This book provides the fundamental concepts required for the development of an efficient smallscale wind turbine. For c
278 58 24MB
English Pages 173 [174] Year 2018
Table of contents :
Preface
Contents
List of Figures
List of Tables
1. Introduction
2. Wind turbines
3. Components of a smallscale wind turbine
4. Aerodynamics of a wind turbine
5. Applying BEM to smallscale wind turbine blade design
6. CFD analysis of wind turbines: Fundamentals
7. CFD analysis of wind turbines: Practical guidelines
8. DiffuserAugmented SmallScale Wind Turbine
9. Unconventional wind energy harvesters
References
Index
Ravi Anant Kishore, Colin Stewart, Shashank Priya Wind Energy Harvesting
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Ravi Anant Kishore, Colin Stewart, Shashank Priya
Wind Energy Harvesting  Micro to SmallScale Turbines
Authors Ravi Anant Kishore Virginia Polytechnic Institute and State University Dept. of Mechanical Engineering 310 Durham Hall Blacksburg VA 24061, USA [email protected] Colin Stewart Virginia Polytechnic Institute and State University Dept. of Mechanical Engineering 310 Durham Hall Blacksburg VA 24061, USA [email protected] Shashank Priya Virginia Polytechnic Institute and State University Dept. of Mechanical Engineering 301 Durham Hall Blacksburg VA 24061, USA [email protected]
ISBN 9781614515654 eISBN (PDF) 9781614514176 eISBN (EPUB) 9781614519799 Library of Congress Control Number: 2018934809 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2018 Walter de Gruyter GmbH, Berlin/Boston Cover image: pedrosala/iStock/Getty Images Typesetting: letex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck www.degruyter.com
Preface For decades, engineers and scientists have recognized wind energy as one of the most promising alternative energy sources and have harvested it using wind turbines of all shapes and sizes. The largest turbines today achieve impressive outputs of several megawatts, enough to power thousands of homes at once. In spite of this progress, the overall contribution of wind energy towards the global electricity supply remains quite small, even with substantial government efforts and incentives. Factors inﬂuencing the low adoption rate are the high wind speeds in excess of 10 m/s needed for the operation of most turbines and the tall towers required to access these winds. Smaller wind turbines that operate closer to the ground are often dismissed as toys because their generally poor efficiencies makes them economically unviable for practical use. Recent developments in the ﬁeld of power electronics have reduced the power requirements of many common devices by orders of magnitude. Our evergrowing reliance on electronic devices now desperately requires smallscale renewable power sources that can make electronic components selfsustainable, especially in remote applications – e.g., wireless sensor nodes installed in difficulttoreach areas. Smallscale wind turbines are an excellent option due to the ubiquity of wind, with the caveat that they need to be efficient at low wind speeds to be practical. Engineers have attempted to build such smallscale efficient turbines but the fundamental design principles are still missing because the tools and techniques used for largescale wind turbines are not directly applicable. Small turbines have quite different aerodynamic behavior. For instance, they operate near the ground where wind is not fully developed but is instead weak and turbulent due to the presence of surrounding obstacles like buildings, trees, etc. Additionally, the design goals of smallscale wind turbines should not only include achieving the highest possible power output but also the slowest possible startup speed. This book provides the fundamental concepts required for the development of an efficient smallscale wind turbine. We have arranged the content in a systematic and exhaustive manner and have attempted to explain all principles with supporting examples wherever possible. It is our hope that this will be an essential resource for students planning to pursue a career in energy technologies. Educators will appreciate that this book reinforces basic design principles by exploring various fabrication techniques for costeffective and efficient turbines, and reinforces theoretical concepts with numerical problems that we have solved in detail. We also provide unsolved problems at the end of most chapters to allow students to practice. Considering these attributes, this book should be a very useful reference material for classes on energy harvesting, sustainable energy, and ﬂuid dynamics taught at both undergraduate and graduate levels. We believe this book will be most beneﬁcial to students, teachers, and professionals who work in the ﬁeld of wind energy, but we also hope that nontechnical readers https://doi.org/10.1515/9781614514176201
VI  Preface
can use this as a guide to fabricate their own efficient micro wind turbine that can charge common household appliances and personal electronics. Even small transitions to renewable energy can make large differences in reducing our global carbon footprint and transforming this planet into a better place to live. Ravi Anant Kishore, Colin Stewart, and Shashank Priya
Contents Preface  V List of Figures  XI List of Tables  XV 1 1.1 1.2 1.3 1.4
Introduction  1 Wind energy  1 A brief history of wind turbines  2 Current state of wind energy  4 Energy policies for wind power  5
2 2.1 2.1.1 2.1.2 2.1.3 2.2 2.3
Wind turbines  7 Classiﬁcation  7 Vertical axis vs. horizontalaxis wind turbines  7 Dragtype and lifttype wind turbines  11 Largescale vs. smallscale wind turbines  14 Need and application of smallscale wind turbines  15 Challenges with smallscale wind turbine designs  15
3 3.1 3.2 3.3 3.4
Components of a smallscale wind turbine  21 Wind turbine rotor  21 Transmission mechanism  25 Generator  25 Auxiliary components  27
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7
Aerodynamics of a wind turbine  31 Froude–Rankine theorem  32 Betz’s law  33 Aerodynamics of a wind turbine rotor  34 Blade element theory  37 Blade element momentum theory  39 Blade losses  43 Buhl correction  44
5 5.1 5.2 5.3
Applying BEM to smallscale wind turbine blade design  47 Iterative scheme for BEM theory  47 Size of the wind turbine  49 Airfoil selection  49
VIII  Contents
5.4 5.5 5.6 5.7
Blade twist angle  51 Number of blades, chord length, and solidity  53 Tapering angle  57 Wind turbine performance  58
6.5.1 6.5.2
CFD analysis of wind turbines: Fundamentals  61 Introduction  61 The need for highﬁdelity modeling techniques  61 Computational ﬂuid dynamics (CFD)  61 Capabilities and tradeoffs  62 Goals of this chapter (and Chapter 7)  64 Continuous model of wind turbine ﬂuid dynamics  64 Discretization techniques  67 Finite difference method (FDM)  68 Finite volume method (FVM)  71 Time discretization  79 Solution methods for linear systems  83 Direct methods  83 Iterative methods  85 Solution methods for the incompressible Navier–Stokes equations  89 Pressure coupling problem  90 Pressurecorrection methods  91
7 7.1 7.1.1 7.1.2 7.1.3 7.2 7.3 7.3.1 7.3.2 7.4 7.4.1 7.4.2 7.4.3 7.5 7.5.1 7.5.2 7.5.3 7.5.4
CFD analysis of wind turbines: Practical guidelines  101 Building the computational domain  101 Turbine geometry and dimensionality  101 Boundary conditions, spacing, and blockage  105 Rotational subdomains  107 Mesh generation and reﬁnement  109 Modeling rotation  113 Multiple reference frame (MRF) model  114 Moving mesh models  114 Choosing a turbulence method  116 ReynoldsAveraged Navier–Stokes (RANS) models  117 ScaleResolving Simulation (SRS)  119 Further reading  120 Computing the solution  121 Spatial discretization  121 Temporal discretization  121 Solver algorithm  122 Parallel computing  122
6 6.1 6.1.1 6.1.2 6.1.3 6.1.4 6.2 6.3 6.3.1 6.3.2 6.3.3 6.4 6.4.1 6.4.2 6.5
Contents  IX
7.5.5 7.6 7.6.1 7.6.2
Convergence criteria  123 Postprocessing  123 Visualization  124 Veriﬁcation and validation  127
8.6 8.7
DiffuserAugmented SmallScale Wind Turbine  131 Flow inside the diffuser without a wind turbine  131 Flow inside the diffuser with a wind turbine  132 Diffuser design optimization  135 Solution strategy  135 Effect of geometrical parameters on the velocity augmentation factor  138 Some other diffuser designs  140 Pros and cons of the diffuser  142
9 9.1 9.2
Unconventional wind energy harvesters  143 Piezoelectric wind turbine  144 Wind power from controlled aerodynamic instability phenomena  147
8 8.1 8.2 8.3 8.4 8.5
References  151 Index  157
List of Figures Fig. 1.1 Fig. 1.2
Fig. 1.3
Fig. 1.4
Fig. 2.1
Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 Fig. 2.7
Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 Fig. 3.7 Fig. 4.1 Fig. 4.2 Fig. 4.3
Wind ﬂowing through a cylindrical differential volume.  1 (a) 1000yearold windmills in Nashtifan, Iran, still in use today. © Tasnim News Agency / CC BY 4.0. (b) Persian windmill. © user: Kaboldy / Wikimedia Commons / CC BYSA 3.0. (c) Chinese windmill with ﬂapping sails. © user: Carl von Canstein / Wikimedia Commons / CC BYSA 3.0. (d) Remains of an old verticalaxis windmill at Camargue, France. © user: Maarten Sepp / Wikimedia Commons / CC BYSA 4.0. (e) Dutch windmill in Golden Gate Park, San Francisco © user: BrokenSphere / Wikimedia Commons / CC BYSA 3.0. (f) Mediterranean tower mill Licensed under CC0 1.0.  3 (a) An American windmill (Western windmill). © Ben Franske / Wikimedia Commons / CC BYSA 4.0. (b) A wind farm utilizing the American windmills for water pumping. © user: denisbin / Flickr / CC BYND 2.0.  4 Total installed wind power capacity (MW) and world wind power market growth rate from 1996 to 2012. Information taken from reference [3].  5 Verticalaxis wind turbine designs. (a) Savonius rotor. © user: Frommcc / Wikimedia Commons / CC BYSA 3.0. (b) Curvedblade Darrieus rotor. © Dietrich Krieger / Wikimedia Commons / CC BYSA 3.0. (c) Straightblade Darrieus rotor. © Hannes Grobe/AWI / CCBY3.0.  8 A horizontalaxis wind turbine. © user: KoeppiK / Wikimedia Commons / CC BYSA 3.0.  9 Typical conﬁguration of a modern largescale HAWT. © Office of Energy Efficiency and Renewable Energy, US Department of Energy.  10 Drag and lift components of the aerodynamic force.  11 Simpliﬁed model for Savonius wind turbine.  11 Trend in lift and drag forces calculated for the wing in Example 2.2.  14 Inﬂuence of Reynolds number on airfoil behavior (NACA0012). Reprinted from [21] with permission. Copyright © 1988 ASME.  17 A smallscale wind turbine rotor (SSWT).  21 CAD model of a smallscale wind turbine blade.  23 Some basic terminologies related to an airfoil. “Wing proﬁle nomenclature” by Olivier Cleynen / Wikimedia Commons / Public Domain.  23 Velocity triangle at the section of a wind turbine blade at radius r.  24 The transmission mechanism of a smallscale wind turbine.  25 A double rotorsingle stator axial ﬂux generator design for a smallscale wind turbine.  27 Tail used in a smallscale wind turbine.  27 The streamtube representing wind ﬂow through the actuator disc.  31 Power coefficient of a wind turbine versus the farupstream and fardownstream velocity ratio.  34 A CAD model of a smallscale wind turbine blade illustrating the airfoil elements.  35
https://doi.org/10.1515/9781614514176202
XII  List of Figures
Fig. 4.4 Fig. 4.5 Fig. 4.6 Fig. 4.7 Fig. 4.8 Fig. 4.9
Flow over an airfoil shown by (a) velocity contours and (b) pressure contours. From [28] licensed under CC BY 4.0.  35 Lift and drag forces on an airfoil.  36 Lift and coefficient versus angle of attack, NACA 0012 at Re = 106 .  36 Blade element theory divides wind turbine blades into differential elements.  37 Velocity triangle and aerodynamic forces at a differential blade element.  38 Flow around a differential blade element of the wind turbine.  40
Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Fig. 5.9 Fig. 5.10 Fig. 5.11 Fig. 5.12
Lift and drag coefficients of the NACA 0012 airfoil at Re = 4 × 104 .  50 Iterative scheme to optimize blade parameters at design point wind speed.  52 Blade twist angle at various constant angles of attack.  53 Power coefficient vs. tip speed ratio at various constant angles of attack.  53 Power coefficient vs. tip speed ratio at solidity σ = 10%.  55 Power coefficient vs. tip speed ratio at solidity σ = 20%.  55 Power coefficient vs. tip speed ratio at solidity σ = 30%.  56 Power coefficient vs. tip speed ratio at solidity σ = 40%.  56 Power coefficient vs. tip speed ratio at different positive tapering angles.  57 Power coefficient vs. tip speed ratio at different negative tapering angles.  58 Iterative scheme to predict the power coefficient at variable wind speeds.  59 Performance of a 1 W wind turbine over the operating range of wind speeds.  60
Fig. 6.1 Fig. 6.2
Streamlines of ﬂow around a smallscale wind turbine generated using CFD.  62 A threedimensional rectangular domain Ω. The base of the turbine’s support tower is located at the origin.  65 (a) Onedimensional grid of N uniformly spaced nodes and (b) twodimensional grid of N × M nodes with uniform spacing in each direction.  68 Numerical solutions to the 1D Poisson equation in Example 6.1.  71 A square problem domain divided into four irregular ﬁnite volume cells with ﬂuxes between them.  72 A 2D control volume for node P. Neighboring nodes in the north, south, east, and west directions are denoted N, S, E, and W, respectively, and their cell faces are denoted n, s, e, w.  72 Comparison between Upwind and Quick interpolation schemes showing diffusive errors and oscillations.  74 (a) A onedimensional ﬁnite volume mesh and (b) deﬁnitions of cell notations.  75 Solutions to the steady 1D convectiondiffusion equation in Example 6.2 calculated using the ﬁnite volume method.  78 Time integration methods: (a) Explicit (forward) Euler, (b) implicit (backwards) Euler, (c) midpoint, and (d) Crank–Nicolson.  80 Instantaneous solutions to the unsteady 1D convectiondiffusion equation in Example 6.3 calculated using the ﬁnite volume method.  82
Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6
Fig. 6.7 Fig. 6.8 Fig. 6.9 Fig. 6.10 Fig. 6.11
Fig. 7.1
Fig. 7.2
(a) A verticalaxis wind turbine. “Windgenerator antarktis hg,” by Hannes Grobe/AWI, Wikimedia Commons. CCBY3.0. (b) The 2D representation of a similar turbine (albeit with six blades instead of three) in a mesh used for CFD.  103 Pressure contours and streamlines around a small diffuseraugmented wind turbine modeled using an actuator disk in a 2D axisymmetric domain.  104
List of Figures 
Fig. 7.3 Fig. 7.4 Fig. 7.5
Fig. 7.6 Fig. 7.7 Fig. 7.8 Fig. 7.9
Fig. 7.10 Fig. 7.11 Fig. 7.12 Fig. 7.13
Fig. 7.14 Fig. 7.15 Fig. 7.16
Fig. 7.17
Fig. 7.18
Fig. 7.19 Fig. 7.20
Fig. 7.21
Fig. 8.1 Fig. 8.2 Fig. 8.3
XIII
A periodic domain with rotational and stationary subdomains viewed from the (a) side and (b) top.  105 A 3D computational domain of a horizontalaxis wind turbine labeled with boundary conditions.  106 Rotating blade meshes. (a) Slice of a computational domain using overset meshes around the tower and blades. The nearsurface meshes are cylindrical and shown in blue with edge cells shown in red. Reprinted from [67] with permission. Copyright © 2009 John Wiley & Sons, Ltd. (b) Nonoverlapping subdomains with a rotating inner subdomain.  108 Components of (left) 2D and (right) 3D computational meshes.  109 (a) Regular and (b) deformed structured grids.  110 (a) Unstructured grid of tri cells with nodes numbered. (b) Common cell types used in unstructured grids.  110 An example of meshing around a horizontalaxis wind turbine. (a) 2D slice in the ﬂow direction. (b) Turbine surface mesh. (c) 2D slice across the chord of a rotor blade. Reprinted from [68] with permission. Copyright © 2013 John Wiley & Sons, Ltd.  112 Key metrics for characterizing mesh quality.  113 Energy cascade of turbulent ﬂow.  117 Reynolds decomposition of velocity magnitude into timeaveraged and randomly ﬂuctuating components.  118 Comparison of a turbulent wake behind a cylinder calculated using RANS and LES methods. Contours show levels of vorticity. Reprinted from [84] with permission. Copyright © 2003 Elsevier Science Inc.  119 Wallmodeled LES (WMLES).  120 Characteristic power curve of a wind turbine. Betz limit denotes the theoretical maximum power coefficient.  124 Velocity vectors illustrate differences in ﬂow around two VAWT designs simulated in 2D: (a) one with blade overlap and (b) one without. Reprinted from [91] with permission. Copyright © 2012 Elsevier Ltd.  125 Pressure contours and velocity streamlines around a uniquely shaped VAWT simulated in 2D as it rotates through rotor orientations of (a) 0°, (b) 30°, (c) 90°, and (d) 270°. Reprinted from [92] with permission. Copyright © 2011 Elsevier Ltd.  125 Vorticity isosurfaces from a LES model of a turbine with a 112 m rotor diameter at (a) t = 30 s and (b) t = 150 s after start. Reprinted from [93] with permission. Copyright © 2011 American Institute of Physics.  126 Isosurfaces of Qcriterion show vortical structures shedding from a HAWT. Reprinted from [86] with permission. Copyright © 2011 Elsevier Ltd.  126 Vortex shedding from a rotating VAWT blade shown by contours of vorticity magnitude. Reprinted from [94] with permission. Copyright © 2012 Elsevier Ltd.  126 (a) Timestep and (b) mesh reﬁnement studies for a VAWT indicates that the calculated torque is insensitive to meshes with greater than 450,000 cells (M2) and time steps smaller than 0.0002 s. Reprinted from [96] with permission. Copyright © 2015 Elsevier Ltd.  127 Diffuser illustrating onedimensional ﬂow.  131 An initial diffuser design for a smallscale wind turbine.  136 Twodimensional computational domain with boundary conditions.  137
XIV  List of Figures
Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 8.8 Fig. 8.9 Fig. 8.10 Fig. 8.11 Fig. 8.12
Fig. 9.1
Fig. 9.2 Fig. 9.3
Fig. 9.4 Fig. 9.5
Twodimensional grid system.  137 Mesh independence test. Change in result by increasing the element count from 26,500 to 59,500 is less than 0.5%.  137 Effect of converging section length L 1 and halfcone angle θ 1 on velocity augmentation factor u/u 0 for a diffuser with θ 2 = 10° and L 2 = 1.0D.  138 Effect of diverging section length L 2 and halfcone angle θ 2 on velocity augmentation factor u/u 0 (L 1 = 0.125D and halfcone angle θ 1 = 15°).  139 Velocity contours at different θ 2 (L 2 = 1.0D, L 1 = 0.125D, θ 1 = 15°).  139 Velocity contours at different L 2 with θ 2 = 12°, L 1 = 0.125D, θ 1 = 15°.  139 A diffuser with a ﬂange at the exit.  141 Wind ﬂow mechanism around a ﬂanged diffuser. Reprinted from [103] with permission. Copyright © 2008 Elsevier Ltd.  141 A diffuser with a nonlinear proﬁle. Reprinted from [98] with permission. Copyright © 2007 Elsevier Ltd.  141 Piezoelectric wind energy harvesters. (a) Concept of a piezoelectric windmill. Reprinted from [116] licensed under CC BY 4.0. (b, c) Test prototypes of contactless wind turbines. Reprinted from [14] with permission. Copyright © Springer Science+Businees Media, LLC 2011.  144 Piezoelectric wind turbine developed at Virginia Tech.  145 (a) A piezowindmill. Reprinted from [118] with permission. Copyright © 2016 Elsevier Ltd. (b) A horizontal piezoelectric wind turbine. Reprinted from [119] with permission. Copyright © 2016 Elsevier Ltd.  146 Aeroelastic phenomenon for wind energy harvesting: (a) vortex shedding, (b) ﬂutter, (c) galloping, and (d) ﬂapping leaf. Figure redrawn from [120].  147 Schematic of a wind energy harvester based on wake galloping. Redrawn from [105].  148
List of Tables Tab. 1.1
Cumulative wind power capacity outlook from 2008 to 2012 [3, 4].  5
Tab. 2.1 Tab. 2.2 Tab. 2.3 Tab. 2.4
Main differences between vertical and horizontalaxis wind turbines [8].  10 Lift and drag coefficient data for Example 2.1.  13 Lift and drag forces on the given wing in Example 2.1.  13 Some of the currently available smalltomidscale horizontalaxis wind turbines. Information taken from NE100 S: [12], NE200 S: [13], NE300 S: [14], Energy Ball V100: [15], Micro Wind Turbine: [16], Bornay 600: [17], AC 120/240: [18], µF500: [19], Superwind 350: [20].  16
Tab. 5.1
Chord length (cm) at different solidity and number of blades.  54
Tab. 8.1
Diffuser design parameters.  140
https://doi.org/10.1515/9781614514176203
1 Introduction 1.1 Wind energy Wind is the largescale movement of air caused by differences in atmospheric pressure. The uneven heating of the earth by the sun, primarily between the equator and the poles, leads to these pressure differences and causes air to rush from highpressure regions to lowpressure regions. However, wind does not ﬂow in a straight path; the effects of Coriolis and centrifugal forces due to rotation of the earth cause wind to deﬂect. The local wind condition is ﬁnally determined by the geographical features in and around the given region, such as plains, mountains, forests, oceans, rivers, and buildings. It should also be noted that although the rotational forces and regional geography remain more or less constant over small periods of time, the solar heating changes throughout the year, resulting in the seasonal change of wind ﬂow. Intuitively, one can expect that wind power is a function of wind speed. Mathematically, the power P of wind ﬂowing at a speed u ∞ through a region of crosssectional area A is given by 1 (1.1) P = ρAu 3∞ , 2 where ρ is the density of air. It is interesting to note from Eq. (1.1) that wind power is proportional to the cube of wind speed. To derive this equation, let us consider a disc with crosssectional area A through which wind ﬂows with a constant velocity u ∞ . The air that ﬂows through the disc in an inﬁnitesimal time step dt travels a distance dx = u ∞ dt, creating a cylinder of volume dV = Adx as shown in Fig. 1.1. The mass dm of air inside the cylinder can be calculated using dm = ρdV = ρAu ∞ dt .
