

W

Wa

Wt

φ

φ

dL

dD

dT

dQ / r

dLdD

Figure 5: Blade lift and drag resolved into thrust and torque components.

dL = B1

2ρW 2 cℓ c dr (48)

dD = B1

2ρW 2 cd c dr (49)

dT = dL cosφ − dD sinφ

= B1

2ρW 2 (cℓ cosφ − cd sinφ) c dr (50)

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dQ = (dL sinφ + dD cosφ) r

= B1

2ρW 2 (cℓ sinφ + cd cosφ) c r dr (51)

From the velocity triangle in Figure 5 we see that

W cosφ = Wt (52)

W sinφ = Wa (53)

so the thrust and torque components can be alternatively expressed in terms of the circulation andnet velocity components.

dT = ρBΓ (Wt − ǫWa) dr (54)

dQ = ρBΓ (Wa + ǫWt) r dr (55)

where ǫ =cdcℓ

(56)

The local efficiency is

η =V dT

Ω dQ(57)

=V

Ωr

cℓ cosφ − cd sinφ

cℓ sinφ + cd cosφ(58)

=V

Ωr

Wt − ǫWa

Wa + ǫWt

(59)

which can be decomposed into induced and profile efficiencies.

η = ηi ηp (60)

ηi =1− vt/Ut1 + va/Ua

(61)

ηp =1− ǫWa/Wt

1 + ǫWt/Wa

(62)

In the limits V/ΩR→0, ǫ→0, and with ua=ut=0 (no externally-induced velocity), the efficiencyreduces to

η →1

1 + va/V(63)

which exactly corresponds to the actuator disk limit, even for arbitrarily large disk loadings.

3.2 Total loads and efficiency

The total thrust and torque are obtained by integrating (54) and (55) along the blade.

T = ρB

∫ R

0Γ (Wt − ǫWa) dr ≃ ρB

∑

r

Γ (Wt − ǫWa) ∆r (64)

Q = ρB

∫ R

0Γ (Wa + ǫWt) r dr ≃ ρB

∑

r

Γ (Wa + ǫWt) r ∆r (65)

The simple midpoint rule is used for the integral summations. The overall efficiency is then

η =V T

ΩQ(66)

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3.3 Parametric derivatives

In order to drive an analysis solution to a specified thrust or torque, their values defined by (64) and(65) must be considered to be functions of the form T (V,Ω,∆β), Q(V,Ω,∆β). Their derivatives, whichare required for Newton iteration, are obtained by implicit differentiation inside the summations,and by using the previously derived parametric derivatives of Γ, Wa, and Wt. For example, for theV -derivatives of T and Q we have

TV = ρB∑

r

[

ΓV (Wt − ǫWa) + Γ (WtV − ǫWaV − ǫVWa)]

∆r (67)

QV = ρB∑

r

[

ΓV (Wa + ǫWt) + Γ (WaV + ǫWtV + ǫVWt)]

r ∆r (68)

where ǫV =cdVcℓ− ǫ

cℓVcℓ

(69)

The Ω- and ∆β-derivatives of T and Q are computed in the same manner.

In order to drive a design solution to a specified thrust or torque, their values defined by (64)and (65) must be considered to be functions of the form T (β,c), Q(β,c). Their β- and c-derivatives,which are defined for each radial station, are obtained simply by taking the one summation termfor that radius, e.g.

Tβ = ρB[

Γβ (Wt − ǫWa) + Γ(

Wtβ − ǫWaβ − ǫβWa

)]

∆r (70)

and likewise for Qβ, Tc, Qc.

4 Analysis

The analysis problem is to determine the loading on a rotor of given geometry and airfoil properties,with some suitable imposed operating conditions. The unknowns are taken to be Γ(r), V , Ω, ∆β.The constraints on Γ(r) are always the Newton residuals defined previously by (32) at each radialstation.

