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WIND ENERGY Wind Energ. 2004; 7:177–188 (DOI: 10.1002/we.118) Research Article A General Momentum Theory Applied to an Energy-extracting Actuator Disc D. J. Sharpe*, CREST, Department of Electronic and Electrical Engineering, Loughborough University, Loughborough, UK Key words: wind turbine; general momentum theory; wake rotation A general momentum theory for an energy-extracting actuator disc modelling a rotor with a multiplicity of blades having radially uniform circulation is presented that includes the effects of wake rotation and expansion. A parallel theory directed at the propeller has been published elsewhere, but not one intended for the wind turbine.The rotation of the wake is shown to be accompanied by a fall in static pressure that is additional to that which occurs across the actuator disc and accounts for the energy extraction from the wind. Energy extraction is recognized in the fully developed wake by the flow regaining the static pressure of the undisturbed wind at the expense of the kinetic energy in the wake. Because the wake is still rotating in the fully developed wake, the additional fall in static pressure appears there also and so does not augment the energy extraction.However, the additional fall in pressure would cause a discontinuity in pressure across the fully developed wake but is prevented by a further slowing down of the wake. The additional slowing down extracts a little more energy from the flow than that predicted by the simple momentum theory, which does not include wake rotation.The overall effect of wake rotation on energy extraction is very small for wind turbines operating at high tip speed ratios but can significantly increase the predicted power output for turbines operating at low tip speed ratios. Copyright © 2004 John Wiley & Sons, Ltd. Introduction Simple actuator disc momentum theory, which is the basis of most blade element/momentum theory codes for wind turbine performance prediction, is a simplification of a general momentum theory developed first by Lanchester,1 Joukowski,2 Betz3 and Glauert4 for application to propellers. The simple momentum theory that regards only the rate of change of axial momentum predicts that the maximum possible power that can be extracted from a flow is 16/27 times the power available; this is known as the Lanchester–Betz limit. Glauert found that, for propellers, the general momentum theory, which accounts for wake rotation and expansion, could be reduced to the simple theory without any significant loss of accuracy by ignoring the effects of wake rotation in the determination of the rate of change of axial momentum, and, in his treatment of windmills, Glauert5 assumed that the same simplification applied. It was common at the time Glauert was writing to assume that the kinetic energy associated with wake rotation is drawn from the translational kinetic energy of the freestream, Goldstein6 specifically states ‘that the kinetic energy contained in the rotation of the wake has to * Correspondence to: D. J. Sharpe, CREST, Department of Electronic and Electrical Engineering, Loughborough University, Loughborough, UK E-mail: [email protected] Published online 7 July 2004 Copyright © 2004 John Wiley & Sons, Ltd. Received 29 August 2003 Revised 13 January 2004 Accepted 23 March 2004 178 D. J. Sharpe be provided by the motor driving an airscrew’. Accordingly, Glauert incorporated the assumed wake energy in his development of the simple momentum theory, and this has continued to be the case for latter-day authors such as Wilson and Lissaman7 and Eggleston and Stoddard.8 The general momentum theory, however, demonstrates that the assumption is not correct; the kinetic energy of wake rotation is acquired by a reduction in the static pressure in the wake. The rotation of the wake requires a pressure gradient to balance the centrifugal force on a rotating mass of air, causing a fall in pressure with decreasing radius. The fall in static pressure exactly matches the kinetic energy of rotation. De Vries9 questions the validity of ignoring the static pressure drop caused by wake rotation, but it still remains unrecognized. Wilson and Lissaman7 and Eggleston and Stoddard8 adapt Glauert’s results for the general momentum theory for the wind turbine and produce an expression for the power coefficient but then follow Glauert in assuming that the pressure in the far wake is uniformly atmospheric, which does not reflect the loss of pressure associated with wake rotation. Wilson and Lissaman make the same specific statement concerning the kinetic energy of the wake rotation. This article reiterates the full analysis of the general momentum theory, which, it is hoped, will facilitate an improved understanding of the physics of the energy-extracting actuator disc, enabling some fresh conclusions to be drawn. The