~ Pergamon 0960-1481(94)E0012-T

Renewable Energy, Vol. 4, No. 5, pp. 505 518, 1994 Copyright (C) 1994 Elsevier Science Ltd

Printed in Great Britain. All rights reserved 096~ 1481/94 $7.00 + 0.00

AERODYNAMIC PERFORMANCE ANALYSIS OF HORIZONTAL AXIS WIND TURBINES

V. H. MORCOS Mechanical Engineering Department, Assiut University, Assiut, Egypt

(Received 22 November 1993 ; accepted 3 March 1994)

Abstract--The present work studies the wind energy resources in Egypt and the aerodynamic performance of propeller type and multi-bladed horizontal axis wind turbines with three different airfoil blade sections (flat-plate, symmetric and circular-arc airfoils). Power, thrust and torque coefficients were investigated as functions of wind turbine design parameters (blade angle, rotor solidity, drag-to-lift coefficient ratio and blade section) and operating conditions (tip-speed ratios). Axial and tangential induction factors and drag coefficient were introduced in the calculations. Recommended design and operating values are given for each wind turbine. The analysis of the theoretical results shows that flat-plate and symmetric airfoil blade sections operate at a wider range of tip-speed ratios than that of circular-arc airfoil blade sections, so they are recommended for small and large sized wind turbines, respectively. Also, the analysis of the available local wind speed measurements indicates that the future potential of wind energy conversion systems in Egypt is promising. Several sites along the Mediterranean and Red Sea coasts have high annual average wind speed and power density of 6.4 m s -1 and 160 W m -2, respectively.

INTRODUCTION

Egypt has begun to show an interest in using wind as an alternative source of energy [1, 2, 3]. To erect wind turbines that best suit the wind climatology of Egypt and satisfy simplicity in manufacturing and cost reduction, many steps and studies are needed to be carried out towards the large-scale local wind- machine industry in Egypt.

The detailed design-relevant parameters of a wind turbine are the number of blades, the variation of the chord, the variation in blade angle and the shape of the blade sections [1,4].

Glauert [5] defined the configuration and per- formance of an optimum actuator disk by developing a closed-form solution to the variation problem using strip theory equations. Glauert's solution, however, neglected drag and tip loss. Rohrbach and Worobel [6] investigated the effect of blade number and section lift/drag ratio on the maximum performance that can be obtained from wind turbines. They reported that increasing number of blades and section lift/drag ratio increase the maximum extracted power. Wilson et al. [7] determined the optimum blade shape of a hori- zontal axis wind turbine. They neglected drag and included tip loss.

Griffiths [8] established a design procedure for the optimum blades using the blade element theory and neglecting tip loss, but including the drag. Mansour

et al. [9] determined the optimum aerodynamic design of a three-bladed propeller type wind turbine includ- ing both drag and tip loss factors. The NACA 4415 airfoil was used in their study. The above review indi- cates that the previous studies neglected some par- ameters in evaluating the performance of wind turbines.

The objectives of the present research are: (i) to study the wind energy resources in Egypt based on the analysis of available wind speed reports of Egyptian locations [10] and (ii) to determine the aerodynamic design of the horizontal axis propeller and multi- bladed wind turbines including drag, axial and tan- gential induction factors for different airfoil blade sec- tions. The power, thrust and torque coefficients were examined as functions of blade section, drag-to-lift coefficient ratio, blade angle, rotor solidity and tip- speed ratio.

THEORETICAL ANALYSIS

Variation o f wind speed with height Surface wind speeds are reduced by comparison

with wind speeds higher in the upper atmosphere, due to friction at the Earth's surface. The variation of wind speed with height may be represented (approxi- mately) by a power law, e.g.

505

506

V(h)/V(IO) = (h/10)", (1)

where V(h) is the wind speed at height h and V(10) is the wind speed at h - - 10 m. The magnitude of the exponent n depends on the roughness of the terrain, but for flat, open country n is approximately equal to one-seventh. Most modern wind turbines typically have hub heights of about 25-50 m [1].

General momentum theory for horizontal axis wind tur- bines (HA WT)

Conventional analysis of HAWT begins with an axial-momentum balance using the control volume depicted in Fig. 1. The upstream wind speed V is decelerated to V ( 1 - a ) at the turbine disk and to V(1 - 2 a ) in the wake of the turbine [11]. Momentum analysis predicts the axial thrust on the turbine of radius R to be

T = 2nRZp V2a(1 - a), (2)

and thrust coefficient is defined as

CT = T/O.5p~zR 2 V z = 4a(1 -- a). (3)

Application of the energy equation to the control vol- ume depicted in Fig. 1 yields the prediction of mech- anical power produced by the turbine as

P = 2nRZpV3a(1-a) 2. (4)

This mechanical power can be nondimensionalized with the wind power in the upstream wind covering an area equal to the rotor disk, i.e.

Pw = 0.5p V3nR2. (5)

The resulting power coefficient is

Cp = P/Pw = 4 a ( 1 - a ) 2. (6)

This power coefficient has a theoretical maximum at a = 1/3 of Cp = 0.593. This result was first predicted by Betz [12]. The torque coefficient is defined as

CQ = Q/O.5pnR 3 V 2 = Cp/(U/V). (7)

This derivation includes the following important assumptions : (i) the turbine must be a horizontal axis

A v (1-2a)

Turbine disk

Fig. 1. Control volume.

