turbines is presented. that includes tip-loss and blade
TRANSCRIPT
AN ABSTRACT OF THE THESIS OF
STEL NATHAN WALKER for the degree of DOCTOR OF PHILOSOPHY
in MECHANICAL ENGINEERING presented on May 10, 1976
Title: PERFORMANCE AND OPTIMUM DESIGN ANALYSIS!
COMPUTATION FOR OPELLER TYPE WIND TURBINES
Abstract approved by:Ro!ert E. Wilson
Development of a performance analysis for propeller type wind
turbines is presented. The analysis uses a strip theory approach
that includes tip-loss and blade interference. A comparison is
presented between analysis and test data. The comparison indicates
that the analysis predicts wind turbine performance to an accuracy
that is within experimental uncertainty.
Design procedures for optimum wind turbines are shown to be
different than those used for propellers. An optimum design genera-
tion approach for wind turbines is developed from a modified strip
theory that includes tip-loss. The approach entails a local optimization
of b'ade element parameters to maximize power output. Examples are
presented that illustrate the optimum design generation procedure and
off- design performance characteristics.
Redacted for Privacy
The design analysis and optimum design generation approaches
have been programmed on a digital computer using the Fortran IV
language. Program listings, operating instructions, and sample out-
puts are included.
Performance and Optimum Design Analysis /Computationfor Propeller Type Wind Turbines
by
Stel Nathan Walker
A THESIS
submitted to
Oregon State University
in partial fulfillment ofthe requirements for the
degree of
Doctor of Philosophy
Completed May 1976
Commencement June 1976
lb.
APPROVED:
Profesor of Mechanical Engineering
in charge of major
Head of Dpartment oi'Mechaiical Engineering
Dean of Graduate School
Date thesis is presented
Typed by Clover Redfern for
W May 1976
Stel Nathan Walker
Redacted for Privacy
Redacted for Privacy
Redacted for Privacy
ACKNOWLEDGMENTS
The completion of a thesis involves the direct assistance of few,
but the indirect assistance of many. It is with pleasure that I
acknowledge their assistance.
Dr. Robert F. Wilson, my major professor, who has been
instrumental in providing guidance, encouragement, criticism, and
financial support.
Dr. James R. Welty, the department head, who provided
teaching assistantships that enabled my pursuit of graduate studies.
Ms. Anita Lang and Mr. Ron Hill, who patiently inked the
figures and Mrs. Clover Redfern, who typed the manuscript.
The Oregon State University Computer Center, which provided
computer time through unsponsored research grants for phases of the
research.
And my wife, Sally, son Bryan, parents Mr. and Mrs. Billie
F. Walker, and friends who provided untiring support and encourage-
ment.
4
TABLE OF CONTENTS
at Page
I. PERFORMANCE AERODYNAMICS OF PROPELLERTYPE WIND TURBINES 1
Introduction 1
1. 1 Axial Momentum Theory 31.2 Momentum Theory in Consideration of Flow Rotation 91.3 Blade Element Theory 131.4 Strip Theory 171.5 Tip-Loss Corrections 21
Prandtl's Tip-Loss Factor 23Goldstein's Tip-Loss Factor 27Effective-Radius Tip-Loss Concept 28Tip-Loss Factor Application 30Hub-Loss (Inner Blade Tip-Loss) Effects 32
1.6 Cascade Theory 33Effect of Thickness 36
1. 7 Sectional Aerodynamic Data 371. 8 Windmill Brake State 39
II. PERFORMANCE PREDICTION 46Introduction 46
2. 1 NASA's MOD 0 Wind Turbine Performance Analysis 472, 2 Chalk Wind Turbine Performance Analysis
Compared with Field Data 552. 3 Experimental Performance Verification 57
lIt. OPTIMUM PROPELLER-TYPE WIND TURBINES 75Introduction 75
3. 1 Local Optimization 82Standard Tip-Loss Application 83Second Order Method of Tip-Loss Application 85
3. 2 Constant Wake Displacement and Constant AxialVelocity Methods of Optimization 90
3. 3 Optimum Rotor Configuration and Performance 92Example Cases 104
IV. SUMMARY AND CONCLUSIONS 121
BIBLIOGRAPHY 123
Page
APPENDICES 130Appendix A: Design Analysis Program 130
Program Operation 130Program PROP Listing 139Program PROP Output Listing 149
Appendix B: Optimum Design Generation Programs 157Program Operation 157Program OPTO Listing and
Sample Output 159Program AXOPTO Listing and
Sample Output 168Appendix C: General Momentum Theory 178Appendix D: Glauert's Optimum Wind Turbine Analysis 189Appendix E: Theodorsen-Lerbs Approach Applied
to Wind Turbine Optimization 196Introduction 196Flow Geometry 197Power Equation Development Z03Thrust Equation Development 208Slipstream Expansion 209Approach Inconsistency 213
>
LIST OF FIGURES
Figure Page
1. 1. 1. Control volume used for a wind turbine. 4
1. 1.2. Induced axial velocity variation with distance (y) fromrotor plane (r = 0). 7
1.2.1. Flow diagram of a wind turbine. 11
1.2.2. Rotor blade element. 11
1.3. 1. Blade coning configuration. 14
1.3.2. Velocity diagram. 16
1.5.1. Tip and hub-losses--flowdiagram. 22
1.5.2. Circulation distribution along a blade for a two bladedpropeller. 24
1.5.3. Performance variation with tip-loss model. 25
1.5.4. Variation function for lift coefficient using theeffective radius concept. 29
1.6.1. Cascade of airfoil sections. 34
1.6.2. Flow geometry of a cascade of airfoil sections. 34
1.7. 1. Sectional airfoil data for a NACA 4418 airfoil section(NACA TR 824). 38
1.8. 1. Graphic display of equations used in strip theory. 40
1.8.2. Vortex-ring-windmill brake flow fields. 42
1.8.3. Windmill brake state performance. 43
1. 8. 4. Yarn tuffs illustrate a turbulent wind turbine wakeat a low blade angle. 45
2.1. 1. NASAs MOD 0--100-kilowatt wind turbine. 48
Figure Page
2. 1.2. NASA's MOD 0 developed power variation with bladeangie. 49
2.1.3. NASA's MOD 0 chord distribution. 50
2. 1.4. NASA's MOD 0 blade thickness distribution. 51
2. 1. 5. NASA's MOD 0 blade angle distribution. 52
2. 1.6. Performance variation with tip speed ratio for NASA'sMOD 0 wind turbine. 54
2.2. 1. Chalk wind turbine. 56
2. 2. 2. Comparison of theory to experimental field data forthe Chalk wind turbine. 58
2. 2.3. Power versus RPM at constant wind velocity for theChalk wind turbine. 59
2. 2.4. Power versus wind speed at constant RPM for theChalk wind turbine. 60
2.3. 1. Three foot diameter OSU experimental wind turbine. 61
2.3.2. Flow model for wind tunnel blockage corrections. 63
2.3.3. Blockage corrections for wind turbine testing in windtunnels. 68
2.3.4. OSU environmental wind tunnel. 70
2.3.5. Performance comparison between theory and experimen-tal data for a three foot diameter wind turbine. 71
2. 3. 6. Blockage corrections for the three foot diameter windturbine tested. 72
3.0. 1. Wake flow geometry for a propeller. 78
3.0.2. Rotor wake model. 78
3.0.3. Wake flow geometry for a wind turbine. 81
Figure Page
3.0.4. Vortex system of a wind turbine. 81
3. 2. 1. Effect of LID on peak performance of optimum2-bladed wind turbines. 91
3. 2. 2. Effect of number of blades on peak performance ofoptimum wind turbines. 91
3.3.1. Tip-loss factor variation with blade number. 93
3.3.2. Optimum blade chord-lift distributions for threebiaded wind turbines. 95
3. 3. 3. Optimum angle of relative wind for three bladedwind turbines. 96
3.3.4. Power coefficient variation as affected by drag foroptimum three bladed wind turbines. 97
3.3.5. Thrust coefficient variation as affected by drag foroptimum three bladed wind turbines. 98
3. 3. 6. Off -design tip speed ratio performance of optimumthree bladed wind turbines, designed to operate at tipspeed ratios of six and ten. 101
3.3.7. Blade pitch effects on an optimum three bladed turbine. 102
3.3. 8. Blade pitch effects for operation at different windvelocities at constant RPM for a three bladed turbine. 103
3.3.9. Optimum blade chord-lift distribution for a two bladedwind turbine at a tip speed ratio of ten. 106
3.3. 10. Optimum angle of relative wind for a two bladed windturbine at a tip speed ratio of ten. 107
3.3. 11. Sectional airfoil data for NACA 23018 (NACA TB 824). 108
3.3. 12. Blade setting angle as a function of radius for designsI and II compared to NASA's MOD 0. 112
Figure Page
3.3. 13. Blade chord distribution for designs I and II comparedto NASATs MOD 0 design. 113
3.3. 14. Off-design tip speed ratio performance for designsland II. 114
3.3. 15. Performance comparison between MOD 0 design andan optimum design at off-design tip speed ratios. 115
3. 3. 16. Optimum angle of relative wind for Chalk -type windturbines at atip speed ratio of 1.75. 117
3.3. 17. Optimum blade chord-lift distribution for a Chalk-type wind turbine at a tip speed ratio of 1.75. 118
3.3.18. Optimum blade design for a Chalk-type wind turbine. 119
Appendix
A. 1. Program flow diagram. 133
A.2. Programdecklayout. 138
C. 1. Stream tube geometry. 180
F. 1. Velocity diagram for a wind turbine. 198
LIST OF TABLES
Table Pg e
22. 1. NASA MOD 0 wind turbine design and operatingspecifications. 53
2. 2. 1. Bicycle wheel wind turbine specifications. 55
3.3. 1. Blade geometry for design I. 99
3.3.2. Blade geometry for design I. 100
3.3.3. Blade geometry for design I. 109
3.3.4. Blade geometry for design II. 111
3.3. 5. Optimum geometry for Chalk wind turbine. 120
Appendix
D. 1. Flow conditions and blade parameters for optimumactuator disk. 192
NOMENCLATURE
Symbols Description
T rotor thrust
mass flow rate
V wind or approach velocity
V2 downwind or wake velocity
U velocity of air flow through rotor
A rotor swept area
AT wind tunnel cross sectional area
Aw slipstream wake area
pressure upwind of rotor
p pressure downwind of rotor
p free-stream static pressure
a axial interference factor
a' angular interference factor
r radial distance from rotor axis to blade element
rL radial distance from rotor axis to blade element in coned
position
dT differential thrust
drL differential radial distance with blade in coned position
Vt tangential velocity
K.E. kinetic energy
P power
Symbols Description
Q rotor torque
dQ differential torque
R maximum rotor radius
x local tip speed ratio r12/V,,
XL local tip speed ratio in coned position defined by Equation
(1.2.4)
X tip speed ratio = R2/V
X tip speed ratio in coned position defined by Equation (1.2.5)
CL sectional coefficient of lift
CD sectional coefficient of drag
C coefficient of force--y--directiony
C' coefficient of force--y--direction with coning
C" coefficient of force--y--direction neglecting drag
C coefficient of force- -x- -directionx
C" coefficient of force--x--direction neglecting drag
B number of blades
c blade chord
W relative velocity
C power coefficient P/pVACT coefficient of thrust = T/ pVA
r hub radiushubt airfoil thickness
Description
ZLL zero lift line
RE effective radius (NASA's Tip-Loss Model)
V2 wake velocity inside slipstream (Chapter II)
V3 wake velocity outside slipstream (Chapter II)
V1 equivalent free-stream air speed (Chapter II)
V00 wind tunnel air velocity (Chapter II)
r1 radial distance in wake from rotor axis (Appendix C)
angular velocity at radius r1 in wake (Appendix C)
u axial velocity at radius r in rotor plane (Appendix C)
u1 axial velocity at radius r1 in rotor wake (Appendix C)
v radial velocity at radius r in rotor plane (Appendix C)
p1 pressure in final wake (Appendix C)
p pressure in front of rotor (Appendix C)
pressure increase behind rotor (Appendix C)
V axial component of induced velocity (Appendix E)
VT tangential component of induced velocity (Appendix E)
VR radial component of induced velocity (Appendix E)
v interference velocity (Appendix E)
F Prandtl's tip-loss factorp
K Goldstein's tip-loss factor
w advance velocity of helical vortex surfaces
l(x) modified Bessel function of 1st kind
Symbols Description
T (x) modified Lomme function or Goldstein function defined1, n
in reference (23)
v advance velocity of rotor
a coefficient in r distributionmA approximation coefficient to am mBO constant (input) tip-loss factor (NASA's Tip-Loss Model)
CL (R ) lift coefficient factor (NASA's Tip-Loss Model)feGreek Symbols
p air density
w angular velocity of wind
angular velocity of rotor
a angle of attack measured from zero lift line
0 blade angle defined by Figure 1. 3.2
angle of relative wind defined by Figure 1.. 3. 2
aLO angle of attack for zero lift
t4i coning angle
local solidity in coned position defined by Equation (1.4. 2)
maximum solidity in coned position defined by Equation
(1.4.3)
local tip speed ratio
tip speed ratio of rotor
Greek Symbols Description
F bound circulation
E correction factor to approximation coefficient Am ma ratio of A/AT (Chapter II)
ratio of Aw/A (Chapter II)
PERFORMANCE AND OPTIMUM DESIGN ANALYSIS/COMPUTATIONFOR PROPELLER-TYPE WIND TURBINES
I. PERFORMANCE AERODYNAMICS OF PROPELLER-TYPEWIND TURBINES
Introduction
The propeller-type wind turbine can be considered as an
airscrew which extracts energy from the driving air and converts it
into a mechanical form, in contrast to the propeller which expels
energy into the air from another energy source. This similarity
between the propeller and the wind turbine enables similar theoretical
development to be followed in performance analysis. Propeller
theory development followed two independent methods of approach,
one of which has been called momentum theory and the other blade
element theory.
Momentum theory was developed first by W. J. M. Rankine (52)
in 1865 and later improved by R.E. Froude (16). The basis of the
theory is to determine the forces acting on the blade to produce the
motion of the fluid. The theory has been useful in predicting ideal
efficiency, and flow velocity but it gives no information concerning
the rotor shape necessary to generate the fluid motion. Rotational
effects of the wake were included in the theory by A. Betz (5) in 1920.
2
Blade element theory was originated by W. Froude (15) in 1878
and developed byS. Drzewiecki (11) in 1892. The approach of blade
element theory is opposite that of momentum theory in that the con-
cern is with the forces produced by the blades as a result of the motion
of the fluid. It was hampered in its original development by lack of
knowledge of sectional aerodynamics.
Modern propeller theory has developed from the concept of free
vortices being shed from the rotating blades. These vortices define a
slipstream and generate induced velocities. The theory can be
attributed to the work of Lanchester (33) and Fiamm (14) in original
conception; to Joukowski (30) in induced velocity analysis; to A. Betz
(4) in optimization; to L. Prandtl (50), and S. Goldstein (23) for
circulation distribution or tip-loss analysis; and to H. Glauert (18)
and (19), F. Pistolesti (49), and S. Kawada (31) for general overall
development. The theory has been referred to by a number of names:
vortex theory, modified blade element theory, and strip theory.
Strip theory is a design analysis type approach which requires
the geometric configuration of the wind turbine and the airfoil sectional
aerodynamics since locally 2-D flow at each radial station is assumed.
The theoretical development of a modified strip theory, in which
tip-loss corrections and blade interference have been accounted for,
is presented in this chapter for propeller-type wind turbines.
3
1. 1 Axial Momentum Theory
The function of a wind turbine is to extract energy from the air
and to produce mechanical energy which later may be transformed
into other forms of energy. Energy losses are attributed to the
rotational motion of the fluid that is imparted by the blades and their
frictional drag. As a first approximation to determine the maximum
possible output of a wind turbine, the following assumptions are made:
1. Blades operate without frictional drag.
2. A slipstream that is well defined separates the flow passing
through the rotor disc from that outside the disc.
3. The static pressure in and out of the slipstream far ahead of
and behind the rotor are equal to the undisturbed free-stream
static pressure (p2 = p).
4. Thrust loading is uniform.
5. No rotation is imparted to the flow.
Applying the momentum theorem to the control volume in
Figure 1. 1. 1, where the upstream and aft control volume planes are
infinitely far removed from the turbine, the thrust, T is obtained:
- T = Momentum flux out momentum flux in
T xi(V -v ) pAU(V-V2) (1. 1. 1)2
Also from pressure conditions, the thrust can be expressed as:
4
cv
a.-
LI,-streamfube
4V214
4i
P2 P
a-- .- -
a--
a- PROPELLERa-a.-
a 4Ia
--------
Figure 1. 1. 1. Control volume used for a wind turbine.
+ -T = A(p -p ) (1. 1.2)
Now applying Bernoulli's equation:
Upstream of the wind turbine
fPV + p f pU2 + p (1. 1.3)
and downstream of the wind turbine
pV + p00 = pU2 + p (1. 1.4)
or by subtracting (1. 1.4) from (1. 1.3)
+ -
p p = p(V-V)
Substituting the above equation into (1. 1.2) yields
T pA(V-V) (1. 1.5)
Now equating (1. 1.5) with (1. 1. 1)
1 2 2pA(V00-V2) = pAU(V00-V2)
orV +Vcc 2 (1.1.6)
2
This result states that the velocity through the turbine is the average
of the wind velocity ahead of the turbine and the wake velocity aft of
the turbine. An axial interference factor a can be defined by
U = V00(l-a)
Using the definition of a and Equation (1. 1. 6)
v+v00V00(l-a)
2
the wake velocity can be expressed as:
V = V (l-Za)00
The wake induced velocity is twice that of the induced velocity
in the plane of the rotor. This also may be concluded by constructing
a system of helical vortices and calculating the induced velocity by
utilizing the Biot-Savart relation. Figure 1. 1.2 shows the variation
of induced axial velocity with a nondimensionalized distance ii from
the rotor plane, using the Biot-Savart relation.
Therefore the axial interference factor is expressed as:
V00+v2
a = 1 (1. 1.7)00
which implies that if the rotor absorbs all the energy, i.e. , V2 = 0,
then a would have a maximum value of 1 /2. Since power is
defined as mass flow rate times the change in kinetic energy, the
power, P, is given by
or
vvP = rK.E. = pAU( -) = pAV4a(1-a)2
2P = 2pAVa(1-a) (1. 1.8)
Maximum power occurs when dP/da = 0
or
2pAV3(14a+3a2) = 0da
a 1 or 1/3
Since a must be less than 1/2, maximum power occurs when
a 1/3 so
P = l6/27(fpAV,)max
or the coefficient of power1 equals approximately 0.593.
The thrust force exerted on the wind turbine in terms of the
axial interference factor, a is:
'C = Power/( pAV).
T = pAU(V-V2)
1 2= ( pAV)4a(1-a) (1. 1.9)
For maximum power at a = 1/3
T 8/9( pAV)
or the coefficient of thrust2 is equal to 0.889.
1. 2 Momentum Theory in Consideration of Flow Rotation
Initial assumptions of axial momentum theory considered no
rotation was imparted to the flow. It is possible to develop simple and
useful relationships if we consider the angular velocity w imparted
to the slipstream flow to be small in magnitude when compared with
the angular velocity 2 of the wind turbine. This assumption main-
tains the approximation of axial momentum theory, that the pressureof the wake is equal to the free-stream pressure.
Writing an energy balance for the flow shown in Figure 1.2. 1,
it can be shown that the rotational kinetic energy reduces the power
that can be extracted.
2CT Thrust/(fpAV)
10
ICE. = Power Extracted + K.E.translational( 1) translational(2)
+ K.E. rotational(2)
The power is equal to the product of the torque of the wind on the
propeller, Q, and the angular velocity of the rotor, 2. In order
to obtain maximum power it is necessary to have a high angular
velocity of the propeller and low torque since high torque will result
in large wake rotational energy. An angular interference factor is
defined as:
a' angular velocity of the wind at the rotor (1 1)2 x angular velocity of the rotor
Using the annular ring in Figure 1.2.2, and writing element thrust
and torque equations we obtain: (taking into account blade coning)
where
dT = pUZTrrLdrL(VVZ)
rL = r cos L4i
drL = dr cos
coning angle.
Assuming that
= +ZaV
dT = +4rLpVa(1a)drL (1.2.2)
11
-
l\I .-
2
Rotational Flow H Non-Rotational Flow 1 V
a-a-
a-Rotating Device
a-a-
Figure 1. Z. 1. Flow diagram of a wind turbine.
(y2r
I
.1 I..dr------------ I I- - -I I
/ I'\I\i
('I \I\I
l\ I
I ,4 'l
l II
"N" ;i,)-
I Iv
Figure . 2.2. Rotor blade element.
12I
The torque equation becomes
dQ = d(Vr) = ZrLpUrLdrL
= (1.2.3)
Since
dP 5dQ
P= 2dQ0
Now, substituting (1.2.3) into the integral equationfor power:
PSRcos4i 2 34pV (la)a'rLdrL
defining
r2XL
L (local tip speed ratio) (1.2.4)V00
andcos JiX= (1.2.5)
where R is the radius of propeller we obtain
P pAV cos250
(1a4dxL (1.2.6)
where A = TrR2, the coefficient of power becomes
13
x8 2
C = cos 4i XL a'(l_a)dxL (1. 7)P X
1.3 Blade Element Theory
By determining the forces acting on a differential element of a
blade, the torque and thrust loading of a rotor may be determined by
integration of the forces along the blade. This is based on the funda-
mental assumption that there is no interaction between adjacent blade
elements along the blade.
In an actual wind turbine design, the blades are made to cone,
in order to obtain relief from the bending moment due to gusts.
Without coning, the force coefficients can be represented as
shown in Figure 1. 3.1 (4i = 0) and with coning as in Figure 1.3. 1
(4i 0) so that the force coefficients in the coned position may be
written as
C' CL cos cos i + CD sin cos4.'
C CLsin Ccos
Also the radial distance must include the effects of coning. Therefore
C
00+
U)0C)
>0->0
Figure 1.3. 1. Blade coning configuration.
'.4
rL = r cos ti
drL = dr cos
By applying the velocities determined from the momentum
theorem at a blade element, the velocity diagram (Figure 1. 3. 2)
is obtained, where ZLL refers to the zero lift line of theairfoil section. From Figure 1.3.2, one can see that
(i-a) Vcos qitan (l+a') rL
Q.= e
C = (C cos 4 + CDS1fl )y L
C1 = C cos 4i
y y
C C sin-C coscx L D
By determining the thrust acting on the blade element, one obtains
wh e r e
15
1 2 drLdT = +BcpW C; (1.3.1)
B = number of blades
c = chord
The torque acting on the blade element is given by the following
expression.
y
D'
x
V(a)cos/'
Figure 1. 3. Z. Velocity diagram.
16
17
drdQbld element rLBc(Z pW2)C L (1. 3.2)
x cog Ii
1. 4 Strip Theory
Utilizing both the axial momentum and blade element theories,
developnient. of a series of relationships can be achieved to
determine the performance of a wind turbine.
By equating the thrust determined from the momentum theory
equation (1.2.2) to (1.3. 1), the thrust determined by blade element
theory, one obtains
ordTmomentum dTblade element
1dr
+4TrrLpV2a(1a)drL +Bc pW2C'
which reduces to
where
ci 2 C'(1-a)(2a) (1.4.1)
cos 4i
°LTrrL (1. 4. 2)
Bc (1.4.3)m rrRcosqi
From Figure 1.3.2 one can define
II
(i-a)Vcos qisin 4 = w
substituting into Equation (1.4. 1) for W
2 2o C'(l-a) cos
Za(l-a) L y4 sin cos
orC cos2aLy (1.4. 4)i-a 8 sin
Expression (1.4.4) relates the axial flow conditions to the blade
element geometry. By considering the torque, we can likewise
develop a relationship between rotational flow and blade element
geometry.
Equating the torque determined from the momentum theory,
(1. 2.3) with Equation (1.3.2) of blade element theory one obtains
or
dQangular momentum blade element
4r2 pV (la)a'rLd = rLBcp ! w2c drLL 2 xcosi
which reduces to
cos V(ia)Za'rL C W2 (1.4.5)4Lx
From Figure 1.3. 2
(l-a)Vcos 4i
sincI w
rLcos = (l+a') -w
substituting into Equation (1.4. 5)
or
crC2a' Lxsin 4 cos c
crCa' L x (1.4.6)l+a' 8 sin c cos cj
19
It has been an assumption that the drag terms should be omitted in
calculations of a and a' (Equations (1.4.4) and (1.4. 6)) on the basis
that the retarded air due to drag is confined to thin helical sheets in
the wake and have negligible effect on these factors. Therefore C
and C used in calculation of a and a' are defined as:x
= CL cos 'f'
= CL Sfl
By using the relations developed, a and at can be
determined for a given differential element by the following iterative
process.
20
1. Assume a arid a'
2. Calculate 4): 4) = tan'[(l-a)cos /(1+a')x]
3. Calculate a a 4) - 9
4. Calculate C , C C", C"L D x y
5. Calculate a and a'; by Equations (1.4.4) and (1. 4.6)
using C" and C" in place of C and Cy x K y
6. Compare to previous values of a and a'
a) If equal, stop
b) If not equal, repeat steps two through five.
Having determined a and a' by the above iterative process, C
and C can be determined. Torque, thrust, and power can be
calculated by integration along the blade, from the following relation-
ships:
3 xpVR3cosS m/V2CxXLLQ=
0
1 2 2 2 XpVR cos m2CyLT=0
x8 C
C x a'(la)dxL' 0L
21
As such, the equations developed do not include tip-loss and
blade interference effects. Modifications to account for these effects,
as well as defining sectional aerodynamics are necessary for a mean-
ingful solution to the system of equations. These areas are developed
in subsequent sections of this chapter.
1. 5 Tip-Loss Corrections
Strip theory, as previously developed, does not account for the
interaction of shed vorticity with the blade flow near the blade tip.
This "tip-loss" or circulation reduction near the tip can be explained by
the momentum theory. According to the theory discussed previously,
the wind imparts a rotation to the rotor, thus dissipating some of its
kinetic energy or velocity and creating a pressure difference between
one side of the blade and the other. Since the pressure is greater on
one side, air will flow around the blade tips as shown in Figure 1.5. 1.
This means that the circulation is reduced at the tip and as a conse-
quence the torque is reduced.
Because blade elemeLlts at the tip contribute greatly to the
torque, and thus to the overall performance of the wind turbine, the
tip-loss factor is very important to the analysis. Tip-loss has
been treated in a variety of manners in the propeller and
helicopter rotor industry. The simplest method being to reduce the
maximum rotor radius by some fraction of the actual radius, which in
22
Figure 1.5. 1. Tip and hub-losses--flow diagram.
23
helicopter studies is of the order of 0. 03 R. Other relations have been
developed that calculate this fraction of reduction based on the tip
chord length. A more detailed analysis was done by Prandtl as an
approximate method for estimation of lightly loaded propeller tip-
losses, and later by Goldstein who developed a more rigorous analy-
sis. A comparison of the circulation distribution of a two -bladed
propeller using the Prandtl and Goldstein methods is shown in Figure
1. 5. 2. Figure 1. 5. 3 shows a numerical analysis of a wind turbine
design using different tip-loss models. As can be seen, there are
substantial differences between methods and the validity of such
applications to wind turbines is of question. Therefore further
experimental research and verification is needed in this area. Due to
the lack of such verification, theoretical and computer analysis utilize
the following tip-loss model options. Prandtl, Goldstein, effective
radius, or no tip-loss.
Prandtl's Tip-Loss Factor
L. Prandtl (50) developed a method to approximate the radial
flow effect near the blade tip. The basis of his approximation was to
replace the system of vortex sheets generated by the blades by a
series of parallel planes at a uniform spacing, . . Here I is
defined as the normal distance between successive vortex sheets at
the slipstreain boundary, or
THE DISTRIBUTION OF CIRCULATION ALONG A PROPELLER BLADE
[I.]
rn 06iT L) V00
GOLDSTEIN'S EXACT SOLUTION- - - - PRAND TL 'S A10PROX/MA TION____UU'!
"i-n-i.--'!iUk\1_II_11_11_II_II_11_11
12U_LI_11_Ii_11_11_11_11_11_iiI_11 i_11 1' 1' ii_11 U'
11 Ui IIi_UUIUIUU I0 I 2 3 4 5 6 7 8 9 tO
fir00
Figure 1. 5.2. Circulation distribution along a blade for a two biaded propeller. NJ
220
200k
l80 I-4
0a-wU)
0IUia-0 -
-JUi> I
Ui
2
l20
I 5S.
POWER vs BLADE ANGLE
25
80 I I I I I I
4 6 8 10 12 14 16
BLADE ANGLE AT ROOT (DEGREES)
Figure 1.5.3. Performance variation with tip-loss model.
26
2irR.B S1flcT
remembering B represents blade number and is the angle the
screw surface has at the slipstream boundary or is equal to the
angle 4 at the blade tip. Thus the flow around the edges of the
vortex sheets could then be approximately estimated as the flow around
the edges of a system of parallel planes.
where
Prandtl's factor is defined as
F 2 -f= arc cos ep r
£B R-r2 R sin
The expression for f can be suitably approximated by writing
r sin 4 in place of R sin T' since local angles of 4 are more
convenient in calculation procedures. Prandtl's approximation, as
one can see from Figure 1. 5.2, is sufficiently accurate for high tip
speed ratios, and also when the number of blades exceeds two for a
propeller, Of course, it should be remembered that the approximation
was developed for a lightly loaded propeller (contraction negligible)
and that the vortex system is a rigid helix, an optimum condition for
the propeller. Both of these conditions are not necessarily valid for
wind turbines.
27
Goldstein's Tip-Loss Factor
Goldstein (23) developed a more accurate analysis of tip-loss
for propellers by determining the circulation along the blade in terms
of the induced velocity for a rotating blade, again ignoring contraction
and utilizing the rigid helix vortex representation. He obtained his
solution by solving Laplace's equation, V24 0, with suitable
boundary conditions.
wh e r e
Goldstein's tip-loss factor is defined as:
2
2 rrVwF.L
£2 = angular velocity of rotor
r = local radius
B = maximum blade radius
w advance velocity of helical vortex surfaces
V = velocity of advance
local tip speed ratio
= tip speed ratio of rotor
F = bound circulation around the blade section
a coefficient in F distributionmA = approximation coefficient to am mE correction factor to approximation coefficient, Am m
I (x) = modified Bessel function of ist kindnT (n) = modified Lomme function or Goldstein function defined in
1, n
where
reference (23)
00
o 2m+lr 2I (Zm+lt)
A-E)2 m mTrVw Tr I (2m+lj.mO o 2m+l o
00
2 F2m+i(fJ)2 2
1+L it (2m+l)2m 0
2- T (Zm+l p.)F2+1() 1, Zm+l
As one can note from Figure 1.5.2 Goldsteints analysis should be used
for low tip speed ratios and for the one and two blade cases. A dis-
advantage of this method is the complexity of the solution involving
Bessel functions, but with the use of a digital computer and suitable
approximations it can be easily handled.
Effective-Radius Tip-Loss Concept
The tip-loss model that has been used by NASA and the
helicopter industry is essentially one in which the rotor radius is
reduced to an effective radius, R e
29
R RB0e
where BO is a constant (input) tip-loss factor.
Then with Re defined, radial integration proceeds inward from tip,
setting CL = 0 at all integration steps greater than and
CL CL for stations less than R. For the specific integration
around 1e' the integration step is changed to evaluate at Re and
let CL = CL *C at that station, whereF
R rkeC (R)=
LF e rk+lrk
as shown in Figure 1. 5.4.
CLf
1.0
e
0 rk B re k+1r
Figure 1.5.4. Variation function for lift coefficient using the effec-tive radius concept.
30
Tip-Loss Factor Application
The application of Prandtl's or Goldstein's tip-loss factor to
strip theory is of great importance. The tip correction represents
physically that the maximum decrease of axial velocity in the slip-
stream, 2aV, occurs at the vortex sheets and the average decrease
in axial velocity in the slipstream is only a percentage of this
velocity. Therefore Equations (1. 2. 2) and (1. 2.3) of axial momentum
theory assume the forms:
dT 2- = 4iirpV (l-a)aF (1.2. 2)dr
dQ 3pV c2(l-a)a'F (1.2.3)- = 4iTrdr
Thus in combination with blade element theory, Equations (1. 4. 4) and
(1.4. 6) of general strip theory assume the forms:
C"cos2a L y (1.5.1)1-a 2SFsin
0 C"a' L y (1.5.2)li-a' 8Fsincos
Another application approach has been suggested by Wilson (68).
Since the induced axial velocity is determined by
or
31
dTmomentum = dTblade element
B 2CucdrpUVZTrrLdrL = pW L
Consider the induced axial velocity to be localized at the rotor blade
in a manner similar to the induced rotational velocity. Thus
U = (1 -aF)V and tV = ZaFV so we obtain
whe re
therefore
As F1
As FO
o C"L y (1-a)2 = S(l-a)2(1-aF)aF8 sin24
cr C"S- .28 sin
ZS+F-J F2+4SF(1-F)a= (1.5.3)2(S+F2)
a
a 1
and a' is the same as for general strip theory (1.5.2).
Since there is question of which approach more accurately
corrects the basic strip theory, both methods are considered in the
analysis. The first method discussed being referred to as the first
32
order or standard method of tip-loss application and the second method
suggested by Wilson as the second order method of tip-loss application.
Hub-Loss (Inner Blade Tip-Loss) Effects
Another factor is the inner blade tip-loss, when there is no hub.
Prandtl!s tip-loss formula has been applied in the following way to
account for this, and is considered sufficiently accurate for this pur
p0 s e.
where
2 -fF arc cos elT
rrhb2 rhbsin l
rhub = radius of hub
This is applied to the general strip theory by defining the tip-loss
factor as:
FTotai = FTip*FHub
and applying either the standard or second order correction to the
strip theory.
33
1.6 Cascade Theory
For high solidity wind turbines, it is necessary to account for
flow blockage of the air, as it passes through the turbine blades.
The relative velocity of the air for a blade element is the resultant
of the axial velocity through the wind turbine plane of rotation,
V(1-a), and the tangential velocity c2r(1+a') as shown in Figure
1. 2. 2. If at a given radius r of the turbine, the circumference
can be unrolled and represented as a flat surface, then the blade
elements are represented as a cascade of airfoils along an axis LL,
where the distance between airfoil sections is 2irr/B as shown in
Figure 1.6.1. This cascade of airfoil sections representing B
blade elements of the turbine must be repeated to form an infinite
cascade of airfoils.
Immediately in front of the wind turbine, the effective axial
velocity (free- stream velocity - axial interference velocity) and the
rotational velocity define the local velocity (see Figure 1.6. 2). As the
flow enters the cascade of airfoils the axial velocity must increase to
satisfy continuity, since the cross sectional area of the channel
decreases. As the flow proceeds, the tangential component of
induced velocity increases from zero at the leading edge to (2a'2r)
at the trailing edge. As the flow passes the cascade, the tangential
component of velocity remains unaltered, while the axial component
L
1'
V (I - a)
Figure 1.6. 1. Cascade of airfoil sections.
//
(I-a)
//
r(&)//
/
I
a) //
/ flr(2a')
//
/
34
Figure 1. 6. 2. Flow geometry of a cascade of airfoil sections.
35
must decrease again by continuity. Because of this, the flow traces a
curved path which effectively increases the camber of all sections.
Making the following assumptions:
1. The tangential velocity varies linearly from 0 at the
leading edge to 2a'2r at the trailing edge.
2. 0 is the slope of flow at any point, a distance y from the
leading edge.
ThenV00 (i-a)
tan 0 = (1. 6. 1)c2r(i+2 a')
Now the change in 0 from the leading edge to the trailing edge is:
V (i-a) V (i-a)-i 00 100
0 tan tan (i.6.2)r(i+Za')
The effective change in the camber ratio, therefore becomes
zA
c 8(1.6.3)
For a circular arc airfoil, this corresponds to a reduction in the angle
of attack of the zero lift line of
Aa (1.6.4)
36
Effect of Thickness
To consider the effect of thickness, approximate the airfoil with
an ellipse of the same thickness to chord ratio:
wh e r e
From continuity
The angle
t = t z,i/zmax' c/2 c/2 (1.6.5)
Bc0
TrR
rx
r7x
2rV = V(y){2r-Bt [2() ()Z]1/2} (1. 6. 6)max c/Z c/Z
o tan
0tan' 1/xo-tmax 1(1))2]1/2}xli xc 1c c
The departure of the flow path from being a straight line is
expressed as:
dz -i 10-tandy Xx
or
37
z (OtanL)dymax
The change in the angle of attack is
or
zmax= =
Co
Xx c
10 t4 max
L\a T c(1.6.7)
((i)
In the performance calculations, the sectional angles of attacks
are reduced by the amounts given by expressions (1.6.4) and (1.6.7).
