Wind Power Air Turbo-Propeller Driven Vehicle

An All Direction, Wind Powered Car Driven By Turbine-Propeller Airscrew

Y. Neumeier

Introduction

What is a ParadoxFrom the Wikipedia: A paradox is a true statement or group of statements that leads to a contradiction or a situation which defies intuition. The term is also used for an apparent contradiction that actually expresses a non-dual truth (cf. kōan, Catuskoti). Typically, the statements in question do not really imply the contradiction, the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together.

The brain teaserThe following is from

http://www.boatdesign.net/forums/propulsion/ddwfttw-directly-downwind-faster-than-wind-25527-25.htmlBy all accounts so far it was first done in the the '60s by a team of engineers from Douglas Aircraft, but even they weren't the ones who came up with the idea and as they did it to settle a bet between friends, they didn't document their exploits in a way that satisfied very many critics.

AMO Smith (google him), the Supervisor of Aerodynamics Research and Chief Aerodynamics Engineer at Douglas and one of his wind tunnel engineers, Dr. Andrew Bauer discovered the idea in a paper presented by a midwestern student who was angling for a summer internship.

Unfortunately, no one remembers who the student was (they didn't get the intern position) so credit for the actual invention will apparently remain nebulous for all time.

Bauer believed the student was correct and AMO believed DDWFTTW to be impossible. They made a bet and Bauer assembled a small team, built and sucessfully tested the device. AMO paid off on his bet and by all acounts it was near 40 years before anyone physically tried it again.

Andrew Bauer and his car, left, a drawing from his report right

Recent accomplishmentsRick Caravallo Black built a car with which he set a recognized record for driving a wind power car directly down the wind 2.8 times the wind speed1 . The statement that a wind powered car can move directly down the wind in ground velocity much higher than the wind, (above X3 claimed by the black bird in July 2020 trials), appears to many as paradoxical as a paradox comes since the common sail car, like their close relative the sail boat can tack the wind and make velocity made true down the wind higher than the wind but cannot sail directly downward the wind with velocity higher or even equal to that of the wind. In relating to this subject, this author confirmed by personal e-mail exchange that serious professors could not grasp this possibility. One professor is from Berkley university, he was a member in a group that won the Nobel prize with Al Gore, and has been on an advisory board to the government on energy related technologies. Another one from Louisiana State University, has dedicated significant efforts in his blog to prove that such a claim would violate the laws of physics 2. This author

1 See the Autopia blog http://www.wired.com/autopia/2010/06/downwind-faster-than-the-wind/2 http://scienceblogs.com/dotphysics/2008/12/physics-and-directly-downwind-faster-than-the-wind-dwfttw-vehicles.php

find the subject fascinating and believe that the theory presented below beside proving the particular case can be used to develop some practical applications.

The vehicleFigure 1 shows a photo of the black bird and Fig. 2 shows a schematic of the propeller driven car. In the considered car, at least one pair of the wheels is coupled via transmission that, at least for the sake of the discussion, engage the propeller to the wheels either in forward or reverse direction at any desired gear ratio. When the car travel downward with backwind the wheels drive the propeller that in turn produces force that pull the car forward. When moving upwind the propeller must act as wind turbine producing torque on the shaft that is transmitted to the wheels. It should be noted that the startup of the car with backwind down the wind also benefits greatly if initially the propeller act as turbine by changing the pitch and or using reverse gear transmission and gradually move to propeller mode. When moving with an angle to the wind the propeller may need to be turned around.

Figure 1 Blackbird Built by Rick Cavallaro

F w

F p

Car BodyVcar

Vwind

F D

Figure 2 Schematic of considered car

Propeller TheoryPropeller impart “small” increase of velocity to “large” volume of air. The theory presented herein uses the famous disc actuator theory with L/D correction (believed to be an original contribution). First we assume low Mach number thus, neglecting compressibility effects. Further, in reference to Fig. 3. it is assumed that a streamline that start far away at ambient pressure pa and velocity V ∞ follows a constant total energy Bernoulli Eq. all the way just upstream the propeller where the pressure is lower than the ambient thus implying increase of kinetic energy. Upon crossing through the propeller the pressure jump, the velocity does not change since the density is considered constant. From there down the streamline follow again a constant total energy streamline thus further accelerate from the higher pressure behind the prop to the ambient pressure far downstream. It should be noted that while the pressure along the streamline goes to ambient, the downstream velocity is of course different than V ∞,

we denote this velocity by V d

F p

V p

p up d

Figure 3 schematic of propeller flow geometry

In accordance with the above discussion we can write

(1)