V. H. MoRcos

~ ~ ~ V (1-2a) Turbine disk

Fig. 2. Annular-ring control volume.

configuration such that an average stream tube (Fig. 1) can be identified, (ii) the portion of kinetic energy in the swirl component of velocity in the wake is neglected and (iii) the effect of the radial pressure gradient is excluded.

Blade-element theory for HA W T This theory provides the mechanism for analyzing

the relationship between the individual airfoil proper- ties and the axial induction factor, the power produced and the axial thrust of the turbine. Rather than the stream tube of Fig. 1, the control volume consists of the annular ring bounded by streamlines depicted in Fig. 2. It is assumed that the flow in each annular ring is independent of the flow in all other rings.

A schematic of the velocity and force-vector dia- grams is given in Fig. 3. The elemental torque which acts on all blade elements in an annular ring is

dQ = 0.5Bcrp W 2 (CL sin ~b- CD cos ~b) dr. (8)

The above equation is derived for the case with no turbine conning, which can be accounted for if appro- priate. The sectional lift and drag coefficients are obtained from empirical airfoil data and are unique functions of the local flow angle-of-attack and the local Reynolds number of the flow. Lift and drag coefficients are defined from

dL = CL (0.5p W2)c dr, (9)

dD / d L

. . . .

Air foi ix

v(1-a)

Fig. 3. Velocity and force-vector diagrams.

Horizontal axis wind turbines 507

dD = CD (0.5p W2)c dr. (10)

Power is computed by integrating eq. (8) after mul- tiplying it by the turbine angular velocity f~. The result is

P= 0.5pBf~ crW2(CL sin~--CDcosd~)dr. (11)

Similarly the total thrust force on the turbine is

T = 0.5pB cW2(CLcosd~+Cosin4))dr. (12)

Referring to the vector diagram of Fig. 3

sinq~ = V(1-a)/W, (13)

cos tk = (1 +a')f~r/W, (14)

W = [V2(1-a)2 +(l +a')2f~2r2] °5, (15)

sin~ = sin (q~-O) = sin~bcosO-cosq~sinO. (16)

formance [13]. The lift coefficient of the circular-arc airfoil is [15]

CL = 2n[~+ (2f/c)]. (19)

In the present work the maximum thickness of a circular-arc airfoil is assumed to be 6% of the chord, hence the lift coefficient equation will be

CL = 2n(a+O.12) --~ 0.754+2rising, (20)

where sin c~ ° -~ ~ in radians for small values of ~.

Flat-plate and symmetric airfoil HA WT analysis Substitute eqs (13)-(17) in eq. (11) to get the power

produced by the wind turbine in the form

P = npBf~ V2(1-a)2ccosOrdr

-1 TM0 V(1 - a ) ( 1 +a')cf~ sin 0 r 2 dr

Airfoil blade sections characteristics Flat-plate, symmetric and circular-arc airfoil blade

sections were investigated in the present study. The simplest airfoil section that may be used is the fiat plate, for which the relation between angle-of-attack and lift coefficient is [13]

CL = 2n sin a. (17)

This equation compares quite well for low angle of attack (0-12 ° ) with experimental data.

In most cases strength considerations dictate some profiling and a number of sections have been used. One of them is the symmetric airfoil section. It is an airfoil with no camber, i.e. with the mean camber line and chord line coincident. Clearly, the shape of a symmetric airfoil is the same above and below the chord line. For example, the NACA 0012 airfoil is a symmetric airfoil with a maximum thickness of 12% of the chord. A symmetric airfoil is preferred, because its camber line is a streamline of the flow, which reduces flow losses and gives high lift-to-drag ratios. The lift coefficient equation of a symmetric airfoil for angles of attack range of - 12-16 ° is [14]

CL = 2net. (18)

In practice the simplest camber line is a circular-arc with the maximum thickness as small as practicable. Experimental data suggests that thicknesses up to 12.5% of the chord do not affect performance, and thicknesses above this dictated by strength con- siderations will result in a drastic fall-off in per-

_j,R0 V(1 - a ) ( 1 +a')cf~(CD/CL) cos 0 r 2 dr

f R

+ (1 +a')%fF(CD/CL 0

sinOr3dr}. (21)

To integrate the above terms in eq. (21), the following simplifying assumptions were made : (i) uniform dis- tribution of upstream wind speed along the blade ; (ii) constant chord along the blade (parallel plan form blade) ; (iii) constant blade angle during steady-state operation; (iv) constant drag-to-lift coefficient ratio, because it varies in a narrow range from 0 to 0.05 for all shapes of airfoil sections [13]; and (v) uniform distribution of axial and tangential induction factors along the blade. This assumption agrees well with the results of Mansour et al. [9], especially for low and high tip-speed ratios for axial and tangential induction factors, respectively.

Taking the above assumptions into consideration, integrate eq. (21) and introduce the definitions of sol- idity (projected blade area to rotor-swept area) and power coefficient to get

Cp = ha(1 -- a) (U/V) {[1 -- a - - (2/3)(1 + a') (CD/CL)

× (u/v)] cos o - [(2/3)(1 +a') (u /v)

-0.5(Co/CL)(1 +a')2(U/V)2/(1-a)] sinO}. (22)

Equating eq. (22) with eq. (6) and rearranging to get the axial induction factor

508 V . H . MoRcos

a = [na(U/V)/4(1 --a)]{[1 - -a- - (2/3)(1 +a')(Co/CL)

× (U/V)] cos 0 - [(2/3)(1 +a')(U/V)

--0.5(CD/CL)(1 +a')2(U/V)2/(1--a)] sin0}. (23)

In the same way, substitute eqs (13)-(17) in eq. (12) to get the total thrust force on the turbine in the form

{J0' T= ~pB V(1-a)( l +a')cf~cosOrdr

- f~(1 + a')2cf22 sin 0 r 2 dr

+ fro V 2 (1 - a) 2 c (Co/CL ) COS 0 dr

-- f ; V(1--a)(l +a')cf2(CD/CL)sinOrdr}.