1. 7 Sectional Aerodynamic Data
Sectional aerodynamic data is needed to apply the strip theory.
The predecessor of the National Aeronautics and Space Administration
(NASA); the National Advisory Committee for Aeronautics (NACA),
has published Technical Reports 460, 669, 824 summarizing experi-
mental two dimensional airfoil data on different airfoil families, as
well as Schmitz (55) on model airplane airfoil measurements (low
Reynolds numbers).
Typical sectional airfoil data appears as shown on the next
page, Figure 1.7. 1. As one can see, airfoil data is dependent to a
'a
.1
0
.10
0
'a
-2U
0
0
-4
-55ec to,, OP,g/O of of/ac/c e. dcg
jO2O0'a
.0/6'a
012
0
U
If-.2
U
'a0Ia
AerOdyflfl3iC characWistk o the NJLCA 4418 aIrfoil sectIon. Z4-incb chord.
Secf,o,, lift coeff,ciep,t c1
Figure 1.7. 1. Sectional airfoil data for a NACA 4418 airfoil section(NACA TR 824).
20
39
significant degree on airfoil surface roughness and the Reynolds
number at which the section is operating, Re = Wc/v. Therefore in
considering airfoil data, the following items should be checked:
1. Airfoil data has correct NACA identification number, or other
appropriate identification.
2. Is airfoil data two-dimensional or corrected to an infinite
aspect ratio?
3. Is airfoil data for a rough or smooth surface?
4. Is airfoil data for a Reynolds number that is in the range of
operation?
The maximum lift coefficient, life-curve slope, and drag coefficient,
depend upon the Reynolds number. Depending on the required ac-
curacy and the Reynolds number operating range, it may be necessary
to account for this dependence in developing a suitable sectional
aerodynamic model, since different blade elements of a wind turbine
operate at different Reynolds numbers (see Jacobs and Sharman (29)).
1. 8 Windmill Brake State
The flow model used in strip theory is invalid when the force
developed on the rotor blade is incompatible with the momentum loss
of the fluid. This is illustrated in Figure 1. 8. 1 which shows the
momentum equation and the blade element equations for the special
dCTdA
PROPELLORI
WINDTURBINE
BLADE FORCE EQUATION
MOMENTUMEQUATIONNS
0.5 0 0.5 1.0 1.5
INDUCED AXIAL VELOCITY
Figure 1. 8. 1. Graphic display of equations used in strip theory.
41
case of zero drag and CL = ZTT sin a. The momentum relation is
fixed while the blade force relation changes position with blade angle
and chord. The boundary between the normal operating state and the
windmill brake state, is approximately expressed by the relation (72);
where
1 rU >- -Zx 2Bcx
B = angie of zero lift measured from the plane of rotation
x local tip speed ratio, r/V
r = local radius
c local chord
B = number of blades
The windmill brake state is of importance to wind turbines since the
blade angle for maximum performance is very close to that necessary
for entrance into this state.
The flow field of a helicopter rotor operating in this state has
been observed by Lock (37), Drees and Hendal (46), and Brotherhood
(6). Figure 1. 8. 2 shows the flow patterns for these states. From the
empirical work of Glauert (21), based on previous experiments
(NACA T.N. 221), (R&M 885), and (R&M 1014) we observe the
deviation of the thrust coefficient from that predicted by the momentum
theory as shown in Figure 1. 8.3. An analytical approach, that has
42
VORTEX RINGSTATE
WINDMILL BRAKESTATE
WINDMILL BRAKESTATE
Figure 1.8.2. Vortex-ring-windmill brake flow fields.
4i]
'a
1.2
CT
o NACTN22I-B:4NACTN22I-B:2
L R&M 885-uvA R&M 885-cv
1L)
0
UERT'S CHAR.__ GLAUERT'SCURVE ' EMPIRICAL
-
/1 MOMEN TUMTHEORY
o 0.2 0.4 0.6 0.8
Axial Interference Factor, a
Figure 1.8.3. Windmill brake state performance.
43
LIJ
44
good agreement with experimental work, was developed by Castle (7)
and (8) for the helicopter vortex-ring state.
The wind turbine windmill-brake flow field has not been
reportedly observed other than the noting by Iwasaki (27) of extreme
instability in obtaining performance measurements at low blade angles
for certain tip speed ranges. Preliminary work has been done in
the Oregon State University Environmental Wind Tunnel for this flow
state. It appears that for low blade angle settings the wind turbine
wake is turbulent (separated) while non-turbulent at high blade angle
settings (Figure 1. 8. 4). This turbulent region is characteristic of the
windmill brake state in helicopter work. On the basis of performance
analysis and the experimental work of Iwasaki, the wind turbine enters
this state at tip speed ratios only slight greater than the optimum tip
speed ratio for maximum power. It is expected that a performance
analysis can be developed and incorporated into the numerical pre-
diction program once the flow field model is determined. Flow state
definition for this region of operation will have great importance to
dynamic and structural analysis if this flow state is turbulent or
unsteady.
a) Looking upwind
b) Side view
45
Figure 1. 8. 4 Yarn tuffs illustrate a turbulent wind turbine wake at alow blade angle.
46
II. PERFORMANCE PREDICTION
Introduction
Design analysis of propeller-type wind turbines can be obtained
by a modified strip theory approach developed in Chapter 1. This
approach is believed to be adequate for performance analysis.
Experimental verification has been hampered by the fact that avail-
able test data is for low blade Reynolds number ranges, an area
where only limited sectional aerodynamic data has been obtained, and
which is quite sensitive to free stream turbulence.
For high tip speed ratios, propellers shed strong tip vorticies
in a wake that is contracting. The tip vorticies therefore have an
influence upon the flow in the rotor plane, thus the force distribution
deviates from that predicted by a strip theory approach. For the
wind turbine, the wake expansion is large; therefore tip vorticies move
outboard of the tip and induce velocities of a substantially lower
magnitude than if they moved inboard.
The modified strip theory approach for wind turbines has been
programmed on a digital computer due to its complexity. This
approach uses relatively little computer time (from 10 to 60
seconds/run), while for a propeller lifting surface theory, run times
of the order of two hours are common. This is a distinct advantage
47
of this approach, since many program runs arenecessary to define
the performance spectrum of wind turbines. The performance analy-
sis program is used by the National Aeronautics and Space Admiriistra-
tion, Langley Research Center for design and control analysis for the
NASA MOD 0 wind turbine, as well as, by industry and other
universities engaged in wind power research
This chapter includes a computer performance analysis for tne
NASA MOD 0 wind turbine, a comparison between experimental field
data obtained by Oklahoma State University (74) for the Chalk Wind
Turbine and that predicted by analysis, and the computer projected
performance compared to the experimental performance for a
propeller-type wind turbine tested at Oregon State University
Environmental Wind Tunnel.
2. 1 NASA's MOD 0 Wind Turbine Performance Analysis
A performance analysis of NASA's MOD 0-100 KW wind turbine,
illustrated in Figure 2. 1. 1, has been made utilizing the computer
program package PROP described in Appendix A. Figure 2. 2. 2
shows the variation of horsepower at different pitch angle
settings. The curves numbered two, three, four, and five use
design and operating specifications as depicted in Table 2.1.1,
Figures 2,1.3, 2.1.4 and curve A of Figure 2.1.5. Curve
number two of Figure 2. 2. 2 denotes the use of NASA'S
25 FT
Figure Z. 1. 1. NASAS MOD O--100-kilowatt wind turbine.
220
.7
LU 180
03-LUC')
0ILU3-0J140LU>LU
I20
49
POWER vs BLADE ANGLE
TIP LOSS MODEL
3 R1I6- BEST CALC.
2- Reff=O975- PRANDTL
4 GOLDSTEIN
4 6 8 tO 12 14 16
BLADE ANGLE AT ROOT (DEGREES)
Figure 2. 1.2. NASA's MOD 0 developed power variation with bladeangie.
S.
0.09
0.08
0.07
0.06
0.05
0.040° 003J
0.02.J
I0
z0I-.
U)z4I-.-
ILl0.
I-I
4I-Co.
CHORD DISTRIBUTION FOR NASA'S MOD 0WIND TURBINE
(R62.5ft)
0 0.1 0.2 0.4 0.6 0.8 1.0
Figure Z. 1.3. NASATs MOD 0 chord distribution.u-i
0.5 BLADE THICKNESS DISTRIBUTIONFOR NASA'S MOD 0 WIND TURBINE
0.4 k(NACA 23000 SERIES AIRFOIL SECTION)
!\ R=62.5FEETJC.)o 0.3
U)U)
xO
w I-
z0.2
z2
L..
U)'Z
I-.
d, 0.1 o0_J
4'I-U)
0.1 0.2 0.4 0.6 0.8 1.0
YRFigure 2. 1. 4. NASA's MOD 0 blade thickness distribution. u-i
I-
BLADE ANGLE AS A FUNCTION OF RADIUSFOR NASA'S MOD .0' WIND TURBINE
24
U)Lii
2O
LU I0I I
LUJ
z1201I.
I4 z CURVEBo 0
El(1)
8 ZIs41
w
LU U..
01-S
-S
'-Ia:'4I-.
0 I I I I0.1 0.2 0.3 0.4 0.5
$ BLADE ANGLE BETWEEN BLADECHORD LINE AND PLANE OFROTATION
CURVE A
0.6 0:7 1.0TI 'R
Figure 2. . 5. NASA's MOD 0 blade angle distribution. Ui
53
effective radius tip-loss model. Curve number three represents the
no tip-loss case, while curve number four represents the Goldstein
tip-flow model, and curve number five, the Prandtl tip-flow model.
The tip-loss model is seen to have a noticeable influence on the mag-
nitude of the calculated power output, as mentioned in Chapter I.
Curve number six uses the same design and operating specifications
as curves two through five, except curve B of Figure 2. 1. 5, for non-
linear blade twist, is used. Curve six also uses the Goldstein tip-loss
model utilizing the second order method of tip-loss application rather
than the standard method employed in the calculations for curves four
and five.
Table 2.1. 1. NASA MOD 0 wind turbine designand operating specifications.
Tip radius 62. 5 feetHub radius 6. 25 feetCone angle 7 degreesRotor angular velocity 40 RPMWind velocity 18 MPH
Figure 2. 1. 6 shows how the computer predicted power
coefficient for the MOD 0 varies with tip speed ratio for a blade pitch
angle, measured at the three-quarter radius position, of -3.2°
Appendix A contains a sample computer performance analysis evalua-
tion for the MOD 0 at a tip speed ratio of ten, from which blade load
and moment distributions due to the aerodynamic forces may be
0.6
0.5
0.4
cp
0.3
0.2
0.I
0
PERFORMANCE OF NASA'S MOD 0 WIND TURBINE
0 2 4 6 8 10 12 14 16
x
Figure Z. 1.6. Performance variation with tip speed ratio for NASA's MOD 0 wind turbine.
55
be obtained, as well as other useful information for design and control
analys is.
2. 2 Chalk Wind Turbine Performance AnalysisCompared with Field Data
The Chalk Wind Turbine invented by Thomas 0. Chalk and
marketed by the American Wind Turbine Company as the SST (Super
Speed Turbine) Figure 2.2. 1 and commonly referred to as the Bicycle
Wheel Wind Turbine has been analyzed using the strip theory analysis.
The design specifications used appear in Table 2.2. 1.. Sectional
aerodynamics for the N60 airfoil were used because of the availability
of low Reynolds number data, Schmitz (55), and because of the great
similarity to the Clark Y airfoil that is actually used. The analysis
utilized a Prandtl tip-loss model.
Table 2.2. 1. Bicycle wheel wind turbine specifications.
Outside diameter 15 ft. 3 in.Inside diameter 5 ft. 3 in.Length of blades 5 ft.Number of blades 48Chord of blades 3.475 in.Blade pitch 9° outside rim in relation to the
18° inside rim rotor planeBlade twist 9°Support wires 0. 062 in diameterNumber of wires not covered by blades = 96Length of hub in axial
direction 2 ft.Airfoil Clark Y
56
Figure 2.2.1. Chalk wind turbine.
57
Figure 2. 2. 2 shows a dimensionless plot C versus tip speed
ratio which is equivalent to a plot of power versus RPM at constant
wind ve'ocity. Figure 2.2.3 is a dimensionless plot of C/X3 versus
ilx, which is equivalent to a graph of power versus wind speed at
constant RPM or Figure 2. 2. 4.
When the above analysis was examined it was observed that the
rapid decrease in power at high wind speeds is caused by aerodynamic
stall and not by the drag of the wires, as originally conceived. At
high tip speed ratio operation, the windmill brake flow conditions are
encountered, a state of operation not covered by the analysis.
Preliminary experimental data obtained by Professor Dennis
K. McLaughlin (74), Oklahoma State University, is plotted on Figure
2. 2.2. The data, although scattered, demonstrates the validity of the
analysis.
2.3 Experimental Performance Verification
In addition to the previous discussed experimental verification,
performance data for a three foot diameter wind turbine, Figure 2.3.1,
was obtained at the Oregon State University Environmental Wind
Tunnel (73).
0.5
0.3
0.2
0.I
EXPERIMENTAL FIELD DATA COMPAREDWITH THEORY FOR THE CHALK WIND TURBINE
Theory
0 19 Nov. 1975£ 13 Dec. 1975 Oklahoma StateEl Minimum Loading University Data
Ave. of NumerousMeasurements
/.
L.
.
..
I I I I I
11I I
X
2 3
Figure 2. 2. 2. Comparison of theory to experimental field data forthe Chalk wind turbine.
POWER vs. WIND VELOCITY AT CONSTANTRPM FOR THE CHALK WIND TURBINE
6.0
5.0
4.0
cP
3.0
2.0
1.0
00.2 0.4 0.6 0.8
I/X
Figure Z. Z. 3. Power versus RPM at constant wind velocity forthe Chalk wind turbine.
S
20.0
10.0
POWER vs. WIND VELOCITYAT CONSTANT RPM FORTHE CHALK WINDTURBINE
RPM
I
/
/
\ /50I -/750
/25
- N/00
WIND VELOCITY (MPH)
Figure 2.2.4. Power versus wind speed at constant RPM for the Chalk wind turbine.
61
Figure 2.3.1. Three foot diameter OSU experimental wind turbine.
The wind tunnel imposes a constraint on the wind turbine by
limiting the expansion of the free air stream. The uniform axial veloc-
ity V which occurs in front of the wind turbine differs from that
which would occur for the same torque conditions in an unconstrained
stream. In general, it turns out that the equivalent free- stream
62
velocity is less than that of the wind tunnel. By neglecting the rota-
tional motion of the slipstream, the velocity reduction can be esti-
mated by the axial momentum theory. The basic mathematical
approach as originally applied by Glauert (i 2) for propellers is as
follows:
Defining
AT = wind tunnel cross sectional area
A = rotor swept area
A = slipstream wake areaw
V tunnel wind velocity
V3 = wake velocity outside slipstream
V2 = wake velocity inside slipstream
U = velocity through rotor
a = ratio of A/AT 0 <a< 1
= ratio of A IA or (A IA )1 Ia 0 < (A IA ) < 1w w T w T
Referring to Figure 2. 3. 2 and applying continuity
Slipstream UA = V A (2.3.1)
Outside slipstream A V - UA (A -A )V (2.3.2)TaD T w 3
Tunnel Wall
V3streamtube I
FI
wake
4
P+4.
I V..-
U
414' F.4'
4
41...- -. 44'41
Rotor Area,Ai4
414
Figure 2. 3. 2. Flow model for wind tunnel blockage corrections.
P
Applying momentum balance for entire flow
or
64
T {pA V (Vz)+p(ATA )V3(V3)+pZAT} - {pA V (V )+A p }T°° 00 ToOp wZ
-T A {p00-p2) pAT(VV) + pA (V2 2-V2) (2.3.3)p T
T AT(pzp ) A pV (V -V ) (A -A )pV (V -V (2.3.4)w 2200 T w 3 °° 3
The total pressure head remains constant outside of the slipstream.
The refore
or
1 2 1 2p00 +-pV00 = p2 +-pV
p2 -p00 p(V-V) (2.3.5)
Pressure conditions for slipstream yield
-T A(p-p)
Using Bernoulli's equation.
1 pV2 + p00 p + pU2Ahead of rotor 2
Behind rotor pV + p2 = p + pU2
or
65
+ -
p p = p(V-V) + (p00-p2)
1 2 2 1 2 2= p(V-V2) p(V-V)
T = pA(V-V) (2. 3. 6)
Now defining CT = TI(} APVc) the thrust coefficient and substi-
tuting relations (2.3.1) and (2.3.2), expression (2.3.6) becomes
C V2A2(A -A )2= U2A2(A -A )2
(VATUA)2A2T00w T w T
= 2UA(UA-V A )A (A -A ) (UA-V A )2A2w T T w w T
(2.3.7)
and relation (2.3.4) becomes
C V2AA (A -A )2 = 2UA(UA-V A )A (A -AT °° w T w w T T
(UAVA)2ATA (2.3.8)
Combining relations (2.3.7) and (2.3.8) the following equivalent
equations are obtained
C V2A (A-A )(A -A ) = (UA-V A )2A (2.3.9)T w w T w w T
C V2A (A A-A2) = 2UA(UA-V A )A (2.3. 10)T °° w T w w T
66
Defining an equivalent free -stream air speed V' as the speed which
yields the same values of thrust T and axial velocity U.
For free airT = 2ApU(U-V')
or2 2(2U-V') =CTV+Vt2
Letting Ti = V/IT',
The free air condition is
x = 1 + CTfl2
(x+1)V
Substituting these relations into Equations (2. 3.9) and (2.3. 10) the
equations become:
4(x2-1)(1-)(1-a) (x+1-2)2 (2.3. 11)
2(x-1)(1-a2) (x+l-Zfl) (2.3. 12)
By suitable algebraic manipulation we obtain from (2.3. 11) and
(2.3. 12)
(lcr)(lctcr)x+l 22o(l-acr
and
(2.3. 13)
67
1 + (x-1)a (Z-l)x-1 (2.3. 14)2r
and from the definition of x2
2x -1CT 2
(2. 3. 15)Ti
Now for a given a (ratio of A/AT) and assuming o we may
solve Equation (2. 3.13), (2. 3.14), and (2.3. 15) for the thrust coefficient,
CT
Now the power coefficient for experimental work is defined as
PowerC
expexp pVA
and the power coefficient for free-stream conditions is
or
Cfree - stream
PowerexppV'3A
Cfree-stream 3
Ce xp
Now by solvingEquations (2.3.13), (2.3.14) and (2.3.15)for o
over a range we may obtain Figure 2.3.3 which gives approximate
LU
U)4LU
0.
\
4LU
I-.
U)
Lii
C-)
0.8
0.6
0.4
0.2
Jo
1.25
TUNNEL SECTION AREATURBINE SWEPT AREA
0.1 0.2 0.3 0.4 0.5 0.6 0.7
COEFFICIENT OF THRUST MEASURED
Figure 2.3. 3. Blockage corrections for wind turbine testing in wind tunnels.
0.8 0.9 1.0
power coefficient corrections for various observed thrust coefficients.
The thrust coefficient for experimental work is defined as
ThrustCT l 2exp -pV00A
and the thrust coefficient for free stream conditions is
or
ThrustCT 1 2free-stream pV' A
CTfree-stream 2C Te xp
The wind turbine tested used untwisted constant chord model
helicopter blades. The airfoil section was a Clark -Y and had a
chord of 2. 125 inches. The wind tunnel has cross-section dimensions
of 5 feet wide by 4 feet high (see Figure 2.3.4). The power coefficient
versus tip speed ratio is plotted in Figure 2.3.5 for the data obtained.
This data must be corrected for the wind tunnel constraint effect. The
ratio of the wind turbine swept area to the tunnel area was 0.3 534. In
the experiment conducted, thrust was not measured, therefore to
obtain the power coefficient and velocity corrections, the theory
70
predicted (free-stream) thrust coefficients were used in conjunction
with Figure 2.3.6.
I'
0
Figure 2. 3. 4. OSU environmental wind tunnel.
Figure 2. 3. 5 shows the corrected data fit. For the wind turbine
test, local tip Reynolds numbers varied from 70, 000 at tip speeds of
two to 260, 000 at tip speeds of seven. The available aerodynamic
data for a Clark-Y airfoil is at a Reynolds number in excess of
3, 000, 000. Schmitz (55) obtained section data for an airfoil series
designated N 60 for Reynolds numbers 21, 000 to 165, 000. This air-
foil is very similar to the Clark Y airfoil and is used in the
performance prediction model.
71
EXPERIMENTAL PERFORMANCE DATA COMPAREDWITH THEORY FOR THE OSU MODEL TURBINE
0.20
0.15
cP
0.10
0.05
Re = 3,200,000
Re=/65,000
I
2
!8CL/00
V = 31.3 ft /sec
o UNCORRECTEDo.A..rDATA FIT
?.. .. CORRECTEDFIT
-/05,000
4 6
x?Figure Z. 3. 5. Performance comparison between theory and experimental data
for a three foot diameter wind turbine.
[;]
L&J
U)0.9
I&I
a.
"-0.60.
C-)
0.5
O.40
a = 0.3534
TUNNEL CORRECTIONS FOR A
WIND TUNNEL TEST OF ATHREE FOOT DIAMETER WIND
TURBINE
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
CT FREE-STREAM
Figure 2.3.6. Blockage corrections for the three foot diameter wind turbine tested.
73
Reynolds numbers vary along the blade, as mentioned
previously, and greatly affect local aerodynamics at low values. For
the design analysis of the experimental wind turbine, aerodynamic
data at Reynolds numbers of 105, 000, 165, 000, and 3, 200, 000 were
used. These Reynolds numbers span the tip Reynolds numbers of the
test. The analysis used one set of Reynolds number data for all
evaluations along the blade. A refined analysis, if data was available,
would use the appropriate Reynolds data for each specific blade
station, especially for laminar flow, Reynolds numbers under 500, 000.
The drag coefficient of an airfoil decreases as the Reynolds
number increases, while the lift coefficient remains fairly constant.
In addition, changes in the stall angle and angle of zero lift can be
expected. These effects can be attributed to the transition, from
laminar flow to turbulent flow. After transition, further increases in
Reynolds number produce diminishing effects.
The corrected data fit is shown in Figure 2. 3. 5, which is seen to
agree with the performance predicted. The prediction of the peak
performance is shifted by about one-half of a tip speed ratio. This
may be due to the uncertainty in measurement of pitch angles. For
this experiment pitch angle measurement error could be plus or minus
one degree. If the blade angle was 9. 50 instead of ten as reported,
then the peak performance would agree with theory.
74
It is felt that this program can predict wind turbine performance
to an accuracy that is within experimental uncertainty and limited only
by the availability of sectional aerodynamic data.
75
III. OPTIMUM PROPELLER-TYPE WIND TURBINES
Introduction
Configuration and performance of optimum propeller-type
wind turbines is needed to gauge trade offs in design. The optimum
propeller was originally investigated by Betz (4) who proved that for
the case of light loading, the optimum load distributions along the
blade for maximum tractive power requires that the far wake move
rearward as a rigid helical surface. That is proved by considering the
efficiency of a propeller, which depends on the ability to convert the
power absorbed, 2Q into useful work VT, where 2 is the
angular velocity, V is the forward speed, T is the axial thrust,
and Q is the torque input from the propeller shaft. The optimum
condition is determined by considering the effects of increasing the
circulation by an amount tF for a blade element. The increase in
circulation will result in an increase in torque Q and thrust T.
The ratio of the useful work increase to the power required increase
can be expressed as
Holding the power absorbed Q, to a fixed value by addition of an
increment of circulation A F at a radial position on the blade, and
76
allowing the ratio to vary with radius, would require that the
circulation increase for greater than and decrease for X
smaller than XLV, a condition that would also require the power to
change. Therefore the optimum distribution of circulation along the
blade is that ?. is constant at all blade stations.
By using Munk's displacement theorem, which states that a
small increment of circulation can be added to the ultimate wake
instead of at the propeller, the thrust and torque increments due to
the increment of circulation iir can be determined. Using Figure
3.0. 1 for the wake velocity flow field, the thrust and torque incre-
ments are expressible as
Therefore
ptT(2-)rdrLQ = pzF(V+v)rdr
VT V(c-)2Q (V+v)2 1+
V
Now since the forward speed V is constant and the optimum condi-
tion states is constant, V' commonly referred to as the dis
placement velocity is constant.
( -l)V = constant
77
This means that the velocity of the wake flow is such that the
vortex lines move back as a perfectly rigid sheet, Figure 3. 0. 2.
Goldstein (23) used the Betz rigid wake criteria to determine the
exact distribution of circulation for a lightly loaded propeller.
Theodorsen (59, 60, 61, 62, 63) extended the Betz criteria to include
heavy loading since the far wake has the same appearance regardless
of the loading. By redefining Goldsteints solution for heavy loading
and including wake contraction, Theodorsen developed a theory that
for a specified power coefficient, the wake could be determined (i. e.
displacement velocity) and thereby optimum design configuration and
optimum performance could be calculated. Crigler (9) presented
Theodorsen's theory for the practical design of propellers, and later
Lerbs (35, 36) presented a refined approach, similar to Theodorsen's,
for the design of marine propellers.
For a wind turbine, it is not the condition that the relation
between maximum tractive power and shaft power be made stationary
as for the propeller, but rather the relation between induced power
and shaft power. For a wind turbine the efficiency depends on the
ability of the wind to due useful work on the turbine, liT and to
convert that work into shaft power c2Q, where 2 is the angular
velocity of the turbine, Q the torque, T the axial force exerted
by the wind, and U the axial velocity through the turbine. As for
the propeller, an increase in circulation will result in an increase in
B____
(Uw)r Ut
Figure 3.0.1. Wake flow geometry for a propeller.
Figure 3.0. 2. Rotor wake model.
78
I
79
torque LQ and thrust LT. This may be expressed as the ratio of
power output increase to useful work increase X,,T.
W.T. UT
The procedure used for the propeller to establish that X is a
constant, does not yield results for the wind turbine, since a change
in circulation would not only change the thrust, T and torque, Q
but also the axial velocity, U. But a condition can be established
that the induced tangential velocity must be a minimum along the blade,
by considering the flow field in the plane of the rotor as shown in
Figure 3.0.3. The expression for the thrust and torque increments
due to an increment of circulation F are expressible as
or
Therefore,
or
pF(+)rdr= ptF(V-v)rdr
= ptFUrdr
W.T. UT (2+
1
W.T. 1+a'(1+
For a wind turbine we want X T. to be large, therefore the
induced angular velocity or induced tangential velocity must be a mini-
mum along the blade, a condition that leads to a variable wake dis
placement velocity.
Near the blade tips at high tip speed ratios the optimum wind
turbine experiences a nearly constant displacement velocity, but as
the axis is approached, the displacement velocity increases. There-
fore it is apparent that the wake vortex sheet will change shape with
distance from the rotor, Figure 3.0.4. This fact eliminates the
approaches of Goldstein, Theodorsen, and Lerbs except for high tip
speed ratio designs, where error in design specification of inner blade
stations are not critical to performance. Rohrbach and Worobel (54)
have used a modified Goldstein type approach which yields phenome.-
nally consistent results.
Glauert (12) defined the configuration and performance of an
optimum actuator disk by determining a closed form solution to a van-
ation problem using the strip theory equations (see Appendix D).
Glauert's solution, however, neglected drag and the important
phenomena of tip-loss. As previously mentioned in Chapter II, proper
design of the tip stations to minimize circulation loss is very impor-
tant. This chapter presents an approach using modified strip theory
that incorporated tip-loss and determines the lift/drag effect on
optimum performance.
J1
2
Figure 3.0. 3. Wake flow geometry for a wind turbine,
/ //,I\/
vT'
Figure 3. 0. 4. Vortex system of a wind turbine (note root displace-ment velocity is different than at tip).
3. 1 Local Optimization
Optimum design relations can be developed from the strip theory
equations that include tip-loss. Since there are two methods of tip-
loss approximation to strip theory, the first order or standard and the
second order methods, both will be considered.
Consider the differential-blade element power equation:
= 2rBc pW2C (3. 1. 1)dr
DefiningR&7 Bc
x , X = , and UI =00
and substituting into Equation (3. 1. 1) we obtain
dP 2 X)Zdx L V00 pW2 (3. 1.2)- = (1TR )o C
Now expressing Equation (3. 1.2) in terms of the power coefficient,
yields
Cp 1 3 2
pV00TrR
dC w2dx L x (3. 1.3)
Equation (1. 5.2) expresses conservation of angular momentum.
By defining
C"
83
x8 j
a' (1.5.2)COS F l+a'
Vsin = -(1-a) and cos ci? =
(1. 5.2) becomes:
Va' 00
C" = ( ,)8F()2x(1+a')(1-a)L x l+a
Vo°2
o C" = 8Fxa'(l-a)() (3. 1.4)Lx W
Standard Tip-Loss Application
By utilizing Equations (1.5. 1) and (1. 5.2) (linear and angular
momentum)
We obtain
0 C"a L y (1. 5. 1)1-a 8F sin24
c:r C"a' Lx(1.5.2)1+a' 8F sin cos
0 C" 2a' (1-a))(1+a') a
L X }{8F sin8Fsincos cr C"Ly
C"a{(1a)} xa (l+a') = (-;;-) tan (3. 1. 5)
y
From Figure 1.3. Z, it maybe seen that the angle may be
defined geometrically by
(1-a) = x tan(1+a')
which yields when substituted into expression (3. 1. 5)
C"x aa' (3.1.6)y
Now expression (3. 1. 4) for the standard method of tip-loss application
becomes
aC" = 8Fx{()Lx C x Wy
Substituting expression (3. 1.6) yields
C" Vc020 LL 8F( )a(1-a)) (3. 1.7)
y
Second Order Method of Tip-Loss Application
From the second order method of tip-loss application of Chapter
I we have from Equation (1.5.4):
or rearranging
From Equation (1.5.3)
or by rearranging
o- C"L y (1-a)2 (1.5.4)(l-aF)aF8 sin
cr(1-aF) 1sin L
(1-a)2aFj 8 sin (3. 1.8)
y
o C"a' Lx (1.5.3)l+a' 8F sin 4 cos
a' Fcoscf L (3. 1. 9)1+a' C" 8 sin t:ix
Equating (3. 1.8) and (3. 1.9) we obtain
or
(1-aF)aF sin aT cos c
2 (1+a')C"C"(l-a) xy
C"(l-aF)aF tan =
(i-a) }(1-a)Fa' (3. 1. 10)C" (i+a')x
but from Figure 1.3.2
{(1a) }xtan(1 +a')
Therefore (3.1 . 10) becomes
or
Ct,
(1-aF)aF tan = --} (1-a)Fx tan a'Ct'x
C't
a' {(1_aF) x a -,
(1-a) } -E-(3.1.11)
y
Now substituting expression (3. 1. 11) into Equation (3. 1.4), we obtain
or
V C"C"L x W (1-a) C" x
y
CI' V 2o CI'SF(--)a(1_aF)(-W-) (3.1.12)Lx C
y
for the second order method of tip-loss application. Proceeding on
the basis that the maximum power of the rotor can be obtained by
maximizing elemental contribution to power, the blade configuration
can be determined. The basic procedure is as follows:
At a given station (x/X) or (rIP)
1. Set a = 0.32 and a' = 0 (initially).
2. Calculate ,
-1 (i-a) 1tan (l+a')
3. Determine tip-loss factor, F. Methods--Prandtl,
Goldste, or no tip-loss (F 1).
4 Calculate C", C"x y5. Calculate a',
Standard method a' = () (3. 1.6)Cxy
C"Second order method a' {(i-aF)
}(K) a (3 1. ii)(i-a) C xy
6. Repeat steps two through five, five times.
7. Calculate (W/V)2,
W2() = (i-a)2 + x2(1+a')2
8. Calculate o C",LxStandard method
C" V2C" 8F(+)a(l-a)() (3. 1.7)Lx C
y
Second order method
C'7 Vco 2O C" = 8F(-7r)a(1-aF)( ) (3. 1. 12)Lx C
y
9. Calculate dC /dx,p
dC xZ W 2(r C" )() (3. 1.3)dx Lx
10. Compare dC/dx calculatedwith dC/dx previous.11. Iterate with a, a = a + da until maximum dC /dx is
p
determined- - repeating steps two through ten.
12. Obtain a, a', 4, rLC" and (WIv)2 for maximum power.
13. Calculate TLCL
14. Calculate
15. Calculate c,
(o C"LxLCL Sin
LCLL CL
rLc (BCL)LCL
16. Set CD = 0, FAG = 25.
17. Calculate C , Cx y
C C sin-Ccos4x L
C =C cos+CDsiny L
18. Calculate oLCD
0LCD OLCD
19. Calculate dC /dx,p
dC x W 2dx
00
20. Calculate dCT/dx
dCT x W2dx = LZ)CY(V00)
21. Set CD CL/FAG.
22. Repeat steps 17 through 21, five times.
As would be expected, for the no tip-loss case with CD 0, yields
the same values as the Glauert approach does (Appendix D).
The procedure outlined above has been programmed on a digital
computer. The program package named OPTO is presented in
Appendix B with operating instructions, a listing, and sample output.
3. 2 Constant Wake Displacement and Constant AxialVelocity Methods of Optimization
The constant wake displacement method of propeller theory, as
mentioned in the introduction to this chapter, has been used by
Rohrbach and Worobel (54). Their approach is based on the Goldstein-
Theodorsen theory neglecting wake expansion. The method maxi-
mizes the local power at the 3/4 R position on the blade, and then
determines the corresponding wake displacement velocity. The wake
displacement velocity calculated at the 3/4 R position for maximum
local power is .then considered constant over the length of the blade.
The procedures employed by Theodorsen (59, 60, 61, 62, 63) or Lerbs
(35,36) to define optimum blade geometry and performance for a con-
stant wake displacement velocity are employed. Figures 3. 2. 1 and
3.2.2 show several of Rohrbach and WorobePs performance curves
for optimum wind turbines. Their results appear to be phenomenally
consistent.
The constant axial velocity method assumes the axial velocity
is constant along the blade. This method also maximizes the power at
3/4 R station on the blade and then assumes that the axial velocity,
(1a)\T calculated at this station is the same as at other blade
stations. The basic procedure is outlined as follows:
cp
0.6 GLAUERT IDEALO.5L00.4 80
0.3
40 L/D0.2 L
20VARIATION
0 I I
0 4 8 12 16 20 24
x
91
Figure 3. 2. 1. Effect of L/D on peak performance of optimum2-bladed wind turbines (ohrbach and Worobel (54)).
06 GLAUERT IDEAL BLADES
02LL1
0
00 4 8 12 6 20 24
x
Figure 3. 2. 2. Effect of number of blades on peak performance ofoptimum wind turbines (Rohrbach and Worobel (54)).
92
1. Determine the axial interference velocity at 3/4 R station
using either the standard or the second order method.
2. Calculate the angle of the relative wind by
I(l-a)l= tan r/R .75
(l+a')x
3. Calculate the angular interference factor using the
appropriate standard or the second order method.
4. Determine the tip-loss factor.
5. Iterate on steps two through four until convergence.
6. Now determine the chord-lift coefficient product, power
coefficient and thrust coefficient as outlined in Section 3. 2. 1,
since a, a', and is known at each station.
This approach also appears to yield acceptable results. A computer
program package labeled AXOPTO for this method is also included in
Appendix B.
3. 3 Optimum Rotor Configuration and Performance
In considering a wind turbine design, the question arises as to
how many blades should be used. In general, as the number of blades
increases, so does the cost. The performance advantages of increas -
ing the number of blades are that the starting torque is higher and
less periodic, and tip-loss is lower as s,hown in Figure 3.3. 1. As a
0.9
Li.. 08.
0Io 0.7Li..
U)U)0_J
a-I-
0.5.
0.4.