Actuator Disc Most horizontal axis wind energy converters employ a rotor comprising a number of rotating blades. The blades sweep out a disc and by virtue of their aerodynamic design acquire a bound circulation, which develops a pressure difference across the disc and a driving torque about the rotor axis. To simplify the analysis, it is assumed that the number of blades is infinite, providing circumferential uniformity to the flow, but with a total circulation that is finite. In the wake of the rotor the air rotates in reaction to the torque, attaining angular momentum and changing the static pressure in the wake. By losing some of its kinetic energy, the wind must slow down, but only that mass of air which actually passes through the rotor disc is assumed to be affected and that mass remains separate from the air which does not pass through the rotor disc. A boundary can be drawn containing the affected air mass, and this boundary can be extended upstream as well as downstream, forming the so-called stream-tube which is of circular cross-section. By definition, no air flows across this boundary and so the mass flow rate of the air flowing within the stream-tube will be the same at all streamwise positions. Because the air slows down and is assumed to be incompressible, the cross-sectional area of the stream-tube must expand to accommodate the slower-moving air. Although, ultimately, kinetic energy is extracted from the airflow, a sudden step change in streamwise velocity across the rotor disc is not possible because of the unlimited accelerations and forces that this would require. Pressure energy, however, can be extracted in a step-like manner using the actuator disc as a permeable diaphragm between the two pressure regions. The disc first causes the approaching air to slow down gradually from the freestream speed U• to U•(1 - a), which results in a steady rise in the static pressure above the freestream level p•. Across the disc there is a drop in static pressure such that, having passed through the disc, the air is below p•. As the air proceeds downstream, the static pressure rises to match the freestream level at the wake boundary, causing a further slowing down of the wind to U•(1 - b) in the fully developed wake. The induced velocity factor b in the developed wake is greater than the corresponding factor a at the disc. Wake Rotation and Expansion The exertion of a torque on the rotor disc by the air passing through it requires an equal and opposite torque to be imposed upon the air. The consequence of the reaction torque is to cause the air to rotate in a direction opposite to that of the rotor; the air gains angular momentum and so in the wake of the rotor disc the air particles have a velocity component in a direction which is tangential to the rotation as well as having an axial velocity component. Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 Energy-extracting Actuator Disc 179 The change in tangential velocity is conventionally expressed in terms of a tangential flow induction factor a¢. Upstream of the disc the tangential velocity is zero. Immediately downstream of the disc, a radial distance r from the axis of rotation, the tangential velocity is 2Wra¢, W being the angular velocity of the rotor. At the mid-plane of the disc, however, the induced tangential velocity is assumed to be Wra¢. If the rotor disc is assumed to be an infinitesimally thin disc of circumferential bound vorticity, then the induced tangential velocity on the disc is exactly Wra¢, whereas immediately downstream of the disc the tangential velocity is 2Wra¢. The velocities and pressures at the different stages of the flow are shown Figure 1. Static pressure within the wake The angular velocity of an air particle in the wake will give rise to a centrifugal force, which has to be balanced by a radial pressure difference across the particle, causing a radial pressure gradient that is a feature of the entire wake. The angular momentum of a particle of air remains constant as it travels along the wake, but expansion of the flow from radius r to radius rw will cause the angular velocity of the particle to fall from 2Wa¢ immediately downstream of the disc to 2Wb¢ in the fully developed wake, where b¢ is less than a¢. Therefore the angular momentum per unit volume in the wake is 2 rWa ¢r 2 = 2 rWb ¢rw2 (1) Application of Bernoulli’s theorem The total energy of the flow upstream of the rotor disc is constant everywhere at the freestream level, ignoring potential energy. Downstream of the disc the total head of a particle remains constant but at a lower level than that of the freestream because of the extraction of energy by the rotor. As a fluid particle passes through the rotor disc, it will have a radial velocity w because of the flow expansion. In the fully developed wake the radial velocity is zero but the tangential velocity remains, together