(1.2)
The kinetic energy of this air is equal to 1 1 1 (1.3) dmu 2∞ = (ρAu ∞ dt)u 2∞ = ρAu 3∞ dt . 2 2 2 Assuming that the change in internal and potential energies of the air are negligible, the power of the wind can be calculated as the rate of change of its kinetic energy: KE =
P=
dKE . dt
dx
u∞
A
Fig. 1.1: Wind ﬂowing through a cylindrical differential volume. https://doi.org/10.1515/9781614514176001
(1.4)
2  1 Introduction
We arrive at the desired power equation by substituting Eq. (1.3) into Eq. (1.4): P=
1 ρAu 3∞ . 2
(1.5)
Example 1.1. What is the power of the wind ﬂowing through a circular crosssection of diameter 5 m with velocity 10 m/s normal to the crosssection? Assume the density of air is 1.225 kg/m3 . Solution. The area of the circular crosssection is given by A = π (D/2)2 , where D is the diameter. Setting diameter D = 5 m yields A = π (5/2)2 m2 = 19.64 m2 . The power is then calculated using Eq. (1.5): P = (1/2) (1.225 kg/m3 ) (19.64 m2 ) (10 m/s)3 = 12 kW . Example 1.2. What is the average power output of a wind turbine with a 5 m rotor diameter, deployed on a location with average wind velocity 10 m/s? Assume the net efficiency of the wind turbine is 25%. Solution. The total available wind power is again given by Eq. (1.5). Following Example (1.1), we ﬁnd P = 12 kW. The power output of the wind turbine is then given by Pout = ηP, where η denotes the wind turbine efficiency. Thus, Pout = ηP = (0.25) (12 kW) = 3 kW .
1.2 A brief history of wind turbines Wind energy is one of the most abundant renewable energy resources on Earth and has been targeted as a source of energy for centuries. It is believed that King Hammurabi of Babylon used windpowered scoops in 1700 B.C. to irrigate the plains of Mesopotamia [1]. Wind powered machines were often used to mill or grind grain, which is why wind turbines are still commonly referred to as “windmills” even though they are rarely used for this purpose today. Figure 1.2 displays some of the primitive windmills used by early human civilizations. Most of the world’s oldest windmills had a vertical axis of rotation and a simple construction, as shown in Fig. 1.2a–d. Braided mats and sails were used to catch the wind, generating a torqueproducing drag force that caused the windmills to turn. The verticalaxis design allowed windmills to harvest wind coming from any direction. It also allowed millstones to be directly attached to the vertical drive shaft without any intermediate gearing or mechanism to redirect the rotational movement. Horizontalaxis wind turbines are a newer invention. Though the ﬁrst documentation of a horizontalaxis wind turbine dates back to the 12th century, the theoretical descriptions regarding the driving power of horizontalaxis devices, i.e., lift forces on the blades, were investigated only during the beginning of the 20th century [1]. One of the most
1.2 A brief history of wind turbines  3
(a)
(d)
(b)
(e)
(c)
(f)
Fig. 1.2: (a) 1000yearold windmills in Nashtifan, Iran, still in use today. © Tasnim News Agency / CC BY 4.0. (b) Persian windmill. © user: Kaboldy / Wikimedia Commons / CC BYSA 3.0. (c) Chinese windmill with ﬂapping sails. © user: Carl von Canstein / Wikimedia Commons / CC BYSA 3.0. (d) Remains of an old verticalaxis windmill at Camargue, France. © user: Maarten Sepp / Wikimedia Commons / CC BYSA 4.0. (e) Dutch windmill in Golden Gate Park, San Francisco © user: BrokenSphere / Wikimedia Commons / CC BYSA 3.0. (f) Mediterranean tower mill Licensed under CC0 1.0.
popular early horizontalaxis wind turbines was the tower mill, shown in Fig. 1.2f, which were found in southern Europe. The ﬁrst written evidence of such windmills dates back to the 13th century [1]. Other types of horizontalaxis wind turbines existed in different parts of the world (mainly in the Occident) during different periods of time. Examples include the Post windmill (1100s), Wipmolen Dutch (1400s), Dutch smock mill (1500s, Fig. 1.2e), Paltrock mill (1600s), and Gallery smock mill (1700s). A brief description of these windmills can be found in [1]. However, none of these historical horizontalaxis wind turbines gained as much popularity as the American farm windmill (sometimes also called the Western mill). These windmills were developed in the mid19th century mainly to provide drinking water to people and cattle in North America. Moreover, they were used to assure the water supply for the steam locomotives of the new railways expanding into the West [1]. Figure 1.3a shows an American windmill. The most important component of this windmill is the rotor, which is also called “rotor rosette” because of its structural design. Its diameter varies between 3 m and
4  1 Introduction
(a)
(b)
Fig. 1.3: (a) An American windmill (Western windmill). © Ben Franske / Wikimedia Commons / CC BYSA 4.0. (b) A wind farm utilizing the American windmills for water pumping. © user: denisbin / Flickr / CC BYND 2.0.
5 m and it has more than 20 metal sheet blades. It also consists of a tail that allows the rotor to turn automatically so that it always faces the incident wind. It uses a crank shaft to drive a piston pump. The American mills are still in existence and several of them are installed with a nearly unchanged design in Australia, Argentina, and the USA. Figure 1.3b depicts one such wind farm utilizing American mills for water pumping.
1.3 Current state of wind energy The fourth edition of the Global Wind Energy Outlook, released on Nov 14, 2012 by Greenpeace International and the Global Wind Energy Council, states that wind power currently shares about 3.5% of global electricity demand and could reach up to 12% by 2020 [2]. Figure 1.4 shows the global cumulative installed wind power capacity over the last 17 years [3]. At the end of 2012, the worldwide total wind power capacity was 282 gigawatts (GW), showing a growth of about 18.7% (44 GW) over the preceding year. It is important to note that although 2012 created a new record in total installed wind power capacity, the wind market has cooled down. If we look at the annual growth rate, there was a steady increase from 2004 to 2009, which peaked at 32.1%. However, since then the growth has decreased substantially. In 2012, global growth went down to 18.7%, which is the lowest rate in the last two decades, according to a report by the World Wind Energy Association (WWEA) [4]. Table 1.1 presents the cumulative wind power capacity from 2008 to 2012 in the top ten countries and worldwide [3, 4]. The data indicates that even though the global wind power capacity exhibited a low growth rate (18.7%) in 2012, it nevertheless increased over 133.5% during the last ﬁve years. It is also interesting to note that 73.7%
1.4 Energy policies for wind power 
40% 35%
250,000
30% 200,000
25%
150,000
20% 15%
100,000
Capacity Growth
50,000 0
Growth
Wind power capacity (MW)
300,000
5
10% 5%
1996
2000
2005
2010
0%
Fig. 1.4: Total installed wind power capacity (MW) and world wind power market growth rate from 1996 to 2012. Information taken from reference [3]. Tab. 1.1: Cumulative wind power capacity outlook from 2008 to 2012 [3, 4].
Country China USA Germany Spain India UK Italy France Canada Portugal Worldwide
Total capacity (megawatt)
Growth rate
2008
2009
2010
2011
2012
2012
12,210 25,237 23,897 16,689 9587 3195 3736 3404 2369 2862 120,986
25,810 35,159 25,777 19,149 11,807 4092 4850 4574 3319 3357 159,837
44,733 40,180 27,215 20,676 13,065 5203 5797 5660 4008 3702 197,040
62,364 46,919 29,075 21,673 15,880 6018 6737 6640 5265 4093 238,035
75,564 60,007 31,332 22,796 18,421 8445 8144 7196 6200 4525 282,482
21.2% 27.9% 7.8% 5.2% 16.0% 40.3% 20.9% 8.4% 17.8% 10.6% 18.7%
of the total power capacity (282,430 MW) in 2012 was contributed by ﬁve countries: China, USA, Germany, Spain, and India. China’s wind power capacity continued to grow at the rate of over 21% in 2012. The United States also gained momentum and displayed an annual growth rate of 27.9% in 2012, which is the highest growth rate in the last three consecutive years. The installed wind power capacity in the USA reached up to 60,007 megawatts by the end of 2012.
1.4 Energy policies for wind power One of the biggest hindrances to the expansion of wind energy harvesting technologies is the lack of awareness among the people. To combat this, a number of state
6  1 Introduction
and federal initiatives were recently created. The U.S. Department of Energy (DOE) has launched the Wind and Water Power Program to accelerate the deployment of wind power sources. The Wind Program, which is part of the Wind and Water Power Program, endeavors to promote clean, affordable, and reliable domestic wind power. Additionally, more than 20 states in the USA have established renewable portfolio standards that require electricity suppliers to produce a certain percentage of their power from renewable sources. More than 15 states have established renewable energy funds that offer ﬁnancial incentives and several other provide support for wind energy development. Detailed information can be found at the U.S. Department of Energy website (http://energy.gov/eere/renewables/wind). In addition, the federal government has issued several taxbased policy incentives to stimulate public acceptance of wind energy. According to a report by the U.S. Department of Energy’s Wind and Water Power Program published in May 2011, the federal government allows owners of qualifying renewable energy facilities to receive tax credits of 2.2 cents for each kilowatthour (kWh) of electricity generated by the facility over a tenyear period. The owners of small wind turbines (100 kW or less) are eligible to receive tax credits worth 30% of the value of the facility. In addition, DOE offers loan guarantees to help wind power ﬁrms deploy innovative, clean energy technologies that reduce, avoid, or sequester carbon dioxide and other emissions. Detailed information about the policy incentives and other beneﬁts related to wind energy can be found at the website of the U.S. Department of Energy: Energy Efficiency and Renewable energy (www.eere.energy.gov/ informationcenter).
Exercises Q 1.1. What is the power of the wind ﬂowing through a circular crosssection of diameter 3 m with velocity 7 m/s normal to the crosssection? Q 1.2. What is the average power output of a wind turbine having 3 m diameter, deployed on a location with average wind velocity 7 m/s? Assume the net efficiency of the wind turbine is 20%. Q 1.3. What would be the diameter of a wind turbine required to produce 10 watts of electrical power at an average wind speed of 4 m/s? Assume the net efficiency of the wind turbine is 25%.
2 Wind turbines 2.1 Classiﬁcation There are three broad ways to classify wind turbines: (i) orientation of the axis of rotation (vertical or horizontal), (ii) component of aerodynamic forces that powers the wind turbine (lift or drag), and (iii) energygenerating capacity (micro, small, medium, or large).
2.1.1 Vertical axis vs. horizontalaxis wind turbines Categorization of wind turbines according to the orientation of the axis of rotation leads to two types: VerticalAxis Wind Turbines (VAWTs) and HorizontalAxis Wind Turbines (HAWTs). As the names suggest, the rotor of a VAWT rotates perpendicular to the ground while that of a HAWT spins parallel to the ground. Most early wind turbines were of verticalaxis type because they were relatively simple to construct (especially for milling) and did not require any mechanism to orient themselves towards the direction of the wind. In spite of these attributes, none of the old designs survived to the present day. Currently, there are three popular VAWT designs: (a) Savonius, (b) Curvedblade Darrieus, and (c) Straightblade Darrieus, as shown in Fig. 2.1. Savonius turbines are dragtype while Darrieus turbines are lifttype. The next section will provide greater detail on these designs. In principle, Savonius rotors normally have two cups or halfdrums attached to a central shaft in opposing directions, as shown in Fig. 2.1a. The drum, which is against the wind ﬂow, catches the wind and creates a moment along the axis. The aerodynamic torque by the ﬁrst drum rotates the rotor and brings the opposing drum against the wind ﬂow. The second drum now catches the wind, causes the rotor to rotate even further, and thus completes a full rotation. This process continues until there is sufficient wind to turn the axial shaft, which is normally connected to a pump or a generator [5]. Savonius turbines generally have poor efficiency (less than 25%), which make them less commercially successful compared to HAWT, but they also have favorable attributes for low power applications: simple construction with low cost, high startup and dynamic torque, wind acceptance from any direction, low noise and angular velocity in operation, and reduced wear on moving parts [6]. The Darrieustype VAWTs consist of two or more blades that are attached to a vertical central shaft. These blades can be curved or they can be straight, as shown in Fig. 2.1. Irrespective of the curvature, the blades always have an airfoil proﬁle, which creates aerodynamic lift when exposed to the incident wind. This phenomenon creates torque about the axis and causes the central shaft to rotate, which ultimately runs the generator to produce electricity. The curvedblade Darrieus VAWTs have lower https://doi.org/10.1515/9781614514176002
8  2 Wind turbines
(a)
(b)
(c)
Fig. 2.1: Verticalaxis wind turbine designs. (a) Savonius rotor. © user: Frommcc / Wikimedia Commons / CC BYSA 3.0. (b) Curvedblade Darrieus rotor. © Dietrich Krieger / Wikimedia Commons / CC BYSA 3.0. (c) Straightblade Darrieus rotor. © Hannes Grobe/AWI / CCBY3.0.
bending stress in the blades compared to straightblade Darrieus VAWTs and therefore are more commercially successful [5]. However, for smallscale power production, straightbladed Darrieus VAWTs are more popular because of their simple blade design [6]. It has been found that constant pitch straightbladed Darrieus VAWTs cannot selfstart [7], so straightbladed Darrieus VAWTs may feature a variable pitch angle. Although this allows the turbines to overcome the starting torque problem, it also complicates their design, making them quite impractical for smallscale power generation [5]. At present, horizontalaxis wind turbines (HAWTs) are the most popular among all windmill designs. This is primarily because HAWTs generally have much higher efficiency than VAWTs. The maximum mechanical efficiency of a modern HAWT has been reported to be up to 45% to 50% while that of an efficient VAWT normally lies below 40% [8] (mechanical efficiency of a Savoniustype VAWT is even lower, normally below 25% [6]). Figure 2.2 shows a horizontalaxis wind turbine (HAWT). The rotor shaft of a HAWT is positioned in the horizontal direction, i.e., parallel to the ground. The electric generator that is connected to the turbine rotor via the primary and secondary shafts is stored inside a nacelle box at the top of the tower. HAWTs are lifttype wind turbines and, therefore, are very sensitive to changes in blade proﬁle, design, and surface roughness. Another limitation of HAWTs is that they cannot catch the wind from all directions; they need a special mechanism to turn the rotor so that they always face the wind. This is probably one of the main reasons that the ﬁrst HAWTs were unsuccessful. In fact, the American windmill was the ﬁrst HAWT that had a fully automatically controlled yaw system. The yaw system is an essential component of a HAWT and is responsible for the orientation of the wind turbine rotor towards the wind. In smallsized HAWTs, the yaw system comprises a simple roller
2.1 Classiﬁcation
 9
Fig. 2.2: A horizontalaxis wind turbine. © user: KoeppiK / Wikimedia Commons / CC BYSA 3.0.
bearing connected between the tower and the nacelle. A tail with a ﬁn at the end is mounted on the back of the nacelle to produce a corrective moment and turn the wind turbine rotor into the wind. This type of yaw system is called a passive yaw system. The largescale (megawatt) HAWTs, however, need an active yaw mechanism. The active yaw systems are normally equipped with a sensor that detects the wind direction, and a servo motor that produces the required torque to rotate the nacelle against the stationary tower. This book is primarily focused on HAWT, so more details about these kinds of turbines will be given in later sections. Figure 2.3 details all the major components of a largescale modern HAWT. A HAWT, in general, consists of a rotor, a gearbox, a generator, and a yaw system. The rotor of a HAWT most commonly includes two to three blades connected together with a hub. The hub is attached to a main shaft (sometimes also called the primary shaft or low speed shaft), which passes through bearings and connects to a gear train. The gear train ampliﬁes the rotational speed and provides higher rpm to a secondary shaft (sometimes also called the highspeed shaft). The secondary shaft drives a generator, which produces electricity. The gearbox, the primary and secondary shafts, and the generator are contained inside a nacelle box. The nacelle box also contains a yaw system to orient the rotor, and a heat exchanger to cool down the generator. Table 2.1 summarizes some of the most important differences between vertical and horizontalaxis wind turbines.