Rr(Γ(r)) = Γ −1

2W ccℓ (32)

The three residuals which constrain the remaining three variables V , Ω, ∆β will depend on thetype of analysis problem being solved. Some typical constraint combinations might be:

1) Specify velocity, RPM, pitch:

R1(V,Ω,∆β) = V − VspecR2(V,Ω,∆β) = Ω − Ωspec

R3(V,Ω,∆β) = ∆β − ∆βspec

(71)

2) Specify velocity, RPM, torque (find pitch of constant-speed prop to match engine torque):

R1(V,Ω,∆β) = V − VspecR2(V,Ω,∆β) = Ω − Ωspec

R3(V,Ω,∆β) = Q − Qspec

(72)

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3) Specify velocity, pitch, thrust (find RPM to get a required thrust):

R1(V,Ω,∆β) = V − VspecR2(V,Ω,∆β) = ∆β − ∆βspecR3(V,Ω,∆β) = T − Tspec

(73)

4) Specify RPM, pitch, thrust (find velocity where required thrust occurs):

R1(V,Ω,∆β) = Ω − Ωspec

R2(V,Ω,∆β) = ∆β − ∆βspecR3(V,Ω,∆β) = T − Tspec

(74)

The three chosen residuals are simultaneously driven to zero by multivariable Newton iteration.

δVδΩδ∆β

= −

∂(R1,R2,R3)

∂(V,Ω,∆β)

−1

R1

R2

R3

(75)

If R3 is either a T or Q constraint, its Jacobian matrix entries are already known from the para-metric derivatives of T (V,Ω,∆β) or Q(V,Ω,∆β), computed as described previously.

5 Design

The design problem is to determine the geometry of a rotor which matches specified parameters,typically R, V , Ω, and either T or Q. The blade airfoil properties are also specified. The two un-knowns at each radial station are taken to be β(r) and c(r), which therefore requires two constraintsto be imposed at each radial station. One such constraint, used for all design cases, is simply aspecified local lift coefficient.

Rr1 = cℓ − cℓspec (76)

For the second constraint, different options are used depending on the type of design problem beingsolved.

5.1 Minimum Induced Loss

In this design case, the local induced efficiency is required to be radially constant, and equal tosome initially-unknown value η. Setting ǫ=0 in equation (59) gives a suitable expression for theinduced efficiency for this constraint.

V

Ωr

Wt

Wa= η (77)

Rr2 = η ΩrWa − V Wt (78)

5.2 Overall load constraint

The design residual (78) has introduced η as one additional unknown. This in effect controls theoverall propeller load, since a large thrust or torque typically implies a small induced efficiency,and vice versa. Therefore, a suitable constraint for η is to impose a specified thrust or torque.

Rη = T − Tspec (79)

or Rη = Q − Qspec (80)

10

Since Ω is assumed to be given, specifying Q is equivalent to specifying the shaft power P = ΩQ.

Imposing Rr1 =0 and Rr2 =0 at each radial location, together with the one additional Rη=0,will give a propeller with a Minimum Induced Loss (MIL). Such a propeller will have the maximumpossible overall induced efficiency for its radius and operating conditions. It is the analog of theelliptically-loaded wing, which has a spanwise-constant induced-drag/lift ratio.

5.3 Maximum windmill power

The MIL design residual (78) is applicable to a windmill, provided cℓ(r), and T or Q are set negative.This will give a windmill with the most shaft power for a given tower load, or the minimum towerload for a given shaft power.

An alternative design objective for a windmill is to simply maximize shaft power, regardless ofthe tower load. Actuator disk theory predicts that in the limits λ→0, ǫ→0, the maximum (mostnegative) shaft power is

Pmax = −8

27ρV 3 πR2

In the presence of profile and swirl losses, the actual power will be less than this. It is of greatinterest to maximize the power in the presence of these losses.

With a specified Ω, maximizing P is equivalent to maximizing the torque Q. Because all theradial blade stations are assumed independent, it is sufficient to maximize the differential torquedQ. From equation (55), and the Γ–vt relation (10) we have the following.

dQ = ρBΓ (Wa + ǫWt) r dr (81)

dQ = ρ vt (Wa + ǫWt) 4π r2 F√

1 + (4λwR/πBr)2 dr (82)

Figure 6 illustrates how this dQ varies for the inviscid case, with vt negative as for a windmill. Theproduct dQ ∼ vtWa clearly has an extremum at an intermediate loading. Since all the quantities inthe dQ expression above are parameterized by ψ, we can extremize dQ by setting its ψ–derivativeto zero, taking the log first for algebraic simplicity.

ln(dQ) = ln vt + ln (Wa + ǫWt) + lnF + ln(4π r2 ρ√

1 + (4λwR/πBr)2 dr) (83)

dQ′

dQ=

v′tvt

+W ′

a + ǫW ′

t + ǫ′Wt

Wa + ǫWt+

F ′

F= 0 (84)

where ( )′ ≡∂( )

∂ψ

In practice, both F and ǫ are very nearly independent of ψ, so that F ′≃ 0 and ǫ′≃ 0. Also, from(20), (21), (23) we have

W ′

a = 12U cosψ = Wt −

12Ut (85)

W ′

t = −12U sinψ = 1

2Ua − Wa (86)

v′t = 12U sinψ = Wa −

12Ua (87)

so that (84) can be simplified to the following residual.

