IE010315A

Chapter 8

The Horizontals

Also

Negative Pitch Angles And Their Theory

Now it should be clear what happens when the wind interacts with the blades of a wind generator: deflection, deflection, deflection. Nothing else drives the blades against the load with quite the same refusal to be denied. No vague "circulations" or "Magnus Effects" or, as was postulated in a previous chapter, "styluses acting in a straight line on frictionless surfaces". The blades are, of course, moving while they work and with tangential velocities that vary substantially from root to tip. But with some adroitness aforethought this material has covered subject matter that should do very well, thank you, in allowing us to get a good view of what is happening in the middle of all this complexity.

It would seem that what is being said here is that design improvement is the key to each of these renewables technologies. To some extent this is true. However, nothing will happen without the political persuasion necessary to meet the requirements brought upon nations when important issues are at stake. History teaches us that change often comes hard. That's maybe why Science and Technology have less of an appeal to college-bound young people intent upon entering professional careers. Much else goes on that is of great challenge and is necessary by way of human interaction that has little to do with how an equation or two is put together.

The idea of stimulating competition within the electric energy industry is a good one. The hesitation by the regulators to go forward is not something that the public should accept. Lots of technological stones are unturned as things stand now in the status quo and deregulation will provide many opportunities for new advancements once the process gets moving in earnest. The wind energy people go about nowadays hustling up new projects as the best way to move forward in the time-honored tradition of putting one foot in front of the other. Would that things were that simple.

Blenheim-Gilboa

New York State is a case in point. Renewables technologies where no geothermal power or high concentrations of wind and solar power exist such as this are put to a test that they may not be ever able to pass. But while the rest of the nation has been making plans that allow them to take advantage of naturally-occurring circumstances, this large Eastern state has found that using some strategic planning - a little thinking ahead, in other words - has provided another card in her hand, this being the creation of a state-owned power company, the New York Power Authority.

Some say that the NYPA is just a dusty holdover from the heyday of big hydro and will see less usefulness as time moves forward. But look at the Blenheim-Gilboa pumped storage hydro plant (above) of the NYPA in the Catskills. This may have more of the future about it than many believe and it came about without all the fuss and bother that occurs when the investor-owneds bring to the table every capitalization project, each of which is further ritualized in the annual stockholder reports. Could it be that the modern socio-economic-political idiom requires that change be accommodated only if compensating adjustments of the complete opposite be carried out as well? Those who fear being thrown to the mercies of the power companies should find consolation in the benevolent regard that this oddity has managed to earn over the years. In a state whose rates are among the highest in the nation, higher even that California's on average, special NYPA programs for small businesses in communities in the Mohawk Valley provide rates that are much lower and, in fact, possibly the lowest anywhere in the 50 states.

So maybe something else will turn up, something more in line with the Empire State's own particular circumstances. Meanwhile, no one there need feel that their electrical power requirements will receive short shrift in the years ahead during mayhaps rough-edged transitional periods. This extends to planners in the renewables fields as well. Feeling accepted is important and knowing that quiet professionalism undergirds every move made there is worth more than is admitted in comparison with more visible support such as tax credits and subsidized loan programs that may be available as well.

The Horizontals Airflow Analysis

The airflow vector diagram can now be presented as it applies to, not anemometers and not aircraft wings as were the cases covered previously, but wind generator blades. The case of the horizontal wind machine will be covered first and verticals will be covered in a later chapter. What is to follow has supreme importance to those whose work involves wind energy. How anyone can get by without knowing what the horizontal axis blade aerodynamic vector diagram looks like is a question no one should have to ask.

Again, the keys that help in understanding the process are the mass of the air and its deflection when meeting the blade. The blade sees a force in the same way that a parent feels a force when swinging a child. As the airflow curves around the surface of the blade it pulls (from the back side) or pushes (from the front side) the blade in the opposite direction. Since the blade is moving at the time, the air loses some velocity during this process and hence its kinetic energy is given up. That's a brief synopsis of what the vector diagrams say. Let's now take a detailed look at them.