with the necessary radial pressure gradient, which means that, except at the very edge of the fully developed wake, the static pressure pw will be less than the freestream level p•. Figure 1. Velocities and pressures in the expanding stream-tube Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 180 D. J. Sharpe Bernoulli’s theorem can therefore be applied separately to both the upstream and downstream conditions: 1 1 1 2 rU •2 + r • = rU •2 (1 - a) + rw 2 + pdu 2 2 2 (2) 1 1 1 2 2 rU •2 (1 - a) + r(2 Wa ¢r) + rw 2 + pdd 2 2 2 1 1 2 2 = rU •2 (1 - b) + r(2 Wb ¢rw ) + pw (rw ) 2 2 (3) where pdu and pdd are the static pressures immediately upstream and downstream respectively of the rotor disc and pw(rw) is the static pressure in the fully developed wake at radius rw. Subtraction of the two equations results in the following expression for the drop in pressure across the rotor disc: 1 2 2 r U •2 (2 - b)b - 4 W 2 (b ¢rw ) - (a ¢r) + p• - pw (rw ) (4) 2 2 The angular momentum per unit volume of the flow downwind is 2rWa¢r and remains constant, in this inviscid representation, as the wake develops. Therefore 2rWa¢r2 = 2rWb¢rw2 and hence the pressure drop across the disc is Dpd = pdu - pdd = Dpd = [ { ]} 1 r[U •2 (2 - b)b - 4(b ¢ - a ¢)W 2 a ¢r 2 ] + p• - pw (rw ) 2 (5) The lift forces on the blades forming the rotor disc are normal to the resultant velocity relative to the blades and so no work is done on or by the fluid. Therefore Bernoulli’s theorem can be applied to the flow across the disc, relative to the disc spinning at angular velocity W, to give 1 1 1 2 rU •2 (1 - a) + rW 2 r 2 + rw 2 + pdu 2 2 2 1 1 1 2 2 = rU •2 (1 - a) + rW 2 (1 + 2 a ¢) r 2 + rw 2 - pdd 2 2 2 (6) Consequently, Dpd = 2 rW 2 (1 + a ¢)a ¢r 2 (7) From equations (5) and (7) the overall static pressure drop from far upstream of the disc to the fully developed wake is p• - pw (rw ) = 1 r[ 4 W 2 (1 + b ¢)a ¢r 2 - U •2 (2 - b)b] 2 (8) Differentiation of (8) with respect to rw and using equation (1) to replace a¢r2 with b¢rw2 results in an expression for the radial pressure gradient in the rotating wake: dpw db d (b ¢rw2 ) + 4 rW 2 b ¢ 2 rw = rU •2 (1 - b) - 2 rW 2 (1 + 2 b ¢) drw drw drw (9) However, the radial equilibrium of the rotating wake also provides an expression for the radial pressure gradient: dpw 2 = r(2 Wb ¢) rw drw (10) Consequently, Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 Energy-extracting Actuator Disc 181 2 W 2 (1 + 2 b ¢) d db (b ¢rw2 ) = U •2 (1 - b) drw drw (11) The radial variations of b and b¢ depend upon the radial distribution of blade circulation, which would, in turn, depend upon the geometrical design of the blades. Uniform blade circulation strength It is possible to obtain a solution for equation (11) if it is assumed that the bound circulation is uniform along the blade span. A direct consequence of uniform circulation is that the angular momentum in the wake will also be radially uniform. Radially uniform angular momentum means that 2 Wa ¢r 2 = 2 Wb ¢rw2 = constant If a¢t is the tangential flow factor at the edge of the disc (‘t’ is for blade tip) and b¢t is the corresponding value at the edge of the fully developed wake, then a ¢r 2 = at¢R 2 = b ¢rw2 = bt¢Rw2 = constant (12) where Rw is the radius of the fully developed wake. Inspection of equations (11) and (12) leads directly to the conclusion that the wake flow factor b is radially uniform. Assuming that a is also radially uniform, then, for continuity, rw drw 1 - a Rw2 rw2 a ¢ = = = = rdr 1 - b R2 r 2 b ¢ (13) and this forms a link between the axial and tangential flow induction factors. The assumption of uniformity for a is certainly true at high tip speed ratios, but it has not been rigorously established that the condition is true at all tip speed ratios. As will be shown later, the uniformity of a is an equivalent assumption to that of no radial transfer of axial momentum. Application of the axial momentum theory For the whole stream-tube the axial force due to pressure will be equal to the rate of change of momentum of the air. The sides of the stream-tube and the upstream cross-section are subject to atmospheric pressure, but the static pressure in the fully developed wake is lower than atmospheric because of the rotation of the wake. 2p Ú 0RDpd rdr - 2p Ú 0Rw ( p• - pw )rw drw = 2prÚ 0Rw U •2 (1 - b)bdrw (14) Substitution of equations (7), (8) and (12) into (14) gives (1 - a)b 2 = 4 l2 at¢(b - a) (15) where l = WR/U• is the tip speed ratio of the rotor. Equation (15) could have been obtained by taking the differential form of equation (14), which means that there is no radial transfer of axial momentum, but this relies on the value of a being uniform. At the far wake boundary with the freestream flow the static pressure will be atmospheric, so, from equation (8), 1 r[U •2 (2 - b)b - W 2 4(1 + bt¢)at¢R 2 ] = 0 2 (16) which, combined with