I\TIP SPE
X3RATIO /x
//
/
XIO /,X2
F
/ \/
//
/
//
I/
X6 /X=I
// /
/ /I'ji /
II /
I
X31 //
/
I
//
/ // /
X:2 //
X:I /
/
I
TIP LOSS FACTOR vs. NUMBER OF BLADESFOR 0.9
GOLDSTEIN TIP LOSS FACTORPRANDTL TIP LOSS FACTOR
O 2 4 8 12 16 20 24 28 32 36
NUMBER OF BLADESFigure 3.3. 1. Tip-loss factor variation with blade number.
40
32
44
94
consequence, power output increases, but with diminishing returns.
Ideal blade configuration requires that the blade width or chord
and the blade angle vary continuously and in such a manner along the
blade as to produce maximum power at a given tip speed ratio. There-
fore for different tip speed ratios, the blade chord and angle distribu-
tions change as shown in Figures 3. 3. Z and 3. 3. 3 (three -bladed wind
turbine), noting as tip speed ratio increases, blade solidity and blade
angle decrease.
To specify blade configuration, it is not sufficient to examine the
turbine performance at only design conditions, such as that shown in
Figures 3.3.4 and 3.3.5. Off design operationmust also be considered.
Tables 3.3. land 3.3.a present the blade configurations for three-
bladed optimum rotors designed at different tip speed ratios. As
Figure 3. 3. 6 illustrates, the off-design tip speed ratio performances
are different. Figures 3.3.7 and 3.3.8 illustrate the off-design blade
pitch effects on performance for the blade configuration of Table
3.3. 1. As can be seen, small changes in pitch angles have pro-
nounced effects on off-design performance and delay entrance into the
windmill brake flow state. One may also observe from Figure 3.3.7,
that for a design tip speed ratio of ten, pitching the blade in either
direction from the design blade setting 00, produces smaller power
coefficients, signifying an optimum condition as expected. But we
observe that at a slightly lower tip speed ratio than the design ratio
0.20
0.18
0.16
0.14
0.12
CCL
R
0.10
0.08
0.06
0.04
0.02
00 0.2 0.4 0.6 0.8 1.0
95
UTIONSBLADED3
r1
Figure 3.3. 2. Optimum blade chord-lift distributions for threebladed wind turbines.
40°
300
\ \ ANGLE OF RELATIVE WIND
4, \\ \ DISTRIBUTION FOR OPTIMUM THREEl\ \ BLADED WIND TURBINES
200
100f/
//
6
.8
x
00
0 0.2 0.4 0.6 0.8 1.0
Figure 3.3.3. Optimum angle of relative wind for three bladedwind turbines.
cP
0.6
0.5
0.4
0.3
POWER VARIATION OF OPTIMUMTHREE BLADED WIND TURBINES
0 6 8 (0
L/D = 100.0
L/ 75.0
50.0
L,...250
xFigure 3.3. 4. Power coefficient variation as affected by drag for
optimum three bladed wind turbines.
[.I;I
.I:Iil
0.84
[IL1
[.1:1]
[SI:IiJ
THRUST VARIATION OF OPTIMUMTHREE BLADED WIND TURBINES
0 6 8 10
x
Figure 3.3. 5. Thrust coefficient variation as affected by drag foroptimum three bladed wind turbines.
Table 3.3.1. Blade geometry for design I.(CL = .9, a = 8°, 3-Blades, X = 10.0)
Station Chord(ft)
0twist(d e g)
100 0 -4.80095 1.313 -4.33290 1.420 -3.90185 1.508 -3.57880 1.600 -3.27075 1.703 -2.94570 1.821 -2.58365 1.957 -2. 17060 2.114 -1.69255 2.297 -1.13050 2.515 -0.46045 2.777 0.35340 3.097 1.35735 3.495 2.63030 4.000 4.29025 4.655 6.53420 5.514 9.71015 6.619 14.46010 7.796 21.995
5 7.579 34.254
100
Table 3.3.2. Blade geometry for design I.(CL = .9, a = 8°, 3-Blades, X = 6.0)
Station Chordt
0twist(deg)
100 0 -3.000°95 3.461 -2.26490 3.823 -1.47685 4.092 -0.87180 4.354 -0.31075 4.631 0.26170 4.939 0.87565 5.286 1.55660 5.682 2.33155 6.138 3.22950 6.668 4.28445 7.286 5.54640 8.014 7.07835 8.872 8.97530 9.803 11.36925 11.033 14.46020 12.260 18.53515 13.271 24.00110 13.195 31.355
5 10. 039 40. 590
0.6
0.5
0.4Cp
0.3
0.2
0.1
0
PERFORMANCE COMPARISON BETWEEN THREE BLADE OPTIMUMROTORS DESIGNED FOR OPERATION AT DIFFERENT TIP SPEED RATIOS
X:6.O DESIGN
X=IO.O DESIGN
0 2 4 6 8 10 12 14 16 18
xFigure 3.3. 6. Off-design tip speed ratio performance of optimum three bladed wind turbines, designed to operate
at tip speed ratios of six and ten.
OPTIMUM PERFORMANCE OF A ROTOR AT VARIOUSTIP SPEED RATIOS AND PITCH ANGLES
0.63 BLADE RO TOR FOR
I
OPT/MUM PERFORMANCEAT A TIP SPEED RATIO
0.5iI DESIGN P0/N T
0 DESIGN CO/VOl TION
NORMAL OPERA TING STATE
Cp 0.3'
\' WINDMILL BRAKE STA TE
0.2 d9=-/ 0 H -1-2%
' \
0.1' "
' ''4
\
0 I I i J\ I4 6 8 10 12 14 16 (8
xFigure 3. 3. 7. Blade pitch effects on an optimum three bladed turbine (Design I, 3 blades,
X . 10.0).designI-
C
103
OPTIMUM PERFORMANCE OF A ROTOR AT CONSTANT RPMFOR VARIOUS WIND SPEEDS AND PITCH ANGLES
0.9
0.8
0Q 0.5
DES/Cl
ç, >( 0.4
0.3
0.2
I NORMALo.iI STATE
WINDMILLBRAKESTATE
3 BLADE ROTOR FOROPTIMUM PERFORMANCEAT A TIP SPEED RATIOz/QO
= 0 DES/6N CONDITION
2 =O9,,
0 0.1 0.2V00 L.Rfl X
Figure 3. 3.8. Blade pitch effects for operation at different windvelocities at constant RPM (1/X) for a threebladed turbine (Design 1, 3 blades, X . 10. 0).design
104
(X = 10. 0), the performance is higher. This maybe explained by
Figure 3.3.4, which shows that for LID ranges of 50-75,which
were used, that the power coefficient increases slightly as the tip
speed ratio decreases from ten.
The design procedure for determining the optimum blade
configurations can best be illustrated with several examples.
Example Cases
1) Design a two bladed wind turbine for operation at a tip speed
ratio of ten and compare its predicted off-design
performance with that determined for NASA's MOD 0 turbine.
Pro cedure:
Determine the optimum blade parameters, cCLIR and .
This is accomplished by using the program package OPTO, described
in Appendix B, and inputing the following information:
B2.x = 10.0
CL 1.0
DX = 0.5
TM 1.0
AMOD 1.0
105
From an output listing, as shown in Appendix B, the angle 4)
and the chord/maximum radius ratio can be obtained. (Letting
C = 1.0, cC /R reduces to c/R ). The variation of theseL L max max
parameters with radius are shown in Figures 3. 3.9 and 3. 3. 10 for
this case. Note that from Figure 3. 3. 3, shown for a three- bladed
wind turbine, at a high LID ratios, the performance is higher. The
next step is to decide on an airfoil section. For this example we will
choose NACA Profile 23018. Lift and drag data are shown in Figure
3.3.11 for this airfoil section. By calculating the coefficient of lift
to coefficient of drag ratio at various angles of attack, we notice that
the L/D maximum occurs near CL = 0. 9 and at an angle of
attack equal to 8°. We also notice that L/D ratio varies only
slightly in a CL range of 0.6 to 1.0.
Now we will specify two different blade geometries as
described below.
Design I. Let CL 0. 9 and a = 8° be constant over the
blade.
Design II. Let CL vary from 0. 6 at the tip to 1. 0 at the hub.
Determination of Design 1:
To determine the chord distribution for design I, we must
simply multiply the chord/maximum radius ratio printed out for each
station by the ratio of the maximum radius to a CL of 0.9. For a
0.12
[.J[I
CCLR 0.08
[sI.11
[sX.I!
0.02
CHORD-LIFT DISTRIBUTION FOR 106
AN OPTIMUM TWO BLADED WINDTURBINE AT XlO.O
0 0.2 0.4 0.6 0.8 1.0
Figure 3.3.9. Optimum Made chord-lift distribution for a two bladedwind turbine of a tip speed ratio of ten.
3.6
3.2
2.8
2.4
2.0
'.5
0/2
.8¼)
.1 .4
0 0
-.4
¼)
-.8
¼)
-/2
-.4 -1.6
-.5
'I
.024
.020
.3 .0/6
cf
I
IIiiIiIJiIIiIIIPIIuuuNu
...a.....ai aia..suu.
r'uIuuuuUurF4UUflWSl
.u.u....m..uu.
!iiliiiIiiiiIIii!!!IIHII.uuuua iiUluuuumIuuuSBUIIUUII.u..u.. muu......
,,illilillil:.1
. iii .a...iuu.u..... ......uuuI.uIIUIUlUII
-1,0 -/2 -.8 -4 (1 qSoct,a'i lift coofI,c¼cvt, C,
Figure 3.3. 11. Sectional airfoil data for NACA 23018 (NACA TR 824).
zC)
r')
C,)
0
QOD
109
maximum radius of 62. 5 feet, we multiply each chord/maximum
radius ratio by 69.44 feet.
To determine the blade twist angle for design I, we subtract the
angle of attack, 8°, from the angle of the relative wind at each
station.
&twist
Design I is now specified as shown in Table 3.3.3.
Table 3.3.3. Blade geometry for design I.(CLO.9, a = 8°, 2-Blades)
Station Chord °twiStCL(ft) (deg)
100 0 -5. 000°95 1.914 -4.49490 2. 107 -4. 02385 2.253 -3.66180 2.399 -3.32575 2.557 -2.97870 2.733 -2.61365 2.937 -z. 18260 3.171 -1.69955 3.447 -1.13450 3.773 -0.46245 4.166 +0.35140 4.646 1.35735 5.242 2.63030 6.000 4.29025 6.982 6.53420 8.271 9.71015 9.929 14.46010 11.694 21.995
5 11.369 34.255
110
Determination of Design II:
Design II is determined in the same way design I, except CL
and a vary at each station according to the following relations.
for this case
CL = CL (CL CL )r/Rroot root tip
a = a - (a - a )r/Rroot root tip
CL 1. -0.4 rIB
a = 9.644 4.384 (rIB)
Design II is now specified as shown in Table 3.3.4.
Designs I and II are compared to the blade geometry of NASA's
MOD 0 wind turbine in Figures 3.3.12 and 3.3. 13. Figure 3.3.12
shows that the optimum designs have a greater rate of twist nearer the
hub than the MOD 0 configuration. Figure 3. 3. 13 shows that the
MOD 0 design is a good approximation to optimum design I for chord
distribution, except near the hub.
The design performance of designs I and II may be determined
by utitizing the program package PROP of Appendix A. The power
coefficient for off-design tip speed ratios for designs I and II is pre-
sented in Figure 3.3. 14. Figure 3.3. 15 compares the performance
of design I to that of NASA's MOD 0. Since NASA's MOD 0 design was
a good approximation for the outer blade geometry, which has the
111
greatest influence on performance, it is not surprising to find that the
MOD 0 turbine has a power coefficient only 5% below optimum design
I at the design tip speed ratio. The off-design performance of the
MOD 0 however appears more desirable from this figure, but it would
be necessary to construct figures similar to Figures 3.3.7 and 3.3. 8
to observe the off-design pitch behavior of both designs.
Table 3.3.4. Blade geometry for design II.(CL = 1. -O.4(r/P);a 9.6436 4.3844(r/R), 2-Blades)
Station Chord °tW15tCL(ft) (deg)
100 0 -2.25995 2.779 -1.97290 2.963 -1.72185 3.073 -1.57880 3. 175 -1.46175 3.287 -1.33370 3.417 -1.17865 3.572 -0.97660 3.755 -0.71255 3.977 -0.36650 4.245 0.08745 4.572 0.68040 4.977 1.46735 5.486 2.52030 6.136 3.96225 6.982 5.98720 8.091 8.94315 9.506 13.47410 10.964 20.790
5 10.441 32.831
8CL
24BLADE SETTING ANGLE AS A
FUNCTION OF RADIUS20°
16° 1
BLADE ANGLE MEASUREDFROM CHORD LINE
2° NASA $ MOD .0
Hub'J \\ DESIGN ITranslflon DESIGN 11Zone
8iI \\
\%4°
0° I __----_.-._____-_______
40
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
r/R
Figure 3.3. 12. Blade setting angle as a function of radius for designs I and II compared to NASATs MOD 0.
0.13
0.11
0.09
C0.07
R11
0.05z.0I-.
Cl)
z.
0.03
u-I0I-..
0.01
0
CHORD DISTRIBUTION COMPARISONAS A FUNCTION OF RADIUS BETWEENNASA's MOD 0 AND OPTIMUM DESIGNS i & II
NASA's MOD 0
DESIGN II
DESIGN I
0 0.1 0.2 0.4 0.6 0.8 1.0
r/D'iip -
Figure 3.3. 13. Blade chord distribution for designs land II compared to NASAts MOD 0 design.
cP
0.50
iJ
0.30
114
OFF DESIGN PERFORMANCE OF TWO BLADEDWIND TURBINE DESIGNSDESIGNED FOR XJOO
DESIGN Ia CONSTANT
/
II
I'=DESIGN II iia:a_b(r/R) !/
a9.6436' iib4.3844 ii,'
I'
llII!'IIIIIIII'IIII
I
I
I
I
II
I
It
'I
I
It
It
It
0 2 4 6 8 10 12 14
xFigure 3.3. 14. Off-design tip speed ratio performance for designs
I and II (two blades, X 10. 0).design
0.6
0.5
0.4
cp
0.3
0.2
0.I
0
PERFORMANCE COMPARISON BETWEEN NASA'S MODDESIGN AND AN OPTIMUM DESIGN
NASA'sMODØ
'I
OPTIMUM DESIGN,-_--..-.' X=IO
0 2 4 6 8 10 12 14 16
x
Figure 3.3. 15. Performance comparison between MOD 0 design and an optimum design at off-design tip speedratios.
116
2) Example: Determine an optimum blade configuration for a
Chalk-type wind turbine at an operation tip speed ratio of
1.75. The design should have the same configuration as the
Chalk wind turbine described in Chapter 2. 2 except for a
different constant chord and blade twist distribution.
Procedure:
To generate an optimum design, again we use program OPTO
and obtain the optimum design parameter variations for a 48 blade
wind turbine at a tip speed ratio of 1.75 as shown in Figures 3.3.16
and 3.3. 17. Referring to sectional aerodynamic data, Schmitz (55) at
a Reynolds number of 105, 000 we can determine that the highest or
most suitable LID range is from 12.5 to 13. 5. Now choosing a
chord dimension such that, the lift coefficient calculated at each blade
station (using Figure 3.3. 17), is in the suitable LID range and does
not exceed the maximum lift coefficient; the angle of attack can be
determined. Utilizing Figure 3.3. 16 to determine the angle of the
relative wind at each station and subtracting the angle of attack at
each station, the blade twist angle distribution may be determined as
listed in Table 3.3.5 and shown in Figure 3.3. 18.
Comparing the optimum design specifications, Table 3.3. 5 to
the Chalk wind turbine design in use, Table 2.2. 1, we observe that
the blade chord and blade twist are increased substantially.
60°
500
40°
300
200
l0
00
THEORETICAL OPTIMUM BLADE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
r
RFigure 3.3. 16. Optimum angle of relative wind for Chalk-type wind turbines at a tip speed ratio of 1.75.
0.05
0.04
0.03
0.02
1
THEORETICAL OPTIMUM BLADEDESIGN
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
rR
Figure 3.3. 17. Optimum blade chord-lift distribution for a Chalk-type wind turbine at atip speed ratioof 1.75.
600
500
40°
eGL
30°
20°
10°
0°
THEORETICAL OPTIMUM BLADE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
r
RFigure 3.3. 18. Optimum blade design for a Chalk-type wind turbine -blade angle measured from
chord line).
Table 3.3.5. Optimum geometry for Chalk wind turbine.
r/R x 100% dR CL a° 00zero lift line 0°chord line
100 .0698 0.430 0 19.5 26.290 .0698 0.473 0.7 20.6 27.380 .0698 0.509 1.3 22.2 28.970 .0698 0.540 1.8 24.3 31.060 .0698 0.568 2.2 26.8 33.550 .0698 0.589 2.6 30.0 36.740 .0698 0.595 2.7 33.9 40.630 .0698 0.573 2.3 37.8 44.520 .0698 0.494 1. 1 45.0 51.710 .0698 0.302 -2.0 55.4 62.1
LID range, 12.5 13.5Maximum radius, 7. 625'Hub radius, 2. 625'Chord, 6.38"Tip blade angle, 26.2°Hub blade angle, 42. 5°Blade twist, - 16°
NJ
121
IV. SUMMARY AND CONCLUSIONS
This thesis has been concerned with the development of a
methodical theory for the determination of performance, loads, and
optimum design configurations of propeller-type wind turbines. The
theoretical developments have been programmed on a digital computer
and sample calculations show good agreement with the limited test
data available. The performance prediction program has been used
by the National Aeronautics and Space Administration for design
analysis and is used for control prediction for the experimental
MOD 0 wind turbine. In addition, it is in wide use by other Energy
Res earch and Development Administrator contractors and unive rs ities
engaged in wind power research.
In the development of an optimization theory for wind turbines,
it has been shown that propeller design approaches are not suitable
for wind turbine applications.
There are several areas for which future work is needed, as
outlined below.
1. Large scale wind turbine performance data is needed for
validation of the performance theories.
2. The flow field for the windmill brake state of operation needs
to be defined for wind turbine application and a suitable
performance model developed.
122
3. The determination of blade loading involving dynamic
interactions of blade flapping, aeroelasticity, tower motion,
and gust loading.
4. The determination of low Reynolds number sectional airfoil
data for a variety of potential wind turbine airfoil sections.
This would be of benefit in wind tunnel studies of large scale
wind turbines as well as for the direct design application for
small scale wind turbines.
5. Wind tunnel corrections for wind turbine testing needs
experimental verification.
6. Experimental determination of the tip-flow conditions for a
wind turbine needs investigation.
123
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26. Hough, G. B. and Ordway, P. E., UThe Generalized ActuatorDisk," Developments in Theoretical and Applied Mechanics,Vol. II, Pergammon Press, 1965.
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29. Jacobs, Eastman N. and Sharman, Albert, "Airfoil SectionCharacteristics as Affected by Variations of the Reynolds Num-ber, " NACA Report 586.
30. Joukowski, N.E., Soc. Math. Moscow, 1912, reprinted in"Theorie Tourbillonnaire de l'helice Propulsive, " Paris, 1929.
31. Kawada, S., Tokyo Imperial University, Aero. Res. Inst., No.14, 1926.
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33. Lanchester, F.W., Aerodynamics, London, 1907.
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36. Lerbs, H. W., "Moderately Loaded Propellers with a FiniteNumber of Blades and an Arbitrary Distribution of Circulation,Trans. Soc. Naval Architects and Marine Engrs., 60, 73-117,1952.
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37. Lock, C.N.H., "Photographs of Streamers Illustrating the FlowAround an Airscrew in the 'Vortex Ping State'," Reports andMemoranda, No. 1167, AE.331, April 1928.
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41. Lock, C. N. H., "Note on the Character istic Curve of anAirscrew of Helicopter, " Ministry of Supply, Aeronautical Res.Council, Rept. and Memo. , 2673, 1952.
42. Losch, F., "Calculation of the Induced Efficiency of HeavyLoaded Propellers Having Infinite Number of Blades, "NACA TM884, January 1939.
43. Mangler, K. W. and Squire, H. B., "The Induced Velocity Field ofa Rotor," Reports and Memoranda No. 2642, May 1950.
44. McCormick, B. W., "The Effect of a Finite Hub on the OptimumPropeller, " J. Aero. Sci. , 22, No. 9, September 1955.
45. McCormick, B.W., Aerodynamics ofV/STOL Plight, pp. 73-100,Academic Press, 1967.
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47. Nilberg, R.H., "The American Wind Turbine, " CanadianJournal of Physics, Vol. 32, 1954.
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50. Prandtl, L. , Gottinger Nachr. , p. 193 Appendix, 1919.
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128
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129
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APPENDICES
APPENDIX A
Design Analysis Program
Program Operation
130
The design analysis program utilizes the modified strip theory
discussed in Chapter I. It was written in the Fortran IV program
language and developed under the OS-3 operating package for a CDC
3500 digital computer at Oregon State University. Logical unit num-
bers 60 and 61 in the program refer to the card reader and the line
printer respectively. Library function ACOSF refers to the arc
cosine function. Other small modifications may be necessary for
implementation on other systems.
The program package consists of a main program with 12
subroutines, whose functions are as discussed below.
PROP - The main program performs input-output functions as
well as the numerical integration of various performance
quantities over the rotor blade.
TITLES - A subroutine that outputs the program input information
in a descriptive form for reference.
SEARCH The subroutine that calculates the chord and twist angle
at a given position on the blade utilizing a linear interpo-
lation scheme on geometry data inputed.
131
CALC - The subroutine that determines the axial and angular
interference factors and related parameters.
TIPLOS A subroutine that controls the calculation of tip and
hub-loss factors.
BESSEL - A subroutine that calculates modified Bessel functions.
NACAOO - The subroutine that determines the lift and drag coeffi-
cients at a given angle of attack for a NACA 0012 airfoil
series.
NACA44 - A subroutine that determines the lift and drag coefficients
at a given angle of attack for a NACA 4418 airfoil series.
NACAXX This subroutine is designed to use curvefit
relations for airfoil designations not included by NACAOO
and NACA44, without other program changes. The sub-
routine listing contains airfoil data for a NACA 23018 air-
foil designation.
NACATT - A subroutine that determines the lift and drag coefficients
at a given angle of attack for tabular airfoil data inputed
using linear interpolation.
SOLIDT The subroutine that determines the solidity parameter
of the particular wind turbine design.
R TIP Bcrj drRHUB irR
132
where B is the number of blades, c is the local
chord, and r is the oca1 radius.
ACTIVI A subroutine that determines the activity factor of a
particular wind turbine design.
TIPAF 100, 000
'IUB. (!. )3d()
16
where D = diameter.
GOLDST The subroutine that determines the Goldstein tip-loss
factors for a two bladed wind turbine.
OPTIONAL SUBROUTINE - GOLDST
The optional subroutine determines the Goldstein tip-loss
factors for a wind turbine of any number of blades. It
contains the main subroutine plus ten other supporting
subroutines and requires more time than the standard
GOLDST subroutine.
Figure A. 1 shows a general flow diagram for the program
package.
For design analysis, NACAXX must be rewritten to conform to
the airfoil section used, if other than NACA 4418, 0012, and 23018;
or the data must be imputed. Blade geometry, chord and twist
distributions as a function of percent radius, and design operating
133
Figure A. 1. Program flow diagram.
134
conditions must be specified in input. The parameters to be inputed
are:
Radius of blade--R--ft
Hub radius--HB--ft
Incremental Percentage (percent of radius for integration incre-mentation)- -DR
Pitch angie--THETP--degrees
Number of blades--B
Wind velocity- -V - -mph
Velocity power law exponent - -XE TA
Angular velocity of rotor--OMEGA--rpm
Axial interference model code--AMOD
0--Standard method
1--Second order method
Altitude of hub above sea level--H--ft
Altitude of hub above ground level- -HH- -ft
Coning angle -SI- -degrees
Number of inputed stations for blade geometry specification--NF
NACA profile - -NPROF- -4418--NACA 44180012- -NACA 00128888--Data inserted in tabular form9999--Profile curvefit to be in subroutine
NACAXX
135
Tip-loss model controller--GO--0--Prandti tip model1--Goldstein tip model2--No tip model3 -Effective radius tip model
Hub-loss model controller--HL--0--No hub-loss model1--Prandtl hub-loss model
NASA tip-loss model parameter--BO
Angular interference lockout- -APP- -0--Angular interference factorcalculated
1--Angular interference factor =0Percent radius for stations RR(I)
Chord for Stations CI(I)--ft
Twist angle for stations - -THE TI(I)- -degrees
Maximum thickness/chord ratio of airfoil- -TH
Angle of attack for zero lift--ALO--degrees
Coefficient of lift data (Tabular Input)--CLT(I)
Coefficient of drag data (Tabular Input)--CDT(I)
Angle of attack (Tabular Input)--AAT(I)--degrees
Inputs should be in the following format:
Card 6 Card 7Columns Card 1 Card 2 Card 3 Card 4 Card 5 +NF +NF
1-10 R B H HL TH RR(I) NFS(1-4)11-20 DR V SI BO ALO CI(I)21-30 HB OMEGA NF(21- APP THETI(I)
22) Card 8+31-40 THETP AMOD GO XETA NF+NFS41-50 NPROF HH (1-10) AAT(I)(11-20) CLT(I)(21 -30) CDT(I)
Output will be outputed on the basis of 70 character-field width.
136
There are several operation controllers that must be inputed.
These are AMOD, GO, HL, APP, and NPROF. AMOD determines
which method for calculation of axial and angular interference factors
will be used. An input of 0. 0 means the first order form will be used,
while an input of 1. 0 means the second order form will be used. GO
determines the tip-loss model. An input of 0.0 means Prandtl's model
is to be used, the input 1.0 means Gold steints model is to be used, an
input of 2.0 means no tip-loss model is to be used, and an input of 3.0
means NASA's tip model will be used. The third controller is HL, it
controls the hub-loss model. An input of 0. 0 means no hub-loss model
will be used, while an input of 1.0 means Prandtl's method is to be
used. The fourth controller, APP, controls whether the angular
interference factor is zero or not; 0--usual. method of calculation,
1--angular interference factor set equal to 0. The final controller is
NPROF, it selects the airfoil profile to be used. An input of 0012
means subroutine NACAOO is to be used, which is a subroutine that
calculates sectional lift and drag data for the NACA profile 0012; an
input of 4418 means subroutine NACA44 is to be used to calculate
sectional lift arid drag data for the NACA profile 4418; an input of 9999
means subroutine NACAXX is to be used. NACAXX is a subroutine to
which one must add a curvefit for sectional lift and drag coefficients
as a function of angle of attack (degrees). This enables the user to
137
use other profiles without changes to the program. An input of 8888
means subroutine NACATT will be used to interpolate data from
tabular form. This means that cards 7+NF, 8+NF through 8+NF+NFS
are required on input.
The following are the input Read and Format statements used.
READ (60,39) R,DR,HB,THETP39 FORMAT (4F10.3)
READ (60,39) B,V,OMEGA,AMOD39 FORMAT(4F10.3)
READ (60,41) H,SI,NF,GO,NPROF41 FORMAT (ZF1O. 3,12, 8X, FlO. 2,14)
READ (60,42) HL,BO,APP,XETA,HH42 FORMAT (SF10.3)
READ (60, 10) TH, ALO10 FORMAT (4F10.3)
READ (60, 20) (RR(I), THETI(I), 1=1, NF)20 FORMAT (F5. 1, 5X, FlO. 5, FlO. 5)
READ (60,11) NFS11 FORMAT(14)
READ (60, 12) (AAT(t), CLT(I), CDT(I), I=1,NFS)12 FORMAT (3F10.5)
A sample deck is shown on the next page.
For more specifics on input/output specification, refer to the
program listing and to the output listing for a NASA MOD 0 design
case.
Contro
Deck
Logoff Card
Figure A.Z. Program deck layout.
139
Program PROP Listing
*'J1'lM ' ... .TIEr(Il--TWIST ASGLE O' 31111113 A 71A 71
A A . .. .11--MA) TS1<1ES5/3M1,)r) SArI) A 71A ' A 76
SSSO' - 15 1A5-I 1176 A A ... .ALO--ANGLE 0 ATTACK FQS Z5) LI1T - 1115612 A 75A A AlA
A 5 . T(t)-)2. IF LIFT )ATA A 77IAI'I O5)5*M A A C A Fl
555A IL2ULTES T4 TH)5L1I3AL S1'14NC 'A-?A461155 OF A A 7 2 ....C)T(I)--)E. OF D'A lIlA A 71P5JL5S TY' Ill) TU-A1t5. IT IJTILIZS A 2TI'S3AAS-5tJL6 A I 4 II(ElliS) / T.4.4- 'All TE2H1IAO )F NU1St2AL I sT;sAT1o'4. A 'A . .. .AAI( I) --)N, IF ATTIC.) - )6'?T A 31
-
A SI A 321IIENCIO'l -0<125 21(25) 110<11'S) OAT (25) Ct_I 025) 221 (25) 1 11 A 13241121 5.lA,3,I,V.s,TlET0, I4J),,S3. .21 A, ,SI2.'L.Pl,SX,W, 2 1? 'EA) 55JT TAIl A IA
021P501,A,Tl,T?.IA,T,,TS.T5,.IA.TT,<FTA,'lli.ALO A 13 2 1 53 1. SEA) (53,35) ),3,TSET' 2 36
J5*JT 'A5AT0<SA A 15 SEA) (50,25) 5,S,JM1A,A5O3 A 37A IA SEA) (<0,61) ,ST,1F,G).'l"l A Al....'--SAltJl 2' ALA)r - FT 2 17 sA) (53,6?) 5.., )-1,A?P,XE14,5-I 4 IIA IA REAl ('3.AA( T,ALO 0 30
.. .SA ---125 '13122 - FT 3 II '0<) (F), 61) (55(1) .21(1) ('1I (1), I = 1, IF I A 313 21 IF (I'F)F.A2. '515) 1) 10 A A 32
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....--li!') JE._A2I'V - S'S 2 27 'SPIT 11511 ANT TITAEI E)A OUTJT A 35A 21 3 1111
....,EIA-J1L_l)IT' P').i L3A E)'1N0<' 1 21 CA_I TITLES NS,)I,T-IETI.NF,S)L1( A 13143r 1102.TAL21--A53 114' S-IL3CITY iF -'<0' "5 A 31 IITTIILI?ATI)N ASS CIASTAAT '0<51110< CALCULATIONS 5 13)
4 3? 2 4136))--A)IA t'T F'12C5 'I) EL 2l 2 11 N 201 A 105
1 -- 2"3530'' 3 3 11)1 -1 I 1061 -- IL33'I A 35 11 0.1 A 1)07
A 3, 12 = 0.3 A 10%....-l--OLTII4)< JFliJE7l33AL-41'L-'T A (7 T3=3.0 AlASA 11 71, 3.3 - A 113
-1--ALTSTU31 37 -lA) A105 ISSU,) Lsl'L - A 31 15 3.) 5 1112 .3 TA 0.0 A 11?
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W"ITE 1,1,21) TILT''lx P
CS! Si
4 247O 233A 'N)4 0900 '910 2(20 033A 29'.O 215A 290,A 2974 '133
O '334 335)
0 301I 3021 300O 301.
4 3055 300,
A 337O 3050 3394 3I01 3114 11'O 313O 11'.
A 315I (14A 174 319O 3190 3005 3210 3224 3234 32'.
4 325A 325O 3274 3'N4 3214 330O 3310 3320 330A 33'.-
1 CC
N
5
N
I
101112131'.
15lb1713192)
SI = SO" 1/IRO.CR1. = . 7N'RGALL SEARCH (CRL, R4,C1,TNETI,NF,C,TH"T)SI = CclRANG = THET'IRO./PO,THETPWRITE ('Si.!?) BANGWRITE (61,23) SIWRITE (61.25)WRITE (61,25) 0WRITE (61.26) RWRItE (61.27) 11CAL). 001.100 (R3,OI,NF,A.'1.PI,50L)(WRITE (61,2RP SOLOCAL). OCT13! (R4.C1.NF.3.4,PI.ACFIWRItE (0,1,79) TICF
WRITE (61,30) SPOOFWOIT' (0,1,31) 90WRITE (61,32)WRIT' (61,33)WRITE (0,1,34) (PR(I) ,CI(I),TI-)011(I) ,I1.NF(WRITE (61,35)WRITE (61,36) 04IF (000.00.0.0) 10 tO IWRITE (61,3')
1 IF (AMO).E0.0.0I GO TO 2WRITE 30,t,3R)GO TO 3
2 WRITE (0,1.39)3 IF (R.Gt.2.0) GO TO 1.
IF (10.04.0.0) 5) TO 0IF (G0.EA.1.0) GO TO 5IF )GO.EO.2.0) GO TO AIF (G0.EO.3.0) 1) TO 7GO TO 9
4 IF (00.10.2.0) 5) 00 6IF (00.01.3.0) 00 TO 7GA TO 5
S WRITE (61,61)GO TO N
6 WRITE (0,1,42)GO TO 9
7 WRITE (61,43)GO 10 9
O WRITE (61,43)9 IF )HL.E4.0.I) GO 10 10
WRITE (61,45)GO tO 11
10 WRITE (41,44)11 CONTINUE
WRITE (61.46)WRITE 461,47)RFT URN
12 FORMAT (IHI)13 FORMAT ('575 T'IEIPET 0101. 'ER000M5ICE OF 5 000PELLEO TYPE WINO TURBO
TNT)1). FORMAT )lll,SX,2T5 OPERATING C031JITIONSI)10 F040AT (II, j'Y,235 WIN) VELIIITY MPH ,F7.6)06 FORMAT (/,15X,35'I 4131 VELOCITY GRADIENT EXPONENT ,F7.4)17 FORMAT (1,150,395 II)) HEIGHE ANIOE 130USD LEOEL - Fl- = .012.4)19 FONMAT 1l,15X,43'I ILTITUO' OF PU) AROVE Sri LEVEL - FT .015.4)19 004101 (/,ISV,!4P 050ULOP VELTCITO - PPM = .010.4)-20 FO'MAT (l,15X,193 TI' SPOEO 31111 = .07.4)01 FORMAT (/,551,544 33111,1 SOIL' 04)4 4)40101. 141ST - IEG'EES = .010.
TI.)