(24)

Using the same previous assumptions, integrate eq. (24) and introduce the definition of solidity, so that the total thrust force on the turbine is

T = pTt2~r(1 --a)R 2 V 2 {[0.5(1 +a')(U/V)

+ (CD/CL)(1 --a)] COS 0-- [(1 +a')2(U/V)2/3(1 --a)

+0.5(CD/CL)(I+a')(U/V)] sin0}. (25)

Equate eq. (25) with eq. (2) and rearrange to get the tangential induction factor

a' = [2~(U/V) cos O]{(2a/Tra) -0.5(U/V) cos 0

-- (1 -- a)(CD/CL) COS 0+ [(1 +a')2(U/V)2/3(1 -- a)

+0.5(I+a')(CD/CL)(U/V)]sin0}. (26)

The solution of eqs (23) and (26) for certain assumed values of 0, CD/CL, a and U/V to get the induction factors a and a' cannot be obtained in a closed form, and a trial-and-error technique must therefore be used. The idea is that the blade forces are responsible for blocking the wind and for instilling swirl in the wake. However, to compute the blade forces, the amount of blockage a and swirl a' must be known. Hence the trial-and-error requirement.

Circular-arc airfoil HA WT analysis Substitute eqs (13)-(16) and eq. (20) in eq. (11) to

get the power produced by the wind turbine in the form

P=ztpBff2{f~VZ(1-a)ZccosOrdr

- f ~ V(1 -a ) (1 +a ' )eO sin Or 2 dr

- fo r V(1 -a ) (1 +a')cf~(CD/CL)cosOr 2 dr

+fRo (I +a')2c~2(CD/CL) sinOr3 dr}

+0.12rcpB~ f f~ VZ(1-a)Z c[l + ArZ]° S rdr

- V(1--a)(I+a')cn(CD/CL) 0

x [1 +Ar2] °'s r 2 dr}, (27)

where A = (1 + a')2f~2/(1 - a) 2 V 2. Using the same assumptions as the flat-plate and

symmetric airfoil solution, integrate eq. (27) and introduce the definitions of solidity and power coefficient to get

Cp = ~r(1 -a)(U/V){[1 - a - (2/3)(1 + a')(CD/Q)

× (U/V)] cos 0-- [(2/3)(1 +a')(U/V)

-0.5(1 +a')2(CD/CL)(U/V)2/(1 --a)] sin 0}

+ [0.2513a(1 -a) ' / (U/V)(1 +a ' ) 2]

x (E ~5 - 1)-[0.1885a(1 -a)3(Co/CL)E"5/(1 +a')]

+ [0.0942a(1 --a)3(CD/CL)E°5/(1 +a')]

+ [0.0942a(1 -- a) 4 (Co/CL)/(I + a') 2 (U/V)]

x ln l [(1 +a')(U/V)/(1 - a ) ] + E °5 I, (28)

where E = 1 + [(1 + a')2(U/V)2/(1 -a)2]. Note that the integrations of the magnitudes

Sg (1 +Ar2) °5 rdr and S0 R (1 +Ar2) °5 r2dr are (IDA) (E 15- 1) and A°5{(R/4)[R2+ (l/A)] 15- (R/8A)[R2+ (1/A)] °5 -- (1/SAZ)lnlR + [R2+ (1/A)]°sI +(1/8A2)lnl 1/ A°51 }, respectively. Equating eq. (28) with eq. (6) and rearranging to get the axial induction factor

a = [0.7854a(U/V)/(1 -a)]{[1 - a - (2/3)(1 +a ' )

× (Co/CL)(U/V)] COS 0-- [(2/3)(1 +a')(U/V)

--0.5 (1 +a')2(Co/Q)(U/V)2/(1 --a)] sin 0}

+ [0.0628a(1 -- a) 2/(1 + a') 2 (U/V)](E ~5 _ 1)

-- [0.0471 a(1 -a)(Co/CL)/(1 + a ' ) ] E '5

Horizontal axis wind turbines 509

+ [0.0236a(1 --a)(CD/CL)/(1 + a')]E °5

+ [0.0236a(1 -- a) 2 (CD/CL)/(1 + a') 2 (U/V)]

x ln l [ ( I + a ' ) ( U / V ) / ( 1 - a ) ] + E ° S I . (29)

By the same way, substitute eqs (13)-(16) and eq. (20) in eq. (12) to get the total thrust force on the turbine in the form

T = pn2a(1 --a)R 2 V 2 {[0.5(1 +a') (U/V)

+ (1 - a ) ( C • / Q ) ] cos 0- [ ( (1 +a')2(U/V)2/3(1 - a ) )

+0.5(1 +a')(CD/CO(U/V)] sin 0}

+ [O.1257npaRV3(1 --a)3/(1 + a ')~](E ~5 - 1)

+ O. 18857~paR 2 V 2 (1 -- a) 2 (CD/CL)E 05

+ [0.1885npaR V 3 (1 - a) 3 (Co/CL)/(1 + a')fl]

x ln l [(1 +a')(U/V)/(1-a)] + E °5 I- (30)

Note that the integration of the magnitude ~0 R [1 +Ar2] °5 dr, which appears in the thrust equa- tion, is equal to

0.5A °.5 R[R 2 + (l/A)] ° ' + (0.5/a °5)

x In I [(1 + a')(U/V)/(1 - a ) ] + E °5 I.

