World Renewable Energy Conference Aberdeen 2005
Pi t c h–Cont r olf orVe r t i c al Axi s , Mar i ne Cur r e ntGe ne r at or s . Stephen Salter Engineering and Electronics, University of Edinburgh.
[email protected]
Abstract The Betz theory for optimal performance of wind turbines requires that the momentum of the flow through a rotor should be reduced by two-thirds of the upstream value by its passage through the rotor. Strictly this should apply to all points. But as there is a strong reduction in velocity towards the hub of horizontal-axis machines, a compromise has to be reached with respect to higher chords and increased pitch angles near the hub. The same momentum requirements apply to vertical-axis machines where the problem is the change of relative velocity across the window. Fol l owi ngSt r i c k l a nd’ ss t r e a mt ubea na l y s i sus e df ort heDa r r i e ust r opos k i e nr ot or s ,t hi spa pe rwi l lg i v eame t hodf or calculating the correct pitch angles for a vertical-axis, variable-pitch rotor with straight blades aimed at the generation of energy from marine currents, tidal streams and very low-head hydro sites. Productivity analysis based on lift and drag forces round the circumference of the turbine shows that the performance coefficient of a variable pitch machine with momentum control can equal that of a horizontal-axis one. Keywords: vertical-axis turbine, tidal-stream, marine-current generator, low-head hydro, variable-pitch, Betz momentum theory, Darrieus, troposkien.
Comparisons Although the horizontal-axis configuration is now almost universal for wind turbines, the vertical-axis configuration with rim drive, a bottom ring and cross-bracing, Salter (1998) may have some advantages for tidal streams and marine currents. The arguments are as follows:
A vertical axis allows a large diameter rotor, stable in pitch and roll, even in shallow water. For a given tip velocity the large diameter will have a longer rotation period and so will make lower demands on the pitch torque needed for accelerating the pitch inertia of the blades and the added inertia of the water around them.
Rotors can generate in flows from any direction even in turbulent flows which are not the same across the rotor diameter. Banks of separate short blades allow generation even in deep water with a large velocity shear.
Cross bracing the bottom ring allows blades to be supported at both ends. This reduces bending moments by a factor of four and eases the task of the bearings needed for variable-pitch. The bottom ring can reduce tip vortices and give some flow augmentation equivalent to longer blades.
The bottom ring can contain airbags, which can be inflated to lift the entire structure clear of the water for inspection or the removal of fouling.
The cross-braced structure is strong enough to support its weight if it runs aground on a soft level bottom although we do not plan to do this regularly.
Generation plant can be in the dry, is easily accessible and can be inspected during operation. Blades can have a constant cross section which makes for cheap tooling, even extrusion.
The large radius of the rim drive reduces power conversion forces and the power annulus can be made very strong with radial spokes.
With internal fuel and a power-source, rotors can be self propelled like a Voith-Schneider drive. The main objection to the vertical-axis configuration with fixed-pitch is that the velocities of blades moving upstream are higher than of blades moving downstream. This leads to uneven power production across the flow window and the risk of stall for part of the rotation. However with a sophisticated pitch-
control this problem can be entirely overcome. It could be argued that the compromises are less than those arising from the difference between hub and tip velocities in a horizontal axis machine. Wind turbine designers like quite high tip speed-ratios, five or more, because the requirement for torque in the blade roots, shaft and gearing is inversely proportional to tip-speed ratio and torque, especially in gears is expensive. The only limits to high tip-speed ratios are that noise rises with tip speed and that, at some critical speed, there is a sharp rise in the damage from impacts with rain drops. The torque argument would apply to rotors working in water but the droplet erosion effect is replaced by the much more serious effect of cavitation. Even if we choose foil sections with low pressure-coefficients given by large radii of curvature at the nose, it is still likely that tip-speed ratios will be 2.5 or less in the highest current velocities. We start with the assumption that the momentum objective has been achieved and that the water velocity through the entire rotor window has been reduced to 2/3 of the original value of the distant flow. In obedience to Bernoulli this will have produced a rise in head. The volume of water must be conserved while it is within the swept volume of the rotor. There will be an abrupt loss of head going through both the upstream and downstream arcs of the rotor followed by a slow recovery along the wake. For this analysis we replace the stream tubes used for the troposkien blades of the Darrieus egg-beater wind turbines with vertical stream slits. These are drawn in figure 1, where the slit boundaries are defined by points at equal angles round the circumference of the rotor. This implies that the time taken for a blade axis to pass through a slit is constant. (An analysis using slits of equal width is also possible.)
Figure 1. The geometry of a vertical-axis rotor rotating anti-clockwise seen from above with flow from the top of the page. For a tip-speed ratio of two the chord dimension for this number of blades has been exaggerated by a factor of two for clarity. Momentum forces on the blades are shown with a short T-bar and will be proportional to the width of each slit. The ideal hydrodynamic lift forces are drawn with small circle ends. Blade and current velocity vectors (shown light with arrows) and their resultant (shown bold) are in the true proportion. Blade pitch angles are accurate.