2? FORMAT (1.150,434 'ITEM ONGLE T 0.7, EA)IUS - DEIW005 = .013.6)23 FORMAT I/,15V,26I CCNINE 151LT - !4E5'EEO = .0133.5)74 FO'5401 (///.c0.1.4 PLAI 1335109))25 FORMAT (/.15X.'3M N11l'IFO OF -ILA)ES ,F3.0)2 FORMAT (/,IVY.1)3 TIP SA4IUS - 01 = .07.4)07 001MAT (/,1333,1 Ii HU 40110$ - FT .07.4)
2 FORMAT (I.15X.IIH SOLIST! .F12.50 9 9329 FOMMAT (/,15X.1)H ACTIVITY FACTOR .FI?.5) 3 95SO FORMAT TF.15X,ISH NACA PROFILE = .0'.) 3 95II FORMAT )F,15X,338 NU4AER OF STATIONS ALONG SPAN = .15) 9 9637 FORMAT (.',1OX,Z9H CHORD AND FROST SISTRIAUTI000 9 9733 FORMAT (//,5bX,ISH PERCENT MAOTUS,5X,OHCHORO-FT,tOV,9HYWOST-SEG) P 983'. FORMAl (,,238,F5.1,00,F10.5,I0X,FIO.5) 3 9935 FORMAT (///,SX,33H PVOGRAY OPERATING CONDOTIONSI) 3 10036 FORMAT (//,15X,YSV OICRE4TNTAL PORCENTAGE ,F7.5I 10137 FORMAT )//,15X,39H ANGULAR INTERFERENCE FACTOR, AP = 0.0) 9 10230 FORMAT (/,G5X,0NH WILSON AXIAL INTERFERENCE METUC) USED) 9 10339 FORMAT l/,t5V,RTH STANOOVO AXIAL INTERFERENCE METHOD USED) 3 106NO FORMAT )!,t5Y,31H TIP LOSSES '103ELED 3Y PRANOOLS FORMULA) ION51 FORMAT (II. IRX,41H TIP LOSSES IOOELEO VT GOLOSTEONS FORMULA) 9 10652 FORMAT (//,1RX.lNH NO TOP LOSS MOTEL) 9 10753 FORMAT C/,15X,31H TIP LOSS MOOEL -- NASA VERSION) 3 105.5 FORMAT (F/,ISX,2'H NO HU3LTSS O3EL USED) 3 10965 FOPMAT (//,15X,TMH HUALOSSES MO3ELES VT PRAN)TL) 3 1100.6 FORMAT 0191) 9 lii57 FORMAT (1/1,7TH PERFORMANCE ANALYSIS)) 112
END 3 113-
SUMRDUTINE SIANCH )ML,MR,CI,THETI.NF.C, THET)
SUNROUT ONE SEARCH 0 AL,NR, CO ,THETI, NF,C.TUFT)CC ........ SAOCH - QETERMINES THE IRONS INS THE TWIST AISLE ATC A GIVEN RADIUS ALONG THE SPAN. IT UTILIZES A LINEARC INTERPOLATION TECHNIQUE.C
COMENSION OROY5), 10125), THETII'S)COHON H, IN,HN,3,V.O,TIEFP.A303,4,SI,GO,O'IFGO.VMO,VIS,HL,P1,'X00 1 I1.NF
RRV RLI)RX'COS)SI))'IOO.OF )RAV.EO.EE(1t) GO TI SIF I9RV.GE.RSII)) GO TO 7IF II.ES.NF) 65 1) 3
I CONTTTJ2 1 1*1
(RIP-MR (i-I)) /(RR(i-2)-Ri1 0C = 'E'(CI )J-7)-II (i-I)) PCI (1-1)THET PER' (THEFI(i-T)-THETIIJ100*TVETOIJ-I'GO TO N
3 C CAINE)THEY = THFTI)SF)GO FT S
N C = 10)1)OHET = THFTI(1)
S OMIT = TVET'PTIINS.OF FUR N
NY)
SUNROUT ONE CALC IRL ,C,THET,FXF,FYF.XMFXF, XMFYF,QF,TF, RE,PHIR,CL,CO
SU9ROUTINE CALC (3L,C,THET,FYF,FYF,A'4FXF, UMFVF,SF,TF,RE,PHIV,CL,COT,CX.CY,A,AM.XL,AK,NLPHA,F,CLT,CAT,AAT,CLT,CDT.NFS,SOLO.TfI
CALC - DETERMINES TAT AXIAL INO ANGULAR INTERFERENCEFACTORS AT A GIVEN RADIUS AND DETERMINES FUNCTIONS SEPEROENTUPON THESE PARAMETERS.
SIMFNSION AAT(25), CLT)2V), CJT)25)1O'MON V,TR.N9,3.U, X,THETP,A0S,9,SI,GO,OMEGO,RHI.VIS,HL,PI.P8,W,KPRQF ,A'F ,T 1, T7 ,T I, 1'. .T5, T6 ,T7, TO, TEST, VETA, RH, ALlXL RL'O"EGO/VIF )O,TT.,50 A S.OF (A.IT. .5) OP A.RHDY) 15 i1,50
OTTO =SELlS = UP
CCCCCCCCC
CCCC
C
CCC
C
CC
CCCC
CC
'A
S
N
9to1112131'.15IS171519202122232I.252627-
23'A
56
910111?13IS151617
PHI ATAN)(5._A)COS(SI)l(1.*OX)XL))PHIAA ANSI'HI)XXI COSIHTAAUSON)'HIAOITOLD XXL'RRLPHI' = XliiALPHA = PHI-T-OEE-THETPODTI = ATXN((1.-A)/)XL'(1.*'.'AP)II-ATAN000.-A)/XL)OAt = OATh'..0*2 )I5.l15,)I5OLOTH)/V)/(A1.FV)''2+(RLFRI"2)OALPMA OAIPDAZALPHA = ALPHA-OALPIAALPHNS = ALPMIA3.00I
CALCULATION OF SECTIONAL LIFT AND DRAG COEFFICIENTS
IF )NPROF.EQ.A5iNI GO TO tIF (NPROF.ES.0012) GO TO 2IF (NPROF.EQ.N430I GO TO 3IF (NPROF.EQ.9999) GO TO SWMITT (51,20)CALL NACA'.'. (ALPH*,CL,CO,ALO)CALL NACAS'. COLPHAA,CLI,CSD,ALO)GO TI N
2 CALL NAC000 (ALPHX.CL,CO,OLO)CALL MACOAT (ALPHAO,CLO,COD,ALO)GO TO N
3 CALL NACATT IRL,RX,SI, ALPHA,CL,CO,ALO.W,AAT,CLT.COT,NFS,00LO)CALL MACOTT (SL,RX,SI,ALP'IA),CLT,CCO,ALO,W,ANT,CLT,COT,NFS,SOLO
GO TO SCALL NAC008 IRL,OT,SI,ALPHA.CL,CT,ALO,W)CALL MACWAY )RL,RX,SI,OIPI413,CLO,CIO,ALO,W)
S CONTINUEIF 060.LT.3.P GO TO 5CL CLF'CLCLO CLFCLDF = 1.GO TO 7
CALCULATION CF TIP 0040 HUN LASSES ......
6 IF (CAT.EQ.1.) F 0.0IF )CAT.EO.I.) CO TO 7CALL TIPLOS )XXL,00LO,F, N,GO,HL,PT,R,DL,PMI,RM)
7 CX CL'SIN('HI)-CO'COS(PHI)CT CLCOS(MHI)HCOS101PHI)COY CL'SIN(PlII)CY CLCOS(°NI)SIG (9C)V')INL)OF (AMOO.EQ.O.) GO TO R
HOR = ((.125'SOG'CTYI'(1O5)VI)''200F)SIS(P44II''2)VAR = )O.125'UIGCUX)l(F'TIN(PHO)'COS(pMI))CAN F'F+S.'PDR'F)A.-F)IF !CAN.LT.O.T) CAP) 0.0A = (2.'PRRHF-SSRT)C*N))/)2.')49R,F'F)AX = VA1( 1.-VAR)IF 100F.EQ.1.) AX 0.0GO TO N
N ANN I.125'TtG'CYV'(CIS)SI)2)VAR O.025'SIS'CVAA VOP/TFSIN)PRO)''?HVOR)A' PAR/)T'SIN)PHI)'ITS(HT)-WAR)IF IAPF.E0.I.0 OP 0.0
N PCR RLI(RE'C051SI))TOMPENINS CF AXIUL ON) ANTULAR INTERFERENCE FACTO'
IT FRI IONS.
IF (i-N) 13,12,1010 IF (i-SO) 13,17,1111 OF li-Ic) 13,12,1307 A = (APBETA).P
I/i
OP (AP*OELIN)..513 CONTINUE
C rS1 FO0 CONVENGENCE
IF (OPF.EQ.i.I GO TO 14IF (IRS) (AP-OEL14)/AP) .LE..000i) GO TO 16GO TO 15
14 IF (435) (A-3l*)I3).1..QOO1) GO 10 14
IN CONTINUE16 IF (IK.IE.G.) SI TO 15
FO3 D'EOATION 37 IIRTEU RING STOlE REPLACE TIE FOLLOWING C CGPDO
PC2RWRITE (61.21) 'CEOWRITE (41.22) 0110WRITE )41,'l) U
WIT1 1,1,24) )ELTNWRITE (61,25) 4'WRITE (51.25) 1
I (J.LT.43) 23 rI 17WRITE (61.77)
17 CONIIIU
CULCJLIIION IF FJNCTI050 )PEN)EIT U/ON ISIIL 0502 AMZ3LTP INI'RFEO-ICE FCCTO'.S.
15 CISJTI(UEW = S31((G.-l)'000S(SI))''2'((i.3P('OL'3lE5A("2)SF O'WC/UIOCONSTFX = 21901'3(
3S1 V
4P4XF =UMFYF = FYF(O._93(f'2OS(TI)CII. (3 .50-il' 0C) (W.)3FIF 21 1'IY'OS(SI)
CR = 2/OXCLA= (ILD-_)/i.O1tCDI = ('2"3-CT(/3.O91CX' CLASIS(L)-CJA'2)S(POII2YCYP = CL 305('4I)+CJASIN)P'I)-2.(TAN IN)PSI)/235(PSI)FTFN (2. ''Y.Y' TlN)'(1 -A( 9RQTEST 1455*111 (TECT.71.1.) 23 Ti 19.)OITi (*1.75) 0
WUITE (61,29)1151 3.
ii CO'ITI)LJE11 = T1.ET1'20T2 T?,F** (1 .-1I')PCR13 =TATO =
14 TFTfl.-l)PC12R17 T7X(HT..2)S('SI )2)2L'Sl0(PO1)-COS(P41)'3.'IOS(POT)'?CLI)(1.-4I"7OL"3L(1/2.TO '.(((i..SII)'*4I )? )/713(P-II)2L-C1'SI()POI(*CLISIN)P-III('
5(1. A'( (1.-0(OL.'C))R/2.
-
'5 FOEMIT (9)4 VIU 1006 SPF2IFIEO II MICA '01(16' NIT ShOE) IN IRE P454= R');*A4 ,)ILL JE N420 4,15.)
It FTOIAI (/f,5E,31-)STSIICO - (L12IL -4131Ji/TI' 04)1)3)
1 90 27 FOR4A( (/,iO(..0O IXIAL IsTE'FEO:5C F4CTO' (PRE01'iS). A ,F12.8 1 14?
1 91 1 163
O 92 '3 (30941 )/.I0X,310 AXIAL INTEOFEONCE F..ç15, A ,Fj,M( 3 164
tO 93 2 105541 (,,100.65H INSULAR ISTEREEOEN'tE FACT5.R ('REXI)US), UP , 1 165
o 94 SF11.5) 3 166
1 35 32 FOO'IAI )f,13X,35') SNIULAR IOTERFEOOC1 141101. 3' Fjl3) 1 167
O 96 25 FOXMUI (/,13.?AH NUMUL.' OF 11101110(3 = .04) 1 G65
O 37 37 FOBMOT ...... 4111511)5 - 50 21'EOERTFNCE UT T3IS STATION) 1 169
O 25 FOOIGI (/,130.305 IIUWAL 75536, N .FIT.5( 1 173
O 99 39 FORMAT (/,IOX,?5R TAOGENTIIL F3037. FT .111.51 1 171
o too EN) I G7l
1 101O 102 5180011156 TIPLIS (0, 53, F, 3, GI,4L,PI,2,0L,4I .0(I)
0 1035104 S)J30)JTINE TIPLOAS )U,UI.F,0,G0.-IL,'I,'.RL.PHI.RH) 1 1
31051 GUI
2
2 TIPLOS - 1ETERNINES TIE rIP 431 SUB LISSES5 2
E 3
) 107 2 BASEO U'IN GIL)TIEINUS IIEURY, 10 'RIMOILIS 14(005 S
1 108 2 00 F3( THE CASE OF NO LOSSES I
0)09 1 6
5 71 110 SUN? 0.0
I iii SUM = 0.0 1 ¶
1 112 AK = 15 9
I G13 449 12 10
3 114 AM 3.0 0 ii
O 115 IF (3.51.2.3) =1 TO 1 2 12
1 116 IF (51.73.0.3) 00 10 2 2 GO
I ii? IF (51.155.1.0) 51 13 4 4 14
1 115 IF (S).Ec3.?.3) 30 10 3 1 15
1 119 1 OF (53.10.7.3) 1 10 3 6 G6
0 123 2 CUOTIIUE 4
1121 F = 1 151 19
0 GO?1 123
0)00 1) E 20
3 G2'. 3 F 1.0 5 21
O 125 GO II 5 5 22
j 525 4 I )(lRS)SI9(S1))) .LT..000G) 50 1) 7 1 23
) 127 CA.L SCLST (5,U3,F,I,-0Q,°T,?,RL.PSI,SUMT,SX1.U<.AMO.ON) 7 24
1 121 3 SUILISS CALCJLITIO9S E 29
5129 2 1 26
3 130 5 I ((I.El.1.0) 30 (0 6 2 27
3 131 Fl = 1.0 2 28
GO? GO 13 7 1 23
1133 6 Fl (O./PI(.U315E)EXP((0(XL_RI))/(7.*RH'SIFT(SIN(P5I)'2*.1031I 5 30
3 13 8))) 5 31
1 535 7 F FFI 2 32
3 535 '61505 S 33
1 137 65) 1 36-
O 138 5U36)0J0 LIE 3ESSEL (2.8,01)1 1390 1.0 SURO0JIINL 3ESSEL (2.8,01) 1
1141 2 2
1 1.2 3 OETSEL 3ULCULATES SESSEL 755211055 FIP TIE OIL0STFIS 3
1 160 2 TI' LOSS 53)16 F
3144 2 F
0 165 S 0.5 F 5
1 166 AX = 0.3 7
1.7 C 1 F 5
1 149 00 0 <=1,03 F S
I 541 3 ).25'Z/)"AK F 100 0.4< F 51
1. 1?I 12
1 IX )1. F (3S j3 IF (TK.LE. 0.0) 30 33 3 F GAI F 55
0 1-2. 16156 II 1) 1 F 17
0 157 ' 7183115 S 3/(2)*S F 190 159 5<41. 211 51,1
1 161- F 21
3 CONTIAUCAlET 25 '4
IN)
22F 23F 4
SU3A')UTIIE NACAOD (ALP-lA,CL,T0,ALO)
SU3S'OJTINE 940*30 )ALP44O,CL.0D,ALI) I I
1 2
*300 - 1ETEAMINES TAL 0)EFICIE3TS IF LIFT IN) 0501 1 3
AT A 1031'. A'IGLC IF 01140<, IL.M1; FIA A NACI 3012 AIA0IL.1441 1)UATI)'1S WAE 03141313 344 4 OSIMOGNAL PDLYIOIIAL 1 5
CU50EIT 0 '100) 3014 PU1LIS43 IN ASIA EP05I 3'). 663,PAGE 524. 5 5
A04.73 1 3
A? = 7.A0 A A
SlO 0.00,3 5 10
S21 0.003'. 1 11
SI? .130 5 12SlO = 0.3153 5 13SI'. I. 00)35 1 1'.
S05 12570. 15AMOS 0.213 1 16A A.?AA-AL33.1.1433/140. 1 1'IF (1.51.1344) 0 TO 1 5 03CL 03*0 1 19C') = SDD*)S)111 (A')2441 5 20
00 15 2 1 21
CL )001-(Al')1-AMAX)21 1 2?'0.0) II 10 2 5 23
CL = 3.0 24CT S034'S05(A-A'410)2.S15(0-AMAS)"'. 6 25IF )0).L.1.0( I) TO 3 1 25CI 1.0 5 27.AETUAA S 23EN) 1 29-
S')1AOUTI4E lOCUS. (ALPMA,CL,C).ALO)
SU5(JJT1'JF MACI,, )ALP44A,CL.C),ALO) 4 S
-4 2
NAOO - 1T'51I3* OAF 00EF0'IGN'S JF LIV" 425 3501 44 3
AT A IVTN 155 3* 11100<. AL''4S1 *55 A MACA 4513 AI5F5I.. 4 2
T44E CIUATOIMA A314IA) 1)33 S'I'AOIAIL P1LY401I4. 4 5
CU51'1 0 'SAC) DATA PUOLIS4E) 15 2010 '03051 10. 3?'.. PAl-F 431. '4 5
4 7
AL' 0.'Mi'13O./I.141533-ALI A A
03 (5_*,33) II TA 2 -1 A
CL 3.1293?41L°,-O.3975 A 10I (A_.UE.*2'.0) GO TO I '4 [1Cl ).TlI1jG,4AL'*).)Q)023t'4AS'3ALP'2-0.)00)3370,4Lp'3,3. '4 12000551 -4 13CO TA 5, A IACD = a.)oo15AS355*ALP,'0.qAao732*ALp*z,3.o0o0A2o=24*ALP*.0,l. -4 155004?9752 '4 14GA 13 5, '4 17I (AL.5E.1?.I( 5) 10 3 '4 13CL 0.)731AL'+I.5073 '4 12SI 13 1 A 20
A IF )A,.P.,E.15.,) SO 10 4 A 21CL l.'15l'7AL'-l.3l73311'2-l.3',3 -4 2?GO 1) 1 4 23IF )I_P.A.1G.lI GD TO 5 '4 2'.
CL -T.11A,1.U3 A 2500 1') 1 '4 25CL -T.O?3'AL'l.lI, '4 27CO 'T.Ui3I65,9Nb.MALF-).jMj1335 4 ?'
4 ''4
Ni '4
A'132'0JTI'SE NACAXU )A,A4.50,,.'4A,0L,C),AL2.W)
SUBSUJ'CNP 2*1052 (RL,A44,SI.AL'AA,CL ,C1.ALO,4)
3400055 15 034 EMPTY TU3S)STIN I.4 35* *32 4 *A)*ILLNor *503103503 IIIALJ. INE 3)51 lNS51 'IIj.V *11 F) lOTIONSPO4 SICTIONAL LI*I 413 )SAG SQFICIFNTS AS A 7,32110)3 OF
3* 01101< 12 0500313.
ESJATTU'IS *')< LIFT 440 DAIS C*FIIO03NTA FAA200* '3013 A15)IL '444= 2115 I'4SCATE).
A ALPAAINO./1.121593
00) CJPOE FIT *5)6543 FDA CL 42) 0)SM A/113A.CLAD ,.?3AL) -.02234I 00.51.1,.) 5) TI I
CL = CL*O)1.P.2'X0(2M4'I)/0.I' (OLP.40-ALO)CLASS S.1)1.-.42AXM-.32A4XMSNlIF I_. T.CLIAA) CL =40 1) 2CL . 3'(1.1.522)ALPAA.7323)?)
2 4*30 ABS)4LPA*I00*34 431)0)IF 000035.P.163.) 1) ID 3IF (4003S.E.t3.) GO TA 4CO 0 .378 AL0'402-0.00355A*AS*0.0105SO PD 5
0 CO 0.34*?.0(A4RS-3.142)'lGO 1) 5CD 2.0-1.33 9(I.57-AANS 0303
A RPTLJMAEN)
SUDAOJTINF MACAFT l(L.5X,SI.0L'4A,OL,Cl,ALO,A,AAT,CLI,COT,FJFS.SILI
SUDMIJIIME MOdEl (AL,RX,SI .AL°444,IL,C3,ALO,4,44T,CLT,CJT,NFA,50LAAl
'IACAII - IS AN I'11EP')LA?ING SU'450U11'IE 13 IA4IEAPOLAI'FAISFOIL '1014 IA''JTF0 03 TOAL? *)43
OIIE'1S00N 541)25), 003(25), '31(25)A A_PAA'l 30., 3.1,1593-ALl00 1 'jj,3*5
I' )).LT.AAI(1)) DOT).IF (O.L.A4T(I() DO 1) 2CF (I.E).AFS( 13 10 3
COATI '10!0)31
PTA (A-lOT) J-j)) 111TU-2)-ASI( i-I))CL = '05 (CIT (J-2) -CLI) J-S) I CLI (3-1CO = 'FMIC3I (J-2)-C)T( i-il 4.0)1) 3-1)GO TI 5CL CLI (5S)00GO TA 5CL CLIII)Cl 001)1)CO (C)3..751.Z'.12'.)F3.,'S
ES)
4.
A101112131'.
151?1319202122232'.
25252r232930313233-
131112131'.
15151'Ii1220212223'N2525-
SJ)3O'ITINE S)L131 (3N.C1.5F,5,5,Pt,S)L)ISU3HOI1t ME S)LI 31 I PK,C (.5? .3.5. 1. Sit))
3LI)t - TENI'4(S TI4 13181. SOLI)ITY OF THE11130 TUSMX'4 113155.
011E431)N 55(253. CX(25)NFlSt 1).30 1 11.N
S3. (1C1I1.t)*Ct(Zfl,2.I.)55III-5Q)I.tI)5,I$$.St i6S3L
C3321'IiES3..O 5S1flP152)SET U1EM)
SU33LJT(NE ACII.4I (.X,NF.P,5,PI.4C?)SUSSIJTIIE *111)41 (55.5!. Nc,R.S,31,*CFI
&STI3X OETERNIME3 15 t.CTT4ITV FAcTON or (NE11133 fUSSI'IT iSIf3N.1)!IEIIOI,N 5(20I I(25)5c1Si I.30 1 1t.NX
Ci (CIIl.tIHCIlT))2.5)5 = 1(53(1 4-55II*O/2.55((lt))4tO3.33)5 (55(II-53L1*t))/l01).
4 1' I'. 51) R35"3 05 35SI
CO3(13J1£5' 3t11)0530./56.5(1035EN)
SU13)JTI'4E 3DL1sr 4u.u) .F.1.,PI,5,'L.P441.3U12.SJ'4,*,(.*M'4.A4)SU3N)JTI5E 5)1.350 (U.Ui..O.)).PI.5. 51.lI.SiM2.Su5.8K.AM4,A1I
cOLOSI - )ETEHlIIZS 13 OL)STEM TIP LOSSF*T35S '03 TIlE 2 31.33 U )ESIGS 85E. FOP 315035 CS.SESWITS 3L01E 3311)155 OTHES 11143 (.I) TN P5450(1.N1)E1 15 USE). 11115 548 31 1I44MEfl 1' MECESSASY, 43)1SF 3)L)SIEM 51)11. FOS 431 MIJIIEK 3F IL8OES P8! 51SUSSIITJTE1 '13 TIllS SUJTINE.
30 5 4.1,'.4 C2.'ElPI.IOS U3Il3!
3.42!C4..L 3153EL Il,Il,%II24.1 E5STL (11,5.41015F (l.GE.3.H 53 10 11) 2.'?.
(142 ?2/(8-!2l.(Z2'Z2)/(f3-V2)IV2Il*(Z?''3)lI-V2lI'-V23S 12-4211 (..Il( -421'(3-i', I5-42)(O-V2II
2115!53 13 2TO ii'U)l(j.4U3)12 '.'UU'(I.-U'UI.( (1..JU)"6)TI,
(61011(11. *3i1 "10)
KKKKKKK
KKKKKKK
S
567
9IS1112131115-
I23
5S7P
S
101112131415161718-
23456
93
10It1210141516I?1513202122232425262723
33St
1(10! t1),I2,42,T4IU2"2).1S/(I'"31 N 322 cvi (JJI,(1.,UU)-2T151 1 33
SJI = Si'l''.4J.,l' N 3.3F (45.IE. 1.1) 50 13 3 3 35E -0.3)5/lJfl".6651 '4 36IF (43,91.1.0) 0 Ti . 5 3?E 0.OSt(1)"1.255) '4 39
6 IF (45.1.t.3) E 0.0 '4 35SUI? 2,((U3'(JOAHMII(1.,i0i3)-,)'(AI/811) '4 6384 £4.1. N .14 I (!.'1)5-i.) AKI 24 2.41) N 4?
3 £11 = 4Kf(.'4M.i.I 5 43S (J'J)2(t.IJU)-(l.(I'PI))SUl 'I .6
CIsI U-(?./PI)'SUPI2 5 5F l41.,uu)RJu))cI;2 N 44$ETOPN '4 6
13) '4 4
Line A1.38 should readNN = (R-HB)/(R*DR) + 1.
OPTIONAL SUBROUTINE PACKAGE(changes are necessary for use with PROP)
COMMON statement must be made toagree and sub. CALC must be renamed.
SU3S3IJTT'4E G)L)ST (U.U) ,F ,3,33,F1 ,3.SL,PHI, SUI2.SUI,4K,*N'4, 81$
SIJ3SOJTIIIE G3L)ST (U.U0.F.Q .Gi.PI,R.5L.PIZ.SiM!.SUM.41(.*'Il.Al) 3 13 29155(39 I ----- 3890457 197 3 33 11
5)L0STIT - OTTERMIMES THE 3L)STETN TIP LOSS F8CTOS F33 3 4HISS TUS3IIES .4015 4'IY N349E3 3F 3L8)E3 3 6
3 2501503 3.03.33.' 3U2 PU 3 3UT! U')U) 3 1354113 (i..U2)132 3 11NFL .vATI).5.,(PI.p1) 3 1?SF2 SlTI)7.f'j 1 13C4LL (Si? (U.P,1SUI) 3 14C41.L 31538 (U,j),P,BS3N) 3 1541 SFI'TSUI 'S I,42 3F2B5J3 3 175' 30+52 3 13SI)? 1 11EN) 3 28-
SU4'3UTISE TSUS (U.',TSUI)
SU3'4OJTIME TSU8 IJ,P.TSUI) 11535 000 230 5 3,1,53 ' 3
3M 1-1 PO ''431..;) 0 SFl (?.'3S.1.)"2 S* O.0 7TALL TC4LC 13,5,11)SUs Ill'S 1
I 1535 TSUI.SUI ' 1)FTJOI' 115) 12-
SUA4OUTI>< TC)L3 (1,6.11) IJ3FJUT I1 T3<l ('0,13, 110
SU3IOJTIIE 74L3 t1.X,11) 1 6 5U33)JTISE TEI5 (J,6,T1O> 4 1
C)5MU4 P1 1 2 U2 = JJ I
U 4/40 3 9 U3 J20 1 3
CA..L r<1ls (J,Ii,110) I . 04 J33 1
113 = T10/)4'tOI 3 5 US = J.'J 1 5
IF t43S>TI0).GT..300)I5) ,Q 11 1 1 6 36 35U 1 6
76 1 7 Ui U6'U 1
0/) (1 UUI "10)) UI J7U11. = L6.UU'(1.-1,.U'U121.U",--. .'J"5>F)(I.1U'fl"7) I UI J8'J 3 3
12 5.3J(l. -JJ)/)(1. UU,( 3 13 012 U3L1 1 1010 *U'J)/tt.JU) USlO (1.AU')"Ii 1 1111 3 17 UltI 11.'J2('1t 1 12,O 1) 9 3 13 U312 (1.'U2)"12 1 1302 14 0 1'. U113 )1.AU2>"19 I II.
Il 4 1 15 UI)'. )1.J2("t'. 1 150 = 31/2 3 16 Ull (1.FJ2> 165 412. 1 17 UlU )1..J2)"2 4
IF )1.1.A( 0 71 5 3 15 000 (1.*J2> "0 1 IINV = I 3 19 UI. (1.402)". 1 19
1 20 A 6,.'J2/4I310 1 20- I it UP = (135.U-1152.U3)/Ul11 1 21(F (9.11.0) 32SF 1. 1 22 OPA (12B.-6144.'U142t893.J4(/J51Z 4 22IF (3.63.0) 3791 = .5 3 23 AP'' -15360. J02Ui20.U3-o3776U.3Ol/LP1t 1 2370 = 1. 1 2'. All (1569.1t35160.J2-73j1360.4J41919?93(.'U6)J314 1 24SO'> 0.2 3 23 9 (1.-75.J2+303.'U4-1065.456.b0.U5-16.i13) 4 25
2. '0 25 9P (-150. J*2.12. U3-6390.U510690.'U7-360.131 3 26AC) JI,4I 3 27 'OP' = )-15I.*7203.U2-31300.U47573l.Ub-32,0.'U3) 1 27
A X'1 1 25 IPPP (16.7?.' J-177410.'3315,563.'J5-15920.07) I 26(l"2-42) 3 29 414 = 14472. -0314)O.'U2.772600.'U4-151440.36) 1 29
1 (.E).l.> Ii TI 1 30 C 00551 4 30Fl = F"o 3 31 C' ) 1.-U')/tJ92 4 31AS A'N 1 02 '3P' (-2.U)/Ul?-..'U'(l.-U2)/U73 4 32535 3J9'.5 3 33 1 = J?/'J'l 1 33Fl 75 1 34 EP 2.'U/J32 3 04
3 35 <P' (?.-6.''J?3J13 05CIITI9U< 3 34 3 A''9.A' 3' 1 36C0'ITI3UE '3 37 IP 4F547 l0.3P*l50 3 37CALL 3<212 (IST,l, 1.41) 1 (5 UP. A0'P'413.'2P'373.'UP'0'1A'1'PP 1 30TI (4L'PI'V>/(2.'515(,,'V''I)I-SUI 3 39 F 1' 1 19GO 73 5 49 FP 1 4041 1/7. 3 .1 FPP = AI4'3*5,'0'PP'3P*6.A4''3P'*4.'0p*9ppp4'3IV 4 4153 31 1 42 TI 2)I" 4 4251 FLIAT>1V>f?. 1 43 310 C'>3'')31'*EP'F*'FP)43'(CP''0F2.CP'0P*3'0"EPP'FF2.'EP' 4 4332 51/2. 0 4. 3FP,E.rP*) 4 40.
IF )32.3<.A1> 21 11 7 1 .5 'hUll 4 .5SO = 0. 2 46 ES) I47. 5 47F 1. , SUISOUTIIE 31513 ('J,JU,P,021J1)
3 9'>0 5 I'4=t,IT0 3 53 SU3IIJTINF 51539 )J,J0,P, 9533) 1
A X"1 2 51 203505 P1 5 7
F'> F") 0 52 03 3.3 5 3
F) = 0 0 ,3 5539 = 0.0 5 4
S3 S275/5 1 54 00 1 J1,33 5 5F -)4040-5''7) '0 55 40 P*(5l ,) 5 6
I' (225(7) .1.9.I 0) 13 7 0 14 K 3 7
I 3=7. 3 57 NO 3 S I
535113U 1 55 A FLTST)31) /'. 5 1
CONTISUE 1 39 3 40/2. 0 19CALL 3<530 >4,1. 4<) 1 60 IF )A.9.>I 31>1 = .5 S it
1 61 I (0.73.3) 3111 = 1. 3 1?21 0/2. 3 C1.L "FUll (3IiT,l.4, II> 5 13NY >1 3 33 U U'S) S 14
01 2 6. ALL 37514 I INT,I,X,9I0> 3 153? 1 65 CALL 421SF (41,53,0,13) 5 16IF (Y3.'34,Sy> 743 -1. 1 66 SSI 9353 S 17I )43.1.tY) 1. 1 67 VSJM SSJI.23'>I/5I3 5 1511 1.SD-7l43'J'34 3 69 t (4SS>3S')'l-3St).L1..l031T1( 5) TI 2 5 19 #t.
1 39 >1 = 4541. 5 241
2 73- 1 33011557 3 21
2 CONTI9UE
EN)
SUNNDJTI NT AC)F TXM,A0,U,AC)
SU3NDJTINE ACO! )1M,U3,J,ACICANNON I
IF )9IFQ3.) 0< = 1.
IF )93FQ)) 044 1.IF )XI.EO.0.) .3 10 1.
AKANN
1 AC -(1U01J3)fC1.+UIJUA))A)lRETIJ)1.53
A)-3<DUTINE 35313 (TINT,3,X.'II
003<OJTINE 35013 )1191,V,X,31)COIMOl P0IF (TI14T.E0.l.) 13 13 1SN =GINA 0.(EXP(X)EX°)-5)0N9 0.5(ESPIS)-E.<(-X))F SI1514COLE IEF (F,MLIN.9,30)F -AN-I.MuM FPI.GALL CalF )E,MLI),5,051)NI = IA'IFANISNIA)/SINT(011/2.)10 TI AS 3.)AK 0.0C 1.1)0 (=1,5)
N (.25X91"AK3
1.
2 1< 1-1.IF (Tl(._E.3.3) 10 13 3
P =
3-2.03 F) 7
3 9
Si = S
S
(F (A9S(S1-S) .L1. .000301) 13 TO 5AK AIPI.
AKCONTINUE
5 CTNTIIJE01
S EIAA)EN)
1 225 23S 2'.-
T 1
T 3
T 4
1 61 7
I N
1 3
T 13-
2
I.
S
S
7
N
9
1.0
111213I'.
151517is1923212''32'.
25262'211
293031.
3233
35353?-
SU9RIUTIN COT (F,'ILI4,A,4Nr,)
SU3PTJINE CIC )F,MLIM,9,043) 1) 1
31113513' .3100), IN(1003 1) 2I 1.15 1) 3
IF (F.'3.-1.) SIC = 1. 1 I.
IF (F.E3.1. ) 091 = 1. 3 1
I (.EI.0. I SIC = 0. 3
IF 1'1.-I.) II 1) I. 3 7
I )F.3.1.) 1) Fl I N
IF IR.1.fl.) 3 T5. V 9
LII P1. 3 13IF 1513.51.3) 3j TI 2 1 11.
LII L131 3 1201) 1 <2,LII 1) 13
011) 0. 1) 1'.
1)?) 1. 1) 15
I 1(<*1) I)<-1)-FLIO)2'K-IITl(K) 3 16
ANC G(LIIAI) 1) 17
GO TI 5 3 13
2 NIl V 19
33 0 9.2,011 V 2)
Ill)) 3. V 21
09(7) 1. 9 22
15(9.1) G9l(-1)*FLO0T(-2('3)1G9)() 3 23
3 C0ITI)JE 1) 24
SOC SNINIlPI) 1) 25
4 REIUIN 1) 25EN) 1) 27-
5IJ93)JII3E 3ESKV (0,5,63<)
SUOROJIINE 'OFSKV 19,3,3RK)OII4ENSION 3<1103)IF (3.EQ.O.I 13 TO 2jF (V.E3.1.) GO 10 3A 0.3 1.
CA..L 355<) (9,0,3(11))CAL 3SKO (9,3,1(12))30 1 1=2,0
F L0NT)II1.3<(I*1) = fl2.)IA)3K10)A3<)I-1)
1 CONTINUE53< B<)N1)GO TI
2 0 0.0CALL 3151<0 (1,0.33<)10 13
3 1) 1.
CA.L 055(0 (9,3,33<)A CONTINUE
ET 35 4
EN)
2
4
5
S
9
101112131'.
151617IN1920212223-
SUNF3IIIINE 355(0 (1,35,0<)
SUNNOUT0NE 35SK3 (X,SS,R() 11 1
0 0.0 C 2
BK 0.') A 3
01.)''. 1)
012 Ill?. 1 5
ITS 07/5. 1 6
CAL AL.0 (I,3,SS,X9) * 7
00 1 J1,2,3 C N
2 1*01! 1 9
CA.L TALC (I, 1,SS,X3?) A 10T 1.512 A 11CLL CALC )T,X,SS,XNI) 9 123< -3X,3T6*(X9+4.XN2*X314 1) 03TN 1341 1) 1'.
1 COIlIlIJE C ISFF1319 1 15EN1 1 17-
SUOF)U7IS5 CSLT )I,1,SS,<C)
SU33J)T1N TALC (T,X,SS,AC) 1 1
CSI = 0. (E1')I(*EXP(-T) ) 7 7*5 T(-C3-I) 1 3
IF (SX..5.1.) 51 TI 1 1 4AC = 15 1 1)
GO TI 2 1 5I XC C3'4XT 2 7
? '(10<9 1 N
ENI 4 3-4.-
149
Program PROP Output Listing
PH, C"P F0_ 115' 01 1 0LLF5 FVPI .0159 1JP'TIlF
DFFPA,0lG T1NTT(CIT
w(P9 CCeCI - M'-4 S 1.jS:CWIN] VFLI)CIFY IwAIENT 00205091 = .10P7
.0)7 01) I 5CJN9 LF'L - 70- VS. 1 C
ALTOIU]' Sr 1) A'SV1 SA L005L FT = 210.000104551 0 VSTT -TIP 2101 0rT 0.5599
'OT>! AISLE ]! SIM150L TWO ST - T0100FS -3.0000
pro. 05SIF ST C7c 100005 - TES0005 5 -9.TC'30CON 0151110 - 515 = 0
SL110 lESOSTe
NtJ0V PEP ).r)TOP 50)5] - IT = 57.5000
.ois 0CT,S - = 1.2S0STLO500 = 00TPAUTITIM FPrT]r = 21.507,
NAIL oFrL = qqSill -, 10 STOP TIN P L'N1. SPOOl U
.0 )' P TI 0T '.T5I-1i' 00
T'0'TI5
100.1 1.11300
10. '.07090
?.3'500
'1.1 '.71000
.1 0.05610
'0,. 1.354:2
.1 1. rIO)
1.7');]
1 3.9(920
37,) w.1'503
V.) ,.3111
75.1 5.17000
15.')
,.b0CJ
17.'