10  2 Wind turbines
Pitch
Lowspeed shaft Rotor
Gear box Generator Anemometer
Wind
Controller Brake
Yaw drive Wind vane Highspeed shaft
Yaw motor Blades
Nacelle
Tower
Fig. 2.3: Typical conﬁguration of a modern largescale HAWT. © Office of Energy Efficiency and Renewable Energy, US Department of Energy.
Tab. 2.1: Main differences between vertical and horizontalaxis wind turbines [8].
Blade proﬁle Need for yaw mechanism Possibility of pitch mechanism Tower Guy wires Noise Blade area Generator position Blade load Selfstart Tower interference Foundation Overall structure
Straightblade Darrieus VAWT
Curvedblade Darrieus VAWT
HAWT
Simple No Yes Yes Optional Low Moderate On ground Moderate No Small Moderate Simple
Complicated No No No Yes Moderate Large On ground Low No Small Simple Simple
Complicated Yes Yes Yes No High Small On ground High Yes Large Extensive Complicated

11
Re su lta nt fo rce ,
F
2.1 Classiﬁcation
Angle of attack, α
Lift, L
Drag, D
Incoming wind, u∞ Fig. 2.4: Drag and lift components of the aerodynamic force.
Wind
1
CD1
ω
2
CD2 Fig. 2.5: Simpliﬁed model for Savonius wind turbine.
2.1.2 Dragtype and lifttype wind turbines When a ﬂat object is exposed to an incident wind, it encounters a surface force, commonly known as the aerodynamic force (Fig. 2.4). The component of the aerodynamic force that is parallel to the ﬂow direction is called drag while the one, perpendicular to the direction of the wind, is called lift. The magnitude of the drag force D and the lift force L are determined by the following expressions: ρ D = CD Au 2∞ , 2 ρ L = CL Au 2∞ , 2
(2.1) (2.2)
where A is the reference area, which is often the planform area of the object, ρ is the density of air, u ∞ is the upstream wind speed, and CD and CL are proportional constants called drag and lift coefficients, respectively. The constants depend on the “aerodynamic quality” of the object: the better the aerodynamic quality of the object, the higher is the lift coefficient CL but the lower is the drag coefficient CD , and thus the higher the lift force but lower the drag force. The Savoniustype rotor is a dragbased wind turbine because the drag component of aerodynamic force causes its rotation. We can estimate the torque and mechanical power output of a Savonius rotor using a simpliﬁed model, shown in Fig. 2.5. Let us assume that the rotor has mean radius R and it is rotating with an angular speed ω. The circumferential velocity u of the rotor at the mean radius is equal to u = ωR .
(2.3)
12  2 Wind turbines
The average relative velocities of the wind u r1 and u r2 at the ﬁrst and second rotating drums are given by following expressions: u r1 = u ∞ − u ,
(2.4)
u r2 = u ∞ + u .
(2.5)
The resulting drag forces D1 and D2 on the rotating drums are given as ρ ρ ρ D1 = CD1 Au 2r1 = CD1 A(u ∞ − u)2 = CD1 Au 2∞ (1 − 2 2 2 ρ ρ ρ D2 = CD2 Au 2r2 = CD2 A(u ∞ + u)2 = CD2 Au 2∞ (1 + 2 2 2
u 2 ) , u∞ u 2 ) , u∞
(2.6) (2.7)
where A denotes the projected area of the drums. The aerodynamic torque along the central axis is calculated as τ = (D1 − D2 ) R =
ρ 2 u 2 u 2 ) − CD2 (1 + ) ] . Au ∞ R [CD1 (1 − 2 u∞ u∞
(2.8)
The mechanical power by the turbine can then be determined using the following equation: P = τω =
ρ 2 u 2 u 2 Au ∞ ωR [CD1 (1 − ) − CD2 (1 + ) ] . 2 u∞ u∞
Using Eq. (2.3), P=
u 2 u 2 ρ 2 ) − CD2 (1 + ) ] . Au ∞ u [CD1 (1 − 2 u∞ u∞
This can be rewritten as P=
ρ ρ 1 u u 2 u 2 [CD1 (1 − ) − CD2 (1 + ) ]} = CP (2A)u 3∞ , (2.9) (2A) u 3∞ { 2 2 u∞ u∞ u∞ 2
where the expression 1 u u 2 u 2 [CD1 (1 − ) − CD2 (1 + ) ] 2 u∞ u∞ u∞ is the power coefficient CP . We will study a more formal deﬁnition of this factor in the next chapter. It can be noted from Eq. (2.9) that the mechanical power produced by a Savonius turbine is directly proportion to the total projected area by the rotor and the cube of upstream wind speed u ∞ . HAWTs and Darrieus VAWTs are liftbased wind turbines, i.e., they extract power from the wind mainly using the lift component of the aerodynamic force on their blades. The analytical model to predict the power by a Darrieus wind turbine is quite complex. Normally, people use Computational Fluid Dynamics (CFD) to analyze the characteristics of a Darrieus rotor. There are, however, several analytical theories and models for HAWTs. Some of the major models, which are currently in practice, are Actuator Disc Theory, Rotor Disc Theory, Vortex Cylinder Model of the Actuator Disc, and Rotor Blade Theory. We will discuss some of these theories brieﬂy in Chapter 4.
2.1 Classiﬁcation
 13
Example 2.1. Calculate the lift and drag forces acting on a wing of planform area 1 m2 exposed to a wind speed of 5 m/s at an angle of attack of 2°. Repeat the calculations of lift and drag forces at the angles of attack of 4, 6, 8, and 10°. Use the lift and drag coefficients given in Table 2.2 below. Assume the density of air as 1.225 kg/m3 . Tab. 2.2: Lift and drag coefficient data for Example 2.1. Angle of attack, α
Lift coefficient, C L
Drag coefficient, C D
2° 4° 6° 8° 10°
0.37 0.54 0.69 0.85 0.97
0.0145 0.0152 0.0197 0.0286 0.0456
Solution. It is given that the wing’s planform area A = 1 m2 , density of the air ρ = 1.225 kg/m3 , and wind speed u ∞ = 5 m/s. Using the lift and drag coefficients given in Table 2.2 at various angles of attack, we can calculate the lift and drag forces on the given wing using Eqs. (2.1) and (2.2). Table 2.3 shows the values of the lift and drag forces. Tab. 2.3: Lift and drag forces on the given wing in Example 2.1. Angle of attack, α
Lift coefficient, C L
Drag coefficient, C D
Lift (N)
Drag (N)
2° 4° 6° 8° 10°
0.37 0.54 0.69 0.85 0.97
0.0145 0.0152 0.0197 0.0286 0.0456
5.67 8.27 10.57 13.02 14.85
0.22 0.23 0.30 0.44 0.70
Example 2.2. Calculate the lift over drag ratio using data obtained in Example 2.1. What do you notice about the trend of lift and drag forces with increase in angle of attack? How does the lift over drag ratio vary with angle of attack? Solution. Figure 2.6 below shows the trend in lift and drag forces in the given range of angle of attack. It can be seen that both lift and drag forces continuously increase with increase in the value of angle of attack. However, the lift over drag ratio increases with angle of attack from 2° to 4°, attains a maximum value at 4°, and then decreases with a further increase in angle.
16
40
12
30
8
20 Lift Drag Lift to drag ratio
4
10
0
Lift to drag ratio
Lift, Drag (N)
14  2 Wind turbines
0 0
2 4 6 8 10 Angle of attack, α (degrees)
12
Fig. 2.6: Trend in lift and drag forces calculated for the wing in Example 2.2.
Example 2.3. What is the average mechanical power output of a Savonius wind turbine having height and diameter 1 m, deployed on a location with average wind velocity 10 m/s? Assume the power coefficient of the wind turbine is 15%. Solution. Using Eq. (2.9), the power coefficient of a Savonius wind turbine is given as Pmech = CP 2ρ (2A)u 3∞ . The projected area A of a Savonius wind turbine drum = height (h) × Radius (R) = 1 2. m 2 Therefore, Pmech = (0.15) (
1 1.225 ) (2 × ) (103 ) W = 91.9W . 2 2
2.1.3 Largescale vs. smallscale wind turbines The deﬁnition of “small” and “large”scale wind turbines has remained vague in the literature of wind energy. The small wind turbine was initially deﬁned on the basis of its capability to produce electrical power sufficient enough to cover individual household electricity demands [4]. The problem lies in the fact that the consumption of electricity by a household varies with time and place. For example, an average American family needs a 10 kW turbine to cover their full consumption, while a European household requires a 4 kW turbine and a Chinese household only requires 1 kW [4]. Without a precise deﬁnition, the range for the rated power capacity of smallscale wind turbines varies from a few watts to a few hundred kilowatts. To bring consistency into the discussion, we will use the following deﬁnitions based on the size of the horizontalaxis wind turbine rotors [9]: (i) Microscale wind turbine (µSWT): rotor diameter ≤ 10 cm, (ii) Smallscale wind turbine (SSWT): 10 cm < rotor diameter ≤ 100 cm, (iii) Midscale wind turbine (MSWT): 1 m < rotor diameter ≤ 5 m, and (iv) Largescale wind turbine (LSWT): rotor diameter > 5 m.
2.3 Challenges with smallscale wind turbine designs
 15
2.2 Need and application of smallscale wind turbines The recent advancements in the ﬁeld of microelectronics have not only miniaturized wireless devices but have also decreased their power requirement by more than an order of magnitude. Some wireless sensors nodes, for example, can now be powered with less than 1 mW [10]. Such nodes are used in a variety of applications such as gas and chemical sensors, temperature, pressure and humidity monitoring, motion detectors, structural health monitoring, and explosives detection [10]. However, the everexpanding usage of wireless devices has brought challenges in terms of ﬁnding a suitable power source, especially for the remote applications. In the majority of such cases, lithium cell batteries are used. This presents a maintenance challenge because these batteries need to be regularly monitored and replaced. One of the most convenient methods of supplying the required power to the miniature electronic devices is by harvesting the wind energy. The conventional largescale wind turbines (LSWTs) are efficient and the modern megawatt wind turbines have a power coefficient up to 40–45% [11]. However, they need high wind speed to operate; the typical rated wind speed is around 12–14 m/s. Further, their installations are limited to areas far from a city or township due to practical concerns related to safety hazards and noise emission. In comparison, smallscale wind turbines (SSWTs) can operate at low wind speed, generate minimal noise, and there are no known major safety hazards. In spite of several advantages, very few smallscale wind turbine models have been developed. Table 2.4 shows some of the currently available small to midscale wind turbines. It is interesting to note that most of these wind turbines are in the midscale range. The rated wind speed is typically above 10 m/s. None of the wind turbines except the microwind turbine can operate efficiently at wind speed conditions below 5 m/s. The microwind turbine that operates in a range of 2–7 m/s has an optimal power coefficient of 18%, which is quite low. µF500 is the only SSWT which has a good overall efficiency value of 25%, but its rated wind speed is 12 m/s. The current status of the SSWTs essentially emphasizes the lack of suitable SSWT models that can operate near ground level at wind speeds of the order of a few meters per second.
2.3 Challenges with smallscale wind turbine designs It is clear from the discussion in the previous section that developments in wind energy harvesting technology as of now have been mainly concentrated towards largescale power production. This is mainly due to the economy of scale that justiﬁes the installation and operational costs of LSWTs. The two main reasons for the low adaptability of SSWTs are their poor aerodynamic performance and the low power output. The aerodynamic performance of a wind turbine is primarily inﬂuenced by the Reynolds number of the airfoil used for the turbine blades. The Reynolds number
16  2 Wind turbines
Tab. 2.4: Some of the currently available smalltomidscale horizontalaxis wind turbines. Information taken from NE100 S: [12], NE200 S: [13], NE300 S: [14], Energy Ball V100: [15], Micro Wind Turbine: [16], Bornay 600: [17], AC 120/240: [18], µF500: [19], Superwind 350: [20]. Sr. No.
Description
Rotor diameter (cm)
Rated wind speed (m/s)
Overall efficiencya
1 2 3 4 5 6 7 8 9 10
NE100 S NE200 S NE300 S Energy Ball V100 Micro Wind Turbine Bornay 600 AC 120 AC 240 µF500 Superwind 350
120 130 130 110 23.4 200 120 165 50 120
10 11 13 10 2–7b 11 9 9 12 12.5
14% 18% 17% 17% 18%c 23% 24% 25% 25% 26%
a
Overall efficiency was calculated from manufacturer’s published data, b Not speciﬁed, c Power coefﬁcient.
of an airfoil is given by Re =
ρcu rel , μ
(2.10)
where ρ and μ are the density and dynamic viscosity of air, respectively; c denotes the chord length of the airfoil; and u rel is the relative wind speed. Relative wind velocity (u rel ) is the vector sum of free wind velocity and the blade velocity at midradius of the rotor. Mathematically, it is given as u rel = u ∞ + u ,
(2.11)
where u ∞ is the upstream free wind velocity and u is the blade velocity, whose magnitude equals to rm ω. rm is the midradius of the wind turbine rotor, and ω is the angular velocity of the wind turbine rotor. The Reynolds number of a wind turbine is proportional to the chord length and the wind speed. These two factors have very small value and therefore smallscale wind turbines operate at much lower Reynolds number compared to largescale wind turbines. Figure 2.7 depicts the effect of Reynolds number on the aerodynamic parameters (lift and drag coefficients) for a NACA 0012 airfoil. It is interesting to note that the maximum lift coefficient decreases with a decrease in Reynolds number while the drag coefficient increases. This implies that the lifttodrag ratio reduces sharply with a decrease in Reynolds number, which results in the poor performance of smallscale wind turbines. There are myriad kinds of airfoil available that have been developed for different purposes. Each of these airfoils has a speciﬁc range of Reynolds number where they perform efficiently. The selection of an appropriate airfoil, which is suitable for the operating range of a given Reynolds number, is the ﬁrst and foremost step for achieving good performance of a smallscale wind turbine.
2.3 Challenges with smallscale wind turbine designs  17
Drag coefficient, CD ; Lift coefficent, CL
1.6 Re = 2.4 x 106 Re = 1.3 x 106 Re = 6.6 x 105 Re = 3.3 x 105 Re = 1.7 x 105
1.4 1.2 Lift
1.0 0.8
Eppler model for high angle of attack
Re = 8.0 x 104 Re = 4.0 x 104
0.6 0.4
Re increasing
0.2
Drag
0 0
5
10
15 20 25 30 Angle of attack, α (degrees)
35
40
45
Fig. 2.7: Inﬂuence of Reynolds number on airfoil behavior (NACA0012). Reprinted from [21] with permission. Copyright © 1988 ASME.
The power P generated by a wind turbine is given by following expression [22]: P=
1 ηρπr2 u 3∞ , 2
(2.12)
where r is the radius of the wind turbine and u ∞ is the speed of the wind. η denotes the overall efficiency of the system whose maximum theoretical value is equal to 16/27, deﬁned by Betz’s limit (we will discuss this in greater detail in Chapter 4). The power output is proportional to the square of the radius of the wind turbine and the cube of the wind velocity. As the size of the wind turbine and the wind speed decrease, the power decreases drastically and below a certain limit its magnitude is too low to justify the construction and operational costs. Besides, achieving maximum possible efficiency should not be the only objective in the development of SSWTs. Structural factors such as modularity, portability, and reliability can excessively broaden the utility of SSWTs and thus justify the costeffectiveness of the technology. Another important metric when working at the small scale is the cutin speed. This represents the undisturbed upstream speed of the wind at which the turbine starts producing power. It depends on the total inertia and the internal friction of the system including the rotor, ball bearings, gear train, and generator. The smaller the wind turbine the lower is the cutin speed due to the lower inertia of the wind turbine blades. However, the decrease in the size of the wind turbine blades reduces the aerodynamic torque and thus increases the cutin speed. These two opposing factors should be optimized to design a wind turbine of small size with desired cutin speed and power output. The cutin speed and the efficiency of the wind turbine may also be improved by reducing the frictional losses from the bearings and the gear train. The gear train and bearing can be avoided if the generator is directly connected to the wind turbine.
18  2 Wind turbines
However, this presents challenge in terms of achieving desired revolutions per minute (rpm) of the generator. Therefore, choosing a smallsize generator with a low starting torque and high voltagetorpm ratio is a key step. Determining the gear ratio is another important step that needs to be done when the power transmitting gearbox is unavoidable. Gear ratio depends on the range of speed and torque needed by the generator to perform most efficiently. Sometimes the size of the nacelle also restricts the size of the gear box and thus the gear ratio. Cost, noise, and reliability in operation are the three main factors that should also be taken into account while ﬁnalizing the type and conﬁguration of the gearing system. There are also some geometrical parameters of the wind turbine blades that need to be optimized. The most important among them are the twist angle, blade angle, tapering angle, chord length, solidity, and number of blades. There are numerous design tools and programs available commercially and in the literature that a designer can use to optimize the geometrical parameters of the wind turbine blades. However, the problem is that the majority of these design tools have been developed for LSWTs and SSWTs cannot be designed using the same design tools. The SSWTs have quite different aerodynamic behavior compared to their largescale counterparts because of a few fundamental differences. First, they are much smaller in size and many assumptions that are made at the large scale do not remain valid at the small scale. Second, the SSWTs operate at much lower wind speed than the LSWTs and thus, the design goal for SSWTs is not only to achieve the highest possible power output but also high starting torque at low wind speed. Finally, the application requirements and the operating conditions of the smallscale wind turbines are near the ground where wind is not fully developed but weak and turbulent due to the presence of surrounding obstacles like buildings, trees, etc. We can conclude that there is a critical need to develop a design tool for SSWTs that takes into account their operating conditions and local application requirements. Example 2.4. Calculate the ﬂow Reynolds number for an airfoil at the tip of a stationary wind turbine with a diameter of 1 m. Let the free upstream wind speed be 10 m/s. Also assume, the chord length of the airfoil c = 0.1 m, density of air ρ = 1.225 kg/m3 , and dynamic viscosity of air μ = 18.27 × 10−6 kg/ms. Solution. Since the airfoil is stationary, the wind speed relative to the airfoil is the same as the free upstream wind speed. Therefore, u rel = u ∞ = 10 m/s . Using Eq. (2.10), Re =
ρcu rel = 6.7 × 104 . μ
Example 2.5. What would be the ﬂow Reynolds number of the above airfoil if the wind turbine is rotating with an angular speed 500 rpm?