Equate eq. (30) with eq. (2) and rearrange to get the tangential induction factor

a' = [1.27324/a(U/ V) cos 0] { a - 0.7854a(U/V)

× cos 0 - 1.5708a(CD/CL)(1 --a) cos 0

+ 1.5708a[((1 +a')2(U/V)2/3(1 - a ) )

+0 .5 (Co/C0(1 +a')(U/V)] sin0

--[0.06285a(1 -- a)2/(1 +a')(U/V)](E ~5 - l)

-0.09425a(1 -a)(Ct~/CL)E °5

- [0.09425a(1 --a)2(CD/CL)/(1 +a')(U/V)]

><lnl [ ( I + a ' ) ( U / V ) / ( 1 - a ) ] + E ° S I } . (31)

Again, the solution of eqs (29) and (31) for certain assumed values of 0, CD/CL, a and U/V to get the induction factors a and a' cannot be obtained in a closed form, and a trial-and-error technique must therefore be used.

for Egypt. Annual average wind speeds at the hub height of 25 m AGL were calculated for each site using eq. (1). Wind power density, which is defined as the wind power per unit area of the wind stream, is equal to 0.5 p V 3. Annual average wind power densities were calculated for each site at heights of 10 and 25 m AGL.

The theoretical analysis was devoted to study the aerodynamic performance of three different airfoil blade sections (flat-plate, symmetric and circular-arc) HAWT to choose the best design parameters (blade angle, drag-to-lift coefficient ratio and solidity) at specified operating conditions (tip-speed ratios) for each one, and hence to find out the best one of the three. This choice was based on getting high turbine power coefficients (not less than 0.50) at a wide range of variations of tip-speed ratios to maximize the power developed by the wind turbine. At the same time, this choice was based on getting as low turbine thrust coefficients as possible to minimize the forces on the towers of the wind turbines, and hence reduce their capital costs.

For each airfoil section, the blade angle was varied from 0 to 10 ° (0, 1, 2, 5 and 10°). This variation was dictated by the analysis of the obtained results. For each airfoil section and each value of blade angle, the drag-to-lift coefficient ratio was varied from 0 to 0.10 (0, 0.01, 0.025, 0.05 and 0.10), because this range of variation represents the typical range for all shapes of airfoil sections at different angles of attachment and large values of Reynolds number. For each airfoil section, each value of blade angle and each value of drag-to-lift coefficient ratio, the rotor solidity was varied from 0.01 to 0.30 (0.01, 0.03, 0.05, 0.07, 0.10, 0.15, 0.20, 0.25 and 0.30) to cover the operating range of high and low speed rotors, respectively. For each value of rotor solidity the tip-speed ratio was varied from 2 to 14, to cover the operating range of low and high speed rotors, respectively.

A computer program was constructed to do the trial-and-error technique required to solve eq. (23) with eq. (26) and eq. (29) with eq. (31) to get the axial and tangential induction factors, which were used to calculate the power, thrust and torque coefficients as functions of tip-speed ratio for different values of blade angle, drag-to-lift coefficient ratio, rotor solidity and different airfoil blade sections.

Calculation procedure The annual average wind speeds measured in some

meteorological stations in Egypt at the standard meteorological height of 10 m above ground level (AGL) were used to assess the wind energy potential

RESULTS AND DISCUSSIONS

The annual average wind power densities and direc- tions are shown in Fig. 4 for some meteorological stations in Egypt. The wind power density was cal- culated at the standard meteorological and hub

510 V.H. MoRcos

Rashid I Alex. (53)

(53) [79l

I (75) (106) . [112] [t581

/

i Siwa (18)

i [261

1

I

I

i i I I

I I I i i 0 100 200 km

b_., I I

Wadi El Natrun (49) [731 ~ a

El Minia (23) [33l

(23) [33]

El Kharga (36) [53]

Demita (75)

E1 Arish'~ (14) [2Ol ~, ~ 1 Suweis

(36) [53l

Hurghada I 006) [1581

Kena El Kosir I (75) [1121 ,

Lu xor

x: , :I 1

Aswan

' ~ ¢" " ~"

I' i . I / X • °,~o ~

The Red Sea

Fig. 4. Map of Egypt showing the name and location of some meteorological stations. The annual average wind power densities (Wm -2) and directions are indicated for each station. ( ) Wind power density at the

standard meteorolog:cal height of 10 m AGL, [ ] Wind power density at the hub height of 25 m AGL.

heights of 10 and 25 m AGL, respectively. The figure shows that along the Mediterranean Sea coast f rom Port Said in the East to El Salum in the West, where the annual average wind speeds vary from 4.44 to 5.6 m s -1 at the standard meteorological height of I0 m A G L , the annual average wind power densities vary from 53 to 106 W m -2. The same variations can be observed along the Red Sea coast from EL Toor in the Nor th to E1 Kosir in the South. At wind turbine hub height o f 25 m A G L , the expected variat ion of annual average wind speeds and power densities for the same sites will be in the range o f 5.06--6.38 m s- and 79-158 W m -2, respectively. Power densities of

this magnitude (i.e. 158 W m -2 annual average) com- pare favorably with annual average solar radiation power densities, which typically range from 100 to 250 W m - 2 Moreover , modern wind turbines convert the kinetic energy in the wind into a useful electrical power output at high efficiency, compared with the con- version efficiency of solar radiation into power.

