THE DARRIEUS WIND TURBINE: AN ANALYTICAL PERFORMAfiCE STUDY

by

GOPAL B. REDDY, B.E.

A THESIS

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfil Iment of the Requirements for

the Degree of

MASTER OF SCIENCE

Approved

Accepted

May, 1976

" ^ /

" a

ACKNOWLEDGEMENTS

The author wishes to express his deep sense of gratitude to

Dr. J. H. Strickland for his unreserved aid, encouragement and guidance

His keen sense of criticism helped in overcoming the numerous diffi­

culties that were faced during this study. The author also wishes to

express grateful acknowledgement for the suggestions made by Dr. J. H.

Lawrence and Dr. C. A. Bell which helped to improve the thesis.

Thanks are extended to Mrs. Elsie Hayes for her efforts in

typing the paper.

11

ABSTRACT

The purpose of this thesis is to analyze and predict the per­

formance of the Darrieus Wind Turbine. The analysis employs a multiple

stream tube concept. The standard blade element theory is applied to

each of the stream tubes to calculate the induced flow velocity which

is then used to estimate the force on each of the rotor blade elements.

A computer model, which can predict the rotor performance with

reasonable accuracy, was constructed. This model can be used to study

the effects of rotor geometry variations such as blade solidity and

rotor height to diameter ratios.

In addition, the transient behavior of the Darrieus rotor is

studied for a step change in wind speed and a step change in torque

output.

Ill

TABLE OF CONTENTS

ACKNOWLEDGEMEffTS ii

ABSTRACT iii

LIST OF FIGURES v

LIST OF SYMBOLS vi

I. INTRODUCTION 1

II. AERODYNAMIC MODEL 5

2.1. Kinematic Considerations 7

2.2. Blade Element Forces 11

2.3. Momentum Consideratons 15

2.4. Rotor Power Coefficient 19

III. DISCUSSION OF RESULTS OF AERODYNAMIC MODEL 22

3.1. Comparison of the DART Results 22

3.2. Solidity Effects 28

3.3. Torque Variation of the Rotor Vs. 3 28

3.4. Reynolds' Number Effects 29

IV. TRANSIENT ANALYSIS 33

4.1. Step Change in Wind Velocity 33

4.2. Step Change in Torque Extraction 35

V. CONCLUSIONS AND RECOMMENDATIONS 42

LIST OF REFERENCES 43

APPENDIX 44

A. COORDINATE TRANSFORMATION 44

B. ROTOR GEOMETRY AND INERTIA 45

C. COMPUTER LISTINGS 5O

1v

LIST OF FIGURES

Number ^^^^

1.1. Schematic of Darrieus Rotor 2

2 . 1 . Typical Stream Tube 6

2.2. The Blade and Ground Coordinate Systers 8

2.3. L i f t and Drag Forces on Vertical Blade Element 9

2.4. Force Variat ion in any Typical Stream Tube Due

to a Single Blade Element 14

3 . 1 . Variat ion of "a" for a Vert ical Blade Element 23

3.2. Darrieus Rotor Performance Characteristics 24

3.3. Effect of So l id i t y Ratio on the Rotor Performance. . . . 25

3.4.(a) Torque Variat ion on a Two-Bladed Rotor 26 3.4.(b) Torque Variat ion on a Two-Bladed Rotor 27

3.5. A i r f o i l Data from Ref. [6] ,

4 . 1 . The Performance Characteristics of the Rotor for

Gusts With Dif ferent I n i t i a l Velocity Ratios 36

4.2. Higher I n i t i a l Velocity Ratios 37

4.3. Performance of the Rotor Under Variable Loads

With Di f ferent I n i t i a l Velocity Ratios 39

4.4. Higher I n i t i a l Velocity Ratios 40

B . l . Comparison of Troposkien with Some Standard Curves . . . 46

B.2. Rotor Geometry 47

LIST OF SYMBOLS

a Interference factor

A Swept area of the rotor

A|j Plan area of the blade element

A Cross-sectional area of the stream tube

C Chord length of the a i r fo i l

CQ Drag coefficient

C, L i f t coefficient

C Power coefficient

Cj Torque coefficient

C,,C2 Constants defined by equation 2.2.8

D Drag force on a i r fo i l

D' Skin fr iction

F Force in stream tube in x-directi on

T Average force in stream tube in x-directi on

F* Non-dimensional force defined by equation 2.3.11

g Acceleration due to gravity

H Height of the rotor

I Moment of inertia of the rotor about the axis of rotation

L L i f t force on a i r fo i l

N. Number of blades of the rotor 0

P Pressure of the wind upstream near the rotor

P' Pressure of the wind downstream near the rotor

VI

vl i

P^ Pressure of the wind downstream far from the rotor

P Pressure of the wind upstream far from the rotor

r Radius of rotation of any blade element

R Maximum t ip radius of the rotor

t Time

T or T Torque on the blade element

T Average torque on the rotor

T Torque extracted from the rotor

U Wind velocity through the rotor

U Relative velocity of air with respect to blade element

U. Tangential velocity of a blade element

U Free stream velocity 00 **

z Height of a blade element from the center of the vertical axis

GREEK SYMBOLS

9 Angle of rotation of blade element

6 Angle of inclination of blade element to the vertical axis

a Angle of attack

p Density of air

CO Angular velocity

uj Angular acceleration

CHAPTER I

INTRODUCTION

Several kinds of wind operated contrivances have been mentioned

in ancient Indian and Chinese classics dated about 300 B.C. [ 3 ] . The

ea r l i es t windmil ls in the B r i t i sh islands have been traced back to

1191 A.D. By the seventeenth century, the windmill had long been

an established feature of the English landscape. With the wate rmi l l ,

i t was the only means of providing bread for the rural economy of

medieval t imes.

Most of the seventeenth century windmills were horizontal axis

machines. The ea r l i e r ver t ica l axis windmills were generally very

simple in design but were not used extensively because of t he i r

i ne f f i c i ency .

The age in which we l i v e is one of constant change and is the

one in which we have seen the so-called march of progress. Techno­

logical developments have been so rapid over the las t century that

the watermil l and the windmi l l , man's ear l ies t engines, have become

obsolete and have been discarded. However, the recent in terest in

non-depletable energy sources has again led invest igators to examine

the potent ia l of wind power. One of the more promising configurations

which i s now being examined is the jarr ieus turbine patented by

6. J . M. Darrieus in 1931 [ 9 ] . The Darrieus ro to r , shown in

Figure 1 .1 , is a ver t ica l axis wind machine which has two or three

(1) Airfoil blades (2) Vertical mast (3) Generator (4) Guy wires

Flaure 1.1. Schematic of Darrieus Rotor

airfoil blades bent approximately into a sine curve along their

span [9].

The advantages of the Darrieus rotor over conventional windmills

are numerous. It is simple In design and easy to build. If the rotor

Is used for the purpose of generating electricity, the generator can

be mounted on the ground and the need for a large supporting tower is

eliminated. The cost of the Darrieus rotor has been estimated to be

about 1/6 that of horizontal machines [9]. In propeller type wind­

mills, the blade bending stresses increase as the rotor speed increases

These stresses are eliminated in the Darrieus turbine's blades because

of their shape [2]. This also reduces the cost of the construction of

the blades. Above all, the Darrieus rotor accepts wind from any

horizontal direction.

Because of its potential, at least four institutions are con­

ducting experimental and theoretical research on the Darrieus rotor.

The National Aeronautical Establishment of Canada conducted perfor­

mance tests on a 30 inch diameter model rotor and extended the results

to larger rotors by correcting for the blade Reynolds' number effects.