2.3 Challenges with smallscale wind turbine designs
 19
Solution. From Eq. (2.11), we know that relative wind velocity is the vector sum of free stream wind and blade rotational velocities: u rel = √u 2∞ + (ωr)2 . Angular speed ω = 500 rpm = 500 ×
2π 60 rad/s.
u rel = √102 + (500 × Using Eq. (2.10), Re =
2 2π × 0.5) = 28.02 m/s . 60
ρcu rel = 1.88 × 105 . μ
Exercises Q 2.1. Calculate the lift and drag forces acting on a wing with a planform area of 5 m2 exposed to a wind speed of 10 m/s at an angle of attack of 2°. Repeat the calculations of lift and drag forces at the angles of attack of 4, 6, 8, and 10°. Use the lift and drag coefficients given in Table 2.2 for Example 2.1. Assume the density of air to be 1.225 kg/m3 . Q 2.2. What is the average mechanical power output of a Savonius wind turbine with a height and diameter of 2 m, deployed at a location with an average wind velocity of 10 m/s? Assume the power coefficient of the wind turbine is 20%. Q 2.3. Calculate the ﬂow Reynolds number for an airfoil at the tip of a stationary wind turbine with a diameter of 2 m. Let the free upstream wind speed be 15 m/s. Also assume, the chord length of the airfoil c = 0.1 m, density of air ρ = 1.225 kg/m3 , and dynamic viscosity of air μ = 18.27 × 10−6 kg/ms. Q 2.4. Recalculate the ﬂow Reynolds number of the above airfoil if the wind turbine is rotating with an angular speed 500 rpm.
3 Components of a smallscale wind turbine Section 2.1.1 of Chapter 2 brieﬂy explains the major components of a largescale horizontalaxis wind turbine. Conversely, a smallscale wind turbine has a much simpler design. It broadly consists of three main components: (i) a wind turbine rotor that includes turbine blades and a hub, (ii) a transmission system that consists of connecting shafts, bearings, and the gear train, and (iii) an electric generator. Besides these, an SSWT, depending upon its deployment location and utilization purpose, may have some auxiliary components, such as a tail to orient the wind turbine rotor to the wind direction and a braking mechanism to protect the wind turbine in very high wind speed conditions. Since the power output of SSWTs is quite low, even a small loss in efficiency due to any of these components may greatly affect the overall performance and utility of the turbine. Therefore, each of these components should be designed in such a fashion that they operate at their rated operating condition.
3.1 Wind turbine rotor The rotor is the most important component of a wind turbine. It extracts the kinetic energy of the wind and converts it into rotational mechanical energy of the shaft. As shown in Fig. 3.1, a smallscale wind turbine rotor consists of blades, a hub, and a nose cone piece. The main purpose of the nose cone is to divert the wind ﬂow towards the blades, which also reduces the drag and the axial thrust on the shaft bearings. The hub connects the nose cone and blades to the axial shaft. The blades are the part of a rotor that actually encounter wind and causes the rotor to rotate. We shall see in the subsequent chapters that the most important ratings of a wind turbine, such as the power coefficient, torque coefficient, and tip speed ratio, are primarily dependent on the shape, size, and design of the blades. In Chapter 1, we noted that the power P of the wind
Nose cone Hub
Blades Fig. 3.1: A smallscale wind turbine rotor (SSWT).
https://doi.org/10.1515/9781614514176003
22  3 Components of a smallscale wind turbine
ﬂowing at a speed u ∞ and passing through a cylindrical region of crosssectional area A is given by 1 (3.1) P = ρAu 3∞ . 2 For a horizontalaxis wind turbine of tip radius rt , the crosssectional area of the cylinder is equal to the swept area by the wind turbine rotor, which is given as A = πr2t . Therefore, the power produced by a wind turbine can be expressed as 1 P = η ρπr2t u 3∞ , 2
(3.2)
where η denotes the net efficiency of the wind turbine, which accounts for various losses in the wind turbine, such as blade losses in the rotor, frictional losses in the bearings, and electromagnetic losses in the electric generator. The net efficiency of a wind turbine has two main components: the mechanical efficiency of the wind turbine rotor and the electrical efficiency of the electrical generator. We shall study the electrical generator in the next section. In this section, let us focus on the wind turbine rotor. The mechanical efficiency of the wind turbine rotor is deﬁned as the amount of mechanical power produced by a wind turbine against the total available wind power. Sometimes, it is also called the coefficient of performance or power coefficient. The coefficient of performance CP is calculated using the following expression: CP =
Pmech 1 2 3 ρπr t u∞ 2
,
(3.3)
where Pmech is the mechanical power generated by the wind turbine and the denominator term ( 12 ρπr2t u 3∞ ) denotes the total available wind power passing across the swept area of the wind turbine rotor as derived in Eq. (3.1). There are essentially ﬁve blade parameters that inﬂuence the power coefficient of a wind turbine. These parameters are: (i) airfoil, (ii) twist angle, (iii) chord length, (iv) number of blades, and (v) tapering angle. Figure 3.2 shows the blade of a highefﬁciency smallscale wind turbine developed by the CEHMS lab at Virginia Tech, USA. If we look at its crosssectional area, we note that it does not have just a random shape but a speciﬁc airfoil proﬁle. When an airfoil is exposed to an incident wind, it encounters drag and lift forces, as explained in Section 2.1.2. A horizontalaxis wind turbine is mainly a liftbased wind turbine and therefore it is desirable that the airfoil should have a high lift coefficient but low drag coefficient. The intensity of the lift and drag coefficients for an airfoil is determined by its various geometrical parameters. Figure 3.3 demonstrates some basic geometrical parameters and other terminologies related to the nomenclature of an airfoil. – Leading edge: The point at the front of an airfoil, which has the maximum curvature. – Trailing edge: The point at the rear of an airfoil, which has the maximum curvature.
3.1 Wind turbine rotor

23
Fig. 3.2: CAD model of a smallscale wind turbine blade.
Chord line
Angle of attack
Camber line
α
Suction surface
Incoming wind Leading edge Maximum thickness
Maximum camber
Pressure surface
Trailing edge
Fig. 3.3: Some basic terminologies related to an airfoil. “Wing proﬁle nomenclature” by Olivier Cleynen / Wikimedia Commons / Public Domain.
– – – – – – –
Chord line: The straight line that connects the leading edge and trailing edge of an airfoil. Chord length (c): The length of the chord line. Suction surface: The upper surface of an airfoil and is generally associated with higher velocity and lower static pressure than the pressure surface. Pressure surface: The lower surface of an airfoil and is generally associated with lower velocity and higher static pressure than the suction surface. Mean camber line: The locus of a set of points that lie midway between the upper and lower surfaces. Maximum thickness: The maximum thickness of an airfoil when measured perpendicular to the camber line. Angle of attack: The angle at which the wind is incident at the leading edge of an airfoil. It is the angle between the chord line of the airfoil and the direction of the wind relative to the airfoil.
24  3 Components of a smallscale wind turbine
ωr β α
u∞
urel Fig. 3.4: Velocity triangle at the section of a wind turbine blade at radius r.
As shown in Fig. 2.7, the lift coefficient CL of an airfoil has a strong relationship with the angle of attack α. It ﬁrst increases with α, attains a maximum value at an optimal angle of attack α opt , and then decreases with a further increase in α. In order to produce the maximum lift by an airfoil, it is therefore imperative that the wind should strike the airfoil at its optimal angle of attack. When a wind turbine rotor is stationary, the relative wind speed with respect to the turbine blade is constant from root to tip of the blade and it is equal to the actual wind speed. However, for a wind turbine rotor rotating with an angular velocity ω, the relative wind velocity u rel at a radius r of the rotor is given by the following expression: u rel = u ∞ + r × ω .
(3.4)
Figure 3.4 shows the velocity triangle at an elementary section of the wind turbine blade, situated at a radius r. The magnitude of the relative wind speed u rel and the relative wind inﬂow angle ϕ can then be calculated using the following expressions: u rel = √u 2∞ + (ωr)2 , u∞ ϕ = tan−1 . ωr
(3.5) (3.6)
Equation (3.6) clearly implies that the relative inﬂow angle of the wind on the turbine blade decreases as we move from root to tip of the wind turbine rotor. In order to ensure that the wind turbine rotor generates maximum lift, it is required that the entire section of the blade meets the wind at the optimal angle of attack and therefore the wind turbine blade is twisted from root to tip. If we assume that the wind turbine blade has a constant airfoil proﬁle from root to tip, the optimal angle of attack α opt of the turbine blade throughout its span remains constant. The local twist angle β at a radius r of the rotor is then to be calculated as β(r) = ϕ (r) − α opt .
(3.7)
The net twist angle of the turbine blade is given as β = β tip − β root .
(3.8)
3.3 Generator

25
Hub Drive shaft Gear assembly Driven shaft Bearing
Generator
Fig. 3.5: The transmission mechanism of a smallscale wind turbine.
3.2 Transmission mechanism The rotor of a wind turbine is usually not directly connected to the electric generator. The reason for this is the difference between rated rotational speeds of the wind turbine rotor and the electric generator. Generators are usually rated at a much greater rotational speed. A transmission mechanism is therefore required to amplify the rpm. The drive train of a smallscale wind turbine consists of four main components: (i) the prime shaft, a drive shaft that is connected to the hub of the wind turbine rotor; (ii) the secondary shaft, a driven shaft that is connected to the generator; (iii) a gear assembly that ampliﬁes the rpm of the secondary shaft over the primary shaft; and (iv) bearings that hold all the above three components inside the nacelle box. Figure 3.5 shows a transmission system used in a smallscale wind turbine. It consists of two connecting shafts, four bearings, and a gear assembly with a gear ratio of 80:10. The driving shaft is 1/8 inch (3.175 mm) in diameter while the driven shaft has a diameter of a 1/4 inch (6.35 mm). The bearings used are highload steel ball bearings. They are unshielded from both the sides and oil lubricated to minimize the frictional losses.
3.3 Generator There are several kinds of generators that are used for wind turbines: direct or alternating current types, synchronous or asynchronous, with or without permanent magnets (PM), and self or external electrical ﬁeld excited machines. It has been suggested that the generators equipped with permanent magnets are more suitable for smallsize wind turbines because of their higher efficiency compared with other generators [23]. The permanent magnet electric generators are broadly divided into two categories: radial ﬂux and axial ﬂux machines, depending on the direction of magnetic ﬂux in the air gap between stator and rotor. The radial ﬂux generators can be further subdivided into inner rotor radial ﬂux machines or outer rotor radial ﬂux machines, according to the position of rotor with respect to stator. Similarly, axial ﬂux generators can be subcategorized into double statorsingle rotor machines or double rotorsingle stator
26  3 Components of a smallscale wind turbine
machines, based upon the number of rotors and stators. Normally for smallscale wind turbines, an axial ﬂux generator is preferred because it occupies less space in the radial direction than a radial ﬂux generator of the same power rating [23, 24, 25]. As stated in the previous section, most electric generators operate at a much higher angular speed than that of a smallscale wind turbine and therefore a gear train is normally used in the transmission system to run the generator at its rated angular speed. However, the use of a gear train has several limitations: (i) it increases the mechanical losses and thus reduces the overall efficiency of the device, (ii) the static friction between the gears increases the startup wind speed of the wind turbine, (iii) running gears produce excess noise, especially at high wind speed, (iv) at high wind speed, when gears run at very high rpm, they skid and thus limit the cutout wind speed of the wind turbine, and (v) the wear and tear in the gears reduces the overall reliability and life span of the device. These problems can be avoided if the smallscale wind turbine is directdrive, but then the challenge is the availability of a smallsize generator with the required attributes (high efficiency, low rated angular speed of around 1000 rpm, low starting torque of the order of a few mNm, and high voltagetorpm ratio). A recent study by Marin et al. [26] revealed that the doublerotor singlestator slotless axial ﬂux generator design provides very promising results for microscale wind turbines. The slotless axial ﬂux generator design has several other advantages over other types of generators: (i) The topology of an axial ﬂux machine leads to a short radial length, which makes it a very compact generator suited for small wind turbines; (ii) due to the slotless airgap winding, the values of mutual and leakage inductances are low; (iii) the absence of slots provides for noiseless operation with no cogging torque. A doublerotor singlestator axial ﬂux generator consists of a stator containing a set of coils, and two rotors on either side of the stator. Each of the rotors contains a set of permanent magnets. Holding the coils stationary has both functional and operational advantages. It not only reduces the number of moving parts and thus renders a commutator unnecessary, but also decreases the heating losses in the commutator and thus improves efficiency. The proposed generator design has a slotless stator and the coils are wound around the individual coil holders, unlike a typical DC generator where the core of the machine consists of metallic slots and teeth around which the coils are wound. This design eliminates cogging torque, which is the friction caused by the attraction between permanent magnets and the iron core present in a typical DC motor/generator, thus improving the startup speed of the wind turbine by decreasing the required startup torque and increasing the electrical efficiency at low wind speed. Figure 3.6 demonstrates a fabricated prototype of a double rotorsingle stator axial ﬂux generator set assembled inside the nacelle box of a smallscale wind turbine developed at Virginia Tech. It should be noted that the wind turbine is directdrive.
3.4 Auxiliary components 
Bearing 2 Rotor 2 Stator
27
Rotor 1 Hub
Tail post
Shaft Bearing 1 Thrust bearing Output terminals
Mounting rod Radial bearing
Fig. 3.6: A double rotorsingle stator axial ﬂux generator design for a smallscale wind turbine.
Nacelle
Tail Rotor Tower
Fig. 3.7: Tail used in a smallscale wind turbine.
3.4 Auxiliary components In addition to the abovedescribed active components, which actually participate in power generation by a wind turbine, there are some passive components that are required for a smooth, reliable, and longlasting operation. The most important component among them is the tail. The tail is actually a yaw mechanism used in wind turbines to orient the rotor in the direction perpendicular to the wind. Figure 3.7 shows the tail of a fully assembled smallscale wind turbine. We can see that the shape of the tail is pentagonal with a vertex that attaches to the rod at an angle of about 120°. The arm length of the tail post should not be very long otherwise it creates unnecessary vibration. However, it can’t be made so small that it is completely inside the wake region behind the turbine rotor. Besides the tail, the yaw mechanism of a smallscale wind turbine also requires a radial bearing that actually allows the nacelle box to rotate along the vertical axis. One such radial bearing used in smallscale wind turbines can be seen in Fig. 3.7.
28  3 Components of a smallscale wind turbine
Another most important passive component of a smallscale wind turbine is the tower. A wind turbine must be provided with a very strong tower and a secure foundation. It is mandatory for safety reasons but also for vibrationfree and efficient operation of the wind turbine. The length of the tower and type of foundation are determined by the application and environment where the wind turbine will be used. A high wind speed application requires a stronger wind turbine foundation than that necessary at low wind speed. Also, a taller tower causes more vibration than a shorter tower. However, the wind turbine rotor should not be kept very close to the base because this increases blade losses. Most importantly, for safety reasons, the turbine rotor must be installed above human height if the wind turbine is being used for domestic purposes. Lastly, a smallscale wind turbine requires a braking mechanism to protect itself in higher than desired wind speed conditions. A braking mechanism may also be required if the wind turbine needs to be stopped for some reason, for example maintenance. There are both mechanical and electrical braking systems available that can be used. A mechanical braking system typically incorporates frictional disc brakes. The advantage of a mechanical braking system is that it is cheaper but it has a very short life expectancy and thus requires frequent maintenance. Conversely, an electrical braking system uses a very large dummy load that is connected directly to the generator to greatly reduce rpm of the wind turbine causing its blades to stall. Electrical brakes are generally more effective than mechanical brakes. However, at very high wind speeds, the heat losses in the dummy load and the windings of the generator may exceed the desired limit and cause the generator to fail. A failure in the generator destroys the electric braking circuit causing the wind turbine to run uncontrollably. This is one of the main reasons why a smallscale wind turbine should be light and portable so that it can be quickly dismantled when environmental conditions are very extreme.
Exercises Q 3.1. What are the main components of a smallscale wind turbine? Q 3.2. What is the primary function of a nose cone piece and hub in an SSWT rotor? Q 3.3. Name the ﬁve main parameters that affect the power coefficient of an SSWT. Q 3.4. Why is the blade of a wind turbine twisted? Q 3.5. Calculate the relative wind velocity and relative wind inﬂow angle for an airfoil rotating at 500 rpm, making a circle of diameter 50 cm, when the wind velocity is 5 m/s. Assume the airfoil rotation does not affect the wind ﬂow. Q 3.6. Calculate the local twist angle at hub and tip of a wind turbine rotor, if the relative wind inﬂow angle at hub and tip is 40° and 8°, respectively. Assume the optimal
3.4 Auxiliary components  29
angle of attack of the airfoil used in the wind turbine blade is 6°. What should be the twist angle of the wind turbine blade? Q 3.7. What are the main components of the transmission mechanism of an SSWT? Q 3.8. What is the function of a gear train and why is it necessary for an SSWT? Q 3.9. What are the limitations of using a gear train in an SSWT? Q 3.10. What are the attributes of a generator used in an SSWT? Q 3.11. Name the main auxiliary components of an SSWT? Q 3.12. What is the yaw mechanism? What is the main function of the tail of an SSWT?