Three cases will be presented. Each case covers the airflow at one particular point along the length of the blade, perhaps about one fourth of the way from the root to the tip. In these cases, the blade is moving with a tangential velocity, V, that is about twice the velocity of the incident wind, W, and, in the usual configuration, is at right angles to it. The first case is that of a positive pitch angle of 15 degrees, the second a pitch angle of zero degrees, and the third a pitch angle of minus 15 degrees. No apologies will be made for looking at these latter two, more unusual cases. In fact some important lessons are to be learned from them. In an earlier chapter it was decided that current popular conceptualizing has all but eliminated negative pitch angles from consideration as being of any use and takes a dim view of zero degree pitch angles as well, even though they are often to be seen at the tips of most wind generator blades operating under load.

The Basic Equation

The three cases are presented with the assistance of the three vector diagrams below. The goal here is to find the driving force on the blade at the particular blade segment represented by them. For this purpose we make use, again, of the Basic Aerodynamic Equation derived in Chapter 1:

The dm/dt term can be arrived at in about the same way as was done for the case of aircraft airfoil lift. It can be formulated as: rCA, where A is now used in place of V to represent the airflow the blade sees.

A, of course, is the sum of two vectors, V, the tangential velocity of the blade at the point under consideration, and W, the velocity of the wind. Then A is deflected by the blade to become the vector B, a vector of the same length as A but pointing in a new direction, the direction given it by the pitch angle of the blade. In order to perform the vector algebra on vectors A and B, the vector B is displaced to the left and downwards so that it starts from the same point that A does. But all along it is plain to see in each of the diagrams that the vector B is parallel to the line, just under the x-axis and to the right of the y-axis, depicting the blade. So far, so good.

Now, ÑV in the above equation is the vector difference between the tips of vectors A and B and is not shown in the three diagrams as presented below. (Remember previously for the case of the aircraft airfoils it was shown as the purple vector.) But it is there in spirit nevertheless and represents a force and its direction (up and to the left) acting on the blade segment. This length of this invisible vector is equal to: 2A sin (d/2), where d is the deflection angle (the angle between vector A and vector B).

But to convert it to a force it must be multiplied by the dm/dt term, which is equal to rCA, and hence the length of the A vector must again be considered when converting this vector to a force.

Only the horizontal component of this vector acts to propel the blade forwards, the vector shown in green in the diagrams below. The other component, also not shown but very much present, acts in a vertical direction upwards and forms what is known as the lateral thrust on the blade. This does no work but the blade must be designed to accept this force and so enters blade design calculations. The green vector is rather short but remember that, as is similar to the longer vector, it must be multiplied by a term that is proportional to the length of vector A in obtaining the driving force on the blade. The term "R" is assigned to the velocity ratio, V/W.

The expression for the driving force acting on the blade segment, taking all of this into consideration, now drops out of the geometry of the diagrams as follows:

The expression for the lateral force acting on the blade segment also is available and can be written as follows:

A more conceptually satisfying way of writing these equations involves first defining the apparent wind acting on the moving blade (often several times greater than the actual wind velocity) and substituting this term into them to obtain simpler expressions as follows:

Three Vector Diagram Cases

Well this is basically what is needed. Now the pitch angle of the blade can be changed in going from one diagram to the other and notice taken of the effect of this change on the length of the green vector.

In going to the second case, that of the zero degree pitch angle, the length of the green vector increases. Far from diminishing, the driving force on the blade has reached a maximum at zero degree pitch angle.

Then in the third case, when the blade has been pitched to a negative 15 degree angle, the green vector has diminished somewhat but is still long enough to be about the same length as it was in the case of a positive 15 degree pitch angle in case one. This is a considerable driving force and yet the blade is running at a pitch angle that seems to be completely backwards. The air being driven by the wind is being kicked backwards and back upwind by the blade's passage, to a reverse velocity of a good fraction of its original velocity.

This is theory but the physical laws being applied are based on reality. The reader is invited to go through the steps of the above analysis again and a sufficient number of times to understand in detail what is happening. The green vector is the greatest in length at a zero degree pitch angle and the blade continues to provide good power production well into the region of negative pitch. It isn't until the pitch angle is negative enough to completely turn the wind around and cause it to move backwards with its original speed forwards that the driving force drops to zero.