These were later verified on a 12 foot rotor [7]. Personnel at Sandia

Laboratories are presently conducting experiments on a 15 foot three-

bladed rotor with Savonius rotors coupled to it for the purpose of

self-starting [2]. NASA, Langley, tested the performance of a 15 foot

two-bladed rotor in a wind tunnel [9].

The present thesis is the result of work done in the Department

of Mechanical Engineering at Texas Tech University to predict analyti­

cally the performance of the Darrieus rotor under both steady and

transient conditions.

This analysis employs a multiple stream tube model in which a

series of stream tubes are assumed to pass through the rotor. The

blade element theory, in conjunction with momentum considerations.

Is applied to each of the stream tubes to solve for the blade element

forces and hence the power developed by the rotor.

The complex nature of the equations involved in the calculation

of the power coefficient of the rotor suggests the use of a computer.

The associated computer analysis is given the acronym DART for

Darrieus Jurbine.

To establish the accuracy of the results obtained by the DART

model, comparisons are made with the results published by other

research groups [7,10]. The effect of solidity on the performance

of the rotor is then Investigated. The distribution of torque as a

function of rotor position is also plotted. The effect of Reynolds'

number on overall performance is then briefly discussed.

Finally, the transient behavior of the rotor is examined with

step variations in extracted torque and wind velocities.

CHAPTER I I

AERODYNAMIC MODEL

Basically, the present analysis adopts Glauert's blade element

theory [ 4 ] . This theory ut i l i zes the streamwise momentum equation

which equates the streamwise forces on the a i r f o i l blades to the

change in f lu id momentum through the rotor.

The simplest propeller theory assumes a uniform induced velocity

over the actuator disc. This assumption implies that the propeller

disc is enclosed in a single stream tube, which proves to be reason­

ably satisfactory for propellers. However, large variations in the

wind velocity through the Darrieus rotor ex is t . Therefore, a more

sophisticated multi-stream tube theory must be employed. The

multiple-stream tube model is i l lus t ra ted by Figure 2 . 1 . A typical

stream tube is shown passing through the rotor. The width and

height of the stream tube are dy and dz, respectively. The f lu id

velocity through the stream tube at the rotor is denoted by U and is

a function of the angle 0 and the vertical coordinate Z.

There are several simplifying assumptions applicable to the

multi-stream tube model. I t is assumed that there are no in ter ­

actions between the neighboring stream tubes and that the wind flow

is a x i a l . This implies that the presence of the other stream tubes

Is neglected, and also that there is no motion transverse to the

stream tube.

Blade element flight path

Stream tube

Plan View

Upstream View

Figure 2.1. Typical Stream Tube

2.1 Kinematic Considerations

Use of a coordinate system which is stationary with respect to

the blade element is necessary in the kinematic analysis. This

system is called the blade coordinate system. The blade coordinate

system Is to be related to another coordinate system which is fixed

with respect to the ground. This is called the ground coordinate

system. Any expression in the blade coordinate system can be trans­

ferred to the ground coordinate system and the resultant effect on

the rotor can be studied. A detailed derivation of the "coordinate

transformation" is included in the Appendix. The ground and blade

coordinate systems x, y, z and x. , y. , and Zu are as shown in

Figure 2.2. The origin of the ground system is at the center of the

vertical axis of the rotor. Unit vectors along x, y, z and Xc, y^y z^

are t, j", t and t^, J , t^, respectively.

Now, consider a vertical blade element, as shown in Figure 2.3,

rotating in a counter-clockwise direction with an angular speed of uj

in a free wind stream of velocity U . The wind velocity U, as it • ' 0 0 "*

passes through the rotor, is less than the free stream velocity U^.

This is because energy is being extracted from the wind and the wind­

mill exerts a decelerating thrust on the airstream. For a vertical

blade element whose span is parallel to the axis of the rotor, the

relative velocity of air with respect to the blade element is given

by

^bv ^ (Ucose + U ) tjj - Usino j ^

8

Blade element fliaht oath

View AA

A A

Uostream View

Figure 2.2. The Blade and Ground Coordinate Systems

Figure 2.3. Lift and Drag Forces on Vertical Blade Element

10 and the angle of attack a is given by

a = tan sin9 cosO + U^/U

In general, for any blade element whose span is inclined to the

vertical axis by an angle 6, the expressions for the relative velocity

and the angle of attack are obtained as follows.

The relative velocity of air with respect to the blade element,

Uh» is given by the vector sum of the wind velocity near the blade U

and the tangential velocity IJ. of the blade element in the opposite

direction.

U j = U T + U^ T^ (2.1.1)

Substituting the value of i from the "coordinate transformation"

matrix given from Appendix A into Equation 2.1.1, one obtains

Ujj = (U COS0 + U^) tjj - U sine cos3 '3^

(2.1.2) - U sine sin3 k. .

The component of Uu in the Y. direction along the airfoil soan

does not contribute to the lift and drag. Therefore, the relative

velocity in the x ^ and z ^ plane is:

iJ = (U COS0 + U^) tjj - U sine sine t^ (2.1.3)

The magnitude of 3^ is given by

The airfoil angle of attack Is then given by

4.«_-l sinO sins

n

a = tan

2.2. Blade Element f^orces

sinO sinB cose + U^/U

(2.1.5)

The 11ft and drag forces constitute the forces on the blade

element. The l i f t is perpendicular to the relative velocity and the

drag is parallel to - t . Therefore, the unit vector parallel to the

drag force is given Ly

where

P =

In = P TK - Q k,

COS0 + U^/U

(2.2.1)

Q =

[(cosG + U /U)^ + sln^O sin^B] ' ^

sinO sine

[(cos0 + U /U) + sin^e sin^B]^/^

The unit vector parallel to the l i f t f. is perpendicular to "L

in the x z plane.

Therefore,

since

\ = - (Q I'b ^ P ^b' (2.2.2)

The l i f t and drag forces are then given by

L = L 1L = - L (Q Ijj + P k^) (2.2.3)

12

D = D t^ = D (P i, - Q k ) (2.2.4)

Because of skin friction, there is another drag force along the

span of the rotor blade, which is denoted by

5' = D' J (2.2.5)

Substituting the values for i. , j. and k. from Appendix A into

Equations 2.2.3, 2.2.4 and 2.2.5, one obtains

t = - L [Q COS0 - P sin0 sing) 1

+ (Q s1n0 + P COS0 sin3) J + P cosB t]

fi = D [(P COS0 + Q sin0 sinB) "i (2.2.6)

+ (P sin0 - Q COS0 sinB) j" - cosB k]

D' = D' (-sin0 COSB t + cos0 cosB j"- sinB t)

From the above set of equations 2.2.6, one can write the expres­

sions for the forces on the blade element in the x, y and z directions

Letting F , F and F represent these forces in the x, y and z x y z

directions, then in matrix form, they can be written as

F x

V J

Psin0sinB-Qcos0, Pcos0+ Qsin0sinB, - sin0cosB

Qsin0+Pcos0sinB, Psin0-Qcos0sinB, cos0cose

PcosB ,-QcosB , -sinB

D i(2.2.7)

D'

Neglecting the frictional drag D', as it does not contribute to

the torque of the rotor, the expression for F^ can be written as

13 Fj = C L + C2 D (2.2.8)

where

C = (P sin0 SinB - Q cos0)

and

C2 + (P COS0 + Q sine SinB)

The force F acts on the fluid in the stream tube for a fraction

of the total time taken for one revolution. This is only as the

blade element passes through the stream tube. The time average

force In the stream tube is found as follows.

I f N. is the number of rotor blades, the time taken by each of

N. blade elements for one revolution is 2TT/U. The percentage of

time that each of these blade elements spends inside any stream tube

is d0/7r. In other words, the time in one revolution for which the

force F acts in any stream tube is dG/ir. Therefore, the force

exerted by the blade elements on the fluid in the stream tube as

a function of time can be shown as in Figure 2.4.