4 Aerodynamics of a wind turbine In Chapter 1 we studied the total available power contained in a wind ﬂow passing through a circular section of area A with wind speed u ∞ , given by 1 (4.1) ρAu 3∞ . 2 Unfortunately, no wind turbine can extract 100% of the wind power and convert it into more usable mechanical or electrical power. Several theories have been proposed that attempt to predict the maximum possible mechanical efficiency of a wind turbine, all based on three basic principles: (i) conservation of mass, (ii) balance of forces (Newton’s second law), and (iii) conservation of energy. Let us start with the most basic actuator disc theory that models the wind turbine as an idealized actuator disc. Let us consider an air stream in a streamtube ﬂowing through an actuator disc as shown in Fig. 4.1. The streamtube has an upstream wind speed equal to u 1 and a crosssectional area equal to A1 . The actuator disc extracts the kinetic energy of the wind and thus causes it to slow down to speed u 2 . Since air ﬂowing within the streamtube does not get compressed (assuming incompressible ﬂow), the crosssectional area of the streamtube must expand to area A2 in order to accommodate the slower moving air. Also, because of the static pressure drop (p+ − p− ) across the actuator disc, the downstream wind continues to expand till the point where the static pressure of the ﬂow returns to atmospheric level p∞ and equilibrium is achieved. The fardownstream ﬂow has crosssectional area A3 and a wind speed equal to u 3 . If we assume that the mass ﬂow rate of the wind ﬂow through the streamtube is ̇ by conservation of mass we can write it as equal to m, P=
ṁ = ρu 1 A1 = ρu 2 A2 = ρu 3 A3 ,
(4.2)
where ρ denotes the density of air. Also, from Eq. (4.1), the total available wind power at wind speed u 1 , passing through the actuator disc of crosssectional area A2 , is given by Pmax =
(4.3)
u1
p∞ u1 A1
1 ρA2 u 31 . 2
p+ A2
p– u2
A3
u3
Fig. 4.1: The streamtube representing wind ﬂow through the actuator disc. https://doi.org/10.1515/9781614514176004
32  4 Aerodynamics of a wind turbine
This is the maximum available wind power to the wind turbine. The extracted power by the wind turbine can thus be calculated as Pext =
1 1 1 ̇ 21 − mu ̇ 23 = ρA2 u 2 (u 21 − u 23 ) . mu 2 2 2
(4.4)
To express Eq. (4.4) in a more useful form, we need to derive a relationship between velocities u 1 , u 2 , and u 3 .
4.1 Froude–Rankine theorem Froude–Rankine theorem derives the relationship between velocities u 1 , u 2 , and u 3 using Bernoulli’s equation and the conservation of momentum. Looking back at Fig. 4.1, assuming there are no other energy losses in the wind except that extracted by the wind turbine or the actuator disc, we can use Bernoulli’s equation for both the upstream and downstream wind ﬂow. Bernoulli’s equation for upstream (before wind meets the actuator disc): 1 1 2 ρu + p∞ = ρu 22 + p+ . 2 1 2
(4.5)
Bernoulli’s equation for downstream (after wind passes through the actuator disc): 1 1 2 ρu + p− = ρu 23 + p∞ . 2 2 2
(4.6)
Subtracting Eq. (4.6) from (4.5), we can derive p+ − p− =
1 2 1 2 ρu − ρu . 2 1 2 3
(4.7)
Now, using the conservation of momentum for the force balance, we know that the net axial thrust F = (p+ − p− )A2 exerted by the turbine mounting on the wind ﬂow is equal to its rate of change in momentum. ̇ 1 − u3 ) . F = (p+ − p− )A2 = m(u
(4.8)
Since ṁ = ρA2 u 2 , we can rewrite Eq. (4.8) as (p+ − p− )A2 = ρA2 u 2 (u 1 − u 3 ) .
(4.9)
Combining Eqs. (4.7) and (4.9), we can write 1 1 ρA2 u 2 (u 1 − u 3 ) = A2 ( ρu 21 − ρu 23 ) . 2 2
(4.10)
If we solve Eq. (4.10), we obtain two conditions. First, u 1 = u 3 . This is a trivial solution of the equation because in this condition, no power could be extracted by the wind turbine.
4.2 Betz’s law  33
The second solution of Eq. (4.10) gives us following expression: 1 (4.11) (u 1 + u 3 ) . 2 This implies that the wind velocity u 2 near the wind turbine rotor is the arithmetic mean of upstream free wind speed u 1 and fardownstream wind speed u 3 . This is called the Froude–Rankine Theorem. 2 . Now let us deﬁne an axial induction factor a, which is given as a = u1u−u 1 It essentially denotes the fraction by which wind speed is reduced near the rotor after some of the air is deﬂected away by the wind turbine. This implies that u2 =
u 2 = (1 − a)u 1 .
(4.12)
Also, using Eq. (4.11) and Eq. (4.12), we can derive that u 3 = (1 − 2a)u 1 .
(4.13)
4.2 Betz’s law Betz’s law provides a maximum theoretical limit to the mechanical efficiency of a wind turbine. As explained in previous chapters, the mechanical efficiency of a wind turbine is measured in terms of the power coefficient or coefficient of performance CP , which is given by Pext CP = , (4.14) Pmax where Pext denotes power extracted by the wind turbine from the wind and Pmax denotes maximum available wind power. From Eqs. (4.3) and (4.4), we know that Pmax = 3 1 1 2 2 2 ρA 2 u 1 and P ext = 2 ρA 2 u 2 (u 1 − u 3 ), therefore, CP =
Pext = Pmax
1 2 2 ρA 2 u 2 (u 1 − 3 1 2 ρA 2 u 1
u 23 )
.
(4.15)
Substituting Eq. (4.11) for u 2 and simplifying yields CP =
u2 1 u3 ) (1 − 32 ) . (1 + 2 u1 u1
(4.16)
From Eq. (4.13), we know that u 3 /u 1 = (1 − 2a). Therefore, CP = 4a (1 − a)2 .
(4.17)
In Eq. (4.17), the power coefficient is a function of the axial induction factor. The maximum value of the power coefficient occurs when dC P /da = 0. Therefore, dCP = 4 (1 − a)2 − 8a (1 − a) = 4 (1 − a) (1 − 3a) , da dCP = 4 (1 − a) (1 − 3a) = 0 . da
(4.18) (4.19)
Power coefficient, Cp
34  4 Aerodynamics of a wind turbine
60 40 20 0
0.0
0.2 0.4 0.6 0.8 Wind speed ratio, u3/u1
1.0
Fig. 4.2: Power coefficient of a wind turbine versus the farupstream and fardownstream velocity ratio.
Equation (4.19) has two solutions: a = 1 and a = 1/3. The ﬁrst solution leads to u 3 /u 1 = −1, which denotes a negative wake velocity. It is therefore a trivial solution. Using a = 1/3, we can derive that u3 1 = (1 − 2a) = . u1 3
(4.20)
This implies that the fardownstream wind speed is onethird of the farupstream wind speed, when the wind turbine is running at its maximum possible power coefficient. If we substitute u 3 /u 1 = 1/3 into Eq. (4.16), we obtain the maximum value of the power coefficient as u3 1 16 CP,max = CP ( = )= . (4.21) u1 3 27 If we plot Eq. (4.16), it gives Fig. 4.2, which is Betz’s wellknown power coefficient plot. This clearly indicates that the power coefficient of a wind turbine cannot be more than 59.3%, which is called Betz’s limit. This leads to the basic question: why can’t the power coefficient of a wind turbine be 100%? The answer to this lies in Fig. 4.1. As explained in previous sections, the sudden pressure drop across the wind turbine rotor causes the wind streamtube to expand around the turbine rotor. This means, there will always be some spillover of the wind around the wind turbine rotor, i.e., the wind turbine never meets all the wind coming from far upstream inside the streamtube.
4.3 Aerodynamics of a wind turbine rotor The discussion in the previous section considers the wind ﬂow across the wind turbine as a twodimensional ﬂow. Also, we assumed the wind turbine to be a simple actuator disc that causes a pressure drop from upstream to downstream and leads to ﬂow expansion. In reality, however, the wind ﬂow across and around the wind turbine is not so straightforward. Several factors, such as blade proﬁle, number of blades, blade rotation, and blade losses, make the wind a complex, threedimensional ﬂuid ﬂow. In order to accurately predict the performance of a wind turbine, it is therefore mandatory that we should ﬁrst understand the aerodynamics of the wind turbine rotor. A horizontalaxis wind turbine blade in the most basic form is a collection of airfoils (Fig. 4.3), which by virtue of its shape produces lift and drag forces when exposed
4.3 Aerodynamics of a wind turbine rotor
 35
Fig. 4.3: A CAD model of a smallscale wind turbine blade illustrating the airfoil elements.
(a)
(b)
Fig. 4.4: Flow over an airfoil shown by (a) velocity contours and (b) pressure contours. From [28] licensed under CC BY 4.0.
to wind causing the turbine rotor to rotate. As explained in Sections 2.4 and 3.1, when an airfoil is exposed to an airstream there is generation of aerodynamic forces. It has been found that because of the peculiar proﬁle, an airfoil allows air particles to move faster over its upper surface than below its lower surface. An increase in wind velocity on the upper surface of the airfoil is accompanied by a decrease in air pressure, in accordance with Bernoulli’s principle. Figures 4.4a and 4.4b show the velocity and pressure contours around an airfoil at zero degree angle of attack. It is apparent from the contours that the air ﬂowing over the upper surface of the airfoil has higher velocity and lower pressure than the bottom surface. The upper surface is therefore called the suction surface, while the lower surface is called the pressure surface. This difference in pressure between the pressure and suction surfaces of the airfoil generates a surface force on its body. The intensity of this aerodynamic force primarily depends on three factors: (i) wind speed, (ii) angle of attack, (iii) the proﬁle of the airfoil. The magnitude of the force and its direction changes when the angle of attack is changed for a given airfoil at a ﬁxed wind speed. Similarly, even when the wind speed and the angle of attack are kept the same, airfoils with different proﬁles will experience different forces.
36  4 Aerodynamics of a wind turbine
F α
D
Relative wind speed, urel
Fig. 4.5: Lift and drag forces on an airfoil.
100
1.5 CL/CD
1.25
CL 80
1
60
0.75 40 0.5 20
0.25 0
Lift to drag ratio
Lift and drag coefficients, CL and CD
L
CD
0
2
4
0 6 8 10 12 14 16 18 20 Angle of attack, α (degrees)
Fig. 4.6: Lift and coefficient versus angle of attack, NACA 0012 at Re = 106 .
As shown in Fig. 4.5, the resultant aerodynamic force on an airfoil can be resolved into two components. The ﬁrst component that acts in the direction of wind ﬂow is called drag, while the second component that acts in the direction orthogonal to the direction of wind ﬂow is called lift. Lift and drag are the two most important characteristics of an airfoil and therefore it is important to nondimensionalize these parameters. We introduce the lift coefficient CL and drag CD coefficient which are deﬁned as: CL = CD =
L 1 2 2 ρAu rel
D 1 2 2 ρAu rel
,
(4.22)
,
(4.23)
where L and D denote lift and drag forces respectively, ρ is the density of air, and A is the blade area. Figure 4.6 depicts the variation of lift and drag coefficients versus the angle of attack for an NACA 0012 airfoil at a Reynolds number equal to 106 . It can be noted that the lift coefficient ﬁrst increases with angle of attack, reaches a maximum value, and then decreases with a further increase in the value of the angle. The drag coefficient, however, increases monotonically with increases in angle of attack. For a liftbased wind turbine like a horizontalaxis wind turbine, the lift coefficient of the airfoil of the blade should be as high as possible but at the same time the drag coefﬁcient should be as low as possible. In order to determine the most suitable angle of attack, we calculate the ratio of the lift coefficient to the drag coefficient. The optimal angle of attack α opt for an airfoil is the angle where its lifttodrag ratio is highest.
4.4 Blade element theory
rt
 37
dr rh
r
Fig. 4.7: Blade element theory divides wind turbine blades into differential elements.
4.4 Blade element theory A wind turbine blade is basically a collection of airfoils. If we divide a turbine blade over its span into small differential elements, each element can be considered as a twodimensional airfoil. Blade element theory assumes that the net aerodynamic force acting on a turbine blade is the sum of forces acting on all such differential blade elements. Let us consider an annulus of mean radius r swept by a differential blade element of chord length c and thickness dr, as shown in Fig. 4.7. Also, assume that the blade element makes an angle β with respect to the plane of rotation. The angle β is called the sectional pitch angle or local twist angle of the blade section. If the wind turbine is rotating with an angular speed ω, the element will have a tangential velocity equal to ωr in the plane of rotation. The air streamtube expands as the wind approaches the turbine blade resulting in a decrease in wind speed near the turbine rotor over its farupstream value. We deﬁne an axial induction factor a that denotes the factor by which the axial wind speed near the wind turbine is reduced over the free wind speed. Note that axial induction factor 2 a is given as a = u1u−u , which was calculated in Section 4.2 using the disc actuator 1 theory. We should also note that the disc actuator theory assumes the actuator disc as stationary and therefore it ignores the rotational effect of the wind turbine rotor. Unlike the actuator disc theory, blade element theory assumes that the blade element, which has a tangential speed of ωr, imparts some angular momentum to the wind causing it to gain a tangential speed of 2a ωr at the wake, where a is called the tangential induction factor. Near the wind turbine, the tangential speed of the wind can be assumed to be the average of inlet and exit conditions. At the inlet, the ﬂow is not rotating whereas at the exit, the ﬂow has a tangential speed of 2a ωr. Therefore, the tangential speed of the wind near the turbine rotor can be calculated to be a ωr. Note that a ωr is the absolute tangential wind speed near the rotor, but we are more interested in the tangential speed of the wind with respect to the rotating element, which is equal to (a ωr + ωr).
38  4 Aerodynamics of a wind turbine
u∞(1  a)
ϕ
dL dF
dN
urel
ϕ
α β
dT
ωr(1 + a´)
Plane of rotation
dD
Fig. 4.8: Velocity triangle and aerodynamic forces at a differential blade element.
Figure 4.8 represents the velocity triangle at the given differential blade element. The relative wind velocity has two components, the axial component u z and tangential component: u θ , which are given by u z = (1 − a)u ∞ ,
u θ = (1 + a ) ωr .
(4.24) (4.25)
The magnitude of relative wind speed u rel and its direction (inﬂow angle ϕ) can then be calculated as u rel = √u 2z + u 2θ = √[(1 − a) u ∞ ]2 + [(1 + a ) ωr]2 , ϕ = tan−1
uz (1 − a) u ∞ . = tan−1 uθ (1 + a ) ωr
(4.26) (4.27)
The angle of attack is the angle between relative wind speed and the chord length of the airfoil. From Fig. 4.8, it can be calculated as α=ϕ−β.
(4.28)
Using Eqs. (4.22) and (4.23), the lift force dL and drag force dD on the differential blade element can be calculated as 1 CL ρu 2rel dA , 2 1 dD = CD ρu 2rel dA , 2 dL =
(4.29) (4.30)
where dA denotes the area of the differential blade element. If we assume chord length c is constant over the differential thickness dr, dA = cdr. Therefore, 1 CL ρu 2rel cdr , 2 1 dD = CD ρu 2rel cdr . 2 dL =
(4.31) (4.32)
The net force dF can now be calculated as dF = √dL2 + dD2 .
(4.33)
4.5 Blade element momentum theory

39
The net force dF can now also be decomposed into the tangential force dT in the circumferential direction and the normal thrust dN in the axial direction as below dT = dL sin ϕ − dD cos ϕ ,
(4.34)
dN = dL cos ϕ + dD sin ϕ .
(4.35)
Substituting the values of dL and dD from Eqs. (4.31) and (4.32) into Eqs. (4.34) and (4.35), 1 2 ρu (CL sin ϕ − CD cos ϕ)cdr , 2 rel 1 dN = ρu 2rel (CL cos ϕ + CD sin ϕ) cdr . 2 dT =
(4.36) (4.37)
We can now also calculate the aerodynamic torque dQ developed by the blade element of spanwise thickness dr using the following equation: dQ = rdT , 1 dQ = ρu 2rel (CL sin ϕ − CD cos ϕ)crdr . 2
(4.38) (4.39)
The wind power dP generated by the blade element is, therefore, dP = ωdQ , 1 dP = ρu 2rel (CL sin ϕ − CD cos ϕ)cωrdr . 2
(4.40) (4.41)
If B denotes the number of blades of the wind turbine, the total wind power P produced by the wind turbine can then be calculated using the following integration: rt
P=∫ rh
1 Bρu 2rel (CL sin ϕ − CD cos ϕ)cωrdr , 2
(4.42)
where rh and rt represent the hub and tip radii, respectively. We can predict the power generated by a wind turbine using Eq. (4.42) if the relative velocity u rel and relative inﬂow angle ϕ are known. However, Eqs. (4.26) and (4.27) have two unknown variables a and a , which require two additional equations to ﬁnd their values. These two equations are obtained by balancing force and rate of change in momentum. The next section describes the Blade Element Momentum (BEM) theory, which combines blade element theory and momentum equations.
4.5 Blade element momentum theory The blade element momentum or BEM theory is an extension of blade element theory. It is based on the assumption that the rate of change of momentum of the air passing
40  4 Aerodynamics of a wind turbine
u∞
u∞(1–a) ωra´
u∞(1–2a) 2ωra´ Fig. 4.9: Flow around a differential blade element of the wind turbine.
through the annulus, shown in Fig. 4.7, is only because of the differential blade elements that are inside the annulus. This means that ﬂow through one annulus is not affected by the ﬂow through adjacent annuli and therefore there is no wind velocity in the radial direction. As shown in Fig. 4.9, let us consider the wind ﬂow around an airfoil representing a differential blade element inside the annulus of radius r and thickness dr. The upstream wind speed has an axial velocity of u ∞ and has no tangential component. Also, the wind gains an axial velocity of u ∞ (1 − a) and a tangential velocity of ωra in the midplane of the wind turbine rotor. Lastly, we assume that the axial and tangential velocities of the wind far downstream are u ∞ (1 − 2a) and 2ωra , respectively, as explained in the previous section. If the total number of blades of the wind turbine is B, then by using Eq. (4.37), the normal aerodynamic thrust on the annulus in the axial direction is equal to 1 (4.43) Bρu 2rel (CL cos ϕ + CD sin ϕ)cdr . 2 The normal thrust is equal to the sum of decreases in the rate of axial momentum of air passing through the swept annulus and decreases in the wake pressure caused by the increase in dynamic head by wake rotation multiplied by the crosssectional area of the annulus. Therefore, 1 (4.44) dN = ρu ∞ (1 − a) 2πrdr2au ∞ + ρ(2a ωr)2 2πrdr . 2 From Eqs. (4.43) and (4.44), dN =
1 1 Bρu 2rel (CL cos ϕ + CD sin ϕ) cdr = ρu ∞ (1 − a) 2πrdr2au ∞ + ρ(2a ωr)2 2πrdr . 2 2 (4.45) Equation (4.45) can be rewritten as 2 2 Bc u rel a ωr (CL cos ϕ + CD sin ϕ) = 4 [a (1 − a) + ( ) ] . 2 2πr u ∞ u∞
(4.46)
We deﬁne tip speed ratio λ as the ratio of the velocity at the tip of the blade over the free wind speed. ωrt λ= . (4.47) u∞
4.5 Blade element momentum theory
 41
can therefore be called local tip speed ratio λr , which is deﬁned as The expression uωr ∞ the ratio of tangential speed of the blade element over free wind speed. λr =
ωr . u∞
(4.48)
Blade solidity σ is deﬁned as the ratio of the actual blade area over the total swept area of the wind turbine rotor. We also deﬁne chord solidity σr of the differential blade element at a given radius as the ratio of the chord length over the circumferential length swept by the differential blade element. σr =
Bc . 2πr
(4.49)
Lastly, from the velocity triangle shown in Fig. 4.8, we can note that u rel =
u ∞ (1 − a) . sin ϕ
(4.50)
This simpliﬁes Eq. (4.46) as 2
a (1 − a) + (a λr ) =
(1 − a)2 4ϕ sin2 ϕ
σr C x ,
(4.51)
where C x = CL cos ϕ + CD sin ϕ. Similarly, we can also balance the aerodynamic torque and the rate of change in angular momentum. Using Eq. (4.39), the total aerodynamic torque acting on the annulus having B number of blades can be calculated as dQ =
1 Bρu 2rel (CL sin ϕ − CD cos ϕ)crdr . 2
(4.52)
The rate of change in angular momentum is given by dQ = ρu ∞ (1 − a) 2πrdr2a ω .