Pa+1/2 ρ V ∞2=Pu+1/2 ρ V p

2

(2)

Pa+1/2 ρ V d2=Pd+1/2 ρV p

2

Using the two equations we get

(3)

∆ P≜Pd−Pu=1/2 ρ (V d2−V ∞

2 )

Now conservation of momentum implies that the force extract by the propeller cause the increase in the momentum, thus,

(4)

m (V d−V ∞ )=A p ( Pd−Pu )

wherein m and Ap are the mass flow rate that cross through the propeller and the cross-sectional area respectively. Further we have,

(5)

m=A p ρ V p

Substitute (5) and (3) into (4) gives

Ap ρ V p (V d−V ∞ )=A p1 /2 ρ (V d2−V ∞

2 )=Ap 1/2 ρ (V d−V ∞ ) ( V d+V ∞ )

thus,

(6)

V p=1 /2 (V d+V ∞ )

Equation 6 does not serve us further but it is expressing the fact that the velocity at the propeller is the average of the velocity at infinity and “far” downstream the propeller. It is important to know that all our equations and in particular Eq.6 hold true also when the velocity in infinity come from behind. It is thus clear that a propeller generates thrust also in a backwind. The force that the propeller delivers to the air equals the force the air imparts to the disc and is given by

(7)

F p=Ap ( Pd−Pu )

From (6) we have

(8)

V d=2 V p−V ∞

Substitute (8), (5) and (7) into (4) gives the axial velocity through the propeller disc as function of the propulsive force

(9)

V p=12 (√ 2 F p

ρA p

+V ∞2+V ∞)

Note that Eq. 9 holds for both positive and negative velocities. In particular it shows that when the “wind” comes behind then, for the same delivered propulsive force the velocity through the propeller disc is lesser. The quantity that is most important for us is the shaft power required for a specific thrust. In ideal actuator the only power deliver to the air is at the propeller disc and is given by

(10)

W p=Fp ∙ V p

Thus

(11)

W p

F p

=12 (√ 2 F p

ρA p

+V ∞2+V ∞)

Equation (11) provides the minimum shaft power that is required to produce propulsive force at given ambient velocity. In reality, however the shaft power is greater. We thus, make a first order correction to the above formula using the L/D parameter of the blades.

Consider the blade segment diagram shown in Fig.4. The propeller produces propulsion force F pxfrom

left to right and the shaft provides tangential force F pt to keep the propeller rotating. Note that we analyze here a segment but if all of the segments operate optimally the analysis is valid for the entire propeller. Now,

(12)

F px=L sin β−D cos β

F pt=−( L cos β+D sin β )

Progress direction

Blade Rotation direction

β

LD

Vpt

Apparent wind

Vpx

Fpx

Fpt α

Figure 4. Blade segment velocity-force diagram

The relationship between the lift and drag of a typical profile are shown in Fig. 5 which is taken from the famous NACA report. It shows that at certain angle of attack, typically between 4-5 degrees the L/D ratio reaches a sharp maximum. This is usually the conditions for the propeller blade. Assuming maximum L/D we can write now as follows

(13)

( Fpx

F pt)

actual

=− ( L/D )max sin β−cos β

( L/ D )max cos β+sin β

The values of ( L/ D )max may be 20 or higher, refine profiles may get as high as 60.

Note that if the propeller rotate “slowly” so that β → 0 the ratio of the produced thrust to invested

power is low since F px

F pt

= 1( L/ D )max

the positive sign indicate that the propeller generate retarding

rather the propelling force. When the propeller rotate fast so that β → 90 the ratio is the highest

possible F px

F pt

=−( L/ D )max, the negative sign is indicative that an external force is required to balance

the tangential aerodynamic force F pt. Similar to Eq. 10 the power required is given by

(14)

W p=Fpt ∙ V pt

Accordingly similar to (11) we have

(15)

( W p

Fpx)actual

=V pt

(L /D )max cos β+sin β

( L/ D )max sin β−cos β

Figure 5 L/D of NACA airfoil 0006, taken from “REPORT No. 460 ,The Characteristics of 78 Related Airfoil Sections from Tests in the Variable-Density Wind Tunnel by Eastman N. Jacobs, Kenneth E. Ward, and Robert M. Pinkerton 1933 Web link http://www.aeronautics.nasa.gov/docs/rpt460/index.htm.