The time average of the force F for a single blade element is

then given by

F = F ~ (2 2 9)

From Figure 2 .1 , the width of the stream tube corresponding to

a blade rotation through an angle d0 is given by

dy = r d0 |sin0|. (2.2.10)

/

14

o C CO

1 1 1 ' 1 1 1 1 ' • 1

T O

i c CSJ

o o 00

o o <r>

.i o .+ e i

o

O) ^ 3

H-E (O OJ &. + J t / )

r— n3 o . ^

4-» C o; Q. E

> 1—

>> c (O

c • ^

c o • ^

4-> *o • ^ I .

«o >

0) u &-o u.

CVJ

0) L.

<u r— UJ

a; •o <o ^—

CO

0) r—

cr. c • r -

«/)

03

O 4 J

<U 3

O

o>

15 From Equations 2.2.9 and 2.2.10, for N ^ blades of the rotor.

•.. ~ T T : — I _ j_/-v"r , . . . (2.2.11) X irr SinO ^

The vectorial expressions for lift and drag forces allow one to

derive an expression for torque on the rotor as follows.

If F jj is the force on the blade element along the chord line and

r is the radius of rotation, the torque T is given by

T = F^^ r (2.2.12)

An expression for F . is obtained by adding vectorially the com­

ponents of lift and drag along the blade element chord line-

From Equations 2.2.3 and 2.2.4,

^ b = PD - QL.

Therefore, the torque exerted by each of the blade elements at

any angular position is:

T = (PD - QL)r. (2.2.13)

2.3. Momentum Considerations

To obtain the forces on the blade elements, the wind speed passing

through the rotor must be obtained using momentum theory. This theory

was established by W. I. M. Rankine and W. P. Froude for propellers

and was applied to windmills by A. Betz of the Institute of Gottingen

In 1927 [10].

The time averaged streamwise momentum equation is used along with

Bernoulli's equation to relate the average streamwise force in the

16 stream tube to the local velocity U through the rotor.

Let P^ and U^ represent the pressure and velocity of the wind

far upstream from the rotor, and P and U represent the pressure and

velocity of the wind upstream near the rotor. Let also P' represent

the pressure downstream near the rotor and P., and U the pressure and WW

velocity far downstream from the rotor.

The time average momentum equation is used to obtain an expres­

sion for the average force T .

where p denotes the density of air and A denotes the area of the

stream tube.

The thrust on the rotor stream tube is then related to the

pressure difference Immediately before and after the rotor and is

given by

^x = (P - P')A3. (2.3.2)

Applying Bernoulli's equation between two points, one of which

is upstream far from the rotor and the other upstream near the rotor,

one obtains

POO + I P U«= P + ^ P ^ - (2.3.3a)

Similarly, applying the Bernoulli equation downstream far from

and near to the rotor,

P' + ^ P U = P + i u ^ . (2.3.3b)

17 Equations 2.3.3a and 2.3.3b yield

P - P' = 1 P (uf - U ) (2.3.4)

since P = P . w 00

Inserting Equations 2.3.4 into 2.3.2 and then comparing with

Equation 2.3,1, one obtains

U = 2 ""' ^ • • '

Thus, the retardation of the wind (U - U) in front of the rotor 00

is equal to the retardation (U - U ) behind it. w

An axial interference factor "a" can be defined as

(1 - a) U = U. (2.3.6) 00

From Equation 2.3.5,

U = 2U - U . (2.3.7)

Substituting Equation 2.3.6 into 2.3.7, one obtains

U^ = (1 - 2a) U^. (2.3.8)

Inserting Equations 2.3.6 and 2.3.7 into 2.3.1, the average force

in the stream tube becomes

Tj = 2 p A a (1 - a) uf. (2.3.9)

*

Defining F as

* 1 , x Fv = ^ H — (2.3.10)

^ 2 p U" A

18

then, for the differential stream tube of cross sectional area dy • dz.

F^/d2 F = ^ . (2.3.11)

2Tr p r U" sInO

Substituting F^ from Equation 2.2.8 Into 2.3.11

* (C, L + Cp D) dz ^^ = —^ ^ . (2.3.12)

2Tr p r U" SinO 00

For a blade element of length dl, from Equation B.6 of the

Appendix,

dz = dl sin 3-

The plan area of the blade element is given by

Aj = C dl

where C Is the chord length of the airfoil.

The lift and drag forces acting on the airfoil are given by

L = ^ C A u^

0 2 (2.3.13)

Inserting this set of equations Into Equation 2.3.12

*_i_S_v^^^ l^^\^ N^ , X " 47r sinO sin3 I U j R r ^Z.3.14}

where R Is the tip radius of the rotor. The ratio N.C/R is defined

as the solidity factor.

To solve for "a," Equation 2.3.9 is inserted in Equation 2.3.10

19

which gives

a = F* + a^ . (2.3.15) x

Because of the complex nature of Equation 2.3.15, It is solved

for "a" by the following iteration technique. A value for "a" is

first assumed. F is calculated using Equation 2.3.14, which is then

Inserted into the right hand side of Equation 2.3.15 along with the

assumed value of "a." Thus, one obtains a new value of "a." This

new value of "a" is then used and the above process is repeated until

"a" converges to a suitable value within a specified limit of accuracy.

This final value of "a" allows one to determine the force on the

blade element which passes through the stream tube. The DART computer

analysis uses an initial value of zero for "a." "a" is calculated

within an accuracy of 2%.

2.4. Rotor Power Coefficient

From Equation 2.2.13, the torque by H^ blade elements is

T = 2 Njj (PD - QL) r. (2.4.1)

From the set of equations, substituting the lift (L) and drag (D)

Into Equation 2.2.13,

T = Njj p U^ C dl r (P Cjj - Q C^) . (2.4.2)

Defining the coefficient of torque as

p Torque exerted on the blades

20 then

h-\iw] t r ^ (P "D - QCL) • (2.4.3)

The rotor is divided Into 10 parts along its vertical axis.

This divides the rotor blades into a number of blade elements. The

torque coefficient for the complete rotor for any angular position is

given by summing Cj values for all the blade elements making up the

blade.

Thus, Cj Is calculated for different angular positions of the

rotor during one revolution. By adding the torque coefficients at

all angular positions in one revolution and dividing by the number of

positions, one obtains the time average torque T of the rotor. Calcula-

The power developed by the rotor is given by

U.

i tlons are made at every 10" in 0. J

i

J - "t C

I

Here, U. is the tangential velocity of the blade element, where I

r is the radius of rotation of the blade element. ^

If U ^ Is the tip velocity where the radius r is maximum and

equal to R, then U.

P = T - ^ . (2.4.4)

From the definition of the torque coefficient,

T = C^ p U^ R^ . (2.4-5)

The total power available, Pj, in the wind stream of cross-

section A Is 1/2 pAU^ But A. Betz of Gottingen, In 1927, showed

21 that the maximum power that can be extracted by an ideal aerometer is

(16/27) Pj [5]. The power coefficient C is defined as the ratio of

the rotor power to wind power. Some researchers [7] use (16/27) P.j.

as the basis for calculating C . In this analysis P^ = (1/2) pAU^

is used for the power In the wind. Therefore,

r = Rotor power (2.4.6) P 1/2 p A U^

An expression for the frontal area A of the rotor is given by

Equation B.IO obtained in Appendix B.

Inserting Equations 2.4.4, 2.4.5, and B.IO of Appendix B Into

2.4.6, the power coefficient C becomes

r - r ^^^ ^ 'P T U^ 4 •

oo

Program 1, listed in the Appendix, is the actual computer pro­

gram used to compute C of the entire rotor.