TWIST- TOG
-2.00000
-1.03000
-1.50000
-1.21021
-0.19000
.5)000
2.00000
7.210)1
5.21010
5.. I:
7. AL 001
9.00090
11.2' Oil
I 3.0 000
17.1' 331
'1.503002.' ]3
L""' ')TAV .0503
WTLS"T TXT7L 1NF'FTP0C1 OFTrIOT '(OFT
TIP 150215 'TOSLE' 20 PIAN'lTLS ITAPULT
51) '4!TLTST 'S VL 1SF)
PFC0M0]C 0OlALYTIS
- (LTCPL T'0Il)5/UPSAJ0!J5) .5500
0000L T1FI15N11 IACTCO' IFVI9U0). A
TOtAL TFTROE0CPi12' FTC 01°, 0 = . '00,73142
0'45)JL(O T°TE"'19TC 7OCT17 (005'109U00 OP
ATI21LA1 IITE'5°FN'2 F0'2907. (P = .lC'31457
CTT'0T TIN' S 10
I1'CL F0C.F. N = 1.53339
00CFNTIAL FCCF. 71 .30515
LICOL 12TU0 - FT 59.1750
TO' ITT' 7ACT'V. 0 = .7105
Ao]Lt HI - C=Y°71S 3.6270
AISLE "F OTTO'!!, TL00 - TFC2F'S = 0.5109
C')IIFITTF')T 0 S00011NIL LI7", CL .0402
T1'FI°001T SF SFCTT2EV loll, Cl = .0(73
.35070347
03315553
"0 7015' I' SIECTI0N, CO = .0577
'TOPFITTIiT C FC!!57 IS V S!PECTIOS. 'Y = .94'.
1.0C' IS SI1"CTTTN "15 'LOPE. Fp - 19 = 5.5747
P lr, ISV '0''10U'l P SLATE. FT - CS = 122.F,526
'I!F'.T TI 'T""T"N "" "LATE. PlO - 0L1 = 656.4150
"1 IF TT!ivrI'Ff TO 501 O0. °15T. 00 - FT-LS = 1661,.6710
1555"L'O U°''-'.'I ST '.77271 05
'0T0'!0 VL"IOTTP,l - T/OFT ' 250.0011
- '-L''1 'U"T.T - C ' 1. 7721
In 0 7'IOT'T' '0, .07,1w
- <IL'w'.'T'0 = 575
=1,:- 1''. .3215
- IL"VL 0=1 5/V° =1,1551=. 1q000111) 0'P"'I 1"!VTV''')), P
ViOL Tl1-L-'1 I'''1, I = .
7550 (P2'nJI)LTI. 17
0
(31C
T55'JT10L FC°'T, 1 .12077L)I0 10700510 - 05.2500TIP LOST FACTO33A, F
AISLE I - O°1 ES = 3.9962ALE OF 17107K, OLFIA - ')FGPEFS = 0.3976C3SFFITIE..JT CE SECTTONAL LIFT, CL = .9519C1TEFIIIIENI 'F STC'ITINTL ORAl, CO = .0172CDSFFT7I'IT C FIOCE 10 7 000EOT100, 0 = .0405C)EFFICIE'IT OF ElOISE IS V TIREC110S. ISV .9505FIOCE 15 9 AlOFT ION 070 SLATE. FX - 1' = 5.6'.06FIXCE IN V 005ECTIOS P70 SLATE, FT - 15 = 125.0107"OIENT TI X 001FITION PT' PLAIT. X - Fl_IF 1631.3964
D33lE6V IN V 21°ECTTIN VP LA0E. TOY - FT-IS = 36717.0003°EYNCLOS IJU'°, XC NO 3.0026E 05
LATIVF VELTCII'l.W - FT/SET = '35.51341)90195. 3 - FT-Lu 15SQ393I.T.400TT,T - L° 1330.32571910751 COFFICTFHT, CT = .1397AllIES, P - KILOWATTS 20.90PDWE COEFFICIENT. CF .0507
1101109 - IL000L TATIUS/TIP 001IUS) .950000 OIL INTEPFEFTNCC F1'TTF TPSFII100ST, A .34575557AXIAL ONTE°FFNTF FICTOP. A = .35066347AI1ULAA I3TERFrNCr rAdIO (POFVIIUT), OP .0333256251310100 I'JTEHAETiNCE F1CTTH, OP = .0132'.553'3JMAFS IF ITFSATI'NS = S
'IT '97L F30(, EN = .4347001563 NITAL FT"IE. FT .01553L)'SCL 0071151 - 11 53.1251
1" L aSS FACT00.F = .3351
o3u;LE PHI - 'U56ES 4.36301561' 9F 111105. ALPHA IEGOFES 5.35S1SETCTE' C' SICTIONIL LIET, CL = .5367
EF'00' 'IT 5F SEI011TTIAL DEAl, C) = .0171"F Ffl77 II A OIEC,TTON, CX = .0540
"11FF 1C17OT CE 'C°1' I V 011EISTION, Op
S'CU IA' V S1°ErTTTN P97 31016. FU - L9 = 6.3051F3 lEE 019 V IIECTISA' P' SLATE. FT - LB 119.6211)OIAT I 'lI'0701)N 33'59 'VITAE, III - FT-LI 2666.5612
'111651 040 SI"FuTTI'l4 SLIDE, TOY - FT_IS 55242.5302'45 3.11009E 05
71(5TJ lUE""IV. 4 - 1/SFC = 22T.700505 '9915. -L" 50391.2573l-l'IIsT,I - L' 150.5' 2
0095'T 10FF IC1 40, "1 .21'5C - XTEOWITT'O 36.21
"15 F,'11,, 9' -,3974
STATI"II - TL"SAL AOC4STIF 0' JO) .0000IXLCL I"TTPFE'''9(F FOOTOO (°97915U3). A .15120730AXIAL TNT-VFFO'VJFE 7OCT07, A .35031596Ts;1-Loo I'4TEF'2' '0"' FC'TOP (PHEVISLI). OP = .00150075A3'iULA' T'ITE4F17lI"E FOlIOS. OP = .00361056'lJ'IFEO OF IT00TOO'IC 4
'IDSOAL 63'CE. Fl = 1.36415TlNG391'IAL ElOISE, FT .310391)101 305100 - FT 51.0000TOO LOIS 7OCT15, F .3706AISLE PHI - 'SCG°UEi = 4.6265AISLE SF ATTACK, ALFHA - )E5OEFS 0 0.2221(')UFFIISIE'IT 50 SECTIONAL LIFT. CL = .9152C')FFICTE'JT CE IEISTTV'46L DRAG, CO = .0159112FF 101,9'IT SE F305F IN U ITTPECTION, CX = .0572'0DEFCICT5 CE 75050 335 V T1°EC1100. Cl = .9166E)99CE TN V SOP"CTION CEO SLIDE. '0 - LA 0 7.1510ASOCE IN V 010SCTI(0F PEP SLATE, FT - 15 = 116.5527'lO'IFNI IN 019EIIIIN PEP SLAIF. MO - FT-LB 3650.959IDlEST ON S 010571106 7FF ILADE. MY - FT_LA 710 31.7565°EVNOLIS XU""E°. 01 NO = 3.1737E 06551011fF IELOC100,W - FT/SEC = 212.60215)4094", 3 - FT_LI = 0216.01StTX OUST. I - L' = 2907.7023040111 CIVFF111ELT, CT = .2901V'T0FO, P - KILOWITTI = 6F,.99
15Ek CUOFFICIENT. C .1319
5101100 - (LOCAL '05155/TIP 01)300) 0 .7500AXIAL INTFPFEPEPOCT FACTOO (PPEHIOUS), 33 = .14661565AXIAL 1510F503460 FAC1OF A .346562751317110 IITE6FEF7NCF FICTOF )PPE000IJS), 00 = .13401029A'IGELI0 I'JIEHFUVCOISF FACT)0. AR = .03633551N) 33060 OF 1120001041 ='912°AL F"055 F5 = 1.23219TI'47INTISLFCIF FT .29372L1ICL OCflTUT - FT '.5.9750T0 LSS'7 'ACTC" F = .0567
A'15L7 04'i_J757[E5 0 4.353111511 SF ATIA"F, ALEO1 - OFGCES = 5.1551CDIF'1191"'T IF "EITT'INAL LIEI, CL = .5151CSOFFICI"IT C" S"CTI'N1L 17AS Cl 0
C5"FEI"I"'IT "FF101'" I'I SI°ECTIO'l. C Y=.05'.°"F C11'"9 1' '4 "IECTION, CV .3061
519 ['5C'V''! 09 VL1'OE. EU - 10 =
F5,5 ' 0 ST"""'Tl F'V 310"E. 1 LI 137.6F37"LII-'. V _ FT"LV
33740=631
I-.
Ui3-'
4EYNULOS NUFSF, 10 = . 21 391 TA.
A'ELATIVE JELOCITV.W - FT/DEC = 195.51.56
T3GUF. 2 - FT-L° = 11399.11.I.0
T-IBUST.T - LB = 3582.1.1.90
T-)BUST COEFFICIENT, CT = .3599
PO.EP. P - KILOWATTS = 59.59
POWER COEFFICIENT, CF = .1671
STATION - (LOCAL RADIuS/TIP R001US) = .7000
AXIAL TNT'SFEAENCE FACTOR (PEA0OUD), I .33332900
AXIAL INTERFETENCE FACTOR, A .33330346
ANGULOS INTEP°FAFNCE FACTOR (PREBIOUS), UP .00452971
ANGULAP INTEPEEFENCE FACIAl, 0° = .00452993
NOISES OF ITF°ATITNS = N
NORMAL Fold. F(4 = 1.20439
TANGENTIAL FCNCF. FT = .29641
LOCAL RADIUS - FT 43.7500
TIP LOST ACTO°, F .9930
ANGLE PHI - CEG'ES = 5.1.160
ANGLE OF ATTACK, ALPBA - IEDAEES = 9.0122
CIEFPICICNT )P SFCTIOAIAL LIFT. CL = .9513
COEFFICIEIT TF TFCTIONAL SPAT,, CI .0161.
COEFFICIENT CF FO4CF IN A OIRFCTION. CX .0676
COEFFICIENT C FOPCF IN S TIPECT105. CS
EDICE IN N IDECTION PER ILATE, EU - 15 7.5205
FORCE TN V DIPECTIIIN PP SLATE. FT - 19 91.9296
MO NEAT IN A TIBECTION PEP BLADE. MX - FT-I_B 5503.6996
°3'IENT IN I TIACTTON PEP BLADE, MY - FT-LA = 99110.1444
IVNOLTS AlJMPFP. 00 NT = 3.2223E 06
°ELATIVE ICLCCTTy,. - FT/SEC = 196.4696
TONGUE, 0 - FT-LI = 12505.0207
- LB .229.1227
1-I 'lUST CIFFF!1IrxT, CT = .471.7
- KILOWATTO 71.1.1
POIFE COEFFICIENT, CP .2010
STATIlI - (LCXL PAOIUS/TIP RAOIUS) = .6500
ANIAL TNTPFEOFNCE FICTOR (PSEVIDUS), A .21259071.
AXIAL INT4F4EI1EE FSCTD, A = .31257101
AICUL 09 IJTEBFEBENCC FACTIP (PPF010USI IF .00506930
TxISLJLIR T'TtFE4F51E FSCTON, UP = .00501.954
'II 4FER (F IT!5AITT1S = B
NO/PAL FCCE, " = 1.001.1.'.
ToIrl 90111 FOCF, FT .27117
LOCAL BOOTNI - PT 40.A.20
TIP LTSC OCT00. F = .5960
A'li,LE P/I - = 6.0067
I ;LF IF ITTA'A, OL''-A - OEGPC5 = 7.5091
T1EFETIETT CF SA"TTONOL LI CL = . R92
COEFFICIENT QF CECTIN0L IllS, CI .0162
COEFFICIENT CF FOPCE IN A TIPECTION. CU = .071.7
C3EFFICIENT C FOPCT I'I P 000ECTION, CV .0551
FONCF TN A OIPECTITH PE SLATE, Fl - LB 7.6057
FOBCF IN V DT'FCTION °F' ILADE, °V - LI 00.9751
DIENT IN A CIPECTION PEP BLADE. MX - FT-LB = 6350.4774
PO'IFNT IN V CIAFCTION °E° SLAOE, MY - FT-LB = 109466.5103
°EVNOLTS 'lUMPER, BE NO = 3.2017E 06
°ELATIVE VELCCITV.W - FT/SEC = 173.4210
TOSQUE, S - FT-LI = 11.511.9352
T'$'lLJST.T - L = 1.915. 1570
TN'UST COEFFICIENT, CT .4937
P1110. P - PILYWAITS = 93.10
OEP COEFFICIENT, CF .2332
STATION - (LOCAL BAQIUS/TIP RADIUS) .6000
OXIAL INTTBFEP!NrIO CACIGU (PSEVTO)JC). 0
AXIAL INTEAFEBENCE FACTOR. A = .29710930
ANAULA' INTENFEPENCE FACTOR PREVIOUS, OP .00571295
ANGULAR I'XTEVFFPENCE FACTOR. IF .00577330
NJ'I'EB OF ITEBATTONS = 7
NORMAL FOSCE, FAX = 1.11017
TANGENTIAL FCPCE, FT .2561.0
LOCAL BATTUD - FT = 3.R000TIP LOSS FACTD, F = .9990
ANGLE °HI - DEGREES 6.642?
ANGLE OF ATTACK, ALPHA - DEGREES = 7.61.01
COEFFICIENT CF SECTIONAL LIFT. CL = .0496
COEFFICIETT CE TFCTIONAL SPAT, CO = .I1S9
COEFFICIENT CF FOSCF IN A DIRECTION, CX .0024
COEFFICIENT OF FOPCF IN V !OIRECTION, CV .0457
°)NCE IN U 2T4CTICIN PC' ILATE, FR - LB 7.7909
'ONCE 00/ TIBECTION IFS BLADE, FT - L9 = 19.9250
°OIENT II U DIPECTION PEO PLATE, MA - FT-LB 7152.1691MOlEST IN V CIRECTION E5 BLADE, MV - FT-LB = 119330.541.9
'EYNIOLOS NUMPC. HE NO 3.IBONE IN
PLATIAE IFLCCIT,) - FT/SEC = 160.3901.
TO lAUD, 2 - FFLH = 161.01.7097
TNNUTT.T - LI /T6'.8T13
TI/lIST COFEICTENT, rT = .5367
O)E. F - AILIWITTS = 0393'OXl' COFFI'T'NT, CF = .2636
SVACIOO - (LECIL lI'I/OS/TTP ITIUS) .5931
Ix 201 1'T /'FFACTCP (POFUIDUT), A = .77395F'.2 __A IEL I'OT''F=ANC OCT00, A = .27194146 (JO
A;IJLUl 1TE/FE/ N DCT 0' (P°E/I0C) OP = . 01653493
AOCULA" IJTE0rFPNT! rCTO0, AP = .00153527
NiMl'EO OF ITESATI100 7
'ORAUL FlACK. EN = .V9717
'XGESTIAL FCCE, FT = .23N77
LOCAL PADIUS - FT = 4.3750
ToP LOSS ACTO, F =
A'IGLF Pill - DEGREES 7.47jN
AMGLE OF ATTACK, ALPHA - SEGREES = 7.4702
COEFFICIENT 5F SECTIONAL LIFT. Cl = .0306
CO!FFTCIENT OF SFCTTONAL OPAl, CD .0157
COEFFICIENT CF FORCE IN A DIRECTION, CX .1525
COOFFICIFIT CF FcJBCF IN V DiRECTION, CV 0256
FORCE IN N TIPECIION PEN 9LAPF, El - L9 = 7.0196
FORCE IN V DIRECTION PF RLAOE, FT - 19 6.0202NONFAT TA A IIPECTION PEP BLADE, MX - FT-LB 7076.7729
401E1T IN V STRECTION EP BLADE, MY - FT-LB 125270.252A
PEYNCLOS NUN'F°, RE NO = 3.0974E 16
RELATIVE VFLCCSTV.V - FT/SEC = 167.3997
TONGUE, I - FT-Li 1B156.023'.
T.4OUST,T - LA = 5910.3532
Tl-XUT COEFFICIENT. CT = .5036
POWEC. P - KILOWATTS = 1C3.97
FlUFF COEFFICIENT. CF = .2910
STATION - (LOCAL P10105/TIP RATIUS) .5000
AXIAL 1N1FPF!NENCE TAITOB (PSETOUS), A .IGI6OEI
AXIAL INTEFERENCE FACTOR. A = .29114466
ANGULAR INTERFERENCE FACTOR (PREVIOUS). AP = .00746793
ViTALS' AITERFEPENIE FACTOR, AP = , 10746063
ATlIBEN OF TTATIONS = 6
NORRAL FOACE, FF1 .7979R
TAAGINTISL FCRTF. FT = .22115
LOCAL RICTUS - FT 31.7500
TOP LOSS FACIPT, F =
G'l;LE PRO - I!GPEES = V.4556
TITLE OF STATES. ALNUA - DEGREES = 7.755R
T3F'ICIEN1 OF SICTIONAL LIFT, CL = .MI76
COEFFICTEFIT T SECTIONAL SPAS, CT .0154
cOEFFIr.IEIT 5 FORCE I'! A 2IECTIOR. CX = .1036
COCFFTTICIT r FCCF II V TIPECTION. CV . 0011
FOOT' III TIFTTI0N P5 SLATE. FA - L9 = '.7640
OAC' Ti' V TIECTTT'J P55 SLATE. FT - 19 60.0563
"TIE NT IN A TIRECTI 'N PER °LAOF. MX - FT-L9 = 0523.9776
'bElT IS V OINECTI)F E SLAVE. MY - FT-IF = 13O6F9.Ii5
E YNCLTT IU"flEP, P NO = 3. 0102E 06
°ELOTIVE IELC!TTA,W - FT/SEC 134.4456
'O'CliE. 0 - FT-Li = 10794.9660
- L' E'15.510I
TIRUST CT1FFICIF'T. IT = .6243
NOlE, P - KILOWAT'S = 113.13
POWEP COEFICIENT, CF -, .3175
STATION - (LCCAL "ASIUS/TIP "AOIUS) .4500
AXIAL INTFPFFAENCE FACTOP (PREVIOUS). A = .23115391
AXIAL INT7NFFPENCE FACTOR, A = .23116900
INSULAR INTEAFEPFNTF FACTOR )PREVIOAS), PP .0A970131
ANGULAR I'ITERFIPFNCF FACTOR. AR . 0A01A166
NUMBER OF OlEPATIONS 6
NOIP'AL FOTCE, FN 6703q
TANGENTIAL FCSCE, FT .21191
LOCAL RADIUS - FT = 2R.1250
TIP LOSS FACTOP, F = 99%ANGIE PHI - CEGREES = 0.6136
ANGLE OF ATTACK, ALPHA - OEGPEES = 7.2167
CTEFFICTEJT CF SECTIONAL LOFT. CL .0013
COEFFICIENT CF SECTIONAL ORAC, CO = .0153
COEFFICIENT 0' FANC' ON A DIRECTION, IX = . 110?
COEFFICIENT 0' FENCE IN S DIRECTION, CV = .7926
FORCE IN N TITECTIOS PER BLADE, FX - 19 = 7.6561
'ONCE IN Y OTP!CTIIN PEA SLASE, FT - 19 = 55.1054
MOMENT IN A SIPUCTION PE RLAAE. MX - FT-LB 9308.0516
"39E#T IN V CISECTION "ER RLAOE. MV - FT-LR 134739.501W
"EYNOLOS NUMPF, "N NO 2.5F40E 06
°ELATIVF VELCCITV.W - FT/SEC = 121.5600
TOAQUE. I - FT-LI = 211N5.9446
T'IRUST,T L9 = 6562.3226
THRUST CO'FFTCIFNT, CT = .5591
POWER, P - KILOWATTS = 121.32
POWER CCEFFICIFNO. CT = .3405
STATIST - (LOCAL PAOTUSITT" RADIUS) = .4000
AXIAL INTCFFPINCI FACT)B (PREVIOUS), A = .20523023
AXIAL TNTFBFERFNCE FACTON. A = .20522944
TITULAR liTER F!PENCI ACT1T' (PREVIOUS), AP = .01009321
A'IGULAR ITTERFEPENIE 'AdO", AP I110333NJRFR OF ITFNUTTTRT F
NIRPAL FO'CE. EN = .55474
TANGENTIAL FT55E, F' . IMO9R
LICOL RTT'lJS - FT = 00.1 ITO
TI LOST NOCTVR. F .9097
ATTLE OHI - CFPEES = A1.12N5
TITLE OF I TITlE, OL°'A - lEGAtES 7.0357
0oIFFIClT C' S''CTTCI'.L LTET, CL = .7'IH
!OEFIIIETFSTIIDL 1015. CO = .0150
IT 0' '' TI A 'TETTION. CA = .1361
I1FFT'T IT ( '' T OIRF!TIOl. CV = .770100
F39C IN DICTIOS P5R SLATE, FA - L 7.3768
'TICE IN S 0I'!r'TIP! P'S SLATE. FT - LB 41.7201
'ITHFT IN B DIMEETION EP SLA!OE, MX - FT-LB 9566,2067
SilENT IN V TTAFTITE FP SLATE. MV - FT-LB = j37683.398j
EYNTLTS NUMRE5, RE NO 2.6909E OR,
RELATIVE VEL CCI'S ,W - FT/SEC (08.7063
TOEQUE, - FT-LI = 22434.2734
THMUST.T - L9 = 6851. 4311
T(0(JST CDEFFICIENT, CT = .6582
FOVEA, P - KILOWATTS = 128.47
PiWEP CCrFFI'IFNT, CP .3605
STATIPN - (LCCAL PATIUS/TIP A0IUS) = .3500
AXIAL INT5MFEBNCF FIllIP (PREVIOUS). .08354357
AXIAL INT5FEMENCE FACTOR. A = .18354421
VMGI'LAR INTESFFXENCF FACTOR (PSEVI0US). OP = .01208758
AMAULAM INTE"°F'4C FACTOV. AP .01208763
NIJMBFM OF ITEBATIONS = S
NOMSAL FORCE. EN .44349
TANGENTIAL FCOEE, rT = .15978
LOCAL MACCUT - FT = 21.8760
TI LOST FACTOR. F =
AMC,LE PHI - lESSEES 12.9793
A'IXLF OF ATTACK, ALPHA - OEGPEES = 6.9940
TIEFFITICIT OF SECTIONAL LIFT. CL .7754
C)EFFICI'JT T TFOTITSAL TRAG, CO = .0150
COEFFICIENT C' TOAC IS A SIBECTIOPI, CX = .1596
FOEFFICIENT CF FORCE IN V TIECTI0N, CS = .7591
FTOCE III S TI!CTI0S SLATE, Fl - L9 = 7.0558
OTC6 16 0 DIQFCTITN PM SLATE. FT - LB = 33.5637
POTENT IA A TIOFCTITN OfF SLIDE, MA - FT-IS S 9953.7395
'lOIEF.T I'1 V CI'ECTITN FP MLAOE, MV - Fl-LB = 1!57C2.9102
PEINCLTS MUS"'M, 5 NT 0.4'76E 04
'S1tTIE /FLOCTT'.A - FI/TEC = 95.9652
li-AQUE, I - PT-LI 23400.9016
1(0551,1 - 1' 7V1.f'NT1-45551 ClFFICIPN1, Cl .7117
S3(fP, P - KILTW2TC = 114.52
STAFF CN-FPICINT. CO = .3775
STATITA - (LOCOL 40'IUS/TIP 0ATIUS) = .3100
ISlAL I T'°5F5 FJPP(CTTP (VSFVIIUS), A .16611606
ABIEL INSPSEPTFNPP FAIlS, A .16611636
135100 INTE ACE FACTIP (PREVITUT), OP = .01516290
l4ULOM ITES'P'N'E FTCTTF. AR .01E16240
"('065 55 ITEMATIXET P.
TsP'AL FO'E. EN =
P5'jGEPjl1L FcRfE, Fl .13517
LOCAL A3'')S - FT = 10.7500
T1 LTC1' ACTTO,
UNCLE 0H1 - lESSEES = 15.3129
APAL 3F I 11155, OL°AA - lESSEES 7.0421
CIEFFICIEPT 35 TFC'TTSXL LIFT. CL = .7869
COEFFICIENT 5T TFSTTSBAL OPAl, Cl .0152
COEFFICIENT CF FORCE IN A 118ECTIOM. CX .1532
C)CFFITIENT CF FTPCE 1.1 V IIRESTION, CV = .7530
F3.MCF IN OTSECTIO'I PF 3LAOE, FA - LB 6.7357
FOTEF IN V DT'CT1TN RE0 hAlE. FE - (.5 = 26.6A64
'POTENT IS TIREC'TON ES SLATE, MX - FT-LB 0 10256.5071
'POTENT IN V CASECTITS P50 BLASE, MY - FT-LB = 161023.3161
CYFOLTS NASSER, SF 40 = 7.35935 06
IlLATIVE ELCC1TS,W - FT/SEC 83.3599
TORQUE. 0 - FT-LB = 26365.4234
T-IITCT,T - I' = 7777.6434
TP451J5T COPFFICIEJT. CT = .1305
POVEF, P - KILOWATTS = 119.53
°OVFR COEFFICIENT. CF .3916
STATIO9 - (LOCh PAOIUS/TIP PATIUS) = .2500
AXIAL TNT!MFEPENCF FACTOR (PREVIOUS). = .15234211
AXIAL IPATFBFEBENCF FACTOR. A = .15234071
ANGULAR ISTEPF!PFNCF FACTOR (PREVIOUS). OP = .0202510',
ANGULAR INTERFUVFNCF FACTOR, OP = .02025155
531865 OF ITERATIONS 4
NORMAL FORCE, EN = .24566
'ONGENTIAL FCPCF. FT = .11652
LOCAL MAOIST - FT = A5.6250
TIP LTSS FACTOR. F 1.1000
ASSLT PHI - rFGRFFS = 18.3834
0951 OF ATTACK. ALPRA - TFAEFS S 7,5535
C)EFFICIEIT CF SECTITNAL LIFT. CL .8219
COEFFICIENT CF SCT1TN2L IPAE, CO 0150
ITEFFISIENT flP 610SF IV A OIPECTION, CO = .2462
COEFFICIENT C FOP5E IN S TIECTIOM, CV .7849
FISCE IN P 110555115 MLASE, Fl - 19 = 6.4414
F)TCE TN V SIP"E'TTTP' P"R SLATE, F5 - 15 20.7001
SIlENT TN A CINFCTITN SEP SLATE. RU - FT_LB = 10481.1755'11651 15 V )IRFTTTTN 055 SLATE. MV - FT-LB = 141810.5812
SV5OL]T NUN'S', OF NO 2. 6E 26RLATIVP 0!LTCITV.W - PT/U = 79,9514
'i0.EE, 0 - 51-13 S 2571.2473T4-(USl,T - L '.10.179?
1-IROIST CIPFITTEFV, PT = .7452
"101' F - RILTA'E'T" = 163.57
1AF TOEFIit.'E'. 'F .0029 U-I
1010109 - (LOCAL R101US/TIP A3IUS) = .2030AXIAL IETQFEPFNC5 FACTOP (PREXIOUS). A .13°14010AXIAL INTEPFFRUNCU FACTOR, A = .13913763A')(JLA9 I4TEPFEREP)CE FACTOR (PREVIOUS), OP = .00909536A90LAR I9TEHFEPNCE FACTOR, OP = .02909653A1(J1f9 O ITEHATIT9 4
NOXPIOL FORCE, ON = .15719TANGENTIAL FORCE, FT .09121LOCAL RATIOS - FT = 12.5000TIP LOSS FACTOO. F = 1.0000ANGLE PHI - 051PFES 22.6977A(IGLF OF ATTACK, ALPWI - OEGREES = 8.0266COEFFICIENT OF SECTIONAL LIFT, CL = .9105COEFFICIENT OF SET.TDTHIL 0001, 00 = .0175COEFFICIENT ((F FO'OCF OS 2 OTRECT100, CX .3352(COEFFICIENT CF FORCE 05 9 OOECTION. CV = .0467FORCE IN X IIRFCTION PE 91A0E, Fl - 19 6.0915FORCE IN 0 DIRECTION P50 BLADE, FT - LB 15.305540'4ENT IN 0 DIRECTION PEP SLIDE. MX - FT-IS = 10633.565740'4E57 19 0 OTREC770N PEP SLIDE, NY - FTLB 142270.4350
REYNOLOS NUMSF°, 00 NO = 1.6655E 06OTLITIVE VELOCTTV,W - FT/SEC = 50.0974T300UE, (3 - FT-LA = 25619.3609THXUST.T - LB 7530.9733THRUST COEFFICIENT, CT .7564POWEE, P - KILOWATTS = 166.71POWFF COEFFICIENT. (CF .4117
STATION - (LOCAL PAOIIIS/TIP 801100) = .1500AXIAL INTPFFP!N('F FACTOR (PPE0100S), A = .1342719316111 INI'AFFFNCF FACTOR. A = .13437714A'lC,)JLAS INTEHFPMCF F0'CTOR (PREVIOUS), AR = . 04928199TO9LjLAX INTEFERENEE FUC000, OP .04927911SOIPEP ((F 1159011045 = 4
NOOMOL FOOCE, F)) = .01042TI"IGFNTIAL FOPSE, T .07342L3CAL QATIUS - FT = 9.3750TIP LOSS FACIlE, F = 1.0000A'IOLE PHI - CEOP5S 29.01271)51) ('F 510054, ALPHA - OFE,PTES = 10.6359c)ErrO'CIENT (F SCTITNAL LIFT, CI = 1.0771'17EFTTIEIT C TCTIONAL SPAS. CO .0210
ESEFF101((T CF F155 IN A OIPECTION. CX .5001'OEFFITIE'(T ('F FTATC IN A IIRECTION, CV .9544FIXES IF 5I0CTI0F P0 SLATE, ES - 19 = 5.8911F1(' IN OTAECIION PCS SLA'E, FT - LR = 11.2409"3'PEFT I) A EI"TIT'J °X' SLATE. MO - FT-IS = 1A721.I9N71(F SO T'( S (T'ECTT"(J ('E SLATE, MV - FT-LB 142467.2506
REYNOLTS lI"". ' (40 = 1.3410E 06ELATTXE VELCCTTV.W - T/TFf = 47.4216
T10UE, S - FT-LI = 26027.9765T'IUST.T - L0 = 7613.4294THOUTT C3FFICrFNT. (CI .7647524FF. F - KILOWATTS = 149.05POWER COEFFICIENT. OP = .4103
TT0111N - (LOCAL P10125/TIP R001US) = .1000AXIAL INTERFEPE)ICF FICTOP PPEVIOUS(, A = .05007573AXIAL INTTRFFPENCE FICTOP. A = .05097500INSULAR IJTE8FEPFNCF FACTOR (PREVIOUS), OP = .04623091ANGULAR INTEXFEBEWCF FACTOR. OP = .04623904(Ri IRER OF ITESATI'INS = 6
(404811 FOXCE, F)) = .02844TANGENTIAL FORCE, FT -0.00202LOCAL SAOI()5 - FT = 6.2500III LOST EACTI°. 0 1.0000ANILE PHI DEGREES = 42.2100ANGLE OF ATTICE. OLPHO - lESSEES = 20.9419COEFFICIENT OF SECTIONAL LIFT, CL .5113COEFFICTEPIT OF S!CT100III 0001, 00 .WZ01COEFFICIENT OF FO000 TN V ()0SECTION, CX .0727COEFFICIENT CF FORCE IN OORECT100, CV = .7054FOXCF IN 1 OIRECTOON PER 9LAOE, FT - LB .5297FOOCE IN V 0T°ECTT0S PEA BLADE, FT - LB 9.1302MOlEST IN A 0I°ECTITN P55 9LAOE, MO - FT-LB = 10740.1440MOlEST 04 1 CISECTION POE BLADE, MV - Fl-LB = 142513.2248EV(OIOS SUMnER, E NO 1.IO4SE 06
RELATIVE OEL050TV,W - FT/SEC 37.2909TORQUE, I - FT-IS '6262.4743TX IUST,T - LX '665. 0559THRUST ('1('FFICIENI. CT = .7700POWER. P - KILOWATTS = 150.40P3WE COEFFICIENT. P = .4221
PEPFARMA'((' XU'4APV)
TOTAL POWER = 150.65 KILOWATTS101.69 HORSEPOWFP
TOTAL PS44FP CT(PFICIENT .4220'OXC115E PS4 COEFFICTENT WITH SAlTIEST .417041(111 THSST "('FFICTENT = .7700
SAW FOCCSS S IF TO 515J#S / "L(SE( / (STN(THETK) 451qYAW FO_'CES 1)1 ri ()1- / 'LIE) / (IN(THETK('2) = -0.1330CONING MS0NT T) T( FLA0'NC / (10SF) -1.2009
511
UT
CoIr, E'1T T) SkA / LA)r) / tSTN(TOIK)vAING '11T OU t1 r1TPrNO / 100DE) / (rOSTFT)V.IG 1T 'IJO T OHD / TLAIF) /(SI(THTKICCD(TOQ)E VT'.DTTON DU TO r1'I4 / TLADI =TOQ'J VDO.DTTCN 01fr TO -A/ / LAI) / (STNTTK))TOo, vT/rTr=N 'j 1) 'VD1' S'TN'T TEjvTTT/ETOQJE VTID nON Tn nnn-T/ 00011 1tVDTDV
1.2CqFIETK)) -D.24°3
=
02c
FT
U-i
157
APPENDIX B
Optimum Design Generation Programs
Program Operation
For speed and accuracy the Local Optimum Theory and the
Constant Axial Velocity Method have been programmed on a high
speed digital computer. The program is written in the Fortran pro-
gram language using a CDC 3300 computer under the OS-3 operating
package at Oregon State University. Logical unit numbers 60 and 61
in the programs refer to the card reader and the line printer units
respectively.
The program package consists of a main program labeled OPTO
or AXOPTO with three subroutines: PRANDT, GOLDST, BESSEL.
Subroutine PBANDT determines the Prandtl tip-loss factor, Subroutine
GOLDST determines the Goldstein tip-loss factor for the two bladed
case, and Subroutine BESSEL determines modified Bessel functions.
For use of the Goldstein tip-loss model for other than the two blade
case will require substituting the optional subroutine GOLDST pre-
sented in Appendix A. The use of the Goldstein tip-loss model
requires considerable computer time for the program developed.
To generate wind turbine design relationships it is necessary
to input the following information to either program package OPTO or
AXOPTO.
158
Number of blades--B
Tip speed ratio--X
Differential tip speed for interval evaluation- -DX
Tip-loss model controller- -TM- -0- -No tip-loss modell--Prandtl tip-loss model2--Goldstein tip-loss model
Axial inte rference model controller -A MOD - -0- -Standard method1 - -Second order method
Coefficient of lift--CL (usually specified as 1.0)
Input should be in the following format:
Columns Card 1 Card 21-10 B DX
11-20 X TM21-30 CL AMOD
The following are the input Read and Format statements used:
READ (60,44) B,X,CLREAD (60,44) DX,TM,AMOD
44 FORMAT (3F10.4)
Program listings and sample cases are provided for further details
and implementation.
159
Program OPTO Listing
and Sample Output
P33SAM OPT3
POSRAP OPTS
EN5ION I - J>JA5Y j370070 - JET0'lI4ES 151 IPTI'IUI SLAO'L GE)METSY FOR
5p11+1r0 OPESAIIOS CONDITIONS 35131 A '1301111) 5143051APN3ACl.
DIMENSION 00L('.3,5>. CTL(4O,A), AOl(S), SURT(S), CP(5), CT(S),A CLCO(5) SISCO)'.)