Rr2 =Wa −

12Ua

Ut −Wt+

Wt −12Ut − ǫ(Wa −

12Ua)

Wa + ǫWt(88)

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ψ

vt

Wa

ψ

dQ v Wt a~

BA C

B

A

B

C

max windmilltorque & power

A

C

propellerwindmill

propeller

windmill

Figure 6: Three different windmill loadings. The windmill torque and power are maximum atloading B.

By taking the usual blade geometry variables β(r) and c(r) as unknowns, and driving the residuals(76) and (88) to zero at each radial location, produces a windmill with a Maximum Total Power

(MTP). Both T and Q are a result of this calculation, and a total load constraint such as (79) or(80) in the MIL case is not required here.

5.4 Moderated maximum windmill power

We now consider designing a windmill which intentionally delivers less power than the theoreticalmaximum at point B in Figure 6. Such a “moderated” optimum is point B’ in Figure 7.

ψ

dQ

B

B’

ψB’

va

av’

dQ’dQ

va

Figure 7: Moderated sub-optimum windmill power at point B’. Axial velocity gradient v′a is usedto normalize nonzero dQ′.

This moderated optimum is selected by specifying some nonzero value for the local fractional

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torque gradient dQ′/dQ. Because a suitable value is not easily determined a priori, it is implementedto be a specified multiple of a reference gradient. A convenient choice for this reference is v′a/va,since v′a varies little over a significant range of ψ values, as can be seen in Figure 7. This referenceratio is readily evaluated from equation (20) and (22).

va = Wa − Ua = −12Ua + 1

2U sinψ (89)

v′a = 12U cosψ = Wt −

12Ut (90)

The maximum-power statement (84) is now replaced by

dQ′/dQ

v′a/va= K (91)

where K is a constant which specifies the degree of deviation away from the optimum. Withsubstitutions for dQ′/dQ and v′a/va, equation (91) is recast as an alternative residual which replaces(88).

Rr2 =

[

Wa −12Ua

Ut −Wt

+Wt −

12Ut − ǫ(Wa −

12Ua)

Wa + ǫWt

]

Wa − Ua

Wt −12Ut

− K (92)

The constant K = O(1) is set by the designer. Choosing K = 0 recovers the true maximum powercase, while choosing K > 0 gives progressively smaller power values.

The advantage of a moderated optimum stems from practical reasons. Because the torque Qand the power P = QΩ are stationary with respect to K, the loss in available power scales as

P − P0 ∼ K2

where P0 is the maximum power at K = 0. In contrast, the thrust T (or tower load) and typicalblade chord c differences change linearly with K.

T − T0 ∼ K

c− c0 ∼ K

Figure 8 shows these variations quantitatively for a typical windmill. Choosing K = 0.2, forinstance, will incur only a 2.3% power penalty, but will give 8.5% smaller tower load and bladechords. The smaller tower load is likely to give the windmill a higher maximum safe operatingwindspeed, so that the moderated-optimum windmill may actually produce more time-averagedpower if maximum safe winds are occasionally encountered. Alternatively, for a given tower load orwindmill material cost, the moderated-optimum windmill can have a slightly larger diameter andhence actually produce more power. It is therefore certain that the moderated optimum may bebetter than the true fixed-diameter and fixed-windspeed optimum, when overall system costs andother operating considerations are taken into account.

References

[1] A. Betz. Airscrews with minimum energy loss. Report, Kaiser Wilhelm Institute for FlowResearch, 1919.

[2] S. Goldstein. On the vortex theory of screw propellers. Proceedings of the Royal Society, 123,1929.

13

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7K

P / P0

T / T0

Figure 8: Variation of power P and thrust T with power-moderating constant K, for four-bladedwindmill with V/ΩR = 0.125. Blade chord ratio c/c0 is nearly equal to T/T0.

[3] T. Theodorsen. Theory of Propellers. McGraw-Hill, New York, 1948.

[4] E.E. Larrabee and S.E. French. Minimum induced loss windmills and propellers. Journal of

Wind Engineering and Industrial Aerodynamics, 15:317–327, 1983.

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