Calculation steps 1. We know both the width of each slit and the ideal flow velocity and so we know the mass flow through it.
2. We choose the tangential velocity of the blades with cavitation in mind and so, for each slit, we know the direction and magnitude of the resultant velocity that would be seen by an observer riding on a blade.
3. We know that in each slit the component of force acting on the water in the upstream direction should produce the desired two-thirds reduction in momentum for that mass flow.
4. We know that this force must be a component of the hydrodynamic forces on the blades moving through the slit and that, if the blades are not stalled, this will be nearly perpendicular to the direction of the resultant velocity on the blades. More accurately, it will be inclined from that direction towards the trailing edge of the foil by a small angle whose tangent is the ratio of lift to drag. Away from stall this angle will be very small, less than one degree.
5. If we know the angle between this resultant force and the upstream direction we can calculate the magnitude of the hydrodynamic force that would have the right upstream component from blades in that slit.
6. There will not always be blades in a slit but we can calculate the fraction of time that any blade axis will be in a given slit. We can therefore calculate what the blade forces should be by dividing the slit force by the fraction of occupancy.
7. If we know the resultant velocity on blades in any slit we can calculate the Reynolds number for that slit and so look up the appropriate lift and drag coefficients. For known lift and drag, a given choice of blade chord and knowledge of the magnitude of the resultant water velocity at any slit, we can calculate the angle of incidence that a blade in a slit should have to the resultant velocity in that slit. The variation of this angle through a rotation cycle is shown in figure 2. 8.
It is then easy to calculate the useful component of the force driving the turbine and subtract the drag on the foils and any appendages. Knowing the useful force and the blade speed we can calculate the efficiency, the output power and the torque which would have to be provided by the power conversion mechanism. The contribution to the power distribution of one blade is shown in figure 3.
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Figure two. The angles of pitch required for the Betz ideal momentum change for an open field turbine. Nott hatt hemax i mum pi t c hc hangei sl at e rt han‘ t opde adc e nt r e ’ande ar l i e rt han’ bot t omde adc e nt r e ’ . Thei de alc ur v e( dot t e d)woul dn e e dv i ol e ntpi t c ha c c e l e r a t i onne ar0and180de gr e e s .Thi s would be impractical to apply especially as it happens at a point of low productivity. A more reasonable approximation (solid) reduces efficiency by only about 1%.
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Figure 3. Useful (dashed) and wasteful (dotted) forces together with their difference and its mean (solid) from one blade. The skewed pitch variation produces two symmetrical half sinusoids of power with small negatives at zero and 180 degrees of rotation. Note the slight kinks caused by the pitch angle compromise. Lift and drag coefficients of a NACA 0018 foil were taken from Hanley Innovations MultiElement software. The 50 metre diameter rotor had 12 blades with 2 metre chord and 7 metre span. The open-stream velocity was 4 metres per second and the tip-speed ratio was 2. The output of 12 blades would be attractively smooth. The mean drag power from the rotor blades reduces output by 4.5% of the power in open stream flow through an area equal to that of the rotor window. The drag from the diagonal ties with well streamlined fairings reduces it by about 1% and the skin drag of an unfouled lower ring by a further 2.5%. The drag loss of the lower ring is probably less than that from the tip vortex which it suppresses. In combination these give a performance coefficient of 0.51 without any claim for augmentation due to the lower ring. This remains steady for other flows and tip-speed ratios provided that the pitch control is allowed to operate and stall angles are avoided. It was interesting and reassuring to note that by setting all the drag coefficients to zero, the software predicted a performance coefficient very close to the 16/27 figure for a perfect Betz rotor. The addition of a blade pitch-error of one degree reduced the performance coefficient by less than 0.2%. There are two limits to the design choices. Firstly we have to be sure that the calculated angle of incidence is never above the stall angle at any point of the rotation. If stall were to arise then the chord would have to be increased so as to allow a smaller angle of incidence for the same force. We can therefore choose blade number and blade chord so that we approach but do not exceed, the stall angle. Higher blade numbers make for a good truss. Higher chords give blades that are good as beams and so can be lighter than slender blades. It would be good to keep foil chords low enough for transport in sea containers. Secondly we must also ensure that no point on any blade ever experiences a negative pressure coefficient enough to induce cavitation. Pressure coefficients depend on the curvature of the foil at the nose, its angle of incidence and the magnitude of the resultant velocity. Fortunately the highest velocities are associated with small angles of incidence. Cavitation can be avoided by modified nose curvature, lower tip-speed ratios or the use of larger chords which need smaller angles of incidence. While wind turbines operate in an open stream flow, tidal stream plant will often operate in channels. The question of the impedance of these channels is discussed in another paper to this conference (Theta islands for Flow Velocity Enhancement for Vertical-Axis Generators) where the application was to power generation in shallow bays. It introduced a ratio of channel-length times the friction-coefficient on the sea bed divided by the channel-depth and the performance-coefficient of the turbines installed. High values of this ratio suggest a high impedance flow which will tend to continue despite obstructions. Water cannot flow over vertical-axis machines. It cannot flow through the sea bed. It will not easily be able to flow round a vertical-axis machine if it is one of a close packed array. Furthermore adjacent members of a c l os e pa c k e da r r a yc a nc a n c e lon ea n ot h e r ’ sf l owdi v e r g e n c ea n dwa k er ot a t i on . We can model the effect of a high-impedance channel by raising the value of the flow of the velocity through the rotor and the corresponding increase of forces that results. Provided that these can be resisted economically, the limitation is set by the changes of head on either side of the array, or even arrays, of turbines. The present design for a non-contacting seal for the power annulus cannot take heads of more than one metre. A likely installation procedure would be the installation of a few turbines that would
initially be operating as if in an open stream. But as more machines are added to a high impedance channel their power capability will rise as they become more like ducted machines.