CO ^ <

\ t I I f 0

llfriln

CHAPTER III

DISCUSSION OF RESULTS OF AERODYNAMIC MODEL

A primary objective of this work is to test and establish the

accuracy of the results obtained by the DART computer model. The

DART computer results were compared with the experimental and theoreti­

cal results published by different research groups. The DART model

was then used to examine the solidity effects on the rotor performance.

The analysis also includes torque variation of a two-bladed rotor as

a function of 0. Finally, the influence of Reynolds' number on the

rotor power coefficient is discussed.

i

3.1. Comparison of the DART Results ll

} The interference factor "a," as given by Wilson and Lissaman [10], C

1s a sinusoidal function of 0 for a vertical blade element. The •> .

B

values obtained by the DART model are plotted along with the values 0

calculated by their formula as shown in Figure 3.1. It is worthwhile \

to mention that the variation of the interference factor gives an idea

about the variation of wind velocity near the rotor.

The torque per unit length of the vertical blade element as

calculated from the Wilson and Lissaman formula is higher than the

value obtained by the DART model. This Is because they neglected the

drag forces on the airfoil which are included in the present analysis.

The National Aeronautical Establishment (NAE) of Canada conducted

22

23

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25

t °°

1 ti c D R

U

=

=

=

3.

1 7

15

5

MPH 00

R = 7 f t .

Torque f t - l b f

80

40

0

-^0

-80

-120 ^

U

t 360

Figure 3.4(a). Torque Variat ion on a Tv.o-Bladed Rotor

27

Torque f t - l b f

120 _

= 6.0

1 7

360

Torque f t - l b f

-20

\ V

270

U.

u" = 10.n 00

M r _ 1

R

360

Fiqure 3-4. (b) . Torqje Variat ion on a Two-Bladed Rotor

28 wind tunnel experiments on a 30 in. diameter rotor [7] and extended

these results to a 14 ft. dieneter r^tor by correcting for the

Reynolds' number efforts. The power coeff-icient of the rotor was

measured with different numbers of blades but the ratio of airfoil

chord length to tip radius was the same in all ca^es. Figure 5.2

shows the relatively .Ijse agreemen- of the result: obtained by the

DART model with the 'JA experimental results.

3.2. Solidity Effects

Figure 3.3 depicts the effects of solidity on the rotor per­

formance. From Figure 3.3, it is obvious that the rotor has to

obtain an initial velocity ratio of 3 to 3.3 before it prodjces ;'o\':er i

» from the wind. Therefore, the rotor has to be started initially by j

some external means, ^he initial starting velocity ratio, as seen '

from the graph, is weekly dependent upon the solidity ratio. Also

seen from the graph 13 the fact that the maximum power coefficient j »

and the velocity ratio, at which the power coefficient is naxi^un, 3

are functions of the solidity ratio.

3.3. Torque Variation of the^ Rpjtori

Figures 3.4a and 3.4b depict the variation of torque on a tv.o-

bladed Darrieus rotor at different angular positions during one

revolution. These graphs are drawn for different velocity ratios

while the solidity ratio remains the same. It can be seen that the

torque on the rotor is dependent on the velocity ratio and the

29 posi t ion of the blades. From Figures 3.4a and 3 . ^ : , one can notice

the approximate anculer positions of the rotor at which t'le rot r

blades s t a l l , namely at 90° and 270"^. The toraue on the rotor beco:res

negative because of t . e drag force on the blades. Vihen the rotor is

operating at higher vo loc i t / ra t ios , for example 10, skin fr ict^ 'on

drag forces become O'-edominant. As seen f c.Ti these graphs, the

average torque on the rotor is greater when the veloci ty ra t io is 6.0.

Inc iden ta l l y , the power coef f ic ient of the two-bladed rotor is maxinum

at veloci ty ra t io of 6.0.

3.4. Reynolds' iJumber Effects

The a i r f o i l data, namely the l i f t and drag coef f ic ients (Table 1)

used in the DART model, were taken from reference [10] . These data

correspond to a blade Reynolds' number of 3.0 x 10 . Hov.-ever, for

free stream veloc i t ies ranging from 15 f t . / s e c . to 30 f t . / s e c , the

blade Reynolds; number varies from 0.05 x 10 to 0.1 x 10 . But, at

the time of the invest igat ion, the a i r f o i l data for Reynolds' numbers

as low as 0.1 x 10 were not avai lable. At a la ter stage, i t was

found that the a i r f o i l data for Reynolds' numbers ranging from 0.042

X 10^ to 3.1 X 10^ are available [ 6 ] .

Figure 3.5, reproduced from the NACA report , explains the

dependence of the l i f t and drag coef f ic ients on Reynolds' numbers.

From Figure 3.5, i t is evident that the l i f t and drag coeff-icior;ts

are greater for higher Reynolds' numbers.

A detai led study of the ef fect of blade Reynolds' number on the

TABLE 1

AIRFOIL DATA (R 3.0 x 13^) ^ e

30

a deg.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

r

0.09999

0.20003

0.30002

0.40001

0.50005

0.60004

0.70003

0.80008

0.90007

1.0000

1.1001

1.20009

1.2969

1.3723

1.4232

1.4496

1.4517

1.4293

1.3824

1.3110

1.2153

1.09523

0.9506

0.7825

0.5882

0.3702

0.1279

0.000

0.000

0.000

s 0.00585

0.00597

0.006187

0.006475

0.006842

0.007298

0.007814

0.00842

0.00910

0.009865

0.010707

0.011629

o.oirs 0.02183

0.0621

0.1928

0.498

1.0915

1.000

1.000

1.00

1.00

1 .00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

31

rr- as

— - — I ' . " C — T ^ •^

I - '

J V 0

ri

~* "v

:. r, -J I.

««. r

..

. ^ _ . , — \ "

C-e c. — ' ! C :i • I,' y

I X. <^ ^ -;: C ::

V —

' , <; "^ "««

'Z . :*^ c

c<

> ' n '>, -^ ' I ;

1 .

, I — • • ^

r <

— C -

I I

i ' ' I I -C > • t- c _

• - v - '

cr, r-i

• .^mm

-i4' ( I

c :: c

.•IJ. aZ.

l" I

J J '

I

t '

I' , ^ u • • I

I I ,

k- I V t h'

<". - 0

V X

• . t : «j

1 ^ - , — > .. --

I - . v _ •• . ^ ,

1 O O C I 'V ^ -;- — , ——.-

t.

^ .

• • I

1

•-.-K

^ ^

^ •

*^ 0

'; -

's "i

"-»-

' ^ V.

• % *

w V

A, c- ^ ^ - I- - - T- r . . ^ vf , • '_ , •

A ^ .

I • « r ^ ^ T , . ,

-i-T-r

N

•? J:

,;. Q ^' > -- r, >- ~ • - •'" O i » ^ s: ^ ^ c C - if ,

- ^ ^ ^

; : i I ^ <

u:

c

CI

o

in

ro

<U

3

32 rotor performance characteristics is presented by J. H. Strickland

in a Sandia Laboratories report ["]. When compared with the resjits

published in this report, it was found that the : A ? 7 rodel, using the

airfoil data for a higher Reynolds' rubber, gave 5 to 8 oercent higher

values for power coefficients.

As a matter of fi'-ther interest, the a:v>lysis o^ steady state

behavior of the rotor' can be extended to include the effects of wind

shear and other spatial variations in free stream velocity orofi'es [8].

I I t <

CHAPTER IV

TRANSIEi;: ANALYSIS

The changes in the wind velocity are quite abrupt and unpredicta­

ble. The freque^.t ve'jcity variations in the atmospheric wind

necessitate a transient study of the wind turbine.