equation (15), reduces to Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 182 D. J. Sharpe Figure 2. The ratio b/a as a function of b and l at¢ = 1 - 2 a b (17) Eliminating b from equations (15) and (17) results in a(1 - a) = l2 at¢(1 - at¢ 2 ) (18) From equations (17) and (18) a relationship between the axial flow factors a and b is found: a2 1 Ê b2 b ˆa 1 + - 12 + - 2 =0 2 2 Ë ¯ 8 l b 2 8l b (19) For a range of values of b from 0·1 to 0·95 and a range of values of l from 1·0 to 6·0 the value of b/a is calculated from equation (19) and plotted in Figure 2. For tip speed ratios greater than 2 the ratio b/a is very close to 2·0 but always greater than this value, asymptotically approaching 2·0 as l approaches infinity. When b/a exactly equals 2, equation (16) shows that there is no rotation in the wake. At tip speed ratios less than 2 the ratio b/a lies well above 2·0. Thus at high tip speed ratios the general momentum theory corresponds closely to the simple momentum theory, but at low tip speed ratios there is a significant difference between the two theories. The rotational velocity at the edge of the wake is not zero and so there is a commensurate loss of static pressure. In the near wake there is a discontinuity of static pressure across the wake boundary, but in the far wake there is no discontinuity even though the rotational velocity is non-zero. To achieve the atmospheric pressure just inside the wake boundary in the far wake, the axial velocity must reduce that much more than if there were no wake rotation, i.e. in the simple momentum theory case. The reason why a/b is less than 0·5 is because of the additional fall in axial velocity required to regain the static pressure lost by wake rotation. The additional fall in axial velocity gives rise to an increased power extraction compared with that predicted by the simple momentum theory. Power extraction A certain way to determine the power extraction is from the rate of change of angular momentum. The angular momentum of the wake does not change as the wake develops and so the rotational conditions immediately behind the rotor plane can be used. Power is equal to the rotor torque times the rotor angular Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 Energy-extracting Actuator Disc 183 velocity W and torque is equal to the rate of change of angular momentum of the fluid passing through the disc. The rate of change of angular momentum of the fluid passing through an annular ring of the disc of radius dr is equal to the mass flow rate through the ring times the change in tangential velocity times the radius of the ring. Integration of the rate of change of angular momentum for the whole disc provides the torque on the rotor. Power = 1 2 r 4(WR) at¢U • (1 - a)pR 2 2 (20) which yields the expression for the power coefficient CP = 4 l2 at¢(1 - a) (21) and, combined with equation (15), gives CP = b 2 (1 - a) b-a 2 (22) Figure 2 shows that, for high tip speed ratios, a/b approaches 0·5 and equation (22) converges on the result of the simple momentum theory. The power extraction can also be derived by considering the rate of loss of total head by the fluid passing through the disc, and the same results as offered by equations (20)–(22) are achieved. Wilson and Lissaman7 and de Vries9 have formed equation (22) from Glauert’s propeller theory4 by changing the signs of a and b and show that values of CP that exceed the Lanchester–Betz limit are predicted at low tip speed ratios. No previous author has established the reason for the high predicted values of CP. Pressure drop across the disc The pressure drop across the disc is given by equation (7), which, combined with equations (12) and (15), can be written as Dpd = r U •2 b 2 (1 - a) Ê R2 ˆ 1 + at¢ 2 2(b - a) Ë r ¯ (23) If the rate of change of axial momentum is employed to determine the power extraction, then it is clear, by comparison with equation (22), that only the left-hand term in the brackets in equation (23) contributes to the power. The second term is the fall in static pressure in the wake immediately behind the disc caused by the wake rotation. Because the wake continues to rotate as it develops, the pressure drop caused by the wake rotation means that the pressure field is not uniform; atmospheric pressure is achieved only at the outer wake boundary. The axial force on the wake caused by the rotational pressure drop is zero. The rotation of the wake does not constitute a loss of power extraction. It is common in the literature5,7,8 to attribute the whole of the pressure drop, given in equation (23), to the change of momentum of the flow through the disc, but the above analysis shows this to be incorrect. Ideal power coefficient The maximum value of CP can be obtained by differentiating the expression for power (equation (21)) with respect to a and setting the result to zero to give dat¢ at¢ = da 1 - a (24) From equation (15) is obtained 1 - 2 a = l2 (1 - 3at¢ 2 ) Copyright © 2004 John Wiley & Sons, Ltd. dat¢ da Wind Energ. 