Pitch actuation If the bearings that allow the foils to rotate about the spars forming the rotor structure are placed forward of the centre of pressure, there will be a pitching moment tending to bring the blade noses in to the resultant velocity. If they are released to align with the local flow, there will be very low drag for towing to site. This gives way to disconnect the power input which is faster, cheaper and much more reliable than any braking system and which may be needed following loss of the land connection. For more than half the rotation period the movement of the foil to the optimum angle is in the same sense as the moment on it so that it so blade pitch movement will be generating rather than absorbing power. The energy produced can be recycled to return a foil to its optimum position during the rest of the cycle. It would be possible to use a combination of switching valves, cross-connections and pressure accumulators to achieve the correct control. However an extremely versatile system, compatible with any computable control strategy, can be implemented by fast digital poppet valve machines discussed in another paper to this conference (also Salter et al 2003, Rampen 2005). The block diagram is shown in figure 4.
Figure 4. Blade pitch control can be done with a set of three radial-piston poppet-valve machines and can be a net generator of power for other on-rotor systems. Digital hydraulic machines have two electro-magnetically controlled poppet valves on each chamber. The radial geometry allows them to share a common shaft, which in this case, might be driven by an electric motor. No magnet can ever overcome the force of hydraulic pressure on a poppet valve but valves can be moved at times when there is no pressure difference across them. The correct timing of valve operation allows individual chambers to be idle, to pump or to motor. The machine can change between these modes in one half revolution of the shaft, much faster than any swash-plate system. Two banks of a machine would go to the cylinders of two rams. A third bank can pump oil from a reservoir to a pressure accumulator or be driven by oil from it. At angles a few degrees either side of the 0 and 180 positions both rams will be allowed to act as pumps sending oil to the low pressure tank and the blade will head directly into the local current. After a few degrees the hydrodynamic pitch moment will rise and bank 1 of the machine will act as a motor giving way reluctantly to a rising pressure from ram A with the level of reluctance set by a computer. Meanwhile bank 2 can pump oil back to the cylinder of ram B at very low pressure. The correct energy balance can be maintained by using the bank 3 to pump oil to the high pressure accumulator and so storing most of the energy generated by the blade movement. At some angle beyond the upstream direction the blade will reach its maximum pitch angle and it will be necessary to move it back against the pitching moment which is fortunately reducing. This will be done by drawing power from the high pressure accumulator to drive the bank 3 as a motor while bank 1 pumps oil back to its ram A and the ram B sends oil back to the low pressure tank. The electrical machines are needed only for the initial start. They can be a mix of DC brushless and AC induction ones. Once the system is running they can be used as generate power for control logic, instrumentation, communications, hydrostatic bearings, compressors for inflating air bags which can lift the rotor clear of the water, dolphin repulsion, antifouling or cathodic protection. The direct connection of two rams to the low pressure tank can be a fail-safe panic control.
Figure 5 shows the instantaneous power to or from one blade through one cycle of rotation together with the mean output. A three layer poppet valve bank and pancake generator for this power rating would easily fit in the rotor rings. Figure 6 shows the cumulative pitch energy produced by one blade. Values could be increased if necessary by moving the blade pitch-axis nearer to the nose. 4000
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Figure 5.The power from one blade over a cycle.
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Figure 6. The accumulated energy from one blade.
Conclusions Vertical-axis rotors can be attractive in shallow water or turbulent flows despite the low tip-speed ratios enforced by cavitation, provided that they have sophisticated pitch control. The pitch angles can be calculated from the lift forces needed to give the required 2/3 momentum change in each flow slit. Pitch changing of any degree of sophistication can be achieved with three banks of poppet-valve machines and is likely be a net generator of power which can be useful on the rotor.
References Salter SH. Proposal for a large, vertical-axis tidal stream generator with ring-cam hydraulics. Third European Wave Energy Conference. Patras September 1998. Salter SH, Taylor, JRM Caldwell N. Power Conversion Mechanisms for Wave Energy. Proc. Instn. Mech. Engrs. Part M vol 216. Journal of Engineering in the Maritime Environment. pp 1-27 July 2003. Rampen W. Artemis technical literature. http://www.artemisip.com/