Before we compare (15) to (11) we will normalized all relevant quantities using a reference velocity, V ref

We will decide later on what this velocity would be. Accordingly

(16)

V ∞=V ∞/V ref

F px=F px

12

ρ A t V ref2

^W p=W p

12

ρ At V ref3

Equation (11) is written accordingly,

(17)

( ^W p

Fpx)

ideal

=12

(√ F px+V ∞2+V ∞ )

Equation (15) becomes

(18)

( ^W p

Fpx)

actual

=V pt

(L /D )max ∙cos β+sin β

( L/ D )max ∙ sin β−cos β

Equation (9) is written in non dimensional form

(19)

V px=12

(√F px+V ∞2+V ∞ )

We note that

(20)

sin β=V pt

√V pt2+ V px

2

cos β=V px

√ V pt2+V px

2

tan β=V pt

V px

Now noting that, see Eqs. 17,19 that

(21)

( ^W p

Fpx)

ideal

=V px

and using equations 18, 20 and 21 we can write

(22)

^W p−actual

^W p−ideal

=tan β1+1/ ( L/D )max ∙ tan β

tan β−1/ ( L/D )max

The above formula express the actual to ideal shaft power per unit thrust force.

Setting

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70 80 90

Rati

o of

act

ual t

o id

eal s

haft

pow

er p

er u

nit t

hrus

t

β

L/D=3

L/D=5

L/D=10

L/D=20

L/D=30

Figure 6: Ratio of actual to ideal shaft power per unit thrust as function of β for various L/D

Figure 6 above shows the dependency of the actual to ideal shaft power per unit thrust upon the velocity angle β for different L/D ratios. Figure 6 shows that for large enough L/D say, 20 the curve is flat within large range of β, say 15 to 75 degrees. This is very important and, in fact this very treat is what makes the propeller tick. Consulting Fig. 4 it can be seen that in a rotating blade the angle β increases along the blade from the hub toward the tip. Very near the center β is small and thus, see Fig. 6, large power is required. However if we size the blade with proper tip to hub ratio we can have all the blade segments at an angle that is nearly optimal. For example 15 to 75 degrees would provide for tip to hub ratio of fourteen3 . Assuming that the lift to drag ratio is large enough, say ten and above, Eq. 22 can be manipulated to provide some interesting results. Denoting

ε≜1/ ( L/ D )max x≜ tan β, y≜^W p−actual

^W p−ideal

Equation (22) becomes

(23)

3 tan(75 deg)/tan(15 deg)=14.

y=x1+ε ∙ x

x−ε

Take the derivative with respect to x

(24)

dydx

=1+2 εxx−ε

−x (1+εx )( x−ε )2

Setting the derivative to zero yield the following equation for x

(25)

x2−2 εx−1=0

Solving for x

(25)

x=ε+√ε2+1

For small value of ε we can further expand (25) in Taylor series to the first order around zero gives

(26)

x ≈1+ε

Substitute (26) into 23 the and eliminating high order terms of ε in numerator

(27)

y ≈ x1+ε ∙ xx−ε

=(1+ε )+ε ∙(1+ε )2

1≈ 1+2 ε

Using Equation (27) we now arrive at the important formula that provides a first order correction to the disc actuator theory

(28)

( ^W p

Fpx)

actual

≈(1+ 2( L/ D )max

) ∙12

(√ F px+V ∞2+V ∞ )

Vehicle performance at directly down the wind operationReferring now to Figure 2 we can write force balance as

(29)

m V car=F px−FW−FD

Dividing both sides by 12

ρ A t V r ef2 we get

(30)

m V car

12

ρ A t V ref2=F px−FW−FD

We further recognize that the drag on the car can be written as

(31)

FD=CD ∙12

ρ AcarV ∞2∙ sign

where Acar is the frontal area of the car and CD the drag coefficient, sign is the +1 or – 1 according to the direction

Thus,

(32)

FD=CD ∙Acar

At

V ∞2 ∙ sign

Now, the propeller power comes from the wheels, thus,

(33)