(4.53)
Equating Eqs. (4.52) and (4.53), 1 Bρu 2rel (CL sin ϕ − CD cos ϕ) crdr = ρu ∞ (1 − a) 2πrdr2a ω . 2
(4.54)
Simplifying the above equation by replacing the terms for chord solidity σr and local tip speed ratio σ r , u 2rel u 2∞
σr (CL sin ϕ − CD cos ϕ) = 4λr a (1 − a) .
Again from Eq. (4.50),
where C y = CL sin ϕ − CD cos ϕ.
(1 − a) σ r C y = λ r a , 4 sin2 ϕ
(4.55)
(4.56)
42  4 Aerodynamics of a wind turbine
This can also be written as σr C y 1 a . = (1 − a) 4 sin2 ϕ λr
(4.57)
Also, from Eq. (4.27), tan ϕ =
(1 − a) 1 (1 − a) u ∞ . = (1 + a ) ωr (1 + a ) λr
(4.58)
Combining Eqs. (4.57) and (4.58), σr C y a = . (1 + a ) 4 sin ϕ cos ϕ This implies that a = (
(4.59)
−1 4 sin ϕ cos ϕ − 1) . σr C y
(4.60)
Substituting the value of a from Eq. (4.60) into Eq. (4.51), the expression for a can be derived: −1
a=(
4 sin2 ϕ σr (C x −
C 2y ) 4 sin2 ϕ
+ 1)
.
(4.61)
C2
Sometimes the term 4 siny2 ϕ is negligible compared to C x and the equation can be simpliﬁed without losing much accuracy: a=(
4 sin2 ϕ + 1) σr C x
−1
.
(4.62)
It can be noted that both the above expressions for induction factors contain the inﬂow angle ϕ as a variable. However, Eq. (4.27) shows that the inﬂow angle ϕ itself is a function of induction factors a and a . Therefore, ﬁnding the values of the induction factors actually requires an iterative scheme that we shall discuss in the next chapter. Once the induction factor and then the relative velocity and the relative inﬂow angle of the differential blade element are known, we can calculate the wind power produced using Eq. (4.52) as dP = ωdT =
1 Bρu 2rel (CL sin ϕ − CD cos ϕ)cωrdr . 2
(4.63)
The total wind power P produced by the wind turbine can then be calculated using the following integration: rt
P=∫ rh
1 Bρu 2rel (CL sin ϕ − CD cos ϕ)cωrdr . 2
(4.64)
4.6 Blade losses
 43
Power coefficient CP can then be calculated as r
CP = CP =
∫r t dP h
,
1 2 3 2 ρπrt u ∞ r ∫r t 12 Bρu 2rel (CL h
(4.65) sin ϕ − CD cos ϕ)cωrdr 1 2 3 2 ρπrt u ∞
.
(4.66)
This equation can be rewritten as rt
CP = 2 ∫ ( rh
Bc r 2 u ∞ (1 − a) ωr(1 + a ) (CL sin ϕ − CD cos ϕ) ωdr . )( ) 2πr rt sin ϕ cos ϕ u 3∞
Some terms of Eq. (4.66) can be written in nondimensional forms: σr = u∞ ω dλ r . This yields
Bc 2πr
(4.67) and dr =
λ
r 2 (1 − a) (1 + a ) CP = 2 ∫ σr λr ( ) (CL sin ϕ − CD cos ϕ)dλr . rt sin ϕ cos ϕ
(4.68)
λr
Equation (4.68) can be modiﬁed into a more usable form by using the expressions for induction factors a and a from Eqs. (4.60) and (4.62). After doing some algebra, we obtain the following equation: λ
CP =
8 CD ∫ sin2 ϕ (cos ϕ − λr sin ϕ) (sin ϕ + λr cos ϕ) [1 − ( ) cot ϕ] λ2r dλr . (4.69) 2 CL λ λr
4.6 Blade losses In the discussion in the previous section, especially in regard to the formulations of BEM theory, we did not account for any kind of losses associated with a practical wind turbine. Also, BEM theory indirectly assumes the blades of the wind turbine are continuous so that all the air particles passing through the turbine rotor have similar trajectories and thus they experience similar forces and loss in momentum. However, this is not true for a real wind turbine where the number of blades is discrete. Normally, the number of blades of a wind turbine varies between two and six, so only a small portion of the wind actually interacts with the turbine blades while the major portion of ﬂow just passes between the blades. Therefore, the aerodynamic force, loss in momentum, and the axial induction factors vary around the crosssection of the annulus depending on the proximity of air particles to the turbine blade. Prandtl proposed a correction factor to account for the discrete number of blades and blade loss. Two kinds of blade losses associated with a wind turbine rotor are deﬁned: tip loss and hub loss. As the
44  4 Aerodynamics of a wind turbine
name suggests, tip loss occurs at the tip of the blade and hub loss occurs at the root of the blade. At the tip and root of a wind turbine blade, there is formation of vortices that induces the local ﬂow ﬁeld past the blade to circulate in the upstream direction. The blade loss is more pronounced near the tip than the hub because the area near the blade tip has the greatest inﬂuence on the power produced by the wind turbine. The Prandtl method expresses tip and hub loss correction factors as below: B(r −r) 2 − t Ftip = ( ) cos−1 [e 2r sin ϕ ] , π B(r−rh ) 2 − Fhub = ( ) cos−1 [e 2rh sin ϕ ] . π
(4.70) (4.71)
The overall blade loss factor is given as F = Ftip × Fhub .
(4.72)
This modiﬁes Eqs. (4.60) and (4.62) for induction factors as a = ( a=(
−1 4F sin ϕ cos ϕ − 1) , σr C y
4F sin2 ϕ + 1) σr C x
(4.73)
−1
.
(4.74)
This also modiﬁes the torque and thus the power and power coefficient of the wind turbine as below: 1 FBρu 2rel (CL sin ϕ − CD cos ϕ)crdr , 2 1 dP = BFρu 2rel (CL sin ϕ − CD cos ϕ)cωrdr , 2
dT =
(4.75) (4.76)
λ
8 CD CP = 2 ∫ F sin2 ϕ (cos ϕ − λr sin ϕ) (sin ϕ + λr cos ϕ) [1 − ( ) cot ϕ] λ2r dλr . CL λ λr
(4.77)
4.7 Buhl correction There is another correction that is required with BEM theory to correctly predict the power coefficient of a wind turbine. When the axial induction factor a is greater than 0.5, the basic BEM theory that we studied in Section 4.5 becomes invalid. We can see from Fig. 4.9 that when axial induction factor a is greater than 0.5, the axial velocity of the wind coming out of the turbine is negative. This wind turbine operating state is known as the turbulent wake state, where part of the ﬂow in the fardownstream wake starts to propagate upstream, which is a violation of the basic assumptions of
4.7 Buhl correction

45
BEM theory. In reality, ﬂow does not travel from downstream to upstream, but the turbulence increases because more ﬂow entrains from outside the wake. To compensate for this effect, Buhl [29] proposed an empirical relationship to predict the wind turbine performance as the axial induction factor exceeds 0.4. This relationship includes tiploss correction factor F and thrust coefficient CT , which is deﬁned as 1 CT = Normal thrust on rotor ( ρu 2∞ ) × (referece area) . 2
(4.78)
Ignoring the force due to the difference in wake pressure, the normal thrust acting on an annulus of radius r and thickness dr is equal to the rate of change in axial momentum of the wind passing through the annulus. dN = Fρu ∞ (1 − a) 2πrdr2au ∞ ,
(4.79)
where the term 2πrdr denotes the reference area of the differential blade element. Therefore, the thrust coefficient CT is equal to CT =
Fρu ∞ (1 − a) 2πrdr2au ∞ ( 12 ρu 2∞ ) (2πrdr)
.
(4.80)
Therefore, CT = 4aF (1 − a) .
(4.81)
However, this relationship of the thrust coefficient is based on classical BEM theory thus it is valid when the induction factor a < 0.5 (more precisely, when a < 0.4 as proposed by Buhl). In the case where the induction factor a ≥ 0.4, Buhl’s empirical relationship for the thrust coefficient is given as CT =
8 40 50 + (4F − − 4F) a2 . )a+( 9 9 9
(4.82)
Using the above relationship, we derive the induction factor as a=
18F − 20 − 3√CT (50 − 36F) + 12F(3F − 4) . 36F − 50
(4.83)
We can now combine Eqs. (4.74) and (4.83) to deﬁne the axial induction factor as [29]: if if
a < 0.4 a ≥ 0.4
4F sin2 ϕ
−1
)
then
a = (1 +
then
18F − 20 − 3√CT (50 − 36F) + 12F(3F − 4) . a= 36F − 50
cB 2πr
(CL cos ϕ + CD sin ϕ)
; (4.84)
46  4 Aerodynamics of a wind turbine
Exercises Q 4.1. Prove that the wind velocity near the wind turbine rotor is equal to the mean of the farupstream and fardownstream wind speed. Q 4.2. Prove that the maximum efficiency of a wind turbine cannot be greater than 59.3%. Explain why a wind turbine cannot be 100% efficient. Q 4.3. What is meant by the pressure surface and suction surface of a wind turbine blade? What are the main factors that affect the intensity of aerodynamic force on a wind turbine blade? Q 4.4. What is optimal angle of attack and why is it important to know for an airfoil? Q 4.5. What are the different kinds of blade losses of a wind turbine?
5 Applying BEM to smallscale wind turbine blade design Essentially ﬁve blade parameters inﬂuence the power coefficient of a wind turbine. These parameters are: (1) airfoil, (2) twist angle, (3) chord length, (4) tapering angle, and (5) number of blades. Each of these parameters needs to be optimized to achieve the best possible power coefficient for a wind turbine. In this chapter, we shall learn how to develop a design method based on the blade element momentum (BEM) theory that we studied in the previous chapter. We shall then use the design method to optimize each of the blade parameters described above. A case study is also provided to make the BEM concept and design procedure easier to understand. The case study aims to develop a 1 watt smallscale wind turbine targeted to operate in a wind speed of 7–10 mph (3.1–4.5 m/s).
5.1 Iterative scheme for BEM theory Even though the formulation of the BEM theory is straightforward, its actual implementation for designing a practical wind turbine requires an iterative method. This is because of the implicit relationship between the blade parameters like twist angle and chord length, operating conditions like wind speed and rotor rotational speed, and the induction factors a and a . The main steps of one of the iterative schemes that can be used to design an SSWT are given below. (a) Divide the blades into 15–20 spanwise differential blade elements. (b) Select a differential blade element i at radius r and thickness dr. (c) Start with some guesstimate values for a and a (e.g., a = 0 and a = 0). (d) Calculate relative wind speed u rel and relative inﬂow angle ϕ using the following equations: u rel = √[(1 − a) u ∞ ]2 + [(1 + a ) ωr] ,
(5.1)
(1 − a) u ∞ . ϕ= (1 + a ) ωr
(5.2)
2
(e) Determine the value of angle of attack α for the given local twist angle β using the following equation: α=ϕ−β.
(5.3)
(f) Calculate the Reynolds number using the following equation: Re =
https://doi.org/10.1515/9781614514176005
ρcu rel . μ
(5.4)
48  5 Applying BEM to smallscale wind turbine blade design
(g) Look up the values of lift and drag coefficients (CL and CD ) for the given airfoil at the calculated values of angle of attack and Reynolds number. CL and CD for a given airfoil at various Reynolds number and angle of attack can be found in the published literature. When the angle of attack is within the attached ﬂow and dynamic stall regimes, import the values of CL and CD directly from the database. For the “Flat plate, fully stalled” regime, use the following equations [30, 31, 32]: CL = 2CL,max sin α cos α ,
(5.5)
CD = CD,max sin 2α .
(5.6)
where CL,max and CD,max are deﬁned as CL,max = CL
at
α = 45° ,
CD,max = CD
at
α = 90° .
(h) Calculate the blade loss factor F using the following equations: B(r −r) 2 − t Ftip = ( ) cos−1 [e 2r sin ϕ ] , π B(r−rh ) 2 − Fhub = ( ) cos−1 [e 2rh sin ϕ ] , π
F = Ftip × Fhub .
(5.7) (5.8) (5.9)
(i) Calculate the value of axial induction factor a using the following equations: if
if
a < 0.4 then
a=
a ≥ 0.4 then
a=
1 [1 +
4F sin2 ϕ ] cB 2πr (C L cos ϕ+C D sin ϕ)
;
18F − 20 − 3√CT (50 − 36F) + 12F(3F − 4) , 36F − 50
(5.10)
(5.11)
where CT is the thrust coefficient and is given as CT = (
cB 1−a 2 )( ) (CL cos ϕ + CD sin ϕ) . 2πr sin ϕ
(5.12)
(j) Calculate the value of tangential induction factor a using the following equation: a =
1 4 sin ϕ cos ϕ σr C y
−1
,
cB and C y = CL sin ϕ − CD cos ϕ. where chord solidity σr = 2πr (k) Repeat steps (d) to (j) until convergence is achieved. (l) Repeat steps (c) to (k) for each of the differential blade elements.
(5.13)
5.3 Airfoil selection 
49
(m) Add the differential power produced by all the differential blade elements to obtain the total wind power of the wind turbine. (n) Calculate the power coefficient of the wind turbine using following equation: λ
8 CD CP = 2 ∫F sin2 ϕ (cos ϕ − λr sin ϕ) (sin ϕ + λr cos ϕ) [1 − ( ) cot ϕ] λ2r dλr . (5.14) CL λ λr
5.2 Size of the wind turbine At least two pieces of information are needed before one starts designing a smallscale wind turbine. First, the required power output and secondly, the average wind speed of the region where it will be installed. The diameter D of the wind turbine is then calculated using the following expression: D=√
8P ηρπu 3∞
,
(5.15)
where η is the overall efficiency of the system (including the generator). The value of η can be assumed to be within the range 20–25% as an initial guess for an SSWT. For example, suppose we want to design a smallscale wind turbine targeted to produce an electrical power output of 1 W in the operating range of wind speeds of 7–10 mph (3.1–4.5 m/s). The average target wind speed in this case is 8.5 mph (3.8 m/s). Using Eq. (5.15), we can calculate that the required size of the wind turbine rotor will be around 40 cm.
5.3 Airfoil selection The selection of an airfoil is the most important step to design an efficient smallscale wind turbine. In Section 2.4, we saw that the lift over drag ratio of an airfoil decreases drastically with a decrease in ﬂow Reynolds number. This means that the performance of an airfoil is strongly inﬂuenced by the ﬂow Reynolds number. It is therefore mandatory that we ﬁrst correctly estimate the ﬂow Reynolds number for the given wind turbine. The Reynolds number can be calculated using the following equation: Re =
ρcu rel . μ
(5.16)
It is important to notice that the expression contains relative wind speed and not the absolute wind speed. Also, there are many variables whose values are not initially known, such as rotation rate ω and chord length c, and induction factors a and a are
25
CL CD CL/CD
0.8
20
0.6
15
0.4
10
0.2 0
CL/CD
Lift and Drag Coefficients, CL and CD
50  5 Applying BEM to smallscale wind turbine blade design
5
0
5 10 15 Angle of attack, α (degrees)
0 20
Fig. 5.1: Lift and drag coefficients of the NACA 0012 airfoil at Re = 4 × 104 .
needed to calculate relative wind speed u rel . Each of these variables needs to be wisely guessed. We assume induction factors a and a are equal to zero at the beginning. Rotation rate ω can be calculated using the following equation, if tip speed ratio λ is known, which is given as ωrt . (5.17) λ= u∞ The optimal tip speed ratio for LSWTs lies in the range of 8 to 10, whereas for SSWTs, values in the range of 2 to 4 have been suggested for improved reliability of performance and lower noise levels [19]. The tip speed ratio can be assumed to be 3 as a reliable initial guess. Chord length c needs to be estimated by considering the structural rigidity. But as an initial guess, we can assume its value is equal to 15% of the diameter of the wind turbine rotor. For example, the Reynolds number of a 1 W SSWT with a diameter of 40 cm and tip speed ratio of 3 can be calculated to be around 40,000 using Eq. 5.16. Once the ﬂow Reynolds number is estimated, the next step is to determine an appropriate airfoil which performs well at the given Reynolds number. Thickness is an important feature that should be kept in mind while ﬁnalizing the airfoil. The airfoil should be thick enough to provide sufficient rigidity to the turbine blades. Various kinds of airfoils have been developed for low Reynolds number operation. One such airfoil is NACA 0012, which has been extensively investigated and reliably used for various applications at low Reynolds numbers [33, 34, 35, 36, 37, 38]. Figure 5.1 shows the lift and drag coefficients (C L and C D ) and lift over drag ratio C L /C D of NACA 0012 at Reynolds number Re = 40, 000 obtained using the program XFLR5 v6.06 [39], an analysis tool for airfoils at low Reynolds number.