Stall

Let's stop a moment before going on. We see here confirmation of what was said earlier. For wind generator blades deflection of the airflow is carried out with much greater emphasis than in previous cases reviewed. The problem of flow separation and stall, a factor limiting the amount of deflection obtainable, becomes a consideration. This treads upon difficult terrain inhabited only by aerodynamicists, technical lingo like "Reynolds Number" is heard in discussions, and experimentally-supported airfoil profiles with their identifying tag numbers are trotted out for inspection on a case-by-case basis.

Let it be said here that flow turbulence, in the understanding that is held and conveyed in the entirety of this material, is not the same thing as flow separation and stall. Laminar flow of most fluids becomes turbulent soon after being directed down the length of a pipe but almost everyone has been on an airplane on a chilly morning with temperatures near the dew point and has looked out the window at the mist-laden flow streaming over the wing at takeoff when no turbulence of any kind is to be seen. For flow tends to both create and also remove the little vortices that form from its passage near by or within a solid object, achieving a balance in any given circumstance that is not exceeded. The Reynolds Number has been invented as a way of determining a limiting velocity beyond which some noticeable turbulence is to be expected. Its formal definition can be conveyed in words as: the flow velocity times a characteristic length times the mass density of the flow medium all divided by its viscosity. As such it is normally used as a sort of "velocity percent" or a generalized, non-dimensional velocity that is in reference to the values of other characteristics of the flow study under way. For the benefit of those involved with wind energy, it is not unlike the wind turbine capacity factor, something that can be set up as a standard for a wide class of machine ratings.

The Reynolds Number, despite its use being reserved to mostly just the experts, may have further applications beyond that of wind velocity characterization within this newer context of wind generator blade design. It stands to reason that flow is less likely to separate from the airfoil surface if its contour is not curved quite so sharply that it can't accept the direction change involved, something concerning which this dimensionless quantity can no doubt assist in the analysis. Wind energy has taken a first cut at understanding stall with its deliberate introduction of stall regulation into blade airflow dynamics. Now that some better control of power production can be obtained with pitch regulation the question of stall becomes different.

If, as is suggested herein, pitching blades less positive and even to zero and beyond carries with it some benefit, stall may be a deterrent to efficient power generation and require special design effort with a view to its understanding and control. It could be, after all, that limitations exist on how much airflow deflection, insofar as airflow deflection is considered to be of value, can be achieved and those limitations are imposed by the inability of the blades to avoid this whole question of stall.

Rides On Packets Of Air

Well, back to the problem at hand.

Now we are set to take a ride, a journey just like the one that was taken in a previous chapter, on a small packet of air that is moving with the wind and is headed for a spot where it will just come in contact with and intercept a wind generator blade approaching on its path from the right. The blade in our story here is pitched to a zero degree pitch angle, the angle seen above as the one in which the greatest driving force is obtained from the motion of air passing close by. Only this time a better interpretation will be given to what happens, one that will hopefully stand us in better stead.

Let's take two cases. In the first case, the blade is moving very slowly or even not at all, the case of starting up a blade after it has been stopped. In the second, the blade is moving at a high rate of speed, say, for example, at its outer tip, where speeds of in excess of five or more times the wind velocity are to be seen. It is heuristically of great benefit to look at this moment of drama in the life of a wind generator blade.