IN'UT PARAMETERS
A - NJ'I'ACR OF 3L4):S
S - TIP 50000 RATIO
DR - OIFFTRONTIAL TIP SPT3 FOR INTEOVAL EVALOATION
TI - TIP LOSS IOJEL CONTROLLER- 90 TIP 1035 M001L
1 - SSNJTL (OP LOSS MOOSE2 - GOLISTLIN 110 LOSS IO)EL
AlTO - ANIOL INTEPFRENCE IIO!L COATROLLENII - STAADARS 311)40)I - >11509 METROS
'.L - S)EF0ICIEAT O LIFT
RFAO (9,0,4'.) S,U,CLSEAS ('.3.44) SR,T1,AIOSWRITE ('.1,45)WRITE (5,1,3>) 3.4WRITE (51,32) 34IF (11.10.3.) 30 13 IIF (11.10.1.) 00 1) 2WRIIE (61.33>SO 15 3
I WRITE (61,3'.)SO TO 3
2 WRITE ('.1,35>3 C0'ITI>U0
IF )S101.E).1.) 33 1) 4WRITE ('.1,3'.)SO TO 3
4 WRITE (41,37)S COATIAUT
DI .01P0 0.14I09?5A.F 1.CP 0.IC = 0A .32AL A
FRI ATAN(1.OXL)AMASS = 0.0
N =
SO 24 1=1,5TO 13 J1,20
6 II = 01*103 12 K1,5
P410 = 095(0)40)XXI COS(PMIA)/SIN(PHIA)OSLO XXL0X/XLIF (13.50.0.> GO TO NIF (13.15.1,> GO TO 7IF (P1.13. Xl F = 0.IF (XL.0.X) SO Tj s
CALL 001351 (APL,XULI,F> A 71GO IT 9 4 72
A 1 7 CALL PRANOT (3,PMIA,X,XL,1) A 13A 2 50303 A 7).A 3 P FI. 4 754 4 9 CONTINUE 5 763 5 CXX = CLSIN(PHI) 4 77A S CVI CLCOS(PHI) A ON
A 7 IF A10.E3.1. GO TO 13 A 79A N AP = (CXXA)/(CYYAL) A 30A A GO TO II 6 31A 13 10 AP (1.FA)(CXIA)I((CYYXL)(1.A)( A 02A 11 Ii CONE 1NU A 33O 12 PHI ATANI(1.A)/((1.4APIXL)) A A'.O 13 12 CONTINUE A (5A 1'. AWl = (I.-5)2+XLXL(1.'A)2 A 36A 15 IF (AMO0.E0.1.) SO TO 13 a 3?A 16 TICS = 0.Fa(1.-4)CXXF(MV2CTY) A 03A 17 13 TO 14 a 39U IX 13 SLIT = N.FA(1.AFICIXI(WV2CXV) A NA
A 19 A'. CONTINUE a qiA 20 olax 5L1XW52/N. A 92A 21 SLOPE 0 ISIASXNAXA)I(AAO) a 93A 22 IF (435 (XMAU.-XMAXA) .LT. 1.00-15) 10 TO 19 A 9lA 23 IF (SLOPE.LT.O..ANJ.CONTROL.EO.1.) GO TI' 16 A 95A 7'. IF (SIOPE.LT.0.) GO TO 15 A 96A 25 IF )CONTROL.EO.1.) IA 0Al2. N 97A 25 TONTROL = 3.0 A 35A 07 41 = A0 A 99A ON AO A A 100A 29 A A*OA A 101A 30 XMAX1 XMAXA A 102A 31 XMAXA = XMAS a
A 37 50 TO A 10'.A 33 15 10 a A 105A 34 XMAXA XMAX A 136A 05 3A OA/2. a 1073 (5 50 TO 17 A ION4 37 16 XMAXA XMAS A 109A 33 00 A A 1104 30 17 5 ADO a iii
A 40 CONTROl. 1. 4 112O '.1 SO TO 6 A 113A 2 II CONTINUE A Oil.A 3 19 CONTINUE 115A 5'. 05SF = Af(IOS(PMII2) A 116A 43 30111 SLCXFSIN(PHI) A 117o ,'. 505 TI5CLCL A 113A '.7 CR PIXLSIG7(X) A 119
49 N 3+1 A >23a so C') 0. A 121A 53 OF 25. A 122A RI 000 20 11=1,5 A 123A 52 CLCD) ML) = OFTN. A 12'.A SIIC)(ML) SISCO A 125
CS = CLSIN>HI)C3CCS)"4I) A 126A 55 CA = CLCOS(PHI)PC'0SIN(PHI> A 127A 56 TPL(M,ML) SIGCO(XLX)2WV2 A 123A 57 CTL (3.31) SISCYXLNV?/(AX) A 129A 53 CI = CL.0 A 1304 59 OF = ')F*25, A 131A 60 2T CONTINUE 0 137A 61 P513 DHI15)./PI A 133A 62 WRITE (61,'.'.) A 13'.A 63 W517 ('.1,47) AL A 135A 6'. WRITE (61,,N( A A 136
55 WRITE (51,49) 0 137A 55 WRITA (61,50) F 0 139O 67 WRITE (61,5>) 01SF 109a 6 WElTS ('.1.5?) °RIS 143A 69 WPITE (51,33) CA 1,1A 70 WRITE (61,13) XI S (.0
0'
C
'3 CONTINUE A 1St SUROCUTINE PRISM (3,HI,X,X1,SF)URAXA 0. 0 152Ac) = A. A 153 5U950u1INE PAN3T IV,PHD,X,XL,GFIAl 0 5. 15'.
SMARt = 3 A 155 204507 - DETERMINES TM' P0A'IATL TIP LOSS FACTOAA A-.05 N 196 C
IC 0 A 157 P1 = 3.1.159265.04 = .01 A 150 FL (3/2.)(X-5L(/(ALSIN(P0EI)XL AL-SO A 159 P0 S -FLIF (IXLX).LT..A4) 10 10 29 A 160 XI = EXP(PO)
24 CONTINUE A 161 IF IXT.GT.j.) XE = 1.25 CONTINUE A 162 IF (OT.LT.-1.) ST = -I.
00 30 JLLI,5 A 163 OF S 12./PI)ACOSF(XTISUM(JLL) S CL(1,JLL) 4 164 RETURNSUMT(JLL) CTL(1,JLI) 4 165 ENSNI = 5-1 A 16600 26 K2.N1,2 A 167
SUM(JLL) SUM(JLL)*4.CPL(K.JL1) ' A 160 SURPOUTINE GOLUST (U,UO,F)1AIT(JLL) S SUMT(JLL) +4.CTL(K. iLL) A 169
24 CONTINUE A ITO IURROUTINE 1OLOST (U,UO,GF)00 27 K3,N1.2 A 171
SUM(JLL) SUM(JLL)+2.'CPL(K,JLL) A 112 C SOLOST - 3ETERMISES THE GOLDSTEIN TI LASS FACTORSU010JLL) = SURT(JLL)*2.CTL(K.JLL) 4 173 C FOR THE TWO 014320 CASE
27 CONTINUE A 17'.
CP(JLL) SUM(JLL)0A/3. A I7S P1 3.14159265'.CT(JLL) SUIT(JLL)0X13. A 176 SUM2 0.0IF (CLCS(JLL).EO.0.( GO TO 25 A 177 suo = 0.0WRITE (61,54) CLC)(JLL) A 170 = 1.GO 7') 29 * 179 AM' = 1.
29 WRITE (61,431 A 101 AM = 0.)29 CONTINUE 100 00 5 M(,3
WRITE ('.1.55) C(JLL) A 102 = (7.AM*j.)WRITE (61,56) CT(JLL) A 103 zo uo.v
3T CONTINUE 104 22 V0A 105 7 1V
Slop A 1)6 72 = ZZ4 507 CALL VESSEL 02,5.41)A (00 CALL 9ESSEL (7).A,AIT)A 159 IF (7.11.3.5) 50 TI A
3A FORMAT (/200 MUIAER OF BLADES ,FR.D,SU.100TIP SPIES RATIO = .F10 4 190 A = 2.7.0.4) A 191 = ...
32 FAI10T (/560 OI'FERENTIAL TIP SPEE' RATIO FOR INTERVAL EVALUATION A 152 C = 6.6.IS F5.4) A 103 0 = A. S.
33 FORMAl (/330 GOLSSTEIN TIP LOSS MODEL ID 0513) 0194 TIVZ 22/(A-V2)PCZ2'72(/((A-22)(9-V2j).(Z2'3l/((A-V2)(B-V2)35 FORMAT (/26H 50 TI LASS 000EL IS USIA) S 195 (C- 22)) O( 22"'.)/ ((5-22) (9-22) (5-52) (0-V2))35 FORMAT (/310 PRANOTL TIP LOSS (OSEL 15 USIA) A 095, CTIV2 = (VPIAI)/(2.SIN).52PI))-TIVZ36 FORMAT (/620 ST050ARO 911000 OF CALCULATION FOR AXIAL INTLPFENENCE A 197 GA TO 2
P IS USEO) A 195 1 TO (UU)(1.PUU)37 FORMAT (/610 WILSON METHOD C' CALCULOTION FOR AXIAL INTERFERENCE A 199 72 = 4.UU(1.-UU)/U1..UcJI"5)
SIT USED) A 200 T'. 16.UU(t.-14.UU*71.U"'.-4.U'6b/((1.+UU)'7)35 FORMAT (/3)0 COORS/MAX. RADIUS - RATIO ,FlX.R) A 201 16 b'..'UU(1.-7S.UUA603.U"5-1065.U65460.U"0-36.039 FORMAT (/1190 LIFT/OSAG RATIO = ,'1I.2( 0 202 10)/U 1.*U0)"10)40 FORMAT (//270 LTFT/054G RATIO INFINITY) A 203 CT1VZ = T0*t2/V2.T./)I27)OT4/(V2'3)41 FORMAT (/55210 LOCAL POWER ClEF. = .F1A.'.,4A,22M LOCAL TORUST CAlF A 20'.
2 IVU S (UU)/(I.5U1)-CTIVZS F1O.6) A 205 SAM = SUMHFVU/2'
'.2 FORMAT (/50,190 SOLID-LIFT P501 = ,F10.6,SX,19H SOLID-DRAG P013 = 5 2)6 IF (AM.NE. 0. 9) GO TO UA,FIO.6) A 201 1 =
43 FORMAT (//27H LIFT/ORAl RATIO = INFINITY) A 200 3 IF (00.51.1.)) GO TT 44'. FORMAT (3F5O.4) 5 1 = 9. 091/(UD"1.'85)'.5 FORMAT (///340 TREORETICIL OPTISUM SLAM lESION ) A 210
4 IF (AM.GT.1.9) E 5 0.146 FORMAT (101) 4 211 SUMD S SUM?R((UOUO'AM)/U.,JOUD)-E)(AI/AIO)57 FORMAT (/1250 LOCAL TIP SPEET RATIO = ,F10.3) A 212 AM = A'j.'.0 FORMAT (/290 AXIAL INTERFERENCE FACTOR = ,F15.N) A 213 AK S I ('.AM-1.)OX)/(2,'AM)49 FORMAT (31H ANGUISH IcIDERFETENCT FACTOR S ,F15.R) A 216 5 0MM = AK/(2.AM.I.)50 FORMAT (/190 TIP LOSS FACTOR ,FID.5) A 715 5 (03)/(1.539)-)M./(PIPI))1U951 FORMAT (/7SH DISLACEMENE VELOCITY = ,FI2.0) 5 216 CISC = G-(?.IPI)53M252 FORMAT (/130 ANGLE P01 = ,FIA.1) 0 21' IF =53 FOSMAT (/250 NURSER OF OTERIATIONS - .15) A 210 SETURN54 FORMAT (//F1951 LIFT/URIS RATIO = ,FR.1) A 213 EN)55 FORMAT (/50,730 TOTAL COEF IF P0010 S ,FII.4) A 22056 FORMAT (/50.2.0 TOTAL CD/F OF THRUST ,F1O.0( A 221
ENS A 222-
6
1)IAID13-
2
3
6
5
67
A
91011121314151617IA19202122232'.
252627
293031323334353637303960'.1
4?434'.
"546'74049SO-
a-'
SU-3HSUTI1E 3SIEL 2.0,41) LOCAL TIP SPEFOAATTT 1.750
AXIAL INTEPFFPFNSF ICTSR . O201D000SUPOJTINE 005001 (Z,V.AI) 0 1
- 3 2 AULAP IXITERFEHENCF FACTO6 . 05535165
C RESSEL CALCULATES 5S5EL FJXOTIOHS FOX THE GOLDSTEIN 3 3 TIP LOSS F4CT0 C
TI' .355 MODEL 30 5 OISPLACEMENT VLLSCITY .06076710
S0.0 3 6 ANGLE PHI 55.635
C = 1. 0 0 CHORD/MAX. RADIUS - PATIO
00 3 <t,1) 3 9 NUMBER OF ITEPIATIONS -3 (.?Z'Z)''AK 0 lB
S VFA< I II.
H j. o to LIFT/DRAG PATIO TOFIFITY
j T< 1-U. 0 13 LOCAL P06)5 COEF. C LOCAL THRUST COEF.IF (TK.L1. 0.1) GO DO 2 1 16
SOLID-LIFt 0501 0 5OL1OTAG 1P55 =H )1<R 1 15
3 = 3-2. 3 16
TI 1 3 1? LIFT/OAG RATIO 75.OT
P E' ItS LOCAL POWER COEF. A LOCAL IRRUPT 00FF. =S 9/(C'E)*S 3 19
AK = BEAt. 3 20 SOLID-LIFT PROS = S SOLIO-APAG FATS =
C RKC 3 25
O COXTISUE I ST LIFT/DRAG PATIO = 50.50
Al ((.57)''0)S 0 23LOCAL POWFR 000F. = 0 LOCAL THRUST CS)9. = 0
RETURM 3 s
EN) 1 25- SOLID-LIFT FPOT I SOLID-OPAl APDX
LIFT/DRAG RATIO 75.00
LOCAL PO.FR CUFF. = 0 LOCAL THRUSI COEF. = 0
SOLID-LIFT PROS = S SOLIO-ORAS R'SO =
LIFT/DRAG PATIO 100.00
LOCAL PObER CTEF. = 0 LOCAL THRUST CORE.
OUTPUT
TPECTFTISAL OPTIRUM 1)550 TSIG9
NURSER O SLACIS sR TIP S!ET 00100 = 1.70DIFFERrNTIAL TIR SrEc RATIO FOP INTERVAL EVALUATICN .1)00
PPANCTL TIP LEES STEL IS USES
HILSON 5)1250 ( CALCULATION FTP AXIAL INTEPFERENCT IS LS)0
SOLID-LIFT PRO) = I SOLIO-I'AG FROT
LOCAL TIP SPEED HATTO = 1.650
AXOAL INTE'F!VENTE FACTOR = . 379R7?75
ANGULAR INTEHFERENCF FACTOR 07555617
TIP LOSS FACTOR . 90906
DISPLACEMENT VELOCITY .37005015
ANGLE PHI = 70.682
CHORO/MAX. RADIUS - RATIO .03220770
NUMBER OF ITEPIATTONS SN
LIFT/DRAG RATIO ITFISITY
LOCAL 'OWES CTF. .590120
SOLID-LIFT PRO) = .500260
LIFT/AlAS RATIO
LOCAL 959)5 CO!). .577776
SILII-LIFT PROJ .52026)
LIFT/DRAG RATIO = ST.CO
LTCAL RIRFO ClEF. - .S1ECSALtS-LIFT PP)
LIFTOUA5 23100 75_IS
LOCAL PEHET EJO. .505.'
SOLI'-LIFT
LOCAL THRUST ClEF. 95771P
SOLID-DRAG PROfl 0
LOCAL TYMU5T STEF. .562065
DOLIO-OAVG TOOT .D2O9I
LOCAL TSUOT"OEr. =SOLO]- 1E2) .TI0Sg5
LOUL T409T F. -.0EJ
I-
LIFT/DRAG RATIO 100.00
LOCAL POWER CORE. .574455 LOCAL THRUST ClEF. .95131W
SOLID-LIFT PROD .522246 SOLII-OPIS FOOD .115222
LOCAL TIF SPEFO RATIO = 1.550
AXIAL !NTEFERFNCE FACTOR = .02502731
ANGULAR INTERFERENCE FACTOR = . 15522770
TIP LOSS FACTOR = . SORB.
DISPLACEMENT VELOCITY 37745520
ORACLE PM! 20.502
CHORD/MAX. RADIUS - RATIO .I3352597
NUMBER OF ITEPIIOIONS - 53
LIFT/DRAG RATIO = INFIRITY
LOCAL POWER COEF. = .552945
SOLID-LIFT PROD = .576500
LIFT/DRAG RATIO = 25.00
LOCAL DOWER eocF. .457076
SOLID-LIFT PROD = .576609
LIFT/DRAG RATIO = 50.00
LOCAL POWER COVE. .525511
SOLID-LIFT PRO) = .576519
LIFTIDRAG RATIO 75.00
LOCAL POWER COEF. 534559
SOLID-LIFT PROD .576610
LIFT/DRAG RATIO 101.10
LOCAL ROWER COVE. = .5iRI5SOLID-LIFT PROD .576609
LOCAL THRUST ClEF. = .RRR2II
SOLID-DRAG FROG =
LOCAL THRUST fUE. = .902501
SOLID-T°OG FOOT = .073065
LOCAL THUTT COVE. = .095346
SOLID-IRAG FACT = .011532
LOCAL THRUST COVE. .092955
SOLIU-DPIO FRITO 007555
LOCAL THRUT 10EV. .091775
SDLIO-DRAD POOl .005766
LOCAL TIP SPEED RATIO = 0.450
AXIAL INTERFERENCE FACTOR .32397717
ANGULAR INTERFERESCE FACTOR .09512061
TIP LOGS FACTOR 1.000AO
DISPLACEMENT VELCCITY .302657R
ANGLE PHI 23.151
CHORO/HAA. RADIUS - RATIO .03567130
NURSER OF ITERIIT100S - 4?
LIFT/DRAG 05100 INFIFITY
LOCAL POWER ('3FF. .502102
SOLID-LIFT PROS .539350
LIFT/DRAG RATIO = 75.00
LOCAL POWER ('OFF. 453007
SOLIO-LAT flVO' = .639361
LOCAL THVUT COVE. .079511
SOLIOOAG 0001 I
LOCAL THRUST C0E.
SOLID-ORAL 0Q' .0?574
LIT/OVAG RATIC 53.00
LOCAL RO.'ER CTF.
SOLID-LIFT PROD = .6093.0
LIFT/DRAG PATIO = 75.00
LOCAL POWER COVE. = .496053
SOLOO-LIET PRO') = .539341
LIFT/DRAG RATIO = 100.00
LOCAL POWER COVE. = .500073
SOLIT-LDFT PROD .639350
LOCAL THRUST TIER. = .835635
TOLlS-TOAD PP'l' = .012707
LOCAL TPAOUTT ClEF. .834200
SOLID-DRAG PRO') .100525
LOCAL THRUST ClEF. = .833103
SOLID-051G FOOT = . I063R3
LOCAL TIP SPEED RATIO = 1.350
AXIAL INTERFERENCE FOCOOR = . 32203090
ANGULAR INTVRFERFNCE FACTOR = .10023745
TIP LOSS FACTOR = 1.00001
011PLACEMEPAT VELOCITY = .38897471
ANGLE PHI 25. 052
CHORD/MAX. RADIUS - PATIO .03593945
NUMBER OF ITFPIATIOHS - 39
LIFT/DRAG 51110 = INFINITY
LOCAL OWEA 70EV. = .571069
SOLID-LIFT PROD .711015
LIFT/DRAG PATIO 25.0)
LOCAL POWER COVE. .429430
SOLID-L OFT PROD DI1RIN
LIFT/DRAG RATIO RI.OD
LOCAL 3WER ('DEE. .451254
SOLOS-LOFT RO3 = .711015
LIFT/DRAG RATIO = 75.00
LOCAL ROWER COVE. = .457092
SOLID-LTET ')PO) = .11RA5
LIFT/DRAG RATIO 133.00
LOCAL POWER COEF. .560641SOLII-LIET PROD = '11515
LOCAL THRUST CUFF. = .771945
SOLID-OARS FOOD = I
LOCAL 09OCT50 CUFF. .7BNRYP
SOLDO-IPAG FOOD = .025573
LOCAL THRUST CUFF. .777920
SOLID-DRAG PROD .114236
LOCAL THRUST ('OFF. = .775501
SOLID-DRAG FOfl') = .009491
LOCAL THRUST TIFF. = .77a.63P
SOLIO-TRAG POT" = .027115
LOCAL Tb SR!EO P0113 = 1.251
AXIAL DNTFRFEVEM('E ICT')R .07151101
ONGULAR INTEFERE5C FOCTOP .12415615
TIP LOSS FAcT(c 1.00001
OISPLATIENENT SOLICITS .7966551R
ANGLE PHI = 1'5.'73
('HOOT/PDX. 911133 - OATTC .0)72)496
NU'IRFR OF IT5XII'1 135 - .2
(j.1
LIFTI00G RATIO = INFINITY
LOCAL POWER COEF. 529595
SOLID-LOFT RPD) .795030
LIFT/DRAG PATIO 25.01
LOCAL POWER 10FF. 305255
SOLID-LIFT PROD .795530
LIFT/DRAG RATIO = 50.00
LOCAL POWER COEF. .512092
SOLID-LiFT PROD = . 7RR3O
LIFT/DRAG PATIO = 75.00
LOCAL POWER COF. .515020
SOIOU-LOFT PROS = .795530
LIFT/DRAG PATIO = 1110.110
LOCAL POWER CUFF. .520995
SOLID-LIFT P500 .795830
LOCAL TAPUOT CODE. = .71225
SOLID-TRIG FY70 = 0
LOCAL THRUST CUFF. = .726055
SOLID-DRAG 0500 .131533
LOCAL THRUST COEF. = .719150
SOLID-DRAG ER01 = .115°17
LOCAL THHUTT CODE. = .715873
SOLIO0A0 FOOTA .010511
LOCAL THRUST ClEF. = .71572?
SOLID-TRIG FPTO = . 0D7c5R
LOCAL TIP SPEED RATIO 1.150
AXIAL INTERFERENCE FACTOR .31997753
ANGULAP INTERFERENCE FACTOR = .15305052
TIP LOSS FACTOR = 1.00000
DISPLACEMENT VELOCITY . 50549194
ANGLE PHI 27. 337
CHORD/MAX. RADIUS - RATED = .03543643
SUPPER OF TTERIATIONS -
LIFT/DRAG PA TIE INFINITY
LOCAL POWER CODE. . 38561R
SOLID-LIFT PROS .593565
LIFT/ORAl RATIO 75.00
LOCAL POWER CODE. . 35R54I
SOLID-LIFT PROS = .593665
LIFT/DRAG RATIO = 50.10
LOCh POWER ClEF. .373574
SOLID-LIFT P801 = . 5)3665
LIET/OPAS RATIC 75.00
LOCAL POWER COVE. .378556
SOLID-LOFT PROS .593665
LIFT/OPAl RATIO = 100.03
LOCAL POSER CC0F. 3R1191
SOLID-LIFT ROOT .593555
LOCAL THRUST CODE. .653564
SOLIP-SPAG FOOT
LOCAL THRUST ETFT. .6671'l
SOLID-T011 0005 .035757
LOCAL 0411USD CODE. .660422
SOLIDSRAG 0501 = .17 873
LOCAL S4UT CUFF. = .458A11
SOIlS-SPIT F8)5T = .011516
LOCAL OOVIJST ToFU. .65714'
SOLID-DRAG 080T .005531
LOCAL hF SF/EU RATIO = 1.053
AOIAL INTERFEREME FACTOR = .00506057
ANGULAR INTEOFEPENTE FACTO .14555155
TIP LOSS FACTTR 1.00000
OISPLACESENT VELOCITY = .41634359
ANGLE °HI 79.365
CHORD/MAO. RACIUS - RATIO .03057237
NUMRFR OF ITERIATIONS - 53
LIFT/ORAl PATIO IMEIFITY
LOCAL 0OWFR lOSE. = . 540702
SOLID-LIFT °RO'O = 5.00 7707
LIFT/DRAG RATIO 75.03
LOCAL POWER CUFF. .322744
SOLID-LIFT PROS = 1.007702
LIFT/ORAl RATIO 50.00
LOCRL POWER CODE. = .334739
SOLID-LIFT POOl 1.007702
LIFT/OPAl RATIO 75.10
LOCAL POWFT CUFF. .338901
SOLID-LIFT PROS = 1.007702
LIFT/DRAG RATIC 100.00
LOCAL POWER 10FF. = .340485
SOLID-LOFT PROS = 1.007702
LOCAL THRUST DOFF. = 594073
SOLID-ORAl PROD = 0
LOCAL T5RUD3 COE. = .6081R3
SOLID-DRAG EROD .040305
LOCAL THRUST CODE. .601537
SOLOO-ORAO PROS .020154
LOCAL THRUST CODE. = .549303
SOLID-URIS ROO = .013535
LOCAL THRUST COED. .548210
SOLID-OPAl PROS = .110077
LOCAL TIP SPEED RATIO .950
AXIAL TNTERFEQENCE FACTOR = .31557597
ANGULAR INIE/FERENCC FACTOR .199611205
TIP LOSS FACTOF A.000DD
DISPLACEMENT OFLOCITO .40979790
ANGLE PHI 31.477
CHORD/MAO. RUDIUS - RATIO = .04054945
SUPPER OF ITFPIAT109S - 54
LIFT/OPAl PATIO ISFISITY
LOCIL POWEX CODE. .005504
SOLID-LIFT PP0I = I.141'75
LIFT/OPAl 01011 25.10
LOCAL ROWER CODE. = .255s55
SOLIDLIET PROS = 0.14175
LIFT/OPUS RATIO 2.01
LOCIL POSE11 CQE.SOLID-LIE TU4fl-_ 1.54j77
LIFT/RAI El TIC '5.0)
LOCAL 154/3STIlT_LIST ER7 I.l4177
LOCAL THRUST COEF. = .536211
SOLIDORAl UROO 0
LDTOL THAUST TDFE. =
SOLOS-ISIS cOST .045/51
LOCAL S4ISI C3F. = .542715
S0LSO-CD; URIS . 007R25
LOCAL 3ITD 105/. .541500
.TA1i7
0'
LDFT/S°AC, ATT0 100.00
LOCAL POWER CUFF. 300761 LOCAL T'4RUOT COEF.
SOLID-LIFT R0D 1.16177', SOLIOD0AG FOOT = .011413
LOCAL TIP SROEO RATIO .PR0
AXIAL TNTFPFEPLNCF ICIOP . 31321971ANGULAR INTERFFR000F FACTOR = .240091M2TIP LOSS FACTOR 1.00010
DISPLXCEMFNT VELOCITY .6461H050ANGLE PHI = 30.096
CHORD/MAX. RADIUS - RATIO .0612542?NUMBER O ITERIATIOBS 50
LIFT/DRAG PATIO D'AFIFITY
LOCAL PIWFI rOOF. .20,4523 LOCAL T'.?UTT C'DFF. .6770,39
TOLIO-LIFT PROD 1.207714 SOLID-TEAS PRO'! =
LIFT/DRAG PATIO 25.00LOCAL P)WFR CUFF. .24*2*4 LOCAL THRUST COOP. .490T*7SOLID-LIFT PROD = 1.29771'. SOLIO-O045 FROG .051009
LIFT/IPAG RATIO = 00.00
LOCAL 0DWFR COOP. .256404 LOCAL 'URUTT CUFF. = .483RF'SOLID-LIFT PROD 1.20771'. DOLTS-SPAS FACT = . 025°5'.
LIFT/DRAG PATIO = 75.00LOCAL POWER CUFF. = .254110 LOCAL THRUST 050F. =
SOLID-LIFT ROOD 1.29771. SOLID-0000 PRO!' .T17 103
LIFT/SPAS PATIO = 103.50
LOCAL POWER CORP. .200663 LOCAL THRUST rOOF. = .4RITNASOLOS-LIFT PROD 1.20771, SOLID-OPTS 0020 .012377
LOCAL TIP 50000 RATIO .750AXIAL TNTF000RFN70 000100 . TIO7IR7IONGULAP ONTFRFERFNCP FACTOR .2941999'.TIP LOTS FACTOR = 1.OT000
DISPLACEMEOT VFLCCITO = .60,740770ANGLO PHI 35.079CHOPO/"AR. P00000 - RATIO .04150090StIRRER 50 ITPDATI!'N'! -
LIFT/DRAG RATIO DSPIFITV
LOCAL POWER COEF. = .22 1'. 7R LOCAL PROUST CDE. .419R01SOLID-LIFT PP3) 1.'.92745 SDLIO-T'AC PACT = II
LIPT/DPAS RATIO =
LCCTL 0)WFH 10F. .710900 LOCh THRUST "DEF.
S"LIA-LIP T PRey! 1.402745 TDLIO-T015 PAID .144515
LIFT/APIS RAIDS 50.00LOCAL POWER CCEF. .2I7t0, LOCAL T '401101 'eDO". .4255P',SOLID-LIFT ROOT 1.492745 SOLID-TROT PROS .029055
LIFT/OPAD RATIO = 75.00LOCAL POWER CO1F. . 71929? LOCAL THRUST rOOF. .423577SOLID-LIFT PROD 1.602745 SOLIO-DRAS FP000 = .119770
LIFT/DRAG PATIO = 100.00
LOCAL POWER CORP. = .220331 LOCAL THRUST CUFF. .422504SOLID-LIFT ROOD = A.402745 SOLIO-OAS ROOT .T1427
LOCAL TIP SPFEC PATIO = .650AXIAL CNTEOFERENCF FACTOR = . 30901964ANGULAR INTERFERENCE FACTOR = . 36*71365TIP LOSS FACTOR = 1.10000
DTSPLACEIENT VELOCITY . '.457N27ANGLE PHI 37.*F,PCHORD/MAX. RADIUS - RATIO .0.13501.3NUMBER O DTEPIOTIONS - 36
LIFT/OPAS RATIO INFINITY
LOCAL ROWER COEF. 1P2003SOLID-LIFT PROD = 1.711297
LIFT/DRAG PATIO = 25.00LOCAL POWER CORP. = .073570SOLID-LIFT PROS = 1.701297
LIFT/DRAG RATIO 5D.AILOCAL ROWER COPE. = .179277SOLID-LIFT PROC = 1.7D1097
LIFT/DRAG PATIO = 75.00LOCAL o54p COPE. = . 179'41SOLOS-LOFT POUT S.701297
LIFT/OPUS RATIO 1010.00
LOCAL POWEA CUFF. .1*0010
SOLID-LIFT APOS 1.'01297
LOCAL THRUST COE. = .O62DDR
SOLID-OPAl FOOD I
LOCAL THRUST 00FF. .373300SOLXD-0010 PROD .060152
LOCAL THAIJDT COFF. .367670SOLOS-DRAG ROOD = .034026
LOCAL THRUST CTF. = .365793SSLIO-0005 FOOD . 122PM'.
LOCAL THOU'!'! !'DE". .304054SOLID-T000 POST = .017013
LOCAL TI SPEFO POTTS .553AXIAL INTEPFC"FNCI PACTDO .TYI'6966ANGUIA° INTESFEP'520 FTCTC4 .4730330',TIP LOSS FACTO' 1.02000
OISPLACEIE'IT ASL002'S
ANGIE H1 40.759CHORD/MAP. 500055 - 0'!
ADORER Dl TT''IATIC'JG - '.5
o'
LIFT/')AG RATIO = ISFII,ITYLOCAL PIWEM CODE. . 151507 LOCAL THTMU0T CoE. .312941SOLID-LIFT PROS 1.295 SOLID-IRIS FOIl
LIFT/0000 RATIO = 25.00LOCAL DOWER 00FF. = . 116945 LOCAL 200000 COOP. .313?7PSOLID-LIFT PPOO 1.946295 SOLID-OPAl PROS = .077052
LIFT/DRAG RATIO = 50.00LOCAL POWER COEF. = . 150176 LOCAL THRUST rEF. = .310050SOLID-LIFT PROS = 1.946295 SOLID-TOAD, 6900 . T3H°26
LIFT/ORAG PATIO = 75.00LOCAL POWER CODF. . 141286 LOCAL THRUST 00FF. = .306320SOLID-LIFT PQO1 = 5.946795 SOLTO-OPAC, FOOlS .025950
LIFT/OPAG RATIO = 100.00LOCAL POWER COEF. . 14105? LOCAL THRUST ClEF. = .305551SOLID-LIFT PRO') 1.946295 SOLID-DRAG PRO)) = .019463
LOCAL TIP SPEED RATIO = .450AXIAL INTERFERENCE YACTOD = . 00196964ANGULAR INTEOFEDENCE FACTOR = .63610155TOP LOSS FA000P = 1.00000DISPLACEMENT VELOCITY .57317042ANGLE PHI 53.472CHIRrO/040. 9001US - RATIO = .O37q9A25NUMBER OF ITEPSATID9S - 4'LTFTIOROG RATIO = INFIIrTTY
LOCAL POWER COEF. = . 105718 LOCAL THMUT 09FF. = .247770SOLID-LIFT PO') 2.25735' S00100OPAG 6001 I
LOFT/DRAG 90100 = 25.00LOCAL POWER ClEF. = . 001248 LOCAL THRUST COOT. = .257171GOLDS-LIFT °O) = 7.757350 SOLIO-fl'OG F'S' .090294
LIFT/DAIS PATIO =LOCAL POWER DOFF. = .103s7M LOCAL THRUST CTEF. = .2524'?SOLID-LIFT 0050 = 2.257)50 SOLID-SPAS FROST .C451'.7
LIFT/OPAl RATJ7 75.00LOCAL 03600 = .000270 LOCAL THRUT COFF. = .251911SOLOS-LIFT O) 7.757150 IOLTS-3005 PROS .030090
(IFT/TRAG 'ATOP 111.0]LOCAL POWER CSF. = . ADASA) LOCAL 1095T COEF. .25012'GOLII-LIFT 959) = 2.257)53 STLIO-0000 POST = .02?'75
LOCAL TOP SEES ROTIC .050AXIAL INTERFEOENCF FACTOrS = .01116965ANGULAR INTERFLRENCE FACTOR = .90391107TIP LOSS FACOOP = 1.00000015PLACEOENT VELOCITY = .63935103ANGLE PHI 46.101CHORD/MAX. RATIOS - PATTI .03432747NUMBER OF ITERI0100'4S - 60
LIFT/DRAG °A110 = INFIElDSLOCAL DOWER CUFF. = .070659 LOCAL THRUST COED. = .192717SOLID-LIFT PRO') 2.622426 SOLID-IPAG FOOT =
LIFT/DRAG P0000 = 25.00LOCAL DOWER DOFF. = .167061 LOCAL THRUST ClEF. = .200792SOLID-LIFT FOOD 2.622526 SOLID-ORBS FOOT .104097
LIFT/DRAG RATIO = 50.00LOCAL POWER COOP. = .069310 LOCAL TMRU1T 0006. = .196756SOLID-LIFT PROS = 2.627426 SOLID-DRAG PROD = .052449
LIFT/DRAG 000IO 75.00LOCAL POWER COFF. = .069760 LOCAL THRUST rOEF. = .19540010(00-LIED 9502 2.622526 SOLID-DRAG 0000 = .036906
LIFT/DRAG RATIO = 100.00LOCAL POWER DOFF. = .069905 LOCAL TUPUOT CUFF. = .194706SOLID-LIFT PPO0 = 2.622526 SOUl_OPAl, FOOD = .026225
LOCAL TIP S'FEO RADIO = .250ADIAL INTEMFERFRCE DOCTOR 0 .29196964ANGULAR INTERFERENCE F000OP = 1.30615321TIP LOSS FASTOR = 1.00000SISPLACEMENT VFLOCDTY .70377540ANGLE P141 = 59.005CH000,'MAX. RADIUS - RATIO = .12799752NUMBER OF ITEPI0010NS - 54
LIFT/D940, 0001C INFISITYLOCAL PORED COF. .040159 LOCAL 100UT DOFF. .195'T?SOLID-LIFT P009 2.995395 SOLID-SPAS FOSS =
LIFT/Tool PATIOLTCSL 095(0 CODE. = .030709 LOCAL THUST COlE0. = .141612SOLOS-LIFT 6009 '095395 53(10-5005 POOP .11° 775
LIFT/OPAl PATIO 50.00LOCAL POWR 09FF. .293Rr. LOCAL THRUST ClEF. = .130010SSLTO-LLFT P9 = 7.014395 SOLIO_TPAS 9"
LIFT/TAOS ATIC= 75.09LOCAL 095)0 CO0. 5095]r LOSOL T")D0 ). .1(710
SCLIS-LIE) rP9' 7.3)5") l'L29-T-) fl)9975
0"0"
LIFT/DRAG RATTO 100.00
LOCAL POWER TIFF. . 019770 LOCAL 1HP(JTT COF. = .136605
SOLID-LIFT PROD = 2.996395 SILID-XPAS FROD = .029564
LOCAL TIP SPEED PATIO = .150
AXIAL INTERFERENCE ACTOQ = . 26Iq6qR,4
ANGULAR INTERFERENCF FACTOR = 2.41371109
TIP LOSS FACTOR = 1.00010
DISPLACEMENT VELOCITY .75753536
ANGLE Fl-Il 54.778
CHORD/MAX. RADIUS - RATIO S .01843303
NUNRER OF ITERIATIONS - 49
LIFT/DRAG RATIO = Il-IFIRITY
LOCAL POWER CAlF. .016096
SOLID-LIFT PROT = 3.285753
LIFT/DRAG RATIO 5 25.00
LOCAL POWER COR. .505641
SOLID-LIFT P500 = 3.285753
LIFT/DRAG PATIO 58.00
LOCAL POWER CIFF. =
SOLID-LOFT PRO') 3.285753
LIFT/DRAG RATIO 75.00
LOCAL POWER ClEF. = .015944
300.10-LIFT PRO') 3.255753
LIFT/DRAG RATIO = 100.00
LOCAL POWER ClEF. = .015557
SOLID-LIFT PPD') 3.285753
LIFT/DRAG RATIO INFINITY
TOTAL COLE DF 50WE5 S .4745
TOTAL COFF OF THRIJTT = .8003
LIFT/OPAl ATIG 25.0
TOTAL COEF CF 500F1 = .4362
TOTAL CDFF OF TI-IXIST = .R489
LIFT/DRAG RATIO = 53.0
TOTAL ClEF OF 5OWER .4554
TOTAL ClEF 5F TH5LT = .8396
LOCAL TI4RIJTT CAFE. .075755
SOLID-OPAl SPOT = I
LOCAL TI4OUTI 'lET. .090091
SO1IO-OAG PROD = .131430
LOCAL T9RUOT CUFF. .077904
SOLID-CRAG P50') .065115
LOCAL TMRUST COOT. .0770*9
SOLOO-OPAO PROD = I43MGT
LOCAL TMRUTT COOT. = .076811
SOLID-DRAG PROC = .132855
LIFT/DRAG AT0D = 75.0
TOTAL TIFF OF 5OWER =
TOTAL 02FF OF THRIST =
LIFT/OPAG PATIOs A,A.O
TOTAL ClEF CF DWER =
TOTAL COEF OF THMLST
.4617
5365
4649
.5 349
Program AXOPTO Listing
and Sample Output
C
C
C
RAARA1 AXD1O IF (XL.0.X) A) t)A 73
CALL G)L)AT (AAL,XXLA,FI A 71
P0)52(9 AX)PTO A I GA TA 3 A 72
A 2 7 CALL ORAMIT (B.0AIA.X.L.2) A 73
!ASIIIA I ----- JA'IUAYY 1976 A 3 0) TA A A 74
A 4 A FI. A 75
IAAPIJ - )EIERMIAEA THE 1OCT13) 3LAAE EOIETHY F)9 A 5 3 C)ATI1JE A 76
SPECI'IEO APERATITAA 109AAITIAAS JATIAG CAISTAIT A CXX CL'OtM!PHI) A 77
AXIAL I3TE'FERE)E FACTOR A9 TIE WIALE ALA)E. A 7 CAY = CL'CIS(PIII A TI
A IIF A9I).E0.1.I GA TO II A 73
A 3OP = (AXX'A)P(CVYXL) A 33
ATIEAAIIN AP. (40.5), GIL) 43,,) 331 (5) SUMT(5) , 10(5), CT(S), A 10 50 1) 11 A Al
3 CAA(5) , SIACA(S) A II 03 IA = )1.-FA)(CXXA)/((CYVXL(11.4)) A 52
IN'AT OARAMETES3 A 12 11 CANTINJE A 53
A 13 PAl ATA)(ll.-A)( (j.A0)'XL) I A Is
A 1. 12 C00XTINUE A 35
- 90013E9 O 3L000S A 15 WW? = )1.-41'200LA011.'A'I''2 A 36
A 15 IF (Al0).EQ.1.) 03 TO 13 A IT
X - TI' A0EEI RATIA A (.7 SLAX I.0F'A'(l.-A)'AXX/(2'CVY) A AS
A II GO TA 14 3 33
DX - 3IFE3E3TIAL TIP S'E,T)2 OR IXTEYHAL EVALUATION A IA 13 SLAT A.000A'(l.-AOF) CXAf(AIU2CYYI A 30
A 70 1. CAITISUE A Ml
TA - TIO LASS MOIE. COATR)LLER A 21 YMA+ SLAXI.?/A. A 32
0 - A) TI LOSS IO)EL A '2 SLOPE = ((MA X-XMAXA) f( A-AT) A 93
I - ORXNIJTL 010 LASS IOJEL A 23 IF (AAS(X'AR-AIAEA).LT.l.AE-IA) 13 TA 13 A 34
' - AILASTEIN TIP LOSS lOOEL A 25 IF (SLAP..LT.fl..453.CANTYAL.EO.l.) GO TO IT A 95
A 25 IF (SLODT.LT.D.) 50 0) 15 A 36
AMA) - AXIAL I)TF 2E!9A IOAEL ATITROLTS A 26 IT (CONTROL. EI.1 .1 Al = 90/2. A 37
I - DTANAARJ YETHAD A 27 AA9TR)L = 0.0 A 95
1 - XILSOS 1TH0A A 23 Al = AT A 99
A 79 A) A A 133
AL - CORTTICIEXT O LIT A 3A A 4+01 A 191.
A 31 SMAXI VIANA A 132
READ (60.50) I,E,CL A 32 RMAYA XIAA A 103
lEA) (60.50) A4,TI,A403 A 30 GO TO N A 104
WRITE I41,A)( A 34 A) A) A A 105
WRITE (AA,37) 0,5 A 35 53054 XAAA 5 106
WPITE (51.39) AX A 36 )A = 14/2. A IA?
IT (T1.EA.3.( 01 TI 1 A AT A) 0) 17 A 103
IF (TM.ES.l.( A) TO A 3M' XMAXA XMAS 4 109
IIOITE (60,39) A 39 A) A A 119
GO TA 3 A 43 IT A A-Al A Ill
A WRITE (61,40) A 41 C)(TRAL = 1. A 112
GO TA 3 A 47 0) TO 6 A 113
P WAITE I41,.1) 53 15 C030I9JE 4 114
O CO9TIAJE A 4. CAATIAUJE A 115
IF (A'OI.EA.l.( 55 01 . A) 4(5 = (1.-Il A 016
WRITE (41.42) A 46 XL A 117A) 10 5 A 47 00 30 1=1,9 A 113
-. W°ITL 051..)) A *5 13 25 5L1,5 A 113COATIIUE A .9 0934 433 0°-OIl A 129DI = .01 A 50 ARL AS(PHIAI/SO9 (OHIA) A 121
PT 0.I4149'65. A 51OSLO = XNLXIXLIF (TI.E).O.) .0 TA 21
A 122A 123
F = 1. A 52 IF (01.E).j.) A TA 20 A 12.CR 3. A 3
I (YL.E3.X( F = 0. A 125IC A A 5 12 (XL.EA.5) 03 TO 77 A 126A .02 A 5 CALL S3LAST IXXL.X(L).) A 127XL = 5°. 75 I S6 51 TA 22 A 125PHI ATAIOI. /+L) A 7
20 CALL 2A(1T 09.PRTA,X.EL,T) A 129XMAXA 3.) A ,1 GA TA 02 A 1308 = 0 21 F I. A 131N = 5f3Xl. A 49 22 C)X1I9U A 132IA 14 Jl.20 A 61 CX XL0AIS(P-1I) A 133
6 IC = ICOI A 5' CAY CLA)S(PlI) A 134'10 12 51.5 A IF )A115.fl.1.0 00 0) 75 A 135
PAIA = A030°XI( A 4 A) = A 136+XL A)A(°IIA(/)IIIPIIA( A A) TA 24 A 137OSLO = SEE'S/AL 0 20 0' (1.-'A)(C.XA(IOICYA'XA)'Ol.-AIl 5 035 0='IF (TM.C).).) 50 0) A 7 2. ANTI A 109 D02 (TA. LA. 1.) GA To 7 A 54
II I = A TA 001 . - A) / 1 1 .0 A +L A 149IF (AL. CI. 1 F = 0. A 69 2= =J( IEJ 1,1
lISP 6/lI)3(P5I)"2) A 1,2 A 716SIL SLCXFSIM(PHI) A 1.3 37 FO'MAT (/235 939919 OF ALAJES ,FA.0,55,ISHTI' SPEE) PATIO ,F1O 0 215512 StGL/CL 0 15' 5.4) 0 215CR = PIXLSIG/(5A) 4 j55 35 FORMAT U514 OIFFEXESTIAL TI' SPELl ATIJ FOP ISTEXUAL SALUOTIOS A 717M M 4 156 5= CA..) 2 '15
0. A 1 39 FORMAT (/335 r,)L)STEIN TIP 1155 IOILL IS USED) A 219OF 25. 4 145 50 FORMAT (/265 ND TIP LOSS MODEL IS 9252) A 22000 2 M=A,5 A 149 41 FORMAT (/315 "AIOTL TIP LASS MDIEL IS USED) A 721
CLCO(ML) OF-25. A 150 52 FORMAT (/625 TTANJUS) METHOD D ALCULAT104 SOS AXIAL TATERFEPESCE 4 222SOGCO(ML) SIG'CO A 151 5 IS USEI) A 225CX = CL'S1'4(PHI)-CO'CIStPHII A 152 53 FORMAT (/415 SILSJA IETHDD O CALCULATIDA FOR AXIAL INTERFEXENE A 224C! = CL'COSI'HI)PCO'SIN(PNI) A 153 S DOT)) A 725CPL(M,PIL) SIGCXIXL/X)'2'W52 A 155 54 FORMST (/305 C1)S)/XDX. RADIUS - RATIO ,F1O.0( A 226CTL(M,ML) = SIG'CY0L'W2/(XXI A 155 45 FORMAT (//191 LI1/ORAG RATIO ,Ft0.2) 0 227CO CLFOF A 156 46 FORMAl (//'71 LIFT/DRAG 95710 = 1MINITY) 4 225OF = SF425. 0 157 47 FORMAT (/55215 LOCAL POWER GOES. = ,F10.6,4x.225 LOCAL THRUST GOlF 4 229
26 CONTINU A 153 0. ,:3f,( 5 230P500 091150.FPI A 159 45 FORMAl (/55,191 SOLID-LI'T PRO) = ,F10.6,5A,195 SOLIA-ORAG PROD = 4 231WRITS (51,521 A 160 5.F1e.,) 4 232WRITE (61,53) XI A 151 49 FORMAl (//TT9 LIFTIOSAG RATIO = INFI'4ITI( A 233WRITE (61,54) 4 A 162 50 FORMAT (3F[3.4) A 234WRITE (61,55) 50 A 163 51 FORMAT (/1/345 TMEORETISAL O'TIIJM 3L4)E OESIGN I 4 235WRITS (61,56) F A AS' 52 FORMAT (151) * 236WRITE (61,57) OISP A 165 53 FORMAT (/F?55 LOCAL TIP SPEEO PATIO 2 ,F10.3) A 237WRITE (61,55) P510 A 166 54 505941 (/295 AXIAL IOTERFFRESCE FACTOR = ,F1R.0) 4 235WRITE (61,44) CR 4 167 55 FORMAT (/315 A9GULAR IRTCRFLREMCC FACTDR = ,Fj5,5( 5 239WRITE (51,59) IC 4 155 So FORMAT 1/195 TI' LOSS 6CTOR = ,1O.5) A 2.000 29 XLL=1,5 0 169 57 FOAMAT (/205 DISPLACEMENT VEL)1TY = ,F12.8( 5 261
IF (CLCO(KLL).EO.X.) GO TO 27 A 170 55 FORMAT (/135 AROLE PHI = .513.3) 5 242WRITE (61.45) CLCO(KLL) A 171 59 FORMAT (/255 939919 OF ITERIAOIOSS - ,IAI 5 263GO TO 20 4 172 63 FORMAT (//t95 LTFT/)RAS RATIO = ,0.t) 0 254
27 WRITS (61,46) A 173 61 FORMAT (/55,231 TOTAL GOLF OF P3ER = ,F13.4l A 25523 CONTINUE A 17 67 FORMAl (/55,245 TOTAL GOLF A THRUST = ,10,'.) A 246
(RITE (61,4?) CPL(M,KLL),CTL(M,ELL) 5 175 END 0 247-WRITE (51,43) SIGCL,SIGC)(<LL) OATS
29 CONTINUE 177XMAWA = 0. o ii, SUBROUTINE 'RSM)T 13,PRI.A,XL,GF)00 0. A 179OS = 0. A 100 SUOROJTINE PR450I (5,PHI,X,XL.G) o I.
59455 = 3. A 151 C 3 2
IC = 0 A 132 ...... PRANOT - DETERMINES THE PXAR)TL TIP LOSS OACTOS B 3
OA=.O1 *103XL = AL-OX 104 P1 = 3,141192635 B 5
IF ((AIXl .LT..O4I 10 TO 35 5 135 FL = (B/2.)'(A-XLU(UL'SIN('MI)) 3 6
30 CONTINUE 154 00 = -FL A 7
30 CONTINUE A 1*7 sT = LXP(P)( 3 0
00 36 JLLI,5 0 INN IF (*T.GT.1.( T 1. 3 9
SIJM(JLL) = C'L(I,JLL) A 199 IF IXT.LT.-1. I ST = -1. 9 10
SUMT(JLL) CTL(t,JLL) 0 190 SF (7./PI)AC)5F(AT( 3 11
NI. = N-I A 131 REFORM 3 12
00 02 52,N1,2 4 19? EN) 3 15-SUM(JLLI = SUM(JLL)*6.'CPL(K.JLL) A 193SUMT(JLL) SUM?(JLL).4.CTL(E,JLL( A 19'.
32 CONTINUE s 135 SUOQOUTI'IE SOLIOT (U,UO,SF(00 33 53,51,7 4 196
SUM(JLL( = SUM(JLL)42.'CPLIK,JIL) A 197 SABROJTINE SOLOS' (U.UO,0F) C I
SUMT(JLLI = SAMT(JLL(+2,'CTL(S,JLL) 4 139 2
31 CONITNUF A 193 SOLOST - OLTERMI3ES T5 ;OL)ST:IN TIP LOSS FACTOR C 3
CP(JLL) = SU4(JLLJOO/3. A 200 5 FOX TI TWO TLA1!) CASE C
CT(JLL) = SUIT(JLL)'TXF0. U 231 5 5
IF ICLC)(JLLI.EO.O.I SO TO 34 0 202 PT = 3.151592654 C 6
WRITE (51,50) CLCD(JL_) 4 703 SUIT = 0.0GA TO 35 A 204 SUM = 0.0 C 0
0'. WRITE (31,49) 4 235 AK = E. C
35 CONTINUE A 206 AIM = S. C 13
WRITE (41,61) SP(JLL) A 707 AM = 3.0 'C It
WRITE (51,6?) CT(JLL) A TOM 00 5 9=1.3 C 1?35 CO9II9UE A 739 (2.A121.) C I!
4 213 20 JO3 5 14STOP 0 211 2 = 15
A 213 2 = U'S C 16
S !13 2? = 3'' C 17
I--J0
CALL. BESSEL (ZS,AII C 10 CALL BESSEL ZB.V.AI,I C 19
IF (z.GE.3.) GO TO I C TI A 2.2. C 21 B 4.'.. C 22 C 6.6. C 23 0 0.3. C 24
Tt7 = Z2,O3.V2I,(Z222)/f(*-V2)(B2))+(Z2"3)/)(AV2)(BV2) C 25 (-J2fl *022"'.) /I(A-92)(0-32)(C-V2P(0-V2)I C 26
ClIME )'JtAC)II2.SSMI.5°S5)-1iVZ C 27 GO TO 2 C 25 TO = (UUI/(t.*UU) C 29 73 4.ULIf1.-UUI/Hi.fUJ)"4) C 30 1'. l6.'JLJ (1. -14.3U+21.J"4-4. LJ6)/) (1..UU)"7) C 31 16 ..U.UPOI._75.U*UI403.U*4-I5.OIJ4*46Q.U"B36.U" C 32
S iO,,U1.*JU1"10I C 33 CTIV2 = T5,T2/V2+1k2"2)*TA/(V2"3) C 3'.
2 FMJ = )i.J)(1.*UPU)-tIV2 C 35 SLIM SUMIFVU/32 C 36 IF (51.07. 0.0) 10 13 c 37 I -0.095(U0".663) C 05
3 IF (5I.'IE. 1.0) 10 13 C 3M I 0.031(J3"I.TB5O C '.0
IF (31.11. 1.0) E = 0.3 C 41 SUIT = SUI2.((U0UO3'IM)F(l.*JOUD)-E)(AI/AIO) C 42
AM 00*1. C 43 AK ((2.AM-1.10K)f(2.A1) C 64
5 0MM = AK/(3.AI*I.) C 45 A (jJ),(1.U)-(5.f(11P1))3Ul C 66
dM1 S-(O.fPIISUM2 C '.7
SF ((1.ALJJII(UU))CIKC C 43 512014 C 49
EMS C 50-
SUBPDUTIFOE 3SSSEL 12.V,AII
SU300JTIME 015511 42.3,05) a I
0 2 315511 CALCULATES MESSEL FUICTI005 733 THE GOLASTEIM 0 3
C TIP LOSS 03071 3 4 O 5
S 3.0 0 6 5(5 1.1) 3 7 7=1
3 3 30 3 <1,IS 3 9
S (.2371)"AK 3 10 O = AK 3 11 P = 1 3 12
TE 0-1 13 IF (1K.LE.0.04 GO 13 2 3 14 A . DTK" 0 15 0 0-2 0 16
10 TI 1 0 17 2 EP 7 13
S 9/tE)*S 3 19 AK AK*t 7 23
AK 0 21 3 COMEIMUE
3 22 Al ((.5Z4I4S 0 23
ETU11 0 2. 10) 3 25-
OUTPUT 23705711001 )PTI"UM (1007 TESTIS
NUMOER 0 SLIDES = 2 TIP SPEES SATIn = 10.0003 OSFFTRFMTI&L TIP SPEED 95250 FOR SOATEAVAL EVAIULICIN .5000
P303011 TIP LOSS (-13OEL IS USED
WILSOPI 07300) SF LALCUIAIION FOI AXIAL INTEMFEPEMEF IS USES
LOCAL TS SPEED 01120 10.005 WOTSI. OWTERcRfMlIf BeT0.
. 33035937
ANGULAP INTEMFESENCE FACEUP = .00337222 TIP LOSS FACTOP
DISPLACEIEST VELOCITy .33503067 *14601 PIj 3.113
CHORD/SOS. ATTUS - PATIO = .06520031 NUMAFI OF ITEPIAT100S
- 33
LIFT/UPAG (TIO = INFINITY
LOCAL PTWER ClEF. = .156267 lOCAL bRUIT 70FF. = .233910 SOLID-LIFT PPTI .041565 SOLID-DRAG PROD = S
LIFT/0130, PATIO 25.00
L0I*t A53f5 10SF. .5f,*73 teC*t ,4I*51 eefp. --- SOLID-LIFT P903 .541565 SOLID-SPAS PRO) = .001663
LIFT/OSAG PATIO = 00.00 LOCAL OWE' ClEF.
. 110572 LOCAL TOQUSI COFF. = .237390
SOLID-LIFT 5O') = .341565 50110-5505 P501 = .000531
IIFT/IPAG TACIT = 75.03
LOCAL °14E5 ClEF. = .124670 LOCAL THRUST COEF. = .237103 SOLIfO-IIFI P'#' .561565 50100-0556 *00') .300556
LIFT/ISUI 70110
LOCAL PTh1 I CUTE. 1375q LOCAL TWOUST ClEF. .237134 17113-LIFT /953 .141545 SOLID-OPUS P570 .000416
LOCAL TI SPElT OTtO = 0,5)3
AXIAL 1STAFEF4CE P7')0 .
31025337 ANGOLA" TOTE-IF- 5*fIr VCTO .10206551
TIP LOSS FTETC' = .'-'tJ4
OTSALOCE 1101 V0LICTY ANGL 941 1.573
CHOA"/"I ,
1AjfY-. OTTO = .05002530
O 'TT2TI'lS - I
-J
L1T/l3S 0171J i015ITYLOCOL 0 175-fl 3. . 151031 LOCAL TOOSOT y'E.SILOT-LIFT P0S = 3'575 TTLID-'l°AG p00° 3
LIFT730S '0110 25.00L1COL 03550 CO-. = .'040I LOCAL 100000 10E0. =
0=LI0-LIFT 103 = .03057'- STLIO-0000 00011 = .O0170
LIFT/1O -i)IIC 50.05I TL)Wf'''1''-. . 051)- LOCAL T0'UST rOLE. .211.551
SPLIT-LI00 "FO' = .03947)- SSLIT..CPAO 0PS .003700
LIFT/S-AS 403p = 7.L 'CAL )Wi P C3. .11 3051. LOCAL 100251 11FF. = .213053
SPLIT-LIFT PSiOP .000.77 SOLIO-TRAG P0O = .000526
LIlT/1000 00011)LTCCL ')7 C1)F. .120724 LOSIIL P07551 (STEF. .213010
S OL :0-LOFT 033 = 5 3i75 STL I A-TRAIl 050)5 = . IOU P05
LOCAL 100 50fl 0TIr
AXIAL 0lF017FPS0 00C11' = .33015337ANOULA IJT!F F '5)5' F 01110 .00215355TIP LOSS FAS1C .4597'OISOLAIC0FT 5_L0IT .35017730AWF.L 701 = 4.19704001/001. 00V - 03111 .03255171N0007 0 T1 IT 00SI5_ 3
LIFT/SOS G011tI0 I5IVL1)1)L 0 )-)I1f .1''-P'F LOCAL 10U10 'SF7. = .i9570
03'Al? SOLIP-0055 FAT) 0
LIFT"' 01)3-_ 10.01SPOiL 'OW' ') . -.rr7'1 I LOCAL 040)100 CTF. = .102515
LI-SIr' 0 .1T').33 SSLJO-°050 FOOl .101406
LI0T/il -All) 50.33
LIIOL 'J1 '1). .351)351 LOCAL 090051 lIE0. j03373SIILIO-LI7T 33 .03751' SILlS-TOO; 00'' = .I0I'5
LIFT/O 1)0111 75.1:1155: 1'- 11). .103552 LOCAL T350 1SF. .10205051: JO A0b030i.j' SOLIO-TOAS 00)15 = .000400
LIFI/5 1 OO 1.-1) i. .1' 0314 LOCAL 140(151 1010. .192013
.1: S0LI3-'S 0050
LOCAL 050 505) O;fl) p.5530010L IHTESFF*FMCF FACTO' = . 33005037ANGULA0 090EOFFR7N1 00510)0 = . 0031416).TIP LOIS FACT0O =
OISPLACEAEOI VFLOCITV = .150395,50*$sGtP '40 5.537CHORD/MAO. 050045 - RATIO = .04720272NUMOER OF 117010111')S - 0
LOFT/OPAG 001100 TOFI)110LOCAL POWET CC'F. = .112003 LOCAL THRUST ClEF. 0 .17l1SOLOS-LIFT 'ES) 0 .035353 SOLID-ORAl, PROS = 0
LIETIORAS Ill FIG = 15.50LOCAL 0)41)-P "OE. .951.700 LOCAL 11409150 COF.-0 .171717'JOLlA-LIFT 000) = .035353 SOLIO-0500 PROD = .0011.I(.
LIFT/0000 PATIO 53.03LOCAL P1500 1910. = .003002 LOCOL THRUST COPE. = .I7ISRASOLOS-LOFT FOI = .305350 SOLID-OPAl, PROS 0 .010707
LOFT/bAIl PATIO 75.00LOCAL P)WCP 110F. = .103502 LOCAL THRUST COPE. = 37j353SIlL AS-LIFT 70003 = .035350 SOL!')-0WA 0*90
LOFT/SPAS 00117 = 100.00LOCAL POWLO C370 = .309352 LOCAL THRUST CAlF. 0 .17131°SOLOS-LIFT 0131 = .035353 SOLID-DRAG PROS .00035'.
LOCIL 11° SPEFT PATIO = 0.0000000L OPTP0ERFl05E O00000 . 33035037ANGULAS IRTEIFF050I:F FACTO' .00353077TIP LOSS FACITF .00965DISPLACE lEST 0000LAT = .35065750*0401)- P141
CHORO/OA(. RTIUS -OA00 . 05105333NUWBFR 00 IT:-AATIOIS - 0
LIFT/AA-; PATIO = I5'I5IOLOCAL P07-P 15F. .100011 LOCAL THRUST CTEF. = .151601SOLTP-L0 7051 .337233 SOLOS-SPOT 0901
LIFTI0005 51111 = 15.00LOCAL 0))-4 73)- . TSA,7P LOCAL 0(40000 65F0. 0 .A52191SLLIS-L170 0=-lI = .00320' TIILIO-10DC 057= .TTI'32
LOFT/b-AS AITC 50.01
LOCAL P1450 1''. .515 P01 LOCAL 0491(01 OSFO. = .151051SOLIS-LIF' ')' = .0102'' '1L135005 0010 = .100)-56
LIFT/T 1 '001 71.L'S IL 0 -7, - ". . 031- 01 LOCAL 0051)55 17FF. 0.15105,S'LIO-LIFl 0133 = .1379' STL7'-11715 0001 OST5.i.
I
-J
LIFT/D'A0 PATIO 100.00LICCIL P1508 COP. = .037575 LOCAL TOPUST COFF. .001PIE
01.11-LIFT 'Jl = O3?9M SSL100-T'AO .000333
LOCAL TIP SFO POTtS 7.500AXIAL TSTF :F3C' FACT1.OAANGIJLAT CSTE-TF[0000 FIT?' TX3q27'TIP LOOT TIC AC'.OISPLUF1F1T VFL30108 .0503'177ANAL! IIT - 20
CH?P1/A. PASTAS - PATIO .03680993009FF? OP I1'°IAST0AS
IIFT/1'0A 001I IrIPITOLOCAL '0503 )'. = . 0P700 LOCAL TOPUOT COOP. = .133353SPLASLTF P5 = .031255 SOLTD-TPIS FACT =
1IT/SPA5 'AlIT 25.00LSC3L )0l'C IF'. .O5786 LOCAL TI40lST 'SEF. .1335lSOLID-LIFT O5l = .031250 S.SLIO-rI'lS TRAIT = .001250
LIrT/O501, PATIOLOCAL PSIOFA COEF. = .067003 LOCAL 1(1DT TOE'. .133617SOLOS-LIFT PC 5.T31'5S SOLIO-JTAO P05 = .000F25
LIFT'OIS ATIO. 75.uuL501L 211 CI'. . T75o62 LOCAL TOPUOT 70FF. = .131519SOLOS-LIFT P-i) .03175' SITLIO-OA5 PASO .0517
LIFT,S'A PATIO 100.03LOCOL '1103 'OF. 777P07 LOCAL TOSUST O5E. = .133500SOLID-LOFT OTTO .T31055 SOLIOO-O?A5 FOES .OTOI1?
LOCAL TIP OPF TTIE 7.00)AXIAL TNTE''FE1ST FACTOP 03535037ANALILA P IpTE2' P055! '0CTO = .00556033TIP LOS SCAC5
DISPLACEACT VFLICITY .351355'9ANCLoT 'AT '.375CHO/T. APSIS - AITI? = .03210070NU'"0.8 II TF'RIATIOS - 1
LIFTS C, 'arT '15105L'CIL 0 50 )r. .T717T LOCAL T8AOI COlE'. .116253
"LI)-LTFO 25' = .00I50 SSLT0OIS TACO = 0
LIFTA-0S SIX 35.35LSr1L010I'''' . .O.'1 LOCAL 1U2T OAF. =
-LI0' ''A = .0195 SOLI0-SPAO TOOT .031160
LIFT/SPAS 60010 =LCAL P1AR CT0'. . 561290SOLII-LIFT P1O = .020105
LIFT/DOlT' OTILO 75.00LOCAL P06)0 CO'F. .065721SOLID-LOFT P60) = .025195
LIFT/DPAS PATIO 100.00LOCAL 210(5 COCF. = .53537
TO-I TFT PROS = .025195
LOCAL 108000 70FF. = .1165SOLID-TRIG FOOT = .000505
LOCAL TRU0T 00FF. = .110503SOLIO-ORAA FOOT .080909
LOCAL THRUST CUFF. 0 .116360SOLAIO-OPAS TROT = .000292
LOCAL TIP TPEFO PATIO 5.500AXIAL 190 F9IIF AeToO . 33035R0'ANGULA IOITESFE800ICF FACTOR = .01507910TIP LOSS FACTO' = .99696OISPLAC.E'IERT VELOCITY = .35102052ANNLF 041 = 5.'MTCHOPS/MAX. RADIUS - PATIO = .02771053NUARR 36 ITEPIXTI035 - C
LIFT/SPAS 01T1)' = ITIFIFTTSLOCAL P151? CTOFF. .066023SPLIT-LIFT PADS = .027150
LIFT/SPAS 00115 = 25.03LOCAL P}WFP FOOT. = .339551SOLID-LIFT °'O1 = .02711.9
LIFT/OPAl 60010 5.13LOCAL AIWLO COUP. = . 052502COLTS-LIFT 601 .327169
LIFT/0AG 60115 75.0)LOCAL P06P CO'F. . 157303SOLTO-LIFO P))) = .027153
LIFT/TRIG 00107 135.13LOCAL P11A 015'. = .053513TOLAO-LIFT P0)0 .017158
LOCAL THAUST 05FF. = .100313SOLID-APIS FOOD 0
LOCAL OlIOSOPT 05FF. .100720SOLIO-0005 FOOT 0 001006
LOCAL THRUST ClEF. = .10A511SOLIO-DPAG FADS = .000563
LOCAL T800TT 73FF. = .1X056CSOILOO-OPAS PRO) .01002
LOCAL THP003 COOP. = .1111.105TL1-'RO5 PTT = .101771
LOCILII0TTEFC'7T10 0.33)AXIAL TITP0FPOP 'A' = .333533'ANC,ULX PIATE-'1F'P'' FICTO .20610500TOP LOSS FSC7TISPLI'F.'10ST 'TL'P TOO
ANAL' API
CHO01/'Sp. ?0C -60115 =NUTS 'I' CT ''T' - 0
-J(.5)
LIT'fl'lO '0IL = IlFtATY
L°CL L' E''. =COLIO-LIFO =o1 = .0'513.
LIT,A ATIO = 75.0
p iwoo
2LI'-LTLT - .02510.
LIFI/O'A5 '101L =
L0CL =W'''- 1V. .0450Sc)L1n-LT'0 'LYO
LIFT0A:, C1Jr '5.00
L'C\L 0'. = .
SOLES-LIF' .029105
LI1/P'O 'ATC 100.70
LOCIL F )I.5O ]r. . J51523
SOLI1L I'T -O' .
LOCAL 0'U1 f2F'. = .OIEESI
0OLIO-0R00 CArlo = I
LOCAL 140950 op0 .055520
SOLID-COAl 00Cr) .001004
OAL TH0UT 700FF. = 15573C
SOLID-7°0C O5 = . CO2502
LOCAL T0UO COF. = .0'ScCLIO-'AO C0n = .000'35
LOCAL 1990)71 COFO. o .055060)
SOLID-I'll FAA') = .010251
LOC4L01rS0O Ar!) 0.510
AXIAL 05lTC0L'f)Cr '1'C = .33015931
ANOULAO It 0- FOESO 000107 . 00730.21,
TIP L210 '151')'
015PLOCL'11')T '/L5L0O
ANGLE POT 0.510
COO/')'. )o- Irlr .o1qqo,o,
OF Il_oIl OjI')-. I
LIF1/S'C) ArT"'rJEI'OTVL000L 'R' :,. .I)472fl
LJFT/0' /010 - ?'.flO
LJCIL '''0'' E/'O
"OIL -LIFO "- .22005'.
LXFT/D1 "0,00 500.02
LOCaL A 0''' 0. 119106
"0'), )0-L'. 0100.','' [40,
LIFT/ ''0 'I 0)"
0"), 10-LI') '"1 .'1'3O5'.
LIFT/'=" '0'" '0.01
00 IL ' ''. ..0112
0' 1"- L'"'.'I' 10)'.
LOCAL 190050 C0'FF. = .071970
!OLIO-"O'DI osor' = I
LOCAL 19010'T 70FF. .07251'.
0SLrD-SA') 000") 000023
LOCAL TOO '00007 '21)00 5.100
AXIAL IN'E'II)r" 0ANGULO' T5T0F'05O FAC100 = .00507747
TIP LOSS FACTO')' = .09971
OISPLACEMF'0T vrLOr:1Tv
ANGLE '41 7.'-.??
CHORO/'A9. 00'1101'0 - 90110 .01651674
NUllS'S OF 1l''I40TO'lS - I
LIFT/I'll, '1100 I'JCISITY
L'CAL 0)900 706'. = . 03906?
SOL 10-LIFT FOOT .021230
LIFI/TPAO RATIO = 25.00
LOCAL 'OAF' 7')FF. = .027153
STLIO')-LIFT "'1) .021030
LIFI/0001 RATIO = 50.00
L000L FlOES 00FF. = .003110
STLIO-Lr'T 9O') = .021030
LIFT/OQAG 00110 = 75.00
LOCAL P1070 F'O'0F. = .0350%SULlO-L100 P00) .025530
LIFTr3001 PATIO = 100.00
LOCAL 70619 CJ'F. = .036090
SCLIO-LIFT 0A'') = .021030
LOCAL THRUST 70FF. .059570
IGLOD-ORAG FOOT = 0
LOCAL Tt-FSUT COPE. .059552
SOLTO-SQAG PRIG .00050.1
LOCAL TH'UOT COP'. = .059775
SOLIO-OPAG "00 .000621
LOCAL IOPUSI roEr. = .0506".
SOLXO-OPAG FROG 000215
LOCAL 105UIT ClEF. = .05960.5
SOLIG-IPAG P'flfl = .000210
LOCAL TOP SPEED A010 4.911
AXIAL 00005"FOOf '9700" . 33A3537
A001JLOT 1NTE-0FF0r$CE FAC170 = .01093655
TIP LOCS FACICO = .09907
OIDPLICFMFNT VELCCIr'Y .04551905
ANGLE COO = 0.715
CH0P1/'A,. 0100)0 - ATTO = .01343100
NURSE' 00 T1'0'IOTIONS - 0
LIFT/S''0 00100 = INFINITY
LOCAL T0)i°r' "OFF. = .072162LTCIL 'OW)' )'P'F. .031694 LOCAL 1000)51 roE'. .045350
10L)0-LIFO P0-jO .010007 SOLID-ORAl P500COLTO-00C 0-O'Er .02101
L1FT/''A 05007 = 25.77
LICAL'''U' 0 7r7 = .0721LOCAL 0)0000',)-'. = .922991 LOCAL 155901 COP'. .095631
OOLIOL001 00/010 .=l:0? COLIO-0001, = .0763SOLOI)1005 P'O" .Orq ACA
LIV)/T0/Al 0'ITI 0
LOt' AL TH000ST r00. .,72L'/L 050 L'O''""' r. = . l'7'93 LOC IL 1000)00 COP". = .045401
S"L0'0- )F'0')" = .710002 OOLTr'-T'0'0 00)0) = .000150SSLTO-"A0 00000. 070)31
LIFT/SI)') 'IT: - 70')O'5= L00")L T00F)TC OF = .741464
""1 .O100I' SOLTO-"'A') 00'
LrT,TT; -TUL'CAL = .UTq.Fo LOCAL T0'UST SPEC. .055571ErLt-L)F1 OpT .012007 TOLl flSS EAST .000190
LOCAL TIO C -4TTT = 4.00005105 TTFSPF rc5o 33595937
0NLJL), JT)) 0CTO .Y)3511.TIP LPS FP07-TITPLAC' FL 2. LCCTV . 14131571AfASL 050
CP01/11. \T 301 -E0TT = .JA0?1TO TT00TTfl' - 0
L0FTf02 Eon (nE1 JA3
LUCOL )5r. 50FSSLTT-L000 0007
LTFT,TEO;EArIo=LOCAL olos5'L T7-L 3F
LI0T/flAO RATIOLOCEL )4
1nLI;LIrT 0400
LIET/Y)A1 EAT)1LTTL )4 C
5=). i0L 340 °'OY'
LOS 1 TI)
I (IlL I
L0(-LIr0 =1
175003.115
flt573.fllEAS.
.0)= .
75.0
)5 F. 491.
0731.70
0 I 119
LOCAL T'U0T 1n.50 .03030030LIT-TTC, O = 0
LIT'OAI_ TAPUOT 011FF. .036550SOLOO-El1.00, FOPS =
L000L TSOU5T 50FF. = .130491STLIS_lflF.A5 PET' = .000050
OTCOL I EuPT'11Fr.SOLOT-IQAS EQS) = 000721
LOCAL T4OIIST FTC). = .035373STLTO-TRII FOIl) .000170
LCC1L' (TTr 3SflflAXTIL prf .( AC
ANC/JLAEEFI( F nfl: 01753?qTIC L(O COOT).
TISPLIE 1 L )fT0 .3500'933ASS). 1111 1- 0). '0
I- -
LI 1,0(1'' r'i- o
OIL . 0. :.(IqI,3 L007L OOUFAFF. .02055700LOT-''A1. FAST 0
LIFT) 00 (rI =
.'T0'r LOCOL oFTr-r. .02'T1.)? OPLT'-'AI F,(7
LOFT/000 PAnE 00.30LPTIL 'TWO ((01E .00701
LOFT' 01 r0fjr 7539LOCAL P0450 CUFF. . 017769SOLOS-LIPS FO" = .00.974
LIFT/TAFT 01135 = 015.53LOCAL P)WE COEF. . 019112SULI]LOFT P'(Y) = .011.279
L000L TEI11I)TT'O))F. =
FOPS .000300
LOCAL 100)353 50FF. = .029535SI)L0O-0000 E5'Ifl = 900200
LOCAL THRUST COFF. = .02950ETILIO-ORAG FOOD = .000150
LOCAL IjO SECT 0031 = 3.090ROYAL YHTFQFEQ CF FOCTOP . 0393T050ULA A INTT" E1C P05120 = .0.210501TOP LOSS FACTO' = .22999OISPL4CCSC'OT 2FLTCtT = .3540'.63200061 0+40 = 12.151CHOR'3/FAx. 000005 - 00100 = .0061327).NUIOISFA OF ITCIEA0TIO9S 1
L0F1/O"A5 "ACID ESFTI,ITYL1'CAL O)WFO FIF .614554SPLID-LIFT PY) = .012093
LOFT/APSE, 'ATOp 25.00I0510L 00010 55FF. . 010950SOLID-LIFT P100 = .1393
LIFI/000 P2100= 50.00lOCAL 0),400 F5'F. .002709SCLI0-LOFT 0)00 = .01)1190
LOFT/IPAT "0350 75.00LOCAL 2)0)0)5' F. .01312205115-LOFT 04-))
LIFT/SPAS 00100 0.LOCAL 2WF 9 COF. . 013410TELIP-L1F' "''('1 . 11t230'
LOCAL THISUST COEF. .021773SOLIO-l1010 EAOO
LOCAL 50400351 CUES. = .020566SOLID-OPAS POlO = .000500
LOCAL IPPUOT COFF. .021066SSLTO-0000 F0Or = .000260
LOCOL 399UST ClEF. .021035STLOO-2005 FOCI'S = .000073
LOCAL T9'UST 5OFF = .021016
STLIO-1000 FElT = .000190
LOCAL TI? 00F' :,r(-. = 7.5310510L S11T)060t00r'S,, .31995037APIC.ULAP T5EFOO E(515 .03452101TIP LOT; '=-r=
TISPLAE,1'JT LflO)rO .
ANr,L "U =33,3397'
NIJ002 oj' 00' --4Ui
L00T10'A 00010 P101111!L°CSL 'lI.-O (1°. .109167S'LTl_LAFO °'1I .011035
LIcT)01 'AT! 25.00LOCAL 0:0rI7csL1°-L:T 0011 .011036
I100/0"flS '0101: 5
LrCSL P 1610 71F. 559003SlIPS-LIFT 01.55 = .011035
LIFT/lIFTS =475. 76.00LOCAL 16)5 C1F. =SOLSIS-LIFT -0fJ = fluAfl3
LIFT/004G '0111 = :. :c
LICIL )".L' 71:-F. = .019'osSILPI-LIFT "P51 = .011036
LOCAL THPUTT COFF. .015272SOLID_DAIS FOflfl =
LOCAL 549)001 70FF. .ot5'oSOLID-SOlO 050fl = .000641
LOCIL "45001 CUE0. = .015351SOLIF1-IPSAS (1 .000221
LOCAL TIOOUGT ClIFF. = .01532600110-0005 6fID .000161
LOCAL T4'IJTT COEF. = .015301SOLID-OSlO FOOD = .000310
LOCAL ITO S0fl SATTI = 2.100AXIAL 55507400 c5C300 33015037ANCUL00 pIT!SrIRolS5 FOCTFT = .05316377TIP LOSS FASICS015PLICE11INT VL11rITo .07176713ANGIF 0441 =CH091/5A9. S111JS - FACTO .017505NIJHOEP OF 51T 11 -
LIFT/C: Al Ot' -
LOCAL 004 o 1:-F. .los';l
SSLTI-L1' 0-)-1 .50012,
L0°T/?OS AT
SOLI1L1I - 1 .011117'
LIFT/"A ; O0TC - 5.-.151L 0 ). = .1050535'). 1 l-L 1''' 11 . 15017
LIT,?--4; -
SCLl"-L'' PS-)) .0120LIFTF:s; or!:-
LICAL 0l.? I
S'L 11-Lll I II
L1CAL 14°1J0 70FF. .0095401ILT00FT5 005" 0
LOCAL 1451)0' 'oFc. 0 .010076S1LII--1f1O1 0071 .000165
LOCAL TH'UII 7SF0. .00001200110 .000193
LOCAL 090001 COPS. .0054310on .1(0122
LOCAL T'IAI.IIT 70FF. .009991TILIO-?°'S 0005 .6191
LOCAL TOo 00167 OCT10 1.530AXIAL 040 0005 MOE 'ACT"! 03935937NGULA0' tSTE-'0E574.cE FlOOD' .0°109324
TIP LOST Ari5s 1.00000DISPLACE'IF9T V5LXIIV .P1364911ANRIA 0143 22.900CN090/N4'(. 90100)0 - RATIO 001.71946NUAAEO 00 TT1101TI141 - 0
L105/IPSAG 04017 I'lFIsIIvLOCAL 03403 7000. .C351LSOLIA-LIFT P01 .00729°
L100/soal oo7 's.
LOCAL 0'}WE5 C0ff. . 063169SOLID-LIFT 000) .007299
1101/0041 RATIO = 50.06bOIL 0341-3 COFF. = .003342SOLID-LIFT 0010 .037299
110113006 5010(1 75.00LOCAL 'OAFS CUFF. = .003400SOLtU-LAFT 0015)
LIFT/OPAl 50110 100.010LOCh PlIES 1O'F. 30342°SOLIO-LIFT 0001 = .007294
LOCAL 104901T 70FF. .00579°00110-0507, 0900 0
LOCAl SW°t25V C000.SOLID-OFIS F9OO = .000292
LOCAL THRUST COEF. = .005060SOLID-DRAG 0500 = .000146
LOCAL THRUST C000. = .00503090610-0060 0000 - .000947
LOCAL THRUST COEF. = .005022SOLID-ORAl 0901) = .000073
LOCAL TIP SPLES '1710 = 1.000Aol At T9TFSFFOFSCF 047015 = .03939401ANGULA ° INTEOFEOISCF F OCT09 .10 335995TIP LOFT 000TOS = 1.00000DISpLArEIEOIT VFLCCIT' .61.32493164461' 0441
CH051/016. 44011)0 - 50110 = .0'l00031240)4900 07 ITE'TS101'tS - I
LIFT/lIFTS '01)0 =LOCAL A)40 C3rF. .0(1563SOLTO-LIFT 051)1 = .305622
LIFT/lIFTS 110111L"OAL 0341-0 7(1°F. .991450SOLIO-LOT 05111 .5060''
LIFT/Ol 0; 05;, c-I. (1LCCIL 0)A 0 = .0015170'L11-'.'' O) .flf 2?
LIFTP1'A.. 01T( 75.50
S"L1I-LA1-1 o' .00' 1?
LOCAL 1400)00 71SF. = .102907II4LIO-000G FOOT 0
LOCAL 'OOUT 7050. .00296°0OLIO-0O coon = .010725
LOCAL bOOST 0110 F = .00293°SSLOJ_lSOr, 0071 .1:4112
LOCAL T90I!0171FO. 0
SOLID-" AS 000' = .00007; Cc
110. A
L0C5[ .1000 LOCAL 10UST C)EF. .0020'25'L T'L r-o = . noc' 'o COLIC-'WAO P020 = .000 qOc
LOCAL ItO C0LEO) 70T!0 .501AXIAL IS OFO0CCF ACT000 .035337AoruLo= T0Ci FTC' = .655123TIE' L200 702100 = 1.ofl7012102Lorr42r V L'ITV = .57179ANf5L 'H1 .7.100ClO''O. 021)0 - 20107 .00033797A02 no 1122 111000OC - 0
LI°T/CA, A115 = 13110100LCOO'OL '3W 2 (2F. = .5.0 0'lSOL10LlrT '623 .006230
LIFTflA5 AIr = 70.01L2C4t °396 2 F2. = . bA 372SOLO '-Ll°l P'53 = .0.730
LI°1/O1-A5 =0010 0.LOL °W ' Cli'. . 000301S'L13-L1Fl 0)00
LIFT/li 03 =ATO = 70.20LOCOL ]-' C'°. .O0015001_1')-LlFI o=nn . 25.20"
L1'T/'PAS 'SII 100.02LCOL 0 0)02 C,1F
020 .200.70"
L0100L T1RIJliT 0)FF. .
ILTO-I001 1020 0
LOCAL ))0T 0366. 0 .900056SCLIB-D°A5 r'3 = .000270
LOCAL 1330)2T )OEP. .000"61SOLID-TAOS FOOT = .000065
LOCAL T9'UT 17F. .000935SOLIOO°0G P0000
LO2AL 900UST 01)67. .000937SLIO-fl°AZ P05' = .060062
LI'T'i AS TTFr'PT000L ilo P CF °227 .0201)1/TOp ç)°2 OF 0020LCI 7-07
LI0T/''OC, 0T0r 25.0r060L COb OF 067' = .20371010 L 0))'1o000 0
LIiAS =0''
00031 .011'0T'YIL ç 'po-0
Ll'000-',T] 70.,1(11)3 0;; r,=)t;=013 0 0,0 0)P'000 LiO .71';.
LIFI/0°A ATTrTr'OAL 2166 26 .0616TCT1L 20FF 0)0 090531 = .7931
-J-J
178
APPENDIX C
General Momentum Theory
With simple axial momentum theory, the assumption of an
irrotational wake was assumed. In general, the wake will have a
rotational motion imparted to it by the wind turbine. This rotational
motion gives rise to an energy loss. To modify the momentum theory
to include the effects of rotation, it will be assumed that the wind
turbine can impart a rotational motion to the fluid velocity without
affecting the axial and radial components.
Defining
r radial distance in plane of rotor
radial distance in rotor wake from rotor axis
angular velocity of fluid behind rotor at radius r
angular velocity at radius r1 in wake
u axial velocity at radius r in rotor plane
u1 axial velocity at radius r1 in rotor wake
v iadial velocity at radius r in rotor plane
A rotor swept area
rotor wake area
p1 pressure in final wake
p pressure in front of rotor
tp pressure increased behind rotor
179
From continuity of flow, considering an angular element (see Figure
C. 1):
or
pu1r1dr1 = purdr
u1r1dr1 = urdr (C. 1)
Using conservation of angular momentum:
2 21r1 or (C.2)
Now by applying Bernoulli's equation ahead of and after the turbine:
Ahead of turbine
22H p + ! pV2 P + p(u +v ) (C. 3)0 2
After the turbine
2 2 22H1 = p + p + p(u1+v +w r ) (C. 4)
or1 2 22H1= p1 +-p(u1+1r1)
Now
For total pressure
(C. 5)
H1 H0 = p + p2 2r (C.6)
Va,U
Rotor
Figure C. 1. Stream tube geometry.
Wake
181
1 2 2 1 22p00 p1 =p(u1-V00) +po1r1 - (H1-H0)
1 2 2 1 22 22= p(u-V00) + r ) Lp (C. 7)
The relative angular velocity increases from to + and
therefore the pressure decrease is:
Lp =
= p(2+ )r2
Now combining Equations (C. 2) and (C. 7):
1 2 2 1 22 2P -P1 p(u1-V00) +p(1r1) + pkr
()1 -V) + p(+p00 p1 = p(u
The centrifugal force exerted on the fluid is balanced by the pressure
dp1 2=
Now by differentiating (C. 9) with respect to r1 and substituting
2 1 d (uV)+ d 2-pw1r1 =p p {(+1)1r1}
or
or
21 21 d 2 2 2 r2 + 21r1 + r - + w(V-u1) 1r1 + 1 dr1 1 1 dr1
2dw1
= (c2+o1){r1 j + 21r1}
1 d 2 2 (2+w)- 21 dr1 l'l (C.11)
The e'ement of torque of the turbine is equal to the angular momentum
imparted to it by the corresponding element of the slipstream per unit
time.
dQ pur2(2Trrdr) (C. 12)
The axial momentum for the wind turbine assuming p1 p is
-T = SpuiuiVdAw -pi).dAw (C. 13)
This equation has been accepted in differential form as:
-dT = pu1(u1-V)dA (ppi)dAw (C. 14)
Now from the pressure difference across the rotor using Equation
(C. 8)
183
-dT = pdA
= -p(+)Wr2dA
dT = )r2dA (C. 15)
Now combining Equations (C. 1), (C. 9), (C. 14), and (C. 15)
or
or
)r2(2rdr) = -pu1(u1-V00)(2r1dr1) + p(u-V)(2r1dr1)
2+ p(?+)W1r1(21rr1dr1)
21)r2 = -u(u -v )+1 0° u1 2 1 00
W1 22 w 2 1 2
uu1(u1-V00) u(u-V00) -u1(2+ )r + u(2+T )W1r1
which may be rearranged into the following form:
1 2(2+--) (2+)
2(u1W1r1) (C.16)
The difference ! (H1-H0) is due to the energy departed from the flow
by the thrust and torque (udT dQ) to the mass Zirrupdr
1 22H1 - H0 = Ep + pw r (C. 6)
From Equation (C. 7)
Now
184
1 2 2 1 22 22= p(u-V) + p(wr1- r ) + p1-p (C. 7)
H) 1 2 22 2
p 1 0(u1+1r1-V) +!(p1p)
Multiplying by mass (Zirrupdr)
2 22 2 2{udT- dQ} = Trrup{(u1+w1r1-V) + (p1-p)}dr (C. l6a)
The equations developed may be used to relate thrust, torque
and flow in the wake if one specifies one of the variables such as
f(r) or = f(r1). Knowledge of this variation, is not known,
therefore information about the flow field can only be gained by making
certain approximations.
If we make the approximation that the angular velocity w
imparted to the slipstream is small in comparison to the angular
velocity of the rotor and neglect terms involving !, we obtain
useful simplifications.
Equation (C. 6) becomes:
H1 H0 (C. 17)
the refore
1 22- p(V-u1) = (C. 19)
Combining Equation (C. 17) and (C. 18)
1 22p00 p1 = p(u1-V00)
Now substituting Equation (C. 19)
p00-p1=O or p00=p1
which yields the condition assumed for axial momentum theory.
Using Equation (C. 14)
and letting
by continuity:
-dT = pu1(u1-V00)2rrr1dr1
U1 V(l-2a)u V00(1-a)
-dT = p{V00-2aV00-V00}2irV00(1a)dr
dT 4nprV(1-a)adr (C.20)
yields Equation (1.2.2) of rotational axial momentum theory.
Equation (C. 15) is given by
2neglecting terms
Defining
187
dT = )or2dA
dT = pr22rdr
I-a
dT pr34a'dr (C. 21)
Now equating (C. 20) and (C. 21)
or
2341TprV(l-a)adr = r 4ira'dr
V(1-a)a = 2r2a'
a' = (l-a)a (C. 22)
This equation gives an approximate relationship between a' and a
for small o.. Using torque Equation (C. 12) and substituting the defi-
nition defined above:
dQ = pV(1-a)Zar22rdr
dQ = 4pVQ(1-a)a'r3dr
yields Equation (1.2.3) of rotational axial momentum theory.
::
Substituting Equation (C. 22) into (1. 2.3) above, yields
4pV(l -a)r3dr(1 -a)aVdQ 22r
41TpV,(1 -a)2ardrdQ=
Now differential power
dP=dQdP = 4rpV(l-a)2ardr
dP 2pV(1-a)2a2Trrdr
which integrates for constant a:
3 2P = 2pV A(1-a) a00
which is expression (1. 1.8) in axial momentum theory.
189
APPENDIX D
Glauert's Optimum Wind Turbine Analysis
Glauert's theory for the optimum wind turbine was derived for
the condition of no drag and no tip-loss, a highly idealized situation.
The basic approach is as follows:
Considering the momentum equations for thrust and torque,
(1.2.2) and (1.2.3)
dQ V2(l-a)a' (1.2.2)- 4irr pdr
dT 2(1-a)a (1.2.3)4TrrpVdr co
The energy equation for a wind turbine may be expressed as:
dQ=UdTrelative
where U is the flow velocity through the rotor disc.
Substituting Equations (1.2.2) and (1.2.3) into the above
relation for energy, we obtain
where
(1+a')ax2 (1-a)a (D. 1)
x-
190
Now considering the relation for the power coefficient
wh e r e
or
p 8c (1-a)a'x3dx
p 2 3 2 Z 32
pV,1TR
xv00
For maximum power, the variation is made stationary so that
da'0 = (1-a) + a'(-l)da
da'(1-a) a' (]J.2)da
Now taking the derivation of Equation (D. 1) with respect to a
da' 2 Zda'(1+a') x + a'x = (1-a) ada daor
2 da'x (1+Za') = 1da
Substituting Equation (D. 2)
2 a'x (l+Za') 1 - Za (D.3)(1-a)
Rewriting Equation (D. 1) as
191
2 (l-a)ax = (1+a')a'
And substituting into expression (D.3) gives
or
(l-a)a a?(l+a' )a (1-a) (1+Za) = 1 - Za
l-3aa 4a-1 (D.4)
Now substituting (D. 4) into (D. 1)
or
(1+ (1-3a) (l-3a) 2(4a-1) (4a-l) x (1-a)a
(1-3a)x2 (4a-1)2(1-a)
which if expanded yields a cubic equation.
2 23 2 93x x-1a - 1. 5a +
16 }a + 16 = (D. 5)
Now from Equation (D. 4) we observe that 1/4 < a < 1/3. By
solving Equation (D. 5) for a and (D. 4) for a?, the variation of
a and a? with local tip speed ratio can be determined (see Table
D. 1).
By utilizing numerical or graphical integration, the ideal power
coefficient can be determined (see Table D. 1). At high tip speed
cfLCL l9Z
X a a' 2irV Cp.25 .2796 1.3635 50.64 .3658 .1755.50 .2983 .5428 42.29 .5205 .2894.75 .3099 .2938 35.42 .5552 .3645
1.00 .3170 .1830 30.00 .5359 .41551.25 .3215 .1242 25.77 .4974 .45131.50 .3245 .0894 22.46 .4551 .47721.75 .3265 .0673 19.83 .4151 .49642.00 .3279 .0524 17.71 .3791 .51122.25 .3289 .0419 15.98 .3476 .52272.50 .3297 .0342 14.53 .3200 .53192.75 .3303 .0284 13.32 .2960 .53933.00 .3307 .0240 12.29 .2750 .54543.25 .3311 .0205 11.40 .2566 .55053.50 .3314 .0178 10.63 .2403 .55473.75 .3316 .0155 9.95 .2258 .55844.00 .3318 .0137 9.36 .2129 .56154.25 .3320 .0121 8.83 .2013 .56424.50 .3321 .0108 8.35 .1989 .56654.75 .3323 .0097 7.93 .1815 .56865.00 .3324 .0088 7.54 .1729 .57045.25 .3325 .0080 7.19 .1651 .57205.50 .3325 .0073 6.87 .1580 .57345.75 .3326 .0067 6.58 .1514 .57476.00 .3327 .0061 6.31 .1453 .57596.25 .3327 .0057 6.06 .1397 .57696.50 .3328 .0052 5.83 .1345 .57786.75 .3328 .0048 5.62 .1297 .57877.00 .3328 .0045 5.42 .1252 .57957.25 .3329 .0042 5.24 .1210 .58027.50 .3329 .0039 5.06 .1171 .58087.75 .3329 .0037 4.90 .1134 .58158.00 .3330 .0035 4.75 .1099 .58208.25 .3330 .0033 4.61 .1066 .58258.50 .3330 .0031 4.47 .1036 .58308.75 .3330 .0029 4.35 .1007 .58349.00 .3330 .0027 4.23 .0979 .58389.25 .3330 .0026 'i.11 .0953 .58429.50 .3331 .0025 4.01 .0928 .58469.75 .3331 .0023 3.90 .0905 .5849
10.00 .3331 .0022 3.81 .0883 .585210.25 .3331 .0021 3.71 .0861 .585510.50 .3331 .0020 3.63 .0841 .585810.75 .3331 .cC1r 2.54 .0822 .5861
.3331 .0'1 3.46 .0323 .586311.25 .3331 .0018 3.39 .0786 .586511.50 .3331 .0017 3.31 .0769 .586811.75 .3332 .0016 3.24 .0753 .587012.00 .3332 .0015 3.18 .0737 .587212.25 .3332 .0015 3.11 .0722 .587312.50 .3332 .0014 3.05 .0708 .5875
Table D. 1. Flow conditions and blade parameters for optimumactuator disk.
193
ratios the power coefficient approaches ideal (16/27) with low values
of wake rotation, while at low tip speed ratios there is considerable
energy loss due to wake rotation.
If the aerodynamic forces on a blade element are considered,
in light of the previous results, the configuration of the rotor may be
determined. Ignoring profile drag and coning, expressions (1.3. 1)
and (1.3.2) of blade element theory becomes
whe r e
= i BcpW2C cos (D. 6)dr 2
- = BcrdQ 1 PWCL sin 4 (D. 7)dr 2
tan {( !} (D.8)(l+at) x
as shown in Figure 1.3.2. Now by equating expressions (D. 6) and
(ID. 7) given above with that determined previously from momentum
consideration, (1.2.2) and (1. 2.3), we obtain
where
a oCLcos1-a .28 sin
rCa Ll+a' 8 cos 4:i
Bc0 =
lTr
orcrC cos cb
8 sin24) TCLCOS 4)
aCL COS 8 sin4) + CL COS 4)1+ -8 sin'4)
8cos4) cTCLa
TCL 8 cos1 TCL
8 cos 4)
by using relationships of a and a' - -Equation (D. 4), we obtain
or
crCL 8(l-cos 4)) (D.
X(TC8 sin 4)(2 cos Ct)- 1)
L l+Zcos4)
194
This relation does not determine the configuration uniquely, but if the
lift coefficient is specified. the chord and blade angle may be
determined.
The basic procedure to be followed in utilizing this method along
the blade is
1. Solve Equation (D. 5) for a
2. Solve Equation (D. 4) for a'
3. Calculate 4) from Equation (D. 8)
4. Calculate crCL from Equation (D. 9)
5. Now having determined cTCL and 4), the twist arid chord,
195
once the variance of CL with r or angle of attack with
r is specified, is
2rrroc 0B twist
Table D. 1 shows the numerical results of utilizing such a procedure.
196
APPENDIX E
Theodorsen- Le rbs App roach Applied to Wind Turbine Optimization
Introduction
As mentioned in Chapter III, the optimum propeller was
investigated by A. Betz. He was able to show that, for maximum
tractive power or thrust, the wake moves rearward as a rigid
helical surface. This required a constant wake displacement
velocity, the flow velocity of the helical spiral for optimum propeller
performance. For the wind turbine case, the requirement for opti-
mum performance is the condition of minimum induced tangential
velocity, which requires the wake displacement velocity to vary. For
high tip speed ratios, however, the displacement velocity near the tip
is fairly constant, but increases as the axis is approached. This lack
of a constant displacement velocity means that the wake vortex sheets
will change shape with distance from the turbine.
Theodorsen (63), Lerbs (36), and Goldstein (23) have used the
assumption of constant wake displacement velocity in their develop-
ments for determining optimum propeller geometry and performance.
The following is the application of such an approach to the wind turbine
case. Only brief details of the development are presented to illustrate
the general approach and show the nonvalidity for wind turbine applica-
tion. Additional details of this approach can be obtained from
197
Crigler (9), who redefined Theodorsen's approach for practical
application to propeller design, as well as from Theodorsen (59, 60, 61,
62,63) and Lerbs (35,36).
Flow Geometry
Figure F. 1 shows the conventional velocity configuration for
wind turbine theory. The symbol V denotes the velocity of the
wind far ahead of the wind turbine; rc2 is the tangential velocity
with respect to the air at rest at a given blade station. The vector
V denotes the angular interference velocity r?a', while the
vector Va denotes the axial interference velocity V,a. The vector
V. indicates the total induced velocity of the air with respect to
ground. As in conventional aerodynamic theory, the sectional lift
force will be perpendicular and the sectional drag force parallel to the
relative velocity, W.
In conventional vortex theory, it is required to determine the
location of point D of Figure F. 1 in order to describe the relative
velocity's magnitude and direction (angle 4). The results of which
are that the angle is given by
VVa V(1-a)tanV r(l+a')
and
n
Figure F. 1. Velocity diagram for a wind turbine.
x
199
VoaVa V(1-a)IwH sin sin
where the interference velocities V and V undergo locala rtip-loss corrections. Theodorsen's approach sets the displacement
velocity w constant with radius. Once this parameter is
determined, the point D can be located by proceeding from point B a
distance w in the direction of V to point E and then down the
velocity vector W a distance ED. The important geometric rela-
tions from Figure F. 1 are:
ED
ES ED sin fw sin2c
Vr ED cos = w sin cos
V1 2 1 2
a 2w sin 4 w cos
V wcosi 2
Now from Figure F. 1
V-'wtan4 (F.1)r?
and the magnitude of the resultant velocity is
or
zoo
WI J(r+w sin 4) cos sin
wi {v 1 cos4)) (F.Z)Sin 4) 00 2
Now since w is constant or all other quantities depend on w
or are functions of w, we can therefore determine an optimum load
coefficient (product of chord and sectional lift coefficient) at each
radius of the rotor blade.
In Goldstein's treatment of the propeller, he confined the
problem to a lightly loaded propeller. The slope of the wake spiral
is therefore constant and given by the tangent of the tip vortex angle.
V00
Xg =
In this application, a wind turbine is considered to be heavily
loaded and wake expansion is large. Therefore the tangent of the tip
vortex angle is given by:
V00-wR2
where w is the rearward displacement velocity of the vortex sheet
infinitely far behind the rotor and R0 is the infinite wake radius.
201
Now the circulation r(x) can be referred to as the potential
difference across the helix surface at a given radius r, therefore
we define a function K(x) as:
BF2K(x) = 2rr(V -w)w00
where B is the number of blades or helix surfaces. Note that this
is not the Goldstein function, since Goldstein assumed the displace-
ment velocity was small in comparison to the velocity, or
K(x) = ZrrV w
Now the required ideal circulation r(x) may be expressed as
ZTr(V -.w)wr(x) B2 K(x) (F. 3)
To determine the load coefficient, we equate the force on a
vortex element and on an element of a lifting surface as
L = pFW = f PW2CLc
where c is the element chord. Therefore
F WCLc
202
or substituting for W, (F. 2)
F CLc{.' (V- w cos24))} (F.4)
Now equating (F. 4) and (F. 3) and solving for CCL yields
2 sin 4)CCL 1 2(Vwcos 4))
2ir(V -w)wK(x)B2
w BcNow defining W = and o = local solidity = Zirr
2 sin 4) (1 -)K(x) (F. 5)TCL=iZV00
From Equation (F. 1)
r2 tan 4)V= (F.6)(1- .L)
Substituting (F. 6) into (F. 5), yields
2(1-) sin24) K(x) (F. 7)CLw)(1- W cos24)) cos 4)
This expression is not exact, since in the expression (F. 2) for the
relative velocity it was assumed that the induced velocity is exactly
perpendicular to the relative velocity and that the displacement
203
velocity at the rotor is one-half of that in the wake.
By determination of the displacement velocity , the rotor
configuration may be established using relations (F. 1) and (F. 7).
This is done in propeller theory by developing power and thrust
equation in terms of the displacement velocity and then by specification
of a power coefficient, the displacement velocity may be calculated.
For wind turbine application we will develop the appropriate power and
thrust equations and other pertinent equations and show that they yield
inconsistent results.
Power Equation Development
To determine the power output, consider the increment of the
total energy iE which the unit volume undergoes i the flow. This
is equal. to
wh e r e
= -{p+ p{(VVz)2+V+V}} + {p,+ pV}
= -{p-pQo} p{V22VVz+V+V+V} + pV
= {p-p} p{V+V+V} + pVVz (F. 8)
Vaxial component of induced velocity
Vtangential component of induced velocity
VR - radial component of induced velocity
204
V--wind velocity
V --interference velocity
To maintain the flow which passes, during the unit of time, through
an element dS' of a section normal to the flow in the wake, we
must remove an element of power.
dP = 4E(VVz)dS' (F. 9)
But the moment of momentum of the mass element per unit time is
equal to
and since
therefore
p(VVZ)VTr'dS' = dQ
dP = dQ,
dP = 2p(VVZ)VTr'dS'
but from geometry of Figure F. 1
or
VT Vwtan 4) =
VTr2 Vz(Vw)
Substituting into expression for dP
205
Vz(V -w)dP = 2p(V00V)r' r'c dS'
= p(VVz)Vz(Vw)dS' (F. 10)
Equating (F. 9) and (F. 10)
LE(V -V )dS' = p(V -V )V (V -w)dS'00 Z 00 Z ZOO
or
Now equating (F.8) and (F. 11)
or
= PVz(Vw) (F. 11)
pVz(V00w) = (p00-p) p{V+V+V} + PV00V
(p00-p) + p{V+V+V} pV V + pV (V -w)00Z Zoo
1 2 2 2(p00-p) = p{Vz+VT+VR} pV00Vz + pV V - pV wZOO Z
1 2+v2+v2}(p00-p) = p{V T RpVw (F. 12)
This relation between pressure, induced velocity, and displacement
velocity exists at any point in the far wake. The pressure at the
boundary of the slipstream depends on the relative order of magnitude
of the displacement velocity and the components of the induced
velocity at the boundary and therefore is not necessarily equal to p00
206
Momentum theory makes the assumption p00 = p at the boundary
and therefore in general is not compatible with the conclusions from
vortex theory.
Now if we integrate one of the expressions (F. 9) or (F. 10) we
can determine the relationship for the power extracted.
dP = p(V00V)V(V00w)dS'= p(V
00
=
p(V2 -wV )V +(wV)V}dSt00 00 Z
Let us change the surface integral to a volume integral.
LetV00-w
H c2/ZTr
L H B = no. of blades
L is the pitch of the vortex spiral and is the axial distance between
two successive vortex sheets. Therefore
(1/B)H
1 1 cwVVdT(1/B)H SVVZdT (1/B)H 00 Z
1p V2dT+ (1/B)H SwVdT (1/B)H 00 Z
207
Now Svzdz for one turn equals cF()dF and
it is seen that
and
1 F(x) K(x, 0)(l/B)H
(1/B)H CVZdT = w SK(x, O)dS = KwF
1 rz 2(1/B)H v dT = KW F
1
(l/B)H SVZdT = w2F
as developed in Theodorsen (63) where
E axial interference loss factor
K = mass coefficient (2 K(x)xdx or 1K( 9)xdOdx)
0 Tr,i0
P p{VKWF -WVKWF+WE W2F -V w2F}
pF{KVw{V -w}+E w2(w-V)}
pF(V-w){KVw-Ew2
2P pFV(1-W){K-eW }
pFV3 E=
00 K(F. 13)
This is the power output based on wake area.
Thrust Equation Development
An expression for the thrust can be determined from the law of
momentum by integrating over a control surface consisting of two
cross sections far in front and far behind the wind turbine, which are
normal to turbine axis and approach infinity in all directions.
EF = (Momentum)dt
-T + (p-p')dS' = p
-TS
Substituting (F. 12)
= p{V+V+V} pVw + p(VV)Vz}dS'
-p{ vZvzw+vvzvZ}dS?
Now changing to a volume integral
209
1-T = (1/B)H v2dT (1/B)H p9vdT
1
(1/B)H PSWVZdT + (1/B)H fV V dTooZ
= -p{ Kw2F-EwF-KWF+KwFV}
= -pFwK w- w-w+V}
1= -pFwK{- w- w+V,}K 2
-pFKV{- w-
= -pFVKW{1-W(+)}
T (F. 14)
Slipstream Expansion
By blade element theory, one has the expression for thrust
given by
T pB (F. 15)
and the relative velocity as given by (F. 2) except the displacement
velocity is defined as a0 instead of w.
w2 1 {V-a0cos2}sin 4
Definingr 1 Bcx dx=dr r-R R 2iTr
and substituting into the expression for thrust, (F. 15) yields,
T = p2R2 SxCW2dx
Now substituting for Cy
C C cos+C sin4y L D
T p(R2) (C cos + sin )
(V-a0cos2)2xdx
2sin
Defining A = rrR2 yields
T =CLCOS (V-a0cos2)2xdx
sin
r (oCDsin )(V-a0cos24)2xdx+pA
2sin cl:
Now22(1-W)WK(x)sin 4
(1-a0)(1-a0cos2)cos
Therefore
210
211
T = pAVSCLcos (1-:0cos2)2xdx
Sin 4
B
cCsin (1a0cos2)Zxdx
+pAVj .2sin 4 -j
C
Integral B is determined as
= ç f 2(1-W)K(x)sin241 t(1-ao)(1-aocos 4)cos
2(1-W)WK(x)(1-a0cos)xdxj (1-as)
22cos 4(1-a.0cos 4) xdx
(1-}w 2K(x)(1-a0cos2)xdx(1a)
.2Sn"
2K(x)xdx - aoS ZK(x) cos2 xdx(laü)
K = ZSK(x)xdx
S SK(x)x cos2 dx
(1-w)w (l-w)
(i-a0) (K-Ka05) (1-a0) K(1-a05)
Integral C is determined as
CDS1fl (1-a0cos2)2dxd2(1 w) 2wK(x)sin 4 _____________________S(l)(lZ) CL } sin
(-w) c2K()()( S1fl) (1-a0cos2)xdx(1-a0) j CL cos 4
V,(1-a0) (1-a0)tan =rw Rx
CD 2(1 -)V(l-a0)2K(x)()(1-a0cos )dx(1-a0)R
V2K(x)()dx a 2K(x)()cos2d}= (1-)()
RA)
CN = 2K(x)()dx
L
V= (1-)()(N-a M)0
Now the thrust is given as
But
CD 2M 2K(x)()cos d
[K(1 -a0S)T = pAV(1-) (1-a0) + ()(N-a0M)
T =
212
(F. 16)
(F. 14)
Equating (F. 16) and (F. 14) yields
(1-w) (i-a0S)1 V
}A (1-a )+-()(N-a0M)
(1-( + )) 02K
Nowfor CDO
213
(i-w)(1-a0S)
(F. 17)A (1
1 +)) (i-a0)-w(2
K
Approach Inconsistency
Let us consider the maximum performance indicated by this
method for an actuator disk (infinite number of blades) at a high tip
speed ratio.
and
Power is expressed as:
P pFV3 E= KW(1_W)(1-W)
00 K
(1 -w)( 1 -a0S)
A
Now as 0 K andK
s=
So
and
or
So
_z.orat w i;
inconsistant siti
Now let us exarr
214
Using continuity and geometric relations for velocity
or
Now
(l-a0)A = (1-)F
(l-a)A (l-)
1 E_w- w2 Ka
for case under consideration
Now
-since 1 a0CT
l_2 1w-w -w(w-a0 3_(1-iw) (1-iw)
1 _2w-w1-
F (1-w)A (1-w)
3 1 21- .w- -w+w
(1-)(1-2 _21-2+w (l-w)
(1-w)(1-w) (1-w)(1-w)
(1-w)
A 3_(1- w)
215
216
which agrees with previously defined , therefore continuity is
obeyed in solution. Therefore by using the assumption of constant
displacement velocities, results are obtained that are inconsistent.
As shown in Chapter III this inconsistency is due to the fact that the
displacement velocity is not constant as assumed.