5.4 Blade twist angle 
51
5.4 Blade twist angle It can be seen from Eq. (5.2) that the relative inﬂow angle ϕ is inversely proportional to the radius of the wind turbine. This implies that the inﬂow angle is greatest at the hub and decreases towards the tip. The blades of the wind turbines need to be twisted in such a way that the wind should always strike the turbine blades at the optimal angle of attack: (5.18) β (r) = ϕ(r) − α opt . The iterative method discussed in the Section 5.1 can be modiﬁed to estimate the twist angle of a smallscale turbine blade. The modiﬁed iterative method is given below. (a) Divide the blades into 15 to 20 spanwise differential blade elements. (b) Select a differential blade element i at radius r and thickness dr. (c) Start with some guesstimate values of a and a (e.g., a = 0 and a = 0). (d) Calculate relative wind speed u rel and relative inﬂow angle ϕ using Eqs. (5.1) and (5.2). (e) Calculate the Reynolds number using the following Eq. (5.4). (f) Look up the values of lift and drag coefficients (C L and C D ) from the published literature for the given airfoil at the calculated Reynolds number and the optimal angle of attack α opt . (g) Calculate the blade loss factor F using Eqs. (5.7), (5.8), and (5.9). (h) Calculate the value of induction factors a and a using Eqs. (5.10)–(5.13). (i) Repeat the steps from (d) to (h) until convergence is achieved. (j) With the converged value of induction factors a and a , calculate the relative inﬂow angle ϕ using Eq. (5.2). (k) Determine the twist angle of the blade element using the following equation: β = ϕ − α opt .
(5.19)
(l) Repeat steps (c) to (k) for each of the differential blade elements and calculate their local twist angle. The twist angle of the turbine blade with respect to wind turbine radius can then be plotted. (m) The power coefficient of the wind turbine at the optimal operating condition can also be calculated using Eq. (5.14). Figure 5.2 shows the ﬂow chart of the iterative scheme. Let us now consider the 1 W SSWT taken as a case study in this chapter. We can use the iterative scheme described above to plot the blade twist angle of the wind turbine vs. its radius. Figure 5.3 shows the variation of blade twist angle vs. radius at various constant angles of attack. It can be seen that the variation in the twist angle is nonlinear from hub to tip and the blade is highly twisted near the hub. This is a special feature of an SSWT blade. The large twist angle near the root generates a high starting torque which helps the wind turbine rotor to start at low wind speeds. A negative twist
52  5 Applying BEM to smallscale wind turbine blade design
Given: Design point wind speed, hub and tip diameters, optimal angle of attack, αopt , airfoil properties at αopt Variables: Chord length, number of blades, solidity, tapering angle
Start
Blade element i = 1 Initial guess for induction factors a = 0, a´ = 0 Calculate relative wind speed and angle Calculate Reynolds number Import lift and drag coefficients (CL and CD) at the optimal angle of attack Calculate blade loss factor F Calculate induction factors a and a´ No
Convergence Yes Next blade element i = i+1 i>N
No
Yes Stop Output: Twist angle, power coefficient, optimal tip speed ratio
Fig. 5.2: Iterative scheme to optimize blade parameters at design point wind speed.
angle at the tip of the blade can be seen at α = 10°; this is not desirable as it creates an inverse tangential force and leads to a decrease in the power coefficient [32]. Figure 5.4 shows the variation of power coefficient CP as a function of tip speed ratio λ at different values of constant angle of attack (α = 2, 4, 6, 8, 10°). All of the simulations are run at the wind speed of 4.0 m/s and at a constant chord length of 6 cm from hub to tip. It can be noted that the maximum value of the power coefficient ﬁrst increases with α from 2° to 4°, reaches a maximum at α = 6°, and then decreases with a further increase in α. Also, the optimal tip speed ratio λopt decreases with an increase in α from 2° to 6° and then remains almost constant at λopt = 3.5. This behavioral
5.5 Number of blades, chord length, and solidity
 53
50 α = 2 deg α = 4 deg α = 6 deg α = 8 deg α = 10 deg
Twist angle (degrees)
40 30 20 10 0 0
5
10 Radius (cm)
10
15
20
Fig. 5.3: Blade twist angle at various constant angles of attack.
Power coefficient, Cp
0.5
α = 2 deg α = 4 deg α = 6 deg α = 8 deg α = 10 deg
0.4 0.3 0.2 0.1 0
0
1
2
3
4
5
6
7
Tip speed ratio Fig. 5.4: Power coefficient vs. tip speed ratio at various constant angles of attack.
variation of optimal power coefficient with constant values of angle of attack is directly related to the change in lift and drag forces with the angle of attack. As explained earlier using Fig. 5.1, the lifttodrag ratio has a maxima at around α = 6° for the given airfoil NACA 0012. Therefore, it is obvious that the optimal power coefficient also has the highest value at α = 6° and it decreases with the change in α in either direction. Also, we can note from Fig. 5.3 that at the optimal angle of attack α = 6°, the local twist angle varies from 44° at the hub to 3.5° at the tip of the blade.
5.5 Number of blades, chord length, and solidity Solidity of a wind turbine is deﬁned as the ratio of surface area of the blades to its total swept area and is given by the following expression: rt
σ=
∫r Bc(r)dr h
πr2t
,
(5.20)
54  5 Applying BEM to smallscale wind turbine blade design
where B is the number of blades and c(r) is the chord length as a function of radius. Solidity of the wind turbine takes into account the combined effect of the chord length and the number of blades. It is therefore a very important parameter because it inﬂuences the power coefficient of a wind turbine. Its value lies in the range of 5–7% for the conventional LSWT. However, for SSWTs, much higher solidity has been reported at the maximum power coefficient. An optimal solidity in the range of 15–25% has been suggested by Duquette and Visser [40] for a wind turbine of 2 m diameter. This ﬁgure increased up to 50% as the diameter decreased to ∼ 23.5 cm [16]. These two studies show that there is a very large variation in the optimal solidity depending on the size of SSWTs and any incorrect assumption may lead to a poor design. We shall use the case study of a 1 W SSWT to describe the procedure to collectively optimize the number of blades, chord length, and solidity. In the previous sections, we already calculated that the diameter of the given 1 W wind turbine is 40 cm. It was also found that the twist angle of its blade varies from 44° at the hub to 3.5° at the tip of the blade. However, while calculating the twist angle we made the assumption that the wind turbine has a constant chord of 6 cm and its number of blades is 3. In this section, we shall vary both and recalculate the optimal twist angle. We will keep the assumption that the wind turbine has a constant chord length from root to tip. The effect of tapering will be discussed in the next section. Using Eq. (5.20), we can derive that the average solidity of a constant chord wind turbine is given by the following expression: σ=
Bc (rt − rh ) . πr2t
(5.21)
Table 5.1 shows a matrix for the calculation of chord length at different solidity and number of blades using Eq. (5.21). For example, the table shows that for a ﬁxed solidity of 25%, the chord length increases from 1.5 cm to 9.2 cm when the number of blades is decreased from 12 to 2. Please note, the tip and hub radii of the given turbine are taken as 20 cm and 3 cm, respectively. Tab. 5.1: Chord length (cm) at different solidity and number of blades.
Solidity
10% 15% 20% 25% 30% 35% 40%
Number of blades 12 9
6
5
4
3
2
0.6 0.9 1.2 1.5 1.8 2.2 2.5
1.2 1.8 2.5 3.1 3.7 4.3 4.9
1.5 2.2 3.0 3.7 4.4 5.2 5.9
1.8 2.8 3.7 4.6 5.5 6.5 7.4
2.5 3.7 4.9 6.2 7.4 8.6 9.9
3.7 5.5 7.4 9.2 11.1 12.9 14.8
0.8 1.2 1.6 2.1 2.5 2.9 3.3
5.5 Number of blades, chord length, and solidity
Power coefficient, Cp
0.5
Chord = 0.7 cm, blades #12 Chord = 1.3 cm, blades #6 Chord = 1.9 cm, blades #4 Chord = 3.7 cm, blades #2
0.4

55
Chord = 0.9 cm, blades #9 Chord = 1.5 cm, blades #5 Chord = 2.5 cm, blades #3
0.3 0.2 0.1 0 0
1
2
3 4 Tip speed ratio
5
6
7
Fig. 5.5: Power coefficient vs. tip speed ratio at solidity σ = 10%.
Power coefficent, Cp
0.5
Chord = 1.3 cm, blades #12 Chord = 2.5 cm, blades #6 Chord = 3.7 cm, blades #4 Chord = 7.4 cm, blades #2
0.4
Chord = 1.7 cm, blades #9 Chord = 3.0 cm, blades #5 Chord = 5.0 cm, blades #3
0.3 0.2 0.1 0 0
1
2
3 4 Tip speed ratio
5
6
7
Fig. 5.6: Power coefficient vs. tip speed ratio at solidity σ = 20%.
Table 5.1 represents 49 sets of data points; each data set indicates a particular blade design with certain values of solidity, chord length, and number of blades. We employ the iteration scheme described in Section 5.4 to compare the power coefficients of the different blade designs from Table 5.1. Figures 5.5 to 5.8 show the power coefficient vs. tip speed ratio at four different values of solidity (10, 20, 30, and 40%) obtained at a constant angle of attack α = 6°. When σ = 10%, the optimal power coefficient CP,opt = 31% occurs at c = 3.7 cm, B = 2, and λ = 6. When σ = 20%, CP,opt increases to 36% at c = 7.4 cm, B = 2, and λ = 4.5. When σ = 30%, CP,opt = 38% can be seen at c = 7.4 cm, B = 3, and λ = 3.5 whereas when σ = 40%, the higher power coefficient CP,opt = 39% is found at c = 7.4 cm, B = 4, and λ = 3. This implies that the optimal power coefficient increases with solidity, though the rate of increase is higher at lower solidity (10–20%) and becomes slower at higher solidity (30–40%). An increase in solidity affects the aerodynamics of the wind turbine in two different ways. A higher solidity provides a greater blade surface area to the wind and thus it
56  5 Applying BEM to smallscale wind turbine blade design Chord = 1.9 cm, blades #12 Chord = 3.7 cm, blades #6 Chord = 5.6 cm, blades #4 Chord = 11.1 cm, blades #2
Power coefficent, Cp
0.5 0.4
Chord = 2.5 cm, blades #9 Chord = 4.5 cm, blades #5 Chord = 7.4 cm, blades #3
0.3 0.2 0.1 0 0
1
2
3 4 Tip speed ratio
5
6
7
Fig. 5.7: Power coefficient vs. tip speed ratio at solidity σ = 30%. Chord = 2.5 cm, blades #12 Chord = 5.0 cm, blades #6 Chord = 7.4 cm, blades #4 Chord = 14.8 cm, blades #2
Power coefficent, Cp
0.5 0.4
Chord = 3.3 cm, blades #9 Chord = 6.0 cm, blades #5 Chord = 9.9 cm, blades #3
0.3 0.2 0.1 0 0
1
2
3 4 Tip speed ratio
5
6
7
Fig. 5.8: Power coefficient vs. tip speed ratio at solidity σ = 40%.
increases the aerodynamic torque. However, draginduced losses also increase with an increase in solidity. When solidity is low (σ < 20%), the former effect dominates which is favorable towards a turbine’s performance and thus the power coefficient increases sharply with an increase in solidity. As the solidity increases beyond 20%, the inﬂuence of draginduced losses increase and the optimal power coefficient starts to saturate. When solidity increases beyond 50%, the power coefficient decreases. Also, the optimal number of blades increases from 2 to 4 when solidity is raised from 10% to 40%. It is interesting to note that the optimal chord length remains almost constant when solidity is increased. It is the number of blades that controls the optimal power coefficient. The higher the solidity the better the power coefficient. Also, a higher solidity has a lower optimal tip speed ratio, thus lower rpm and lower centrifugal stress on the blades. However, note that there is not much gain in terms of power coefficient when σ > 30%. Therefore, a very high solidity should be avoided for two main rea
5.6 Tapering angle
 57
sons. First, an increase in solidity means an increase in the cost of the blade material. Secondly, a higher solidity may block the ﬂow and thus the blades need to be highly twisted near the hub to avoid ﬂow blockage, which may cause manufacturing difficulties.
5.6 Tapering angle The blades of a wind turbine are generally tapered from hub to tip to minimize centrifugal stresses. However, the problem of centrifugal stress is generally low in the case of SSWTs due to their smaller size. This is the reason why some SSWTs have been constructed with constant chord length. Tapering can be either linear or nonlinear. To understand the procedure to optimize the tapering angle, we shall once again use the 40 cm SSWT taken in the case study. For simplicity, we shall consider constant tapering angles, i.e., variation in the chord length from hub to tip is linear. Figures 5.9 and 5.10 show the effect of positive and negative tapering angle on the power coefficient. Positive tapering means the blade has a maximum chord length near the hub and it decreases to the tip. Conversely, the chord length of negative tapered blades (fantype blades) increases from hub to tip. Figure 5.9 shows that the maximum power coefficient remains almost constant till θ ≤ 5° and then it decreases with an increase in positive tapering angle. Also, it can be seen from Fig. 5.10 that the negative tapering angle has almost no effect on the maximum power coefficient. This implies that the blades of the 1 W SSWT should have a constant chord length or a small positive tapering angle in the range of 0–5°.
θ = 0 deg θ = 7.5 deg θ = 15 deg
Power coefficent, Cp
0.5 0.4
θ = 2.5 deg θ = 10 deg θ = 17.5 deg
θ = 5 deg θ = 12.5 deg θ = 20 deg
0.3 0.2 0.1 0 0
1
2
3 4 Tip speed ratio
5
6
7
Fig. 5.9: Power coefficient vs. tip speed ratio at different positive tapering angles.
58  5 Applying BEM to smallscale wind turbine blade design
θ = 0 deg θ = 7.5 deg θ = 15 deg
Power coefficent, Cp
0.5
θ = 2.5 deg θ = 10 deg θ = 17.5 deg
θ = 5 deg θ = 12.5 deg θ = 20 deg
0.4 0.3 0.2 0.1 0 0
1
2
3 4 Tip speed ratio
5
6
7
Fig. 5.10: Power coefficient vs. tip speed ratio at different negative tapering angles.
5.7 Wind turbine performance The discussion presented in the previous sections demonstrated the procedure for designing a smallscale wind turbine at a design point wind speed. For example, we found that a 1 W wind turbine targeted to operate at an average wind speed of 8.5 mph (3.8 m/s) has the following features: (i) Tip radius: 40 cm; Hub radius: 3 cm, (ii) Airfoil: NACA 0012, (iii) Twist angle: 37°; Nonlinearly twisted, 40° at the hub to 3° at the tip, (iv) Chord length: 7.5 cm uniform along the entire span of the blade, and (v) Number of blades: 3. It is important to note that the design procedure described above optimizes the blade parameters at a ﬁxed wind speed, called the design point wind speed. For example, the 1 W wind turbine was designed for a wind speed of 3.8 m/s. In reality, however, a wind turbine would not be operating only at its designed operating conditions. Therefore, it is important that the design procedure should also include methods to predict the performance of a wind turbine in offdesign conditions. We can use BEM theory to develop an iterative scheme that can be employed to estimate the performance of an optimized SSWT in a wide range of wind speeds. The steps of such an iterative scheme are the same as that described in Section 5.1. Figure 5.11 shows the ﬂow chart demonstrating the main steps of the iterative scheme. Lastly, the power coefficient of the 1 W SSWT is shown in Fig. 5.12. It shows that the performance of the wind turbine decreases when wind speed is lower than the design point wind speed of 4.0 m/s. However, for higher winds than the design point wind speed, the power coefficient is almost constant.
5.7 Wind turbine performance 
Given: Optimized blade parameters, optimal tip speed ratio Variable: Wind speed Start Wind speed, u∞ Blade element i= 1 Initial guess for induction factors a = 0, a´ = 0 Calculate relative wind speed and relative inflow angle Calculate angle of attack and Reynolds number Import CL and CD at the calculated angle of attack and Reynolds number Calculate blade loss factor F Calculate induction factors a and a´ No
Convergence Yes Next blade element i = i + 1 No
i>N Yes Calculate power coefficient at u∞ Next wind speed u∞= u∞+ Δu
No
u∞ > umax Yes Stop
Fig. 5.11: Iterative scheme to predict the power coefficient at variable wind speeds.
59
60  5 Applying BEM to smallscale wind turbine blade design
Power coefficient, Cp
0.5 0.4 0.3 0.2 0.1 0
2
3 4 5 Wind speed (m/s)
6
Fig. 5.12: Performance of a 1 W wind turbine over the operating range of wind speeds.
Exercises Q 5.1. Suggest a few airfoils for the SSWT of diameter 1 m intended to operate at an average wind speed of 10 m/s. Q 5.2. Write a program using the iterative scheme deﬁned in this chapter to obtain an optimal blade twist angle for the SSWT given in Q.5.1. Q 5.3. Using the optimal blade twist angle found in Q.5.2, obtain other optimal blade design parameters, such as the number of blades, chord length, and solidity. Q 5.4. Modify the program from Q.5.2 to estimate the performance of the optimized SSWT from Q.5.3 in the wind speed of range 5 m/s to 15 m/s. Q 5.5. Why should the tip speed ratio of an SSWT lie between 2 and 4?
6 CFD analysis of wind turbines: Fundamentals 6.1 Introduction 6.1.1 The need for highﬁdelity modeling techniques The blade element momentum (BEM) theory introduced in Chapter 4 has many strengths as an aerodynamics analysis tool. Its formulation is simple. It is computationally inexpensive. Its accuracy has improved over the years from scores of corrections and submodels. It can even couple with structural solvers for aeroelastic analysis. Maybe then it should come as no surprise that BEM remains the most widely used method for rotor design and analysis, despite being over 80 years old [41], and can be found in a number of modern timedependent codes including HAWC2, FAST, FLEX5, BLADED, PHATAS, and GAST. The simplicity that makes BEM an efficient and popular tool, however, also limits its capabilities. At its heart, BEM remains a lowﬁdelity method that depends on airfoil lift and drag data obtained from wind tunnel experiments. Furthermore, BEM by itself cannot resolve 3D ﬂow complexities that affect turbine performance, such as tip losses, inboard stall delay, and yawed inﬂow. Empirically derived correction models must instead account for these phenomena. The shortcomings and dependencies of BEM have motivated engineers to explore higher ﬁdelity methods that can fully describe the ﬂow ﬁeld as it interacts with turbine geometry; methods that engineers can use not only for design, but also for researchoriented tasks that extend beyond the scope of rotor geometry optimization.