Flat-plate and symmetric airfoil blade sections Figure 5 shows the variat ion of power coefficient

with tip-speed ratio for different values o f solidity at a constant blade angle of 0 ° and a constant drag-to- lift coefficient ratio of 0.05. The figure shows that,

Horizontal axis wind turbines

0.8 Flat-plate and ;0 = O*;CD/C L = 0.05 symmetric airfoil

0.6

Cp 0.4

0.2

o 0 2 4 6 8 I0 12 14

U/V Fig. 5. Variation of power coefficient with tip-speed ratio for different values of solidity at a constant blade angle and

drag-to-lift coefficient ratio.

for a ~< 0.05, Cp increases with the increase of U~ V, meanwhile for 0.05 < a ~< 0.25, Cp increases to a maximum value of 0.5926 and then decreases with the increase of U/V. For 0.25 < ~r ~< 0.30, Cp varies in the same way, but its maximum value ranges from 0.5905 to 0.5631, respectively. The value of U/V at which maximum Cp occurs decreases with the increase of a. This means that turbines with high and low solidities operate with maximum power coefficient at low and high speeds, respectively. Results showed that the variation of Cp with U~ V for different values of solidity had the same tendency to variation for the other values of blade angles and CD/CL ratios.

The effect of the Co/CL ratio on the power coefficient for two different rotor solidities of 0.05 and 0.15 at a blade angle of 1 ° is shown in Fig. 6. As the Co/CL ratio increases the power coefficient decreases. When the CD/CL ratio increases above 0.025 the power coefficient decreases noticeably. The maximum power coefficient occurs at tip-speed ratios equal to or higher than 10 for a rotor solidity of 0.05, which is considered as a high speed rotor, while it occurs at a low tip-speed

511

ratio of 3 for a rotor solidity of 0.15 (low speed rotor) The effect of blade angle on power coefficient for

different rotor solidities and a CD/CL ratio of 0.025 is shown in Fig. 7. Increasing blade angle decreases the power coefficient with a noticeable drop in its maximum value for small values of rotor solidity (high speed rotors), while this increase in blade angle increases power coefficient for large values of rotor solidity at high tip-speed ratios. The same tendency to variation was observed for the other values of CD/CL ratios. The analysis of all figures drawn at different CD/CL ratios and similar to Fig. 7 led to Table 1, defining the preferred range of design and operating values for each wind turbine.

For low speed rotors Table 1 shows that: for all values of CD/CL from 0 to 0.10, the solidities used are 0.10, 0.15 and 0.20. Rotors with solidity of 0.10 oper- ate with a blade angle of 2 ° for all values of CD/CL, except for CD/CL = 0.10 where the blade angle is 0 °, because it gives higher values of power coefficient. The average ranges of tip-speed ratio and power coefficient will be 2.44-9 and 0.504).575, respectively. Rotors with solidities of 0.15 and 0.20 both operate at a blade angle of 5 ° for all values of CD/CL. The average ranges of tip-speed ratio and power coefficient for rotor sol- idity of 0.15 are 1.89-10.4 and 0.504).591, respec- tively. For rotors with solidity of 0.20, the average ranges of tip-speed ratio and power coefficient are 1- 10 and 0.5043.588, respectively. For all these rotors, the ranges of thrust and torque coefficients are 0.35- 0.98 and 0.37~).04, respectively.

For high speed rotors Table 1 shows that: for all values of Co/CL from 0 to 0.10, the solidities used are 0.03, 0.05 and 0.07. The solidity 0.03 is not used for both CD/C L equal to 0.05 and 0.10, because it gives a power coefficient less than 0.50 for all blade angles and tip-speed ratios varied. Rotors with solidity of

Cp

0.6

0.4

0.2

Cp

CD/C L 0 0.01 0.025

~' 0 = 1" ; o = 0 . 0 5

0.6

0.4

0.2

CDIC L

0.I

I -,.-O.Ol

O= 1 ° ; o = 0 . 1 5

0 I I I I 0 I I I 0 4 8 12 16 0 4 8 12

UIV UIV

Fig. 6. Effect of CD/C L ratio on the power coefficient for two different rotor solidities and a constant blade angle.

512 V.H. MoRCOS

0.6 O.6 m

r o.ol oo °o [ o = 0.03 ~ " 10

2 ° 0.4 0.4

0.2 ** 0.2

i - - ' ° 0 I 0 1 ¢ " I I I I 0 4 8 12 16 0 4 8 12 16

u/v u/v

0.6

0.4

Cp

0.2

0 0

0.6

0.4

Cp

0.2

0 0

- o = o . 0 5 ~ a ~ : : = ~ 0 .6 - ~ 2 o 1* ,°°° 0.4

0 5 ° Cp

o = 0.07 0.2

I I I I o I I I I 4 8 12 16 0 4 8 12 16

u/v u/v

~ ~ 8 0 . 2 i O.6 --

1 0 . 4 - -

C p

0 . 2 - - o = 0.10

I I I I o I I I I 4 8 12 16 0 4 8 12 16

u/v u/v

o=0 .15

0.6

0.4

Cp

0.2 0 0 °

o = 0.20

0.6 -- ~ 5 °

0.4

0.2 -- 1° 0 00 o = 0.25

I I I I o I I I I 4 8 12 16 0 4 8 12 16

u/v u/v

Cp

Fig. 7. Effect of blade angle on the power coefficient for a constant CD/C L ratio of 0.025 and different rotor solidities.