W p=FW ∙ V car ∙ ηtrans

Where ηtrans is the transmission efficiency. Dividing both sides of (33) by 12

ρ A t V ref3 one gets

(34)

^W p=FW ∙ V car ∙ ηt rans

Pertaining to our problem it is natural to choose the wind speed as a reference. Being that V ∞ is the apparent ambient velocity, it is given by

(35)

V ∞=V car−V wind

And thus

(36)

V ∞=V car−1

We now rewrite Eq 28 in the form

(37)

^W p ≈ F px(1+2

( L/ D )max) ∙

12

(√ F px+( V car−1 )2+ (V car−1 ))

From Eq.33

(38)

FW=^W p ∙1

ηtrans

∙1

V car

The acceleration of the vehicle can be written as

(39)

a=F px−FW−C D ∙A car

A t(V car−1 )2 ∙ sign

Where sign=1 when (V car−1 )>0 and -1 otherwise. The three equations 37,38 and 39 are what we need

to investigate the performance of the car when moving directly down the wind

Figure 7 below shoes the acceleration characteristics corresponding to car speeds in the range 0.1-4. The characteristics is obtained by running the propeller force as a variable and solving Eq. 37 to obtain the shaft power, calculating then the force on the wheel using Eq. 38 and finally using the propeller force and the wheel force in Eq. 39 to calculate the acceleration. The parameters of the car are given in the caption. The characteristics indicate that from low velocity up to 3 times the wind speed there is a range of propeller operation that accelerate the car forward. At car speed of 3 times the wind velocity the positive acceleration range does barely exist and at speed of 4 all propeller force result in deceleration. Thus, for this car 3 times the wind speed is the top performance. It is worth discussing the acceleration process of the car from rest to maximum speed. Recall that we normalize everything by the wind speed.

Assume now that the designer attempt to achieve as close as possible to 3XW speed. Assume now that he uses fix gear and thus design the propeller and gear to provide at this conditions non dimensional propeller force of 0.5 which provides maximum acceleration at the 3XW speed. Now, starting at rest, the wind is pushing behind and thus make the care move forward. As the car start to move forward the propeller provide small force that according to Fig.7 farther push the car forward followed by higher speed and higher force but this propeller force is never too big to cause retardation of the movement. As a result the car will smoothly and gradually increase to its maximum 3XW speed. If the car has a variable gear the acceleration can be augmented by adjusting the transmission to drive the propeller to generate force which result in maximum acceleration. Nevertheless the maximum achievable speed remain the same.

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 1 2 3 4 5

Non

dim

ensi

onal

acc

eler

ation

Non dimensional propeller force

Vcar/Vwind=0.1

Vcar/Vwind=0.5

Vcar/Vwind=1

Vcar/Vwind=2

Vcar/Vwind=3

Vcar/Vwind=4

Figure 7: Acceleration characteristics of a wind power car with a propeller blade lift to drag ratio ( L/ D )max=15, Total car drag coefficient CD=0.3, propeller disc area to car drag area ratio At / Acar=100 and transmission efficiency of 80%

Off wind performanceTo go in an angle to the wind we have to develop the propeller equation for wind in an angle. We approach this problem assuming first, that, as far as the disc actuator theory is concerned the propeller/turbine delivers only axial forces and does not thus, affects the lateral component of the

moving air. A correction to this will be given when the blade element will be considered. Equations (1) and (2) become

Since the component of the velocity in the y direction does not change we can write

(40)

V ∞=V ∞ x i+V ∞ y j V p=V px i+V ∞ y j V d=V dx i+V ∞ y j

(41)

Pa+1/2 ρ (V ∞ x2+V ∞ y

2)=Pu+1/2 ρ (V px2+V ∞ y

2 )

(42)

Pa+1/2 ρ (V dx2+V ∞ y

2 )=Pd+1/2 ρ (V px2+V ∞ y

2 )

Subtracting (41) from (42) gives

(43)

Pd−Pu=1/2 ρ (V d2−V ∞ x

2 )

Now momentum,

(44)

m (V d−V ∞x )=A p ( Pd−Pu )

where

(45)

m=A p ρ V px

We follow the same steps as before to arrive at

(46)

V px=12 (√ 2 F px

ρA p

+V ∞ x2+V ∞x )

and

(47)

θ

VyRotation

D

D

W p

F px

=12 (√ 2 F px

ρA p

+V ∞ x2+V ∞ x)

It follows thus that a first estimate of the propeller performance can be obtained by only considering the ambient velocity component along the propeller shaft and neglecting the lateral component.