4.1. Step Change in l-.'ind Velocity

In this case, the response of the rotor to sharp edged g.sts is

studied. A step variation of wind speed is assumed. The wind velo­

city is lowered or raised by some arbitrary constant C defined by

U. C = >0 u U 00

where

^ooO " ^^^^^ velocity at t<^0

U = const, at t > 0. 00

The extracted torque from the rotor is assumed to be linearly

proportional to U^. Therefore,

Torque extracted = (^.1.1)

For equilibrium.

T = I J) + T^ r e

33

(4.1.2)

34 where I Is the moment of inertia of the rotor about the vertical axis,

0) is the angular acceleration of the rotor which is eaual to U^/R, and

T^ is the torque developed by the rotor. Substituting w into Equation

4.1.2, one obtains

^"t R

Substituting for T^ and 7 into Equation 4.1.3, one obtains

do X ^u ^pO 2 \^-^'^} ^0

where U U^Q ^ R % g A U^ t

^ = r ' 0 = i n : ' " ® = 2 1 00 'ooO

Here, 0 is a non-dimensional time.

The initial condition to solve the differential equation 4.4.1 is:

at 0 = 0, X = x^.

The differential equation 4.1.4 is solved numerically by employing

a Runge-Kutta forward integration method. This equation is solved for

arbitrarily chosen values of C (.1, .5, 1, 5, 10) with different

values of XQ (4, 6, 8, 10).

Program 2, listed in the Appendix C, is the computer program

used to solve Equation 4.1.4.

The results are plotted for x as a function of G for all cases

as shown in Figures 4.1 and 4.2. When C^ is greater than 1, it

35 represents a wind velocity drop to a lower value.

As seen from Figure 4.1, the rotor running initially at a lower

speed eventually comes to rest if the wind velocity drops down to a

very low value (C^ = 5, 10). However, with higher initial speeds, as

shown in Figure 4.2, for the same drop in the wind velocity (C =

5, 10), the rotor does not stop, but reaches a steady-state condition

assymptotically. In all of these cases, it may be observed fron

Figures 4.1 and 4.2 that for the same increases in wind velocity, the

rotor does not accelerate as much as it decelerates for the corres­

ponding decrease in the wind velocity.

4.2. Step Change in Torque Extraction

In this case, the extracted torque from the rotor is varied as

a step function.

Assume that the free stream velocity U^ is constant and the rotor

is rotating at an initial tip speed of U^Q. '"atheratically,

U^ = U.n for t < 0 and U = constant. t tu ^

If C is the power coefficient at time t = 0, the initial torque pO

extracted from the rotor is:

T e o 4 ' ' S o f ' ' ^ D ^ u 2 f o r t < 0 . (4.2.1)

For t > 0, the load on the rotor is changed to T e

Te = f ^ t " S o - ' D ^ U i f o r t > 0 (^.2.2)

where C. is the arbitrary multiplier defined by

36

o cc

o i n

o c

•Ii^

cc

S.-c ..-' c

d

^ .cr • - •

* 4 -

O

in u

•r— 4-> I T

•r— J-C'

+J c; <"3 S «T3 x: t j

o e~ c: fC

:_ .-. C

t * -

S-a;

D .

(i> x:

(/ c

'f—

4-^ fC

cr >

4-> • 1 —

(J o

r—

c > 1—

ro •r—

•^ • 1 —

c »—'

s o ]> C'

1

.r: • * — '

•r—

^ I T

+-> W-! ZT

c: ^ o

J -3 C

37

LO

o II

LS"

I L

CVJ oc

o vn

c c

UD

CO i n ex < — , r ^

II

c cv

c cc

l/> - o o ••-4-> 4 -O (O

od ce:

QJ > . C 4-) 4-> • » -

o «4- O C r—

a; tn > . u

• •— r

4-) «C ^ - f—

• r - 4->

s- -^ c c

4_) .—.

u *C '^ 1 - C" ro x :

J C C -<_; ••-

o 1 ^ '

<r c ^

cr Csl

• 5 .»_ C t/1 •- 4-> ^ to a, r ^ cr w i-

-=: C 1— <+-

cv, •

*

c s :3

- • ^ u.

o j D Q :

T C, = ^ . ^ 'eO

38

I f T^ is the rotor torque,

^^t R 5 f = f ( T ^ - T ^ ) - (4.2.3)

Inserting the values for T and T into Ecjation 4.2.3, r e

dx = J2.. L ^ (4.2.4) do X XQ

where

Y = —— y = ZJL

U ' 0 u 00 00

and

1 R ^ P A , u,t u 2 I

The initial condition to solve the differential equation 4.2.- is

at 0 = 0, X = XQ.

The equation 4.2.4 is solved by Runge-Kutta fonvard intearation

method. The equation is solved with different values o* C., ranging

from 0 to 2 in steps of 0.5 for different values of XQ ranging fron

4 to 10 in steps of 2.

Program 3 of Appendix C is the computer program employed to

solve Equation 4.2.4.

The results are plotted for x as a function of 0 as shown in

Figures 4.3 and 4.4. It can be seen from these graohs that when the

extracted torque (load) is the same as the rotor torque, the turbine

runs at a constant speed. However, as seen from Figure 4.3, for

39

- C C cc -r-C •»-'

Q O.'

re • ( -•<- U s- o

J-

o •— 4->

O '^ en c O) c

C -r-

c u in

c •--

ro

1-

J3 c:

40

cc <c

to c o

o

• c re c

<u

<o •r—

C ra z> s-a;

"O c 13

J-o

.»-> c or

1/1

o •f—

4-> ( —*

>» ••- ) •r-*

u o

1 —

CJ ^

c o

00 vn

OJ «D

»4- c

o •— OJ c U CJ

re cr

c

c —> a . •>-

<3-

GJ

41

lower values of i n i t i a l veloci ty rat ios the rotor stops eventually

i f the load torque on the rotor is greater than the rotor t o rc je .

For example, th is happens when C = 1.5. But, as seen f rcr Figure 4.4,

for higher values of i n i t i a l velocity rat ios the rotor does not stop

even when C. is greater than 1. I t decelerates slowly ano reaches a

steady-state value assymptotically. On the other hand, when C^ is

less than 1 , the rotor accelerates and attains steady-state conditions

assymptotical ly.

CHAPTER V

CONCLUSIONS AND RECOMMENDATIONS

The values of C of the rotor as predicted by the multiple stream

tube model are reasonably accurate in comparison with measured values.

The present model is capable of predicting the distributions in blade

element forces and the rotor near wake velocity distribution. The

DART computer model can be used to study the effects of blade

Reynolds' number and the rotor solidity ratio with minor changes in

the computer program. I t can also be used to estimate the contribu­

tion that various blade elements make to the overall power coefficient,

The most serious deficiency of the present model is that, under con­

dition of large solidity and high t ip to wind speed ratios, the

simple momentum considerations inherent in the model break down [ 8 ] .

In considering recommendations for further study, the analysis of

steady state behavior of the rotor can be extended to include the

effects of wind shear and other spatial variations in free stream

velocity profi les. A study of the effects of non-uniform blade chord

distributions with various span wise blade shapes can also be

included [ 8 ] .

Finally, the transient analysis can be extended to include

various types of time dependent variations in freestream velocity.

42

REFERENCES

1. Banas, J. F.; Kadlec, E. G.; and Sullivan, W. N. "Methods for Performance Evaluation of Synchronous Power Systems Utilizing the Darrieus Vertical-Axis Wind Turbine." Sandia Laboratories Report SAND75-0204, April (1975).

2. Blackwell, B. F.; and Feltz, L. V. "Wind Energy - A Revitalized Pursuit." Sandia Laboratories Report SAND75-0166, March (1975).