2004; 7:177–188 184 D. J. Sharpe giving (1 - 2 a)(1 - a) = l2 (1 - 3at¢ 2 )at¢ (25) From equations (15) and (25) comes the expression at¢ 2 = 3a - 1 5a - 1 (26) Eliminating a¢t from equations (18) and (25) gives an expression for the ideal value of a: 125a 5 - 325a 4 + 290 a 3 - 106 a 2 + (17 - 12 l4 )a + 4 l4 - 1 = 0 (27) For tip speed ratios greater than about 1·5 the optimum value of a is very slightly greater than 1/3, although the value of b becomes close to 2/3 at tip speed ratios greater than about 3·5 (Figure 3) at low tip speed ratios, b rises rapidly, signalling that, in practice, the wake would break up and mix with the freestream flow. This equation does not yield an appropriate real root when the value of l falls below 0·806, but, more significantly, the value of b rises to unity at l = 0·93, which is an absolute limit on the applicability of the theory. Wilson and Lissaman7 suggest that wake mixing begins when a rises above about 1/3 and so it may be assumed in the context of the present theory that mixing becomes significant when b rises above 2/3, providing a limit for the applicability of the theory. Figure 4 shows the variation of the ideal power coefficient with tip speed ratio, comparing values with and without the restriction on the value of b of 2/3. Even with the imposed restriction on b the ideal power coefficient is predicted to exceed the Lanchester–Betz limit at very low tip speed ratios. The results of Figure 4 are significantly in contrast to the ideal power coefficient predicted by Glauert,5 who assumes that kinetic rotational energy of the wake has to be drawn from the kinetic energy of the freestream rather than from a reduction in the static pressure. Almost all other authors have adopted Glauert’s reasoning. Wilson and Lissaman7 discuss the vortex structure of the wake but state that, close to the axis of rotation, tangential velocities would become too large to be sustained and a Rankine vortex core would result. In both References 5 and 7 the assumption is made that the tangential velocity of the fluid cannot exceed that of the blades, but there is no physical argument to support this case. Although the theory predicts infinite velocities at the centre line of the wake, this is no different from all other momentum theo- Figure 3. Variation of the axial flow factors with tip speed ratio as given by equations (19) and (27) Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 Energy-extracting Actuator Disc 185 Figure 4. Variation of the ideal power coefficient with tip speed ratio ries that include wake rotation. In practice the problem of infinite velocity is resolved because the blade root will be at a finite radius. It is clear, however, that at the operational tip speed ratios adopted for the vast majority of wind turbines the simple momentum theory result for axial induced velocities can be used without any significant loss of accuracy. The rotational induced velocities require further examination. Thrust force on the disc The normal thrust force on the disc can be determined directly from the pressure drop expressed in equation (23). The inner radius of the disc is R0 and the rotor thrust is given by R T= = 1 b 2 (1 - a) Ê R2 ˆ rU •2 1 + at¢ 2 2prdr Ú 2 b - a R0 Ë r ¯ 1 b 2 (1 - a) ÈÊ R02 ˆ a R ˘ rU •2 pR 2 1 - 2 + 2Ê 1 - 2 ˆ lnÊ ˆ ˙ Í Ë b - a ÎË b ¯ Ë R0 ¯ ˚ 2 R ¯ If the thrust coefficient is defined as CT = Thrust 1 rU •2 pR 2 2 then CT = b 2 (1 - a) ÈÊ R02 ˆ a R ˘ 1 - 2 + 2Ê 1 - 2 ˆ lnÊ ˆ ˙ Í Ë b - a ÎË b ¯ Ë R0 ¯ ˚ R ¯ (28) Clearly, if the inner radius is zero, the thrust force will be infinite, but this would not occur in practice because the blade roots would have a finite radius. At low tip speed ratios the thrust force will be greater than the simple theory predicts, as illustrated in Figure 5, which shows the variation of the thrust coefficient as a function of a and l assuming a root radius of 10% of the tip radius. Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 186 D. J. Sharpe Figure 5. The thrust coefficient as a function of a and l for R0 = 0·1R The Simple Momentum Theory The simple momentum theory dissociates the axial momentum change from the angular momentum change and so b = 2a. The power developed by the flow through an annular ring of radius r and radial width dr from the rate of change of axial momentum is 2 dP = rU •2 (1 - a) 2 aU • 2prdr (29) whereas the power developed from the rate of change of angular momentum is dP = rU • (1 - a)2 a ¢W 2 r 2 2prdr (30) Equating the two expressions for power gives U •2 a(1 - a) = a ¢(Wr) 2 Thus a¢ = a(1 - a) l2 m 2 (31) where m = r/R. Because a is radially uniform, we have a(1 - a) = l2 at¢ (32) Equation (32) is a simplified version of equation (18) as would be achieved if a¢t were to be considered