To consider the effect that a side velocity component has on a blade segment we consider Fig. 8 below showing a two opposing blades propeller from the front rotating clockwise with a side wind component Vy coming from the left with the angle θ describing the orientation of the a blades to the wind. When θ=0 degrees the Vy express itself as a radial flow along the blade which we consider to have no effect on the blade aerodynamic forces. With θ=90 degrees the left blade advances into the wind thus experiencing increased tangential velocity Vpt and the right blade which retard from the wind experiencing reduced tangential velocity component Vpt, see Fig.4. Assuming that each blade segment operates nominally at the maximum L/D and consulting with Fig.4, the table below describes the situation of each blade. In accordance with this table it can be seen that the lateral wind will create an unbalance drag and lift that will result in side force in the direction of the lateral wind and a reduction of the effective L/D

Figure 7 Geometry of propeller blades and side wind

Apparent velocity

Velocity angle β

Angle of attack

Lift drag L/D

Advancing bladeRetarding blade

Table I Effect of lateral wind upon the aerodynamic of the blades

To obtain a sense to the magnitude of the side force and the amount of reduction in the effective L/D we do the following analysis.

Consider that the blade element operate nominally at an angle β

Recall Eq. 20

tan β=V pt

V px

Let’s express the side wind speed relative to V px, noting that the latter is an indicative of the

propeller loading. High V px corresponds to high forces as indicated by the normalized Eq.46

(48)

V px=12

(√F px+V ∞ x2+V ∞x )

Thus we will write

(49)

V y=ηV px

Denote the nominal wind angle as β0 we can write

(50)

tan β=V pt+ηV px

V px

=tan β0+¿η sin (θ)¿

From Fig. 4 we see that the angle of attack α increases together with β. Thus,

(51)

α=α0+β−β0where α 0 is the nominal angle of attack for which L/D assume a maximum value, for

example, for NACA airfoil 0006 shown in Fig.5 α 0=5deg. For our discussion let’s consider the variation of L/D with α shown in Fig.5 and parameterize it in as a cubic polynomial

(52)

LD

=a α 3+b α2+cα

with a= 0.0522; b= -1.522; c= 11.305 and α in degrees

The resulted L/D characteristics is shown below

Looking in Fig. 5 it can be seen that for a range of α of 0 to 10 degrees the lift can be expressed in term of

(53)

L=CLα α

Hence, in this range we can write

(54)

L=L0(1+β−β0

α 0)

0

5

10

15

20

25

30

0 5 10 15 20

L/D

Angle of attack degrees

Figure 8: Typical L/D as function of α with maximum L/D=25 at α=5 degrees

Equations 50,51,52 and 54 can be used to solve for the average L/D and side force as function of η for β0 as a parameter. We have to remember that as per our assumptions our analysis is limited to

|β−β0|<∝0 above this the local angle of attack becomes negative at certain rotation angle and more so

the assumption behind Eq. 53 may not be valid as α may get to close to stall.

Figure 9 below shows calculations made for two β0 of 75 deg and 45 degrees. The characteristics

indicate that at β0 of 75 degrees even significant side wind of 100% the magnitude of the axial velocity

through the propeller hardly induce side force and hardly diminish the L/D. However, at β0 of 45 deg a side wind of only 20% the magnitude of the propeller axial velocity induce a 50% side force and reduce the L/D to less that half of its nominal value. It was explained above, in conjunction with Fig. 6 that relatively large tip to hub ratio can be used in a propeller still providing for nearly ideal power to thrust ratio while the inner part of the propeller operate with low β0 of say 15 deg while the outer part enjoy

large β0 of say 75 degrees. It is clear now however that side wind will have a profound effect on the inner part of the blade thus generating significant side force and low L/D from the inner part of the propeller. To account for these effect, at list approximately. We will choose

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0% 20% 40% 60% 80% 100%

Vy/Vpx

Side force% beta =75 deg

(L/D)/(L/D)max beta=75 deg

Side force% beta=45 deg

(L/D)/(L/D)max beta=45 deg

Figure 9 Side force and L/D reduction as function of relative side wind at different β angle