3. Freese, Stanley. Windmills and Mill Wrighting. London: The Cambridge University Press, (1957).

4. Glauert, H. The Elements of Aerofoil and Airscrew Theory. New York: The Macmillan Co., (194377

5. Golding, E. W. The Generation of Electricity by Wind Power. E. and F. N. Spon Limited, 17955).

6. Jacobs, E. N.; and Sherman, A. "Airfoil Section Characteristics as Effected by Variation of the Reynold's Number." NACA Report No. 586, 23rd Annual NACA Reports 577-611, (1937).

7. South, P; and Rangi, R. "Preliminary Tests of a Hiah Speed Vertical Axis Windmill Model." National Aeronautical Establishment of Canada Report LTR-LA-74, March (1974).

8. Strickland, J- H. "The Darrieus Turbine: A Performance Prediction Model Using Multiple Streamtubes." Sandia Laboratories Report, SAND75-0431, October (1975).

9. Strickland, J. H.; Ford, S. R.; and Reddy, G. B. "The Darrieus Turbine; A Summary Report." Proceedings of the 2nd U. S. National Conference on Wind Engineering Research, P. V-2-1, Colorado State University, July (1975).

10. Wilson, R. E. and Lissaman, B. S. Applied Aerodynamics of Wind Power Machines. Oregon State University, (1974).

43

APPENDIX

The following analysis includes a brief presentation of the

process of coordinate transformation, which was used in the aero­

dynamic model. Also included are the rotor geometry relations which

are used in the DART computer model followed by the calculation of

moment of inertia of the rotor about the axis of rotation. Finally,

the DART computer programs used for the steady and unsteady-state

analysis are listed.

A. The Coordinate Transformation

The transformation from the ground coordinate system to the

blade coordinate system is attained in two stages. The xy plane Is

once rotated about z axis through an angle 0 into an Intermediate

coordinate system x'y'z' and the transformation matrix is written.

Similarly the y'z' plane is then rotated about x' axis through an

angle 3 into the blade coordinate system XO^J^ZL^ and the corresponding

transformation matrix is written. Combining these two transformation

matrices, one obtains the following transformation matrix from the

ground coordinate system to the blade coordinate system.

1i COS0,

-sInO COSB,

-sin© sine.

sinO 0

cosO cos6, -sinB

cosO sinB, cosB ~i \

J

k

(A.l)

44

45 B. Rotor Geometry dr\6_ Inertia

B. F. Blackwell of Sandia Laboratories has shcv.n th .t the trades

should be bent into the shape of a troposkien (turning rope) [5,r] i"

the bending stresses are to disappear at hich rotational speeds. Fig­

ure B.l compares troposkien with son.e of the standard curves, "o

calculate the rotor parameters used in the DART model, it is assumed

that the rotor blades are bent exactly into the shape of a sine curve

along their span. For a given height of the rotor, the frontal area

of the rotor A is maximum when the height of the rotor is equal to

the diameter of the rotor [7j.

To calculate the lernt^h of the differential tlade ele-^ent. The

expression for the rotor radius at different points along the axis

of the rotor (Figure B.2) is given by

^ = cos (TT ) (B.l)

since 2R = H. - ^ > ic^ ^> —^

The non-dimensional height, z/H, ^rop. the center of the axis to

the i^^ blade element is then obtained by

where H/dz is the number of vertical axis division, namely 10.

Differentiating Equation B.l with respect to z,

^ = f s i n h ^ ) . (5.3)

46

GJ > • • - > > S^

m (O OJ (J O E t— D . CU O O) O +J S- c i - fO ' r - ' I—

H - CJ <_3 oo

i n

11

>JQ:

(/I

>

Z5 CJ

TO s -to

-c c

+J 0 0

OJ E o oo

c: <v

00

o Q . O c

in

c o IT

s. . fC CL E o

CJ

CQ

a-

C7

O

c

N | :

47

(U

> i-

+-> OJ E C

J-O +J o or

CVJ .

oc OJ

cr

x iiil&

Also, from Figure B.2 48

Comparing equations B.3 and B.4

(B.4)

tanB = H/R

TTsin (TT TT) (B.5)

The angle thus obtained is then used to evaluate the ratio dl/dz from

the relation

dz _ r..„o -rr - SinB • (B.6)

Final ly , the length of the blade element dl is given by

d l _ d l dz Fi R • d z H R •

(B.7)

To calculate the frontal area of the rotor. The frontal area of

the rotor is the area under two sine curves placed opposite to each

other

In Figure D.2, considering a differential area under the sine

curve,

dA = 2r dz. (B.8)

Inserting Equation B.l into Equation B.8

dA = 2-|^ [R sin (z 1^)] (^dz) (B.9)

Integrating Equation B.9,

kyfiMt

4Q

A = / sin (z y d ^ z)

2 A = 5i- . (B.IO)

IT

This area is called the frontal area or projected area o' swept are^..

To calculate the moment of inertia of the rotor^ about the axis of

rotation. The moment of inertia of a blade element i, of length dl.

about the axis of rotation

I . = (mdl.) r.^

where

m = mass per unit length of the rotor and r. = the radius of

rotation.

Therefore, the :nonent of inertia of the complete rotor is given

by

1=1 2 (mdl.) r.^ . i = l ^ ^

For the Tech rotor under investigation,

m = 0.6 Ibm/in.

n = 10, and thus the calculated value of I is equal to 479.97

2 Ibm - ft. .

TEXAS TECH UBRAR^

C. Computer Listings

Input. The rotor dimensions along with the vertical ratio at

which the characteristics are to studied form the input to the

program. Namely,

HR = H/R, the ratio of height to maximum radius

NB = the number of blades

CC = D/R, chord length to maximum radius ratio and

UU = U^jnax^U^. the velocity ratio.

Output. The output is tabulated for each blade element for

ewery 10° of rotation. The blade position is identified by the

angle G in degrees. The airfoil data, namely the angle of attack

and ratio of list coefficient to drag coefficient, is then printed.

Then, the ratio U/U^ is printed, followed by torque and power

coefficients for all blade elements. At the end of each table, the

power coefficient of the rotor and the corresponding velocity ratio

are printed.

50

PROGRAM 1 5^

REAL MB W R I T E ( 6 , 5 )

C H R = H / K C UU = iJ ' "J^ X/ 'J INF C CC=C/R

? E ^ r ) ( 5 , i o ) " ; ^ t . ' , B , c c 10 FCR'V.K 3 F i O . 5)

on 60 J= I , 15 U l > J X = ..) . 0 " 0 90 1 = 1 , 5

C ZH=Z/H A : = I Z H = ( 2 « A 1 - 1 ) / 2 0 . Y = 3 . 1-*16*Z^^ f Fv = COS( Y) A?.G=b*^/ ( 3 . 1 - 1 6 * 5 I N I Y ) ) eETA=; .T^ r . { AP.GJ

C PL=DL/R ? L = H ^ / ( 1 0 . 0 * ( S I N ( e E T A ) ) ) T = i . n A A = 0 . 0 GO TO 11

11 THETA = { 3 . 1 ' ^ / 1 8 0 . ) * 7 D=( l . O - A ^ i ) IJ-.= UU^5^VC

GA' " i . - ' = C0 5(THE'*^ ) ^-U-. G A ^"* A 3 = C> A " ^ ^ 1 /G.'. "- ^ i. ALPH^ = A' 'AN( 3.\ " ^ ' ; 3)

u F ' , C = S ' M ' ( ( L " S ( " H E T A ) +0= l ^ -Z -^ i { S r ; ( T ^ t ^ i ) ) * « 2 ) - ( ( / S I N ( P ^ T A ) ) - - 2 ) )

C p= ( C ^ S ( T H : T A M U P . i / C E N ' - ^ O = S : M T H E * A ) X - - S P ; ( C E T A ) / c ) E \ : "

c C1=- ( ' J ^CCS ( T H E T i ) - ( P»s I N ( T ^ — : ) * S I \ ( -"^-"A ) ) ) C2 = P*CnS( ^ H F ' A ) 4-( v-S : N ( T M E - i ) ^ s i - . ( S E ^ M )