as being negligibly small compared with unity. When the axial and tangential induced velocities are superimposed, the pressure drop across the disc is Dpd = r[U •2 2 a(1 - a) + 2 W 2 a ¢ 2 r 2 ] (33) Equation (33) is relevant when the simple momentum theory, is used in conjunction with the blade element theory, because the blade element force, derived from two-dimensional aerofoil data, must balance the whole of the pressure force and not just the force arising from the rate of change of axial momentum. In the fully developed wake there will be a pressure difference across the wake boundary because the wake is rotating and there will be a consequent pressure drop: b = 2a and so there is no additional slowing down of Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 Energy-extracting Actuator Disc 187 the wake to equalize the pressure across the wake boundary. In the near wake it would appear that there is a pressure difference predicted across the wake boundary, as indeed there is in the simple momentum theory, but the momentum theories consider only the conditions far upstream, at the actuator disc and in the fully developed wake: no predictions are made about the conditions elsewhere in the flow field. At the edge of the disc the pressure difference is also non-zero across the wake boundary, and this problem has been addressed by van Kuik.10 For optimal operation the Lanchester–Betz limit applies at all tip speed ratios. High-solidity, Low-tip-speed-ratio Wind Turbines It would appear that wind turbines, designed according to the above theory, intended to operate at very low tip speed ratios could achieve a maximum power coefficient that exceeds the Lanchester–Betz limit. Such turbines would have to be of very high solidity, with a large number of blades. To complete the design, the blade element theory would need to be employed, but the only available blade element data are for isolated, twodimensional aerofoils. The mutual interactions between the closely spaced, probably overlapping, blades would be large and so the available aerofoil data would be useless for the purpose of designing high-solidity turbines. Conclusions A general momentum theory for an energy-extracting actuator disc modelling a rotor with blades having radially uniform circulation that includes the effects of wake rotation and expansion has been presented. The general momentum theory is well known but the fall in pressure caused by the rotation of the wake that the theory predicts is not usually recognized. Accounting for the wake rotational pressure drop changes some of the established conclusions of the momentum theory that appear in the literature. Principally, the theory establishes that there is no loss of efficiency associated with the rotating wake and that, ignoring any losses associated with the high tangential velocities close to the rotor axis, it is possible, theoretically, to exceed the Lanchester–Betz limit. In practical terms the changes determined by the theory are not significant for wind turbine rotors designed to operate at high tip speed ratios. For turbines that operate at low tip speed ratios, however, the effect of the theory upon the blade design may well result in an improved performance compared with designs that have been undertaken ignoring the pressure drop caused by wake rotation. Unfortunately, the necessary blade element data are not available. Acknowledgement Dr David Infield, Director of CREST, is thanked for his encouragement, advice, suggestions and proofreading. References 1. Lanchester FW. A contribution to the theory of propulsion and the screw propeller. Transactions of the Institution of Naval Architects 1915; 57: 98. 2. Joukowski NE. Travaux du Bureau des Calculs et Essais Aéronautiques de l’Ecole Superièure Technique Moscou 1918. 3. Betz A. Schraubenpropeller mit geringstem Energieverlust. Gottinger Nachricht 1919. 4. Glauert H. In Aerodynamic Theory, vol. IV, Durand WF (ed.). Springer: Heidelberg, 1935; div. L, chap. II (reprinted by Dover, New York, 1963). 5. Glauert H. In Aerodynamic Theory, vol. IV, Durand WF (ed.). Springer: Heidelberg, 1935; div. L, chap. X (reprinted by Dover, New York, 1963). Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188 188 D. J. Sharpe 6. Goldstein S. On the vortex theory of screw propellers. Royal Society Proceedings (A) 1929; 123: 440. 7. Wilson RE, Lissaman PBS. Applied aerodynamics of wind power machines. National Science Foundation Report NSFRA-N-74-113, Oregon State University, 1974. 8. Eggleston DM, Stoddard FS. Wind Turbine Engineering Design. Van Nostrand Reinhold: New York, 1987. 9. De Vries O. Fluid dynamic aspects of wind energy conversion. AGARD-AG-243, 1979. 10. Van Kuik GAM. On the limitations of Froude’s actuator disc concept. PhD Thesis, Technical University of Eindhoven, 1991. Copyright © 2004 John Wiley & Sons, Ltd. Wind Energ. 2004; 7:177–188