6.1.2 Computational ﬂuid dynamics (CFD) A natural place to begin looking for highﬁdelity alternatives to BEM is the governing Navier–Stokes (NS) equations. These nonlinear partial differential equations (PDE) describe the motion of viscous ﬂuid ﬂow in a continuum. They can be solved to obtain the velocity ﬁeld u(x, y, z, t) and pressure ﬁeld p(x, y, z, t) around a wind turbine, thus eliminating the dependency on empirical lift and drag data. In an ideal world, the Navier–Stokes equations could always be solved for analytical solutions that describe u and p with unlimited spatial and temporal resolution – e.g., p = xe−3t for all x and t. However, solutions to PDEs are rarely so tidy, and the full threedimensional Navier–Stokes equations are notoriously difficult to solve for practical ﬂow problems due to their nonlinearity and the intricacy of most geometries. We can get around this hurdle using the power of computers to solve numerical approximations to the ﬂow equations for a ﬁnite number of points in space and time. This approach is known as computational ﬂuid dynamics (CFD) and it allows us to predict or simulate the ﬂuid https://doi.org/10.1515/9781614514176006
62  6 CFD analysis of wind turbines: Fundamentals
ﬂow around arbitrary geometry (anything that can be modeled in a CAD program) with the high physical realism offered by the Navier–Stokes equations including the effects of threedimensionality, unsteadiness, viscosity, and even turbulence.
6.1.3 Capabilities and tradeoffs CFD methods are powerful and versatile tools that are able to generate highresolution, highaccuracy ﬂuid simulations of all kinds. We can use these tools to characterize the aerodynamic performance of any turbine geometry in any type of ﬂow without having to fabricate and instrument a prototype. Integration of simulated velocity and pressure ﬁelds over the rotor surface yields values for rotor lift, drag, and torque. Vector plots and streamline plots (e.g., Fig. 6.1) help visualize the ﬂow around the turbine so we can gain an intuitive understanding of the ﬂuid dynamics responsible for its performance, even in obscured regions of the geometry that may be difficult to measure using experimental methods. We can project instantaneous pressure contours onto the turbine surface and animate them to show how the ﬂow separation evolves over the blades. We can even study the velocity deﬁcit in the far wake to choose an optimal spacing for a turbine farm that minimizes interference. This breadth of abilities has been employed extensively by researchers to study a variety of turbine ﬂow problems, including the ﬂow around static airfoils [42], rotating turbine rotors [43], whole horizontalaxis wind turbines (HAWT) [44], whole verticalaxis wind turbines (VAWT) [45], turbines with unusual geometry [46], and large turbine farms [47]. It has also been used to simulate turbines in dynamic weather, like gusts of wind [48], and to study aeroelastic stress on turbine blades [49]. Compared to resources necessary to study these problems experimentally in a wind tunnel, CFD is impressively minimal. All that is required is a sufficiently powerful computer and a software package. Velocity magnitude (m/s) 5 4 3 2 1 0 Fig. 6.1: Streamlines of ﬂow around a smallscale wind turbine generated using CFD.
6.1 Introduction
 63
However, even with this variety of applications and breadth of results, there are sobering realities to CFD: 1. Speed – or lack thereof. Highﬁdelity turbine simulations demand great amounts of computational resources and can take days or weeks to complete even when calculated on highperformance computing clusters (HPCC). In fact, the entire process of setting up (preprocessing), running, and analyzing (postprocessing) a CFD simulation may require the same amount of time as a traditional experiment. A faster, simpler technique such as BEM will suffice for many tasks and is especially favored when many consecutive simulations are required, like for parametric studies. Engineers need to weigh the tradeoff between accuracy, time, and computational cost for every problem. CFD will not always be the correct tool for the job. 2. Price. While not as expensive as a wind tunnel, fully featured commercial CFD software and HPCC still carry a considerable price tag. Frugal engineers can opt for open source software like OpenFOAM and standard performance computers, but they will ﬁnd the software to be less reﬁned and will face reductions in speed as well as model size capabilities due to limited computational resources. 3. Complexity. Mastery of CFD requires an understanding of both ﬂuid mechanics and numerical methods, giving rise to a steep learning curve that deters many. 4. The potential for inaccuracy and unreliability. It is easy to create computationally expensive yet utterly incorrect CFD simulations. Diligent veriﬁcation of the model setup and numerical methods, as well as validation of the results, are required to produce a meaningful simulation that aids research instead of misleading it. This is no different from any other numerical tool but there are greater opportunities for mistakes when dealing with complex models. Wind turbines may seem like trivial systems but their aerodynamic performance is often sensitive to the accuracy of turbulence modeling, particularly the transition from laminar to fully turbulent ﬂow over the blades. The role of CFD in industry and science will continue to expand as computational power increases [50], algorithms become more efficient, and suites of CFD software become more accessible. Software developers will continue to address the problem of complexity by making user interfaces more intuitive. They will also continue to increase the amount of automation in the modelbuilding process. Although few would argue that writing lines of code is preferable to clicking a single button, the convenience of automation and socalled “black box” programs will have the unfortunate consequence of increasing the number of users who lack essential background knowledge and technical proﬁciency. Fewer will know how to troubleshoot their models when automated programs fail to produce sensible results or converge to a result at
64  6 CFD analysis of wind turbines: Fundamentals
all. Many may blankly accept simulation results without critical thought and validation, leading to a dissemination of inaccurate results that ultimately hurts the ﬁeld of CFD. This example of the irony of automation [51] underscores the need for best practice guidelines and, perhaps, a certain amount of complexity in learning to use CFD tools.
6.1.4 Goals of this chapter (and Chapter 7) Our aim is to familiarize the reader with the general process of creating a CFD simulation while providing practical, wind turbinespeciﬁc tips and strategies to save modeling time, increase accuracy, or both. We focus on the nearwake ﬂow of single turbines rather than arrays or farms, but still discuss considerations for both HAWT and VAWT conﬁgurations. We start with a brief introduction to fundamental ﬂuid mechanics and numerical methods theory in Chapter 6 to give the reader proper context. We present the wind turbine ﬂow problem with continuous functions in Section 6.2 before reviewing discretization methods to convert the problem into a form solvable by computers in Section 6.3. We then cover numerical solution methods for linear systems in Section 6.4 before ﬁnally tackling the nonlinear Navier–Stokes equations in Section 6.5. Next, we move on to the general process of creating a CFD simulation in Chapter 7, which includes three steps: preprocessing, solving, and postprocessing. Preprocessing is the model setup and, for wind turbines, we divide this step into problem domain deﬁnition (Section 7.1), mesh generation (Section 7.2), modeling blade rotation (Section 7.3), and turbulence modeling (Section 7.4). We cover numerical methods for solving the model in Section 7.5 and then show some common ways to analyze and display the results, called postprocessing, in Section 7.6. We also emphasize the importance of methodical solution validation and veriﬁcation to give these CFD results credibility. Most of the information presented here should broadly apply to any CFD program of choice, although, at times, the text will cater more to the majority of users who use commercial ﬁnite volume codes like ANSYS Fluent and ANSYS CFX. It is our hope that the diligent reader should be able to extract meaning and functionality from all parts of these chapters.
6.2 Continuous model of wind turbine ﬂuid dynamics The particular form of the ﬂow equations used in a model depend on the dominant ﬂow physics at play. For instance, wind turbines are rotating aerodynamic bodies that operate in medium to high Reynolds number ﬂows. We can infer from this description that the ﬂow is unsteady (“rotating”), boundary layers are important so viscous
6.2 Continuous model of wind turbine ﬂuid dynamics
 65
L
u∞ x
z y L
2L
Fig. 6.2: A threedimensional rectangular domain Ω. The base of the turbine’s support tower is located at the origin.
effects are too (“aerodynamic bodies”), and there is turbulence (“wind” and “medium to high Reynolds number”). If we also assume that the ﬂow is incompressible due to the small local Mach number, which is less than 0.3 even at the rotor tips of the largest turbines,¹ then we can describe the ﬂow around a wind turbine using the unsteady incompressible Navier–Stokes equations, ∇⋅u=0, ρ(
∂u + u ⋅ ∇u) = −∇p + μ∇2 u + ρf , ∂t
(6.1) (6.2)
where p is the pressure, μ is dynamic viscosity, f(x, t) is the speciﬁc body force, and we are solving for the ﬂuid velocity u(x, t). Let us set up a model using these equations. To be mathematically rigorous, we must ﬁrst deﬁne the domains where (and when) we want to solve Eqs. (6.1) and (6.2). Where is x? At what time t are we interested? Suppose the turbine is in a threedimensional rectangular domain Ω with the origin located at the base of the turbine support tower, as shown in Fig. 6.2. Let us deﬁne the dimensions of the domain by Ω = {x ∈ ℝ3  − L < x, y < L; 0 < z < L}, where L > 0 is a constant that is many times larger than the rotor diameter so that the ﬂow can be assumed invariant in the xdirection near the top, sides, and far downstream of the turbine. We denote the boundary of the domain by ∂Ω. Now let Ωt ⊂ ℝ3 be a smaller domain inside of Ω that contains all turbine parts. Then the space in Ω occupied by ﬂowing air where we want to deﬁne Eqs. (6.1) and (6.2) is deﬁned by subtracting this solid turbine domain and its boundary: Ωa = Ω\Ωt where Ωt = Ωt ∪ ∂Ωt . The governing equations for the ﬂow problem can now be restricted to their proper regions so that, for instance, we do not attempt to solve the Navier–Stokes equations
1 It should be noted, however, that the frequency of unsteady phenomena near the blades can be fast enough to make compressible effects signiﬁcant.
66  6 CFD analysis of wind turbines: Fundamentals
inside of the turbine. The ﬂow Eqs. (6.1) and (6.2) remain the same ∇⋅u= 0, ρ(
(6.3)
∂u + u ⋅ ∇u) = −∇p + μ∇2 u + ρf , ∂t
(6.4)
but are now restricted to x ∈ Ωa and t > 0. To describe the problem completely, we also need to model the environment around the turbine using boundary conditions set as the extremes of Ωa . Assuming that wind enters the domain at a steady speed and perpendicular to the rotor, exits behind the turbine, and ﬂows smoothly parallel to the sides of the domain, we deﬁne the following boundary conditions [52]: Inlet velocity :
u ⋅ n = u∞
for
x ∈ Ωa
and
t > 0;
(6.5)
Freeslip sides and top :
u⋅n=0
for
y = ±L
and
z=L;
(6.6)
for
x ∈ ∂Ωt
for
x=L;
(σ ⋅ n) ⋅ s = 0 (σ ⋅ n) ⋅ τ = 0 u=0
Noslip ground, turbine : Outﬂow :
μ
∂u n −p=0 ∂n ∂u τ =0 ∂n ∂u s =0 ∂n
and
z = 0 ; (6.7) (6.8)
where n is a unit vector normal and outward to the boundary ∂Ω, s and t are tangent to ∂Ω and make an orthonormal basis with n, u n = u⋅n, and σ is the ﬂuid stress tensor including the pressure term. The required initial condition to this unsteady model is simply u (x, 0) = u0 ,
(6.9)
such that ∇ ⋅ u0 = 0. While Eqs. (6.3)–(6.9) sufficiently describe the ﬂow around the turbine, we also need to model the rotation and performance of the turbine itself. We can couple the rotor dynamics to these equations via Newton’s second law of motion such that the aerodynamic torque exerted by the wind drives the angular acceleration of the rotor: τaero = J
dω + τgenerator + ∑ τloss , dt
(6.10)
where τ is torque about the shaft axis and J is the moment inertia of the rotor. The instantaneous aerodynamic torque is calculated by integrating the wind pressure and
6.3 Discretization techniques  67
shear stress over the blade rotor surface Γb ⊂ ∂Ωt : τaero = ∬ r (n ⋅ σ) ⋅ eθ dA ,
(6.11)
Γb
where r is the moment arm and eθ is the unit vector in the direction of rotation. This quantity is also used to calculate the wind power captured by the rotor: Paero = τaero ω ,
(6.12)
which is related to the total harvested power by Ptotal = ηt ηg Paero , where ηt and ηg are the transmission and generator efficiencies, respectively. If we assume ω is constant then the ﬁrst term on the righthand side of Eq. (6.10) drops out, leaving a simple torque balance between the rotor, generator, and any frictional losses. This decouples the dynamics and allow us to calculate Eqs. (6.11) and (6.12) after solving Eqs. (6.3)– (6.9). We run into trouble here because the Navier–Stokes equations are prohibitively difficult to solve analytically [mostly because of the nonlinear term in Eq. (6.4)] and, without further simpliﬁcation, we cannot practically obtain an exact solution to Eqs. (6.3)–(6.9). However, instead of trying to ﬁnd an exact solution, we can obtain an approximate solution by discretizing the problem by breaking the continuous domains into a ﬁnite number of points.
6.3 Discretization techniques Analytical solutions to the governing equations describe the variation of ﬂow parameters across an entire continuous domain, e.g., u = f(x, y) for 0 ≤ x ≤ L and 0 ≤ y ≤ L. However, in CFD we seek numerical solutions computed on a discrete domain containing a ﬁnite set of locations, e.g., ui,j = f ̂ (x i , y j ) for i, j ∈ [1, 2, . . . , n]. These locations (x i , y j ) are known as nodes and the collection of nodes is a mesh or grid, used interchangeably (Fig. 6.3). By limiting our analysis to a ﬁnite number of regions in space and time, we are able to approximate the governing partial differential equations as systems of algebraic equations using discretization techniques to replace all derivatives (and integrals) with relatively simple difference equations. In this section, we discuss two of these discretization techniques: the ﬁnite difference method (FDM) and the ﬁnite volume method (FVM). FDM is an academically approachable numerical method for calculating derivatives that will be familiar to many readers from calculus and heat transfer classes. It serves as a foundation as we move on to FVM, which is the most commonly used discretization method in CFD but is more intricate in its formulation. We study how these methods handle the physics of diffusion and convection found in the Navier–Stokes equations by applying them to the more straightforward Poisson’s equation and linear convectiondiffusion equations.
68  6 CFD analysis of wind turbines: Fundamentals
Δx x1
xi−1
xi
xi+1
xN
xi+1
xN
(a) yM
yj+1 i, j
yj yj−1
x1
x1
xi−1
xi
(b)
Fig. 6.3: (a) Onedimensional grid of N uniformly spaced nodes and (b) twodimensional grid of N × M nodes with uniform spacing in each direction.
6.3.1 Finite difference method (FDM) The strategy of the ﬁnite difference method is to approximate derivatives with algebraic difference equations using values at nearby nodes. We can obtain solutions to differential equations by formulating these approximations at each node and solving the resulting system of equations. Let us review the basic difference equations for 1D problems and then show how to use them in an example. Consider a sufficiently smooth function ϕ(x) deﬁned on the 1D domain (0, 1) and assume that we are trying to calculate its derivatives numerically. If we divide the domain into a grid of nodes numbered i = 1, 2, . . ., N as shown in Fig. 6.3a, then we can approximate the ﬁrst derivative ϕ (x) at x = x i from the values of ϕ(x) at nearby nodes using Taylor series expansion. To derive a ﬁrstorder forward difference equation, we expand ϕ(x i+1 ) about the point x i : ϕ i+1 = ϕ i +
2 ∆x n−1 ∆x i+1 ∆x i+1 (n−1) i+1 + O (∆x ni+1 ) , ϕ ϕi + ϕi + . . . + 1! 2! (n − 1)! i
(6.13)
where ϕ i denotes ϕ(x) at x = x i . We can then solve for ϕi by dividing by ∆x i+1 and rearranging: ϕi =
∆x n−2 ϕ i+1 − ϕ i ∆x i+1 (n−1) i+1 − + O (∆x n−1 ϕ ϕi − . . . − i+1 ) . ∆x i+1 2! (n − 1)! i
(6.14)
6.3 Discretization techniques  69
Truncating the equation to the known quantities leaves us with a ﬁrstorder accurate approximation to the ﬁrst derivative at x i : ϕi =
Forward difference, ﬁrst derivative:
ϕ i+1 − ϕ i + O (∆x) . ∆x i+1
(6.15)
Similarly, we can derive the ﬁrstorder backward difference equation for ϕi by expanding ϕ(x i−1 ) about the point x i−1 , ϕ i−1 = ϕ i −
∆x n−1 ∆x i ∆x2i (n−1) i + O (∆x ni ) , ϕ ϕi + ϕ i + . . . + (−1) n−1 1! 2! (n − 1)! i
(6.16)
and then rearranging and truncating to get: Backward difference, ﬁrst derivative:
ϕi =
ϕ i − ϕ i−1 + O (∆x) . ∆x i
(6.17)
We can derive a secondorder accurate central difference equation using information from adjacent nodes on either side of x i provided that ∆x i ≈ ∆x i+1 . The grid shown in Fig. 6.3a is spaced uniformly so, in this case, the condition is satisﬁed and we can set ∆x = ∆x i = ∆x i+1 . Subtracting Eq. (6.16) from Eq. (6.13), rearranging, and truncating leaves us with our third difference equation: Central difference, ﬁrst derivative:
ϕi =
ϕ i+1 − ϕ i−1 + O (∆x2 ) . 2∆x
(6.18)
If we instead add Eqs. (6.13) and (6.16) together and keep up to the fourthorder terms, we get the central ﬁnite difference for the second derivative: Central difference, second derivative:
ϕi =
ϕ i−1 − 2ϕ i + ϕ i+1 + O (∆x2 ) . ∆x2
(6.19)
We could extend this methodology further to formulate approximations to even higher derivatives as well as mixed derivatives. If we conducted an error analysis, we would ﬁnd that all of these approximations converge to the analytical solution as step size decreases. This is an expected result considering the similarity between the equations and the following equivalent deﬁnitions of a derivative: ϕ (x + ∆x) − ϕ (x) , ∆x→0 ∆x ϕ (x) − ϕ (x − ∆x) , ϕ (x) = lim ∆x→0 ∆x ϕ (x + ∆x) − ϕ (x − ∆x) . ϕ (x) = lim ∆x→0 2∆x
ϕ (x) = lim
(6.20) (6.21) (6.22)
More interestingly, we would ﬁnd that the error is proportional to the grid spacing raised to the nth power: (6.23) ϵ ∝ ∆x n ,
70  6 CFD analysis of wind turbines: Fundamentals
where n is the approximation order of accuracy. This means that the error using a central difference on a coarse grid will be similar to a forward difference on a ﬁne grid, and that further grid reﬁnement (i.e., adding more nodes) will cause a central difference to converge at a faster rate. Note that we use information from a greater number of neighboring nodes in a central difference approximation. In general, including more nodes in the formulation (also called the “stencil”) can increase an approximation’s order of accuracy. Now, let us use our difference equations to solve a boundary value problem. Although this example is much simpler than our wind turbine ﬂow problem, it illustrates the key points of the numerical method. The general process of replacing continuous derivatives with discrete difference equations to yield an algebraic system of equations will remain the same even with twodimensional and threedimensional grids, mappings from rectangular grids to arbitrary geometries, and difficult PDEs like Navier– Stokes. Example 6.1. Consider the 1D Poisson equation − ϕ (x) = f (x) ,
0