A O = 5 . ' ' 3 A2=7.O«^A0 S D 0 = 0 . 0 0 5 1 3 0 1 = 0 . 0 0 0 6 ^ 0 2 = 0 . 1 3 S P 3 = 0 . C 1 6 3 5C-t = r. . ( s ) o 6 5-^5= I J : 7 .1 . J ' ' ^ x = o . J l ^

52

A = A L r ^ M I F ( A.GT .A'^** X)Gn ^ r - j CL=A ' ) "A CD=SOU-K S D l - ; ) i - S D 2 * A ' - « 2 GO : n 5 0

20 C L = ( A u - A ) - ( A 2 - { A - A ^ A X ) ) = ^ 2 I F ( CL.GE . J . O J G ' : TO HQ C L = 0 . 0

^ 0 CD=SOi<-Snv* ( * -A MAX )* ^ 2 * S r 5 * ( . - . "^ AX ) » » ^ I'^(C.~).Lf^ . I . O I G O TO 5 0 C D = 1 . 0

50 O L = ' : L / C O

CP= ( ( O L ' ^ c n - c : ) - ^ c c ) / { . - rS( ( " I ' M ^ ^ " ' ^ ^ ' ) ) - { : r . ( c E ' A ) ) ) ) UX= ( ( ( D-'C")S( ' ^ E ^ A ) ) 4-UU-*-- ) ^ * 2 ) M C ' ^Z ) * ( ( S I.*, I T ET A ) )

f-^-^1) * { I S n ( Br i 'A ) ) ^ ^2 ) FXSTAR= ( 1 /12 . 3 - . )*CC*UX»C = ' - . 5 / - -ANE>J-( 1 . 0 - J ) - ^ 2 f r X S T f l C i I F ( ANEW . G T . 1 . ( j )GO T j 70 B = l . 0 - A ^ ; t W I F ( ( D - ? I . G T . J . O D G C TO 100 0 = 3 GO TO 130

70 D = 0 . 0 130 UV^ ( ( (G=«^GnS(THETA) )-»-'JU=»''' )*=>2) + ( C^*2) M ( S I ' M T ^ E T l ) )

/ ^ - 2 ) ^ ( ( S r U EETA) ) ^ ^2 ) ^ ^ = 0 7 ^ : 0 * - L ^ R K * (-Q'^CL-»-P*CP) *V'3 X=X^-C" W R I T E ( 6 , 3 0 ) ^ , C E ^ A , - L ? ^ ^ A , ? L , 3 , V E ' ^ . : T

30 F^R'^AT( 1P7E13 . - ) I '^l " .GE . 17=). ) 3.3 ' C '^i; I - ( ' . ' J . L . ' J . C H . T . = • : . 1 7 J . O ) 3 G ^ i : I I J T = " ^ 1 0 . 0 GG T-. 1 2 0

1 1 ) T = T ^ 9 . 0 1 2 0 ''^^Qi,^'^

r-> - n 11 90 C ^ N T T v j p

CP = ( x / r ^ . ) * u u ^ 3 . 1 ' . / - ^ . >.RITE ( t>, 33)C^ ' ,;JU

30 - G r ^ ' M T i l - i , ^ " C 2 = , EI ^ - b , . S ' - ' J ^ / U : ' i - - » E l ^ . 6 ) 60 CGN^^VJE

W= I T E ( O , 1 - > 0 ) \ E

140 - 3 K V A T ( l - ^ , 3 H \ - = ,E8 .2 ) CALL EX IT EN^

PRCX;RAM 2 53

TIME

10

150

30 60

100 11

PEAL NB REAL rjDT

NOT=N«-'*J-DI.ME\SIOMAL PEAL NOC DIMENSION DUU(250 ) W R I T E ( 6 , 5 )

5 FCPMAT(5X,«N0T» , lOX , • NCT/CU* • 8 X , » L " J » , 10X ,«UU/CU» / f 7 X , ' C P * , l O X , ' C U M . N D T = 0 , 0

J = 0 M^J + 3

R E A 0 ( 5 , 1 0 ) H P , M 5 , C C , 0 T , C U F O R V A T ( 5 F l o , 4 ) C U = ' j r 4 = 0/ 'JI .NP U U I = 1 0 . 0 C P I = 0 . 0 9 R 9 7 4 U U = 1 0 . 0 uun^io.o C P = 0 . 0 8 8 9 7 4 NDC=NnT/ClJ U X = U U / C U W R I T E ( 6 , 3 0 ) N 0 T , N D C , U U f L ' U C , C P , C U F 0 R M A T ( 1 P 6 E 1 3 . 4 ) X = 0 . 0 DO 90 1 = 1 , 5 A I = I Z H = ( 2 = « A I - l ) / 2 0 . Y = 3 . 1 4 1 6 « Z H RQ=C?S{Y) &RG = HP/ ( 3 . U U « S IN'l Y ) ) a E T A = ^ T A \ ( A - ^ G ) RL = HR/ ( 1 0 . 0 * { S i r M E E T A ) ) ) T=1 ,0 A A = 0 . a on T? 11 A i = A N c ^ T H E T A = ( 3 . 1 4 / i e O . ) * T 0 = ( l . n - A A ) UR=UU»FR/0 GAMMA 1=( S I N ( T H E T A ) ) * S I N ( P E T A ) GAMMA2=C3S(THET4 ) 4-UP GAMMA3 = GA -""A 1 / G A ' ^ V A 2

ALPHt = AT^N( G ^ " * ' A 3 )

DEhOv = S;'3,-T( ( C S (ThET-'^ )+0= ) * * 2 «•( ( S IN( T H C T A 1) • • 2 ) « / ( ( S P M S E T A ) i^^Z) )

P = CCnS( THETA ) •uc ) / n e NO'-'

q « S I N ( T H E T A ) » S I N ( F E T A I / n E N O ' -SA

C l = - ( 0 * C C S ( T H E T A ) - ( P » S I N i T H E T i 1 * 5 IN(BET A ) I I C 2 = P * C 3 S ( T H E T A ) * - ( 0 * S I N ( T M E T A ) * S I N ( 5 £ T A ) )

A 0 = 5 . 7 3 A 2 = 7 , 0 * A 0 S D 0 = 0 . 0 0 5 3 S 0 l = 0 . U 0 0 6 S 0 2 = 0 . 1 3 S D 3 = 0 . 0 1 6 S 3 0 4 = 0 . J 0 0 6 S n 5 = 1 2 5 7 0 . 0 A M A X = 0 . 2 1 3 A=ALPHA I F t A .GT.AMAX)GO TQ 20 CL=AO*A CD=S '^U-» - (ST l *A )+S02^A ' - *2

.GO TO 5 0 20 C L = ( A 0 ^ A ) - ( A 2 - ' ( i.-A'^'A X ) ) * « 2

I F ( C L . G E . 0 . 0 ) G n TO 4 0 C L = 0 . 0

40 CD=Sr)3<-SD4«( A - ^ M A X ) * * 2 ' » - S n 5 * ( A - A M 4 X ) * * 4 I F ( C D . L E . 1 . 0 ) G G TO 5 0 C 0 = 1 . 0

50 DL=CL/CO CR=( ( 0 L * C U C 2 ) « C 0 ) / ( APS{ ( S r i ( ^ H E T A ) ) ^ ( S I N I ^ E UX= ( ( (0-* COS (THETA) ) + u U - r R > * * 2 ) + ( D » - 2 ) * ( ( S r > . (

/ * * 2 ) * ( ( S I N ( 3 E T A ) ) « « 2 ) FXST^R= ( 1 / 1 2 . 56) - *CC^UX^C^*NB/RR A N E W = ( I . 0 - ~ ) - - 2 + ^ X S T A R I F ( A \ ' E ' - / . G T . 1 , 0 ) G G TQ 7J !3=1 .0-'^'^'E•>i I F ( ( 0 - 5 ) . G T . 0 . 0 1 )G;l TC 100 n=? GO TO 130

70 D = 0 . 0 130 J V = ( ( (D^C^'S( " "HfTA) ) > U U ^ ^ - 1 * * 2 ) + ( 0 * * 2 ) » ( ( S P : (

/ ' » * 2 ) ^ ( (3 I N ( E E T A ) 1**2 ) CT=UV'*CC='<L^'-r-^ (-C«CL-»-?==CP)«N9 X=X+CT I F ( T . G E - 1 7 9 . ) G n TO 9 0 I F ( T . E J . 1 . 0 . C = . T . E 0 . 1 7 0 . 0 ) G C TC 110 T=T<-10 .a GO TO 120

110 T=T«-9 .0 120 AA=U.O

GO TO 1 1 90 C(>NT!r'UE

r P = ( X / 1 9 , ) - U J » 3 . 1 4 / 4 . C P = - r P IF< J . E g . M ) r , j - 3 i^Q

^A) ) ) ) T- iETA) )

T H E T A ) I

55 J = J * 1 '^ O U U ( J I = ( ( C P / U U ) - ( C U * C P I * U U ) / ( U U I * * 2 ) ) * ^ T I F (J .EO.M)GG TO 80 uu=uun<-(cuu(j))/2.o GO TO 6 0

80 UU=UU04-DUU( J) 60 TO 60

140 J = J<-1 0 U U ( J ) = ( ( C P / U U ) - ( C U » C P I * U U ) / ( ' J U I * * 2 ) ) * " T KsJ^l L = J - 2 N=J-3 DDUU=(DUU(.N)^(2*CUU(L) »*-(2«0UU(K ) )*n'J'J( J) ) / 6 K'DT = NDT-f OT UU0=UUO-».D0UU UU=UUO M = J4.3 GO TO 150 GALL EXIT END

PROGRAM 3 56

REAL N3 PEAL NOT

C NDT=NGr4-DIVENSI0NAL TIME DIMENSr jN OUU(250) W R I T E ( 6 , 5 )

5 FORMAT ( 5X, ' N D T S lOX, 'UU* , l l X , T U U ' , 12X , 'C P» ) NDT=0.0 J=0 M=J^3 PEAD(5 ,10)HO,NS,CC,OT,CTE

10 F0RM4T(5F10 .4 ) C CTE=CQEFFIClENT OF TORQUE E X ^ C A C T E O

DDUU=0.0 U U I = 1 0 . 0

C UUI = I N I T U L VELOCITY R iT IO C C P I = I N ' m A L POWER CCEFrlCIENT CORRESPONDING

/TO UUI C P I = 0 . 0 8 8 9 7 4 U U = 1 0 . ) UUO=10.0 CP=0.Cd8974

150 WPITE(6,35)NDT,UU,D0UU,CP,CTE 30 F0RMAT( IP5E13.4) 60 X = 0 . 0

DO 90 1=1 ,5 AI = I Z H = ( 2 » A I - l ) / 2 0 . Y = 3 . 1 4 1 6 * Z H FR=CO$(Y) ARG = H P / ( 3 . 1 4 1 6 - S I N ( Y ) ) BETA=ATAN(AC,G) FL = H3 / ( l O . O M SIN(BETA)) ) T = 1 . 0 A A = C . O GO TO 11

100 AA = A i.Fw 11 THE'rA = ( 3 . 1 4 / 1 8 0 . )*T

D=( 1 .0 -AA) UR=UU*RR/D GAMMA 1=( S I N ( T H E T A ) ) * S I N ( e E T A ) GAMMA2=C0S(THETA)fUR GAMMA3 = GAMMA1/GA'^MA2 ALPHA=^TAN(GA".VA3) 0 E N 0 ' ' = S C > R T ( ( : C S ( T H E T A ) 4 U K ) * - 2 ^ ( ( S l N ( T H E T & n * - 2 ) »

/ ( ( S I N O E T A ) ) * * 2 ) ) C

P=(COS(THETA)^UP)/OENOv 0=S IN(THcTA)«S IN(bcTA) /DENOV

57

C l= - {0 *COS(THETA) - (P *S IN (THETA ) * S I'M 5 E'A ) ) ) C2=P*C0S(THETA) ^{Q*S I N( THE' A ) * S r . (EE^ A) )

AO=5.73 A2=7«0-^A0 SnO=0.0053 S01=0.0U06 SD2=0 .13 5 0 3 = 0 . 0 1 6 3

. 5 0 4 = 0 . 0 0 0 6 5 0 5 = 1 2 5 7 0 . 0 AMAX=0.218 A=ALPHA IFCA.GT .AMAX)GO ' O 20 CL=AO«A CD=S0O^(S01*A)>S02*A»*2 GO TO 50

20 C L = ( A 0 ^ A ) - ( A 2 * ( A - A M A X ) ) * ^ 2 I F ( C L . G E . 0 . 3 ) G C TO 4o C L = 0 . 0

40 C0=SD3<-S04*( A-AMAX)»*2*-SD54'( 4 -AMAX)* -4 I F ( C n . L E . 1 . 0 ) G 0 TO 50 C 0 = 1 . 0

50 OL=CL/CO CR=( (CL*C1>C2 ) ' C D ) / ( A B S ( ( S I N ( T » - E T A ) ) M S r ' ( B E ' A ) ) ) ) UX=( ( (0*CGS( THETA ) ) -yU'R^* )"* *2 ) •• ( C^-2 ) * ( ( S IN( THETA ) )

/ * * 2 ) « ( ( S I N O E T A ) )*«=2) FXSTAP=(1 /12 .56 ) -CC»UX*C^*N5/RC ANEW=(1 .0 -0 ) - *2^^XSTAR IF(ANEvJ.GT.1.0)GC TO 70 6 = l . 0 - 4 ^ ' E M I F ( ( 0 - B ) . G ^ . U . 0 1 ) G C TO 100 D=B GO TO 130

70 n=o.o 130 UV=( ((0^COS (THETA) )fUU*PR )«*2 ) • ( 0*^ 2 ) • ( ( S ! N C HET A ) I

/ * * 2 ) ^ . ( (S IN ( 3 : T A ) )>»*2) CT=UV-*CC»RL»^K» ( -Q*CL^P*CC)*N9 X = XKT I F ( T . G E . 1 7 9 . ) G 0 TO 90 I F ( T . 2 0 . 1 . 0 . 0 R . T . E Q . 1 7 0 . 0 » G 0 TC 110 T = T + 10.(; GO TO 120

n o T = T ^ 9 . 0 120 AA=0.0

GO TO 11 90 CONTINUE

C P = ( X / 1 9 . ) * U U * 3 , 1 4 / 4 . CP=-CP I F ( J . 5 0 . M I G 0 TO 1^0

ouu( j ) = ( (CP /UU»- :TE*CPI / ' JU I »»OT IFCJ.EQ.M)GO TO eO uu=uun*(cuu(j))/2.o GO TO 60

80 UU=UUO^QU'J( J) GO TO 60

140 J=J + 1 O U U ( J ) = ( ( C P / U U ) - C T E * C P I / U U I ) * 0 T K = J - 1 L = J - 2 N=J-3 0DUU=(D i )U(N)^ (2 *0UU(L ) ) •-( 2-0 L"J(K ) ) OUU( J) ) / 6 NDT = NDr4-DT uuo=uuni-Douu uu=uuo M=J + 3 GO TO 150 CALL EXIT END

58

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