The packet we are riding on now nears the front edge of the blade in the first case. At this point it is cleaved in two, one half continuing on past and headed around the rear of the blade and the other half diverted to pass along the front of the blade. In each case, the air's motion has been altered from a straight line path to a curved one. Since air has mass, conservation of momentum applies and this curvature in its path can only come about if a force is being applied to it, like the sideways force that the road applies to the tires of a car rounding a bend. On the back side of the blade, a lowering of the surface pressure occurs and on the front side, it sees an increase. Since the blade has little or no motion the beginning of this "swing" by the air packet causes a force that is very much directed to the left and along its path of motion. It is only later in the course of this "swing" that the force created becomes more aligned in a lateral direction and "pushes the blade back" but does not contribute to its forward motion. But, in any case, nothing has really slowed the air packet down much and so it continues at practically the full original wind speed along the length of the blade chord and to the trailing edge where the two halves of the packet of air then join up again to form the same packet that we began with. The air packet now continues on a new path after having taken what amounts to a right hand turn of 90 degrees. Little motion has been lost but notice that the driving force imparted to the blade, that which was caused when the air took this turn, was quite large. In fact, in this case, the start-up thrust, again, for a blade pitched to zero degrees angle, is not only large, but, given the benefit of little or no flow separation at this time, it is the largest such start-up thrust that can be achieved, blade pitch angle the variable. Pitching the blade positive means the right hand turn of the air is less than a full 90 degrees and pitching the blade beyond zero and into the negative region means that the swing goes beyond 90 degrees and begins to introduce a reverse thrust force into the blade before leaving the scene.

This story is drawn out at length here but it is because the point needs to be emphasized in any and all treatments of these subjects. From the standpoint of theory, the greatest startup thrust imparted to a motionless blade occurs, under no flow separation, at the rounded front edge of a blade pitched to zero degrees.

Now we look at the case of a fast moving blade. In this circumstance something different happens. The air packet is cleaved in two as before at the front tip of the blade but the blade is moving so fast that the air sees little lowering of the pressure to its right as the blade begins to pass between the two halves of the packet instead. What happens this time is that a huge braking force is applied to the packet halves, the blade essentially stopping them in their tracks. Whatever turn to the right that occurs is rather limited and allows for some resulting motion but at a much reduced velocity. Then the two halves of the packet again join up and move slowly off to the right. So it is plain to see that the kinetic energy of the air has been reduced. Its energy has been almost completely transferred to the blade.

The net result of these two stories is this. In the first case, little energy was transferred to the slow moving blade but the force produced on it was great. In the second case, the force driving the fast moving blade forward was much less but energy was transferred to it with much more efficiency. Interpolation and extrapolation from these two cases are valid and so, in general, as the blade speed, known more formally as the "velocity ratio", increases, the driving force acting on it decreases but not so much as to compromise an increase of the efficiency of energy transfer until it is nearly 100%.

The before and after values of all the parameters in these two stories above, such as the velocities and flow directions, are not as conveniently found in the frame of reference in which they were told. That is why it is so much more convenient to adopt the frame of reference of the blade being fixed near the origin of a vector co-ordinate system and everything else in motion relative to it. The values of these parameters can then all be found with simpler geometry-based calculations.

Stop-And-Go

This, then, is how wind generator blades work in general for air passing by the blades in close proximity to them. Certain conjectures can be made. Since the blades of most wind machines are quite narrow in relation to the spaces between them, something called in more formal terms, the "percent solidity", the air, whose forward motion has essentially been eliminated by the last blade encountered, finds that, in the intervening time period before the next blade arrives, the wind, continuing to blow during this whole time, forces it to get going with full forward momentum. By the time the next blade is ready for it, in other words, the air is moving with its full velocity again and this process is repeated. At one time during an earlier period of wind generator development the opinion was held that only one blade is needed for most horizontals since fast moving blades are highly efficient and fabricating the rotors with more than one blade may not provide enough of this intervening stop-and-go grace period.

As machines grew in size and rotor diameter, however, the empty space between the blades increased and the rates of rotation slowed and so the "one-bladers", seen as lacking upward scaleability therefrom, were discontinued, still to be seen, however, now and again.

This forms the basis of a good understanding of how wind generator blades work insofar as a rudimentary two dimensional approach is concerned and for the air flow passing nearby the blade surfaces. Even in its simplicity, however, some hitherto unexpected discoveries have been made. Energy production not only occurs at pitch angles as small as zero degrees but actually reaches a sort of maximum there. Ditto for the start-up thrust at low blade velocities. Even negative pitch angles work despite their causing the wind to be turned around to flow backwards. Its all simply amazing and falls right out of the airflow vector diagrams.