Horizontal axis wind turbines 513

<

O

r~

E

(3

E E o

m - ~

~2

~5~5~5

~ .

-

~ '

~ d

e d d

0.03 operate at a blade angle of 0 ° at average ranges of tip-speed ratio and power coefficient of 8.93-14 and 0.50--0.579, respectively. Rotors with solidity of 0.05 operate at blade angles of 1 ° for CD/CL <~ 0.01 and 0 ° for CD/CL >~ 0.025. The average ranges of tip-speed ratio and power coefficient will be 6.08-13.4 and 0.50- 0.5724, respectively. Rotors with solidity of 0.07 oper- ate at a blade angle of 2 ° for all values of CD/CL at average ranges of tip-speed ratio and power coefficient of 4.6-13.6 and 0.50-0.5866, respectively. Fo r all these rotors, the ranges of thrust and torque coefficients are 0.61-4).96 and 0.13~).04, respectively.

The effect of tip-speed ratio on thrust coefficient for different values of solidity, with a blade angle of 0" and CD/CL ratio of 0.05, is shown in Fig. 8. As shown, thrust coefficient increases with the increase of tip- speed ratio for different values of solidity ~< 0.10. For 0.10 < a ~< 0.30 the thrust coefficient increases to a maximum value of unity and then decreases with the increase of U/V. The tip-speed ratio of maximum thrust coefficient decreases with the increase of solid- ity. Fortunately, as shown from Figs 5 and 8, the maximum power coefficient occurs at a lower value of tip-speed ratio than that at which maximum thrust coefficient occurs for the same solidity. This helps in getting maximum power at thrust lower than the maximum.

The effect of the CD/CL ratio on thrust coefficient for a blade angle of 1 ° and solidity of 0.05 is shown in Fig. 9. As shown from the figure, increasing the CD/CL ratio decreases the thrust coefficient. The same tend- ency to variation was noticed for the other values of 0 and a.

The effect of the blade angle on the thrust coefficient for a CD/CL ratio of 0.025 and solidity of 0.07 is shown in Fig. 10. Increasing the blade angle decreases the

1.0

0 .8

0 . 6

CT

0 . 4

0 . 2

0 0

Illl 7 / o 3 0

I I I / / Flat-plate and .~ . . . . . . . . . . . r / / / ~ i c ;" ~ - ; ~ _ k ~ . "vJ

o 0 0 ~ I I I I

2 4 6 8 10 12 14

U/V Fig. 8. Effect of tip-speed ratio on the thrust coefficient for different values of solidity and a constant value of Co/CL

ratio.

514 V. H. MORCO~

! 1.0 ]'-" C~C L 0 ~ 0.01

0.8 0.05

CT 0.6

0"4 / ~ 0--1"

0.2 = ~

0 0 4 8 12 16

u / v

Fig. 9. Effect of CD/CL ratio on the thrust coefficient.

0 0

1 o

- - CD/C L =0.025 ~ 2 °

I I I I 4 8 12 16

U/V

1.0

0.8

CT

0.4

Fig. 10. Effect of blade angle on the thrust coefficient.

th rus t coefficient. This was not iced for the o ther values of the CD/CL rat io and solidity.

The effect of the t ip-speed ra t io on the torque coefficient for different values of ro tor solidity at a b lade angle of 0 ° and a CD/CL rat io of 0.05 is shown in Fig. 11. As shown, torque coefficient decreases slowly and rapidly with the increase of tip-speed rat io for small and large values of solidity, respectively• It

o Flat-plate and ;0 - - 0 ° ; C D / C L = 0.05 0.6 - 0.3 symmetric airfoil

0.4 - CQ 0.1

0.2 - 0

2 4 6 8 !0 12 14

U/V

Fig. 11. Effect of tip-speed ratio on the torque coefficient for different solidities and constant values of blade angle and

CD/CL ratio.

b-

J

" 8

e~

D- O

"O

ed

..2 • c ~

° e S ~

~S c,i . o ~ e 5

oi .

O

Horizontal axis wind turbines 515 0.2 - 0.2 -

O= 1 °;0=0.05

CQ 0.1 - ~ ~ CD/CL

0.1 "0 I I t I

0 4 8 12 16

u / v

Fig. 12. Effect of CD/CL ratio on the torque coefficient.

C O 0.1 -

O = 0.05 ; CD/C L = 0.025

0 I I I I 0 4 8 12 16

U/V

Fig. 13. Effect of blade angle on the torque coefficient.

means that low and high speed rotors need large and small values of starting torque, respectively, which are noticed in practice. Figure 12 shows the effect of the CD/CL ratio on the torque coefficient for a rotor sol- idity of 0.05 and blade angle of 1 °. It is seen that

increasing the CD/C L ratio decreases the torque coefficient. Figure 13 shows the effect of the blade angle on the torque coefficient. It decreases with the increase of blade angle. The same variations of torque coefficient were noticed for the other parameters.

Cp

0.6 -- o = 0.01

0 . 4 - - 1 ° o

0.2 - ~ I0 °

4 8 12 U/V

Cp

I 16

0.6

0.4

Cp

0.2

0 0

-- l ° o o o k . ' , , ~ t 2 > . . . - - 5 o

~ ~ 0 ~ 0~010

3

I I I 4 8 12

U/V

0.6 -- 0 0 0 , 1 ~ ~ ' ' - I0° 0.6

0.4 f ~ 2 0 5*

0.2 ~ o = 0 05 Cp

F ~ 1 0 o

0.4 ~ F 0°,10&2°5 * o = 0.07

0.2

o ! I I I 0 4 8 12

U/V

I 16

o I I I I I 0 4 8 12 16 16

U/V

a= 0.15

I I 12 16

06 06 r ,

0 . 4 ~ 0°'l° 0.4 *

Cp Cp

0.2 o=0.10 0.2

0 V I I I I 0 I 0 4 8 12 16 0 4 8

U/V U/V

Fig. 14. Variation of the power coefficient with blade angle for a CD/CL ratio of 0.05 and different rotor solidities (circular-arc airfoil).

516 V. H. MoRcos

Circular-arc airfoil blade section Results analysis showed that the dependence of

power, thrust and torque coefficients on different par- ameters (tip-speed ratio, solidity, CD/CL ratio and blade angle) had the same tendency to variations as that of the fiat-plate and symmetric airfoil, except that the tip-speed ratio range of variation was smaller than that of fiat-plate and symmetric airfoil. Figures 14- 16 show the variations of power, thrust and torque coefficients, respectively, with the blade angle at a certain value of the CD/CL ratio of 0.05 and different values of rotor solidity and tip-speed ratio. The results of the other values of the CD/CL ratio showed the same tendency to variations. The analysis of all these results led to Table 2, defining the preferred range of design and operating values for each turbine.

For low speed rotors Table 2 shows that : for all values of CD/CL from 0 to 0.10, the solidities used are 0.10 and 0.15. Rotors with a solidity of 0.10 operate with a blade angle of 5 ° for CD/C L ~ 0.025 and 10 ° for Co/CL >~ 0.05 at average ranges of tip-speed ratio and power coefficient of 2.72-6.4 and 0.50-0.566,

respectively. Rotors with a solidity of 0.15 operate with a blade angle of 10 ° for all values of CD/CL, except for CD/CL = 0.025, where the rotor operates at a blade angle of 0 °. The average ranges of tip-speed ratio and the power coefficient are 1.84-4.4 and 0.50- 0.582, respectively. For all these rotors, the ranges of thrust and torque coefficients are 0.46-1 and 0.30- 0.07, respectively.

For high speed rotors Table 2 shows that: for all values of CD/CL from 0 to 0.10, the solidities used are 0.03, 0.05 and 0.07. The solidity of 0.03 is not recommended for Co/CL =0.10 , because it gives Cp < 0.50 for all values of blade angles. Rotors with a solidity of 0.03 operate with a blade angle of 5 ° at an average range of tip-speed ratio of 7.825-11 and an average range of power coefficient of 0.50-0.575. Rotors with a solidity of 0.05 operate with a blade angle of 5 ° for the ratio CD/C L ~ 0.025, 10 ° for CD/CL = 0.05 and 0 ° for CD/CL = 0.10. The average ranges of tip-speed ratio and the power coefficient are 5.24-9 and 0.504).584, respectively. Rotors with a solidity of 0.07 operate with a blade angle of 5 ° for

0.8 [ ' o = 0.03 0 00 2 ° / 5°

/ o = 0.01 . ~ f f 10° CT 0.4~- ~ 0 ~ 5 * CT0-4

I -10o 0 I , ~ , , ~ I I I 0 I

0 4 8 12 16 0 4 8 12 16 u / v UlV

0.8

CT 0.4 CT

0.8

Cr 0.4

_ 0 0 ° 2 ° 5 ° 1 0 °

o -~ 0.05

f I I I 4 8 12

U/V

o = 0.07

0 I I I 0 16 12 16

O.8 t ~ °l°°

0 " i / I I 0 4 8

u / v

y oo P/ 0 0o50 10 o

0.8

o = 0 .10 C T o = 0 .15

° ' r / , I I I I I I I 4 8 12 16 0 4 8 12 16

u / v u / v

Fig. 15. Variations of the thrust coefficient with blade angle for a Co/CL ratio of 0.05 and different rotor solidities (circular-arc airfoil).

Horizontal axis wind turbines

0.04 ~ 0.10 V

o = o.oI I I I°° I o.o2 -- 1°° I

, O O ol ,oo6 0 4 8 12

U/V

0.15

0.11

CQ

0 .07

0.03

0.24

0.20

CQ 0.16

0.12

0.08 0

o = 0.05

- ~ 0 °

I I I I 4 8 12 16

U/V

5 ° o = 0.I0 I00

%2 °

/0°,2 °

4 8 12 16

U/V

16 0 4 8 12 16

0.20 --

0.16 --

CQ 0.12

0.08

UIV

0.06 I 0 16

2~o o = 0.07

I I I 4 8 12

UIV

Fig. 16. Variations of torque coefficient with blade angle for a CD/C L ratio of 0.05 and different rotor solidities (circular-arc airfoil).

517

CD/C L ~ 0.025 and 10 ° for CD/CL >1 0.05. The average ranges of tip-speed ratio and the power coefficient are 4.0(~7.8 and 0.50-0.572, respectively. For all these rotors, the ranges of thrust and torque coefficients are 0.60-0.97 and 0.15-0.04, respectively.

CONCLUSIONS

The main conclusions of the present work are : (i) The future potential of using wind energy con-

version systems along the Mediterranean and Red Sea coasts in Egypt is encouraging. The annual average power density for these sites is in the range of 79 to 158 W m -2.

(ii) The power and thrust coefficients of wind tur-

bines increase with the increase of tip-speed ratio and solidity, and with the decrease of CD/CL ratio and blade angle. As the rotor solidity increases, the maximum power and thrust coefficients occur at lower tip-speed ratios. So, low speed wind turbines (multi- bladed type) have a high number of blades, while high speed wind turbines (propeller-type) have a low number of blades. The maximum power coefficient occurs at a lower tip-speed ratio than that of maximum thrust coefficient for the same solidity. This helps in getting maximum power at a lower thrust than the maximum.

(iii) The torque coefficient decreases with the increase of tip-speed ratio, CD/CL ratio and blade angle. It increases with the increase of solidity at low

518 V.H. MORCOS

tip-speed ratios. So, low and high speed wind turbines need high and low starting torques, respectively.

(iv) Recommended design and operating values with the corresponding power, thrust and torque coefficients are given in Tables 1 and 2 for different airfoil blade sections HAWT.

(v) The tip-speed ratio range of variation for cir- cular-arc airfoil blade section is smaller than that of flat-plate and symmetric airfoil blade sections. There- fore, flat-plate and symmetric airfoil blade sections are recommended for wind energy conversion systems. 1.

Simple flat-plate airfoil blade sections are easy to manufacture, and small size wind turbines can use

2. them without marked loss of performance. If strength considerations dictate some profiling for large size wind turbines, symmetric airfoil blade sections are recommended. 3.

NOMENCLATURE

a axial induction factor a' tangential induction factor = ~0/2f~ B number of blades c chord (m)

Co drag coefficient CL lift coefficient Cp turbine power coefficient C O turbine torque coefficient Cx turbine thrust coefficient

dD drag force on the element of blade (N) dL lift force on the element of blade (N) dQ elemental torque which acts on all blade elements

in an annular ring (J) f maximum thickness of circular-arc airfoil (m) P mechanical power produced by the wind turbine

(w) Pw wind power in the upstream wind covering an area

equal to the rotor disk (W) Q turbine torque (J) r radius of blade element (m)

R turbine radius (m) Re local Reynolds number = Wc/v

T axial thrust on the wind turbine (N) U tip-speed = ~R (m s -l)

U~ V tip-speed ratio = tip-speed/wind speed V wind speed (m s -l)

W velocity of the wind relative to the airfoil (m s -a) local flow angle of attack (°)

0 blade angle (°) v kinematic viscosity of air (m2 s -I) p air density at sea-level standard-atmosphere con-

ditions (kg m -3) a turbine or rotor solidity = Bc#rR ~b flow angle (°) co angular velocity of the air just behind the turbine

(rad s -I) wind turbine angular velocity (rad s- ~).

REFERENCES

P. J. Musgrove, Wind energy conversion: recent pro- gress and future prospects. Sol. Wind Technol. 4 (1), 37- 49 (1987). A. Mobarak, A. E1-Mallah and M. A. Scrag El-Din, Wind energy in Egypt. Proc. 4th Int. Conf. for Mech- anical Power En#ineeriny, V-51, Cairo, Egypt, pp. 1-12 (1982). H. H. E1-Tamaly, Utilization of wind energy for the agricultural projects in Egypt. Proc. 2nd Minia Conf. on Eneryy, Minia, Egypt, pp. 445-460 (1988).

4. R.E. Wilson and R. W. Thresher, Electrical energy from the wind. Mech. Engng, January, 60-69 (1984).

5. H. Glauert, Aerodynamic Theory, Vol. IV. W. F. Durand, California Institute of Technology (1943).

6. C. Rohrbach and R. Worobel, Performance charac- teristics of aerodynamically optimum turbines for wind energy generators. American Helicopter Society, Pre- print No. S-996, May (1975).

7. R. E. Wilson, P. B. S. Lissman and S. L. Walker, Aero- dynamic Performance of Wind Turbines. Oregon State University, U.S.A. (1976).

8. R. T. Griffiths, The effect of aerofoil characteristics on windmill performance. Aeronaut. J. July (1977).

9. H. Mansour, M. M. Rayan, R. Comolet and M. A. E1- Naggar, Design optimization of HAWT for Egyptian climatology. Proc. 3rd Int. Con#tess of Fluid Mechanics, Vol. 1, Cairo, Egypt, pp. 169-184 (1990).

10. Wind Speed Reports of Egyptiatt Locations, The Meteorological Authority of Arab Republic of Egypt (1970-1986).

11. J. Shapiro, Principles of Helicopter Engineerin 9. McGraw-Hill, New York (1956).

12. A. Betz, Introduction to the Theory of Flow Machines. Pergamon, New York (1966).

13. R. K. Turton, Principles ofTurbomachinery. Spon, Lon- don (1984).

14. J.D. Anderson, Fundamentals of Aerodynamics, 2nd edn. McGraw-Hill, New York (1991).

15. A. Pope, Basic Win 9 and Airfoil Theory. McGraw-Hill, New York (1951).