Objections no doubt can be raised over all of this. A good question may be asked. What is the pressure profile on the surface of a blade pitched to a negative pitch angle, the same pitch angles as those of fan blades, which are not intended for energy capture but its opposite? Well, it is not an easy question to answer and one has to grit one's teeth in reply but just a few words suffice. If the air is diverted to any degree whatsoever to a flow at least partially in line with the motion of the blade and opposite to it in direction then it must impart a driving force to the blade and transfer energy to it. It can look as wrong as can be but it works and satisfies Conservation of Momentum. Don't forget that all bets are off if flow separation occurs.

Even More Can Be Said

There is even more. We were careful to couch all of our statements above within the context of air that is moving in close proximity to the surface of the blades. This is only a small fraction of the air that is affected by their passage. It is pretty obvious that air farther away from and not passing adjacent or near by the surface of the blades is going to be deflected as well. This is true on both sides, the front and the rear. In our vector diagrams, this is seen as vectors of the same length as the B vector and offset from it and at a reduced angle of deflection. Remember that in vector algebra the vectors are all displaced to the same starting point. So we can visualize many vectors in existence with the same length and starting point as the B vector and extending to places on the deflection arc that are closer to the origin of the diagram. All of these vectors have small green vectors proportional to their own contributions to the driving force also. When we remove ourselves to a distance far enough away from the surface of the passing blade and this may be quite some distance, then the deflections are reduced to zero and no further driving forces result. But between the blade surfaces and these points much air passes by and all of it has some impact on the blade despite its distance from it, in accordance with the Basic Aerodynamic Equation.

That is why the space between the y-axis and the deflection arc is filled in with yellow. It is not just the green vector that is responsible for driving force but also green vectors that occupy all the space in this colored area.

Now something even more amazing can be said. Since the yellow area of the case for a zero degree pitch angle is greater than that of the 15 degree positive pitch angle and the yellow area of the case for the 15 degree negative pitch angle is even greater than this (clearly the negative pitch angle is causing quite a bit more deflection in toto of the entire mass of air passing by the blade) then the case of the negative pitch angle now becomes the one that is most to be desired as having the greatest power production efficiency.

Truly amazing. This is theory and little work has been done to our understanding to verify these negative pitch angle concepts but it is in the category of things that, if obtainable in practice, can maybe find a place in the status quo. We are not talking about exceeding the limits of the Lanchester-Betz efficiency of conversion of what is there to begin with in the kinetic energy of the wind but maybe we are talking about getting these blades to do a little better what they are supposed to by being not afraid to pitch the blades into the negative angle region.

A note must be made about startup. While it is permissible to say that, given the absence of flow separation, good thrust is available for the zero degree pitch angle case at low blade velocities, the same cannot be said of negative pitch angles. So, in pursuing this path wherever it leads, it is supposed that machines could be made to run more efficiently if the blades were to be pitched negative but at least some portion of the blades must be pitched zero or positive upon startup. This is beginning to sound vaguely familiar to the case of vertical axis machines but no parallel is intended here and the coverage of the eggbeaters and other styles of verticals is an entirely different subject to be covered in a later chapter.

A note must also be made about lateral forces. As can be seen, or rather as can be surmised in the vector diagram for the case of the negative pitch angle, the "purple" (invisible) vector is quite a bit longer. Remember that the purple vector extends from the starting point of the small green driving force vector on the deflection arc all the way up to the origin of the diagram. This represents a much more substantial force here than in either of the first two cases and current fabrication technology may not allow such increases in the lateral force that the blades must work under. It's an important point.

The Horizontals Blade Aerodynamics Vector Diagram

It's been quite a wild ride through the material in this chapter and not only that associated with rides on small packets of air. It is not clear how the ideas supported herein can find application in the case of the big horizontals. The blades should, maybe, be pitched with less of a positive angle, being careful to realize that pitch angles at blade tips are much more sensitive to change than those near the blade roots. Does this mean a return to flat blades of zero pitch? Not necessarily since at least some curvature may be necessary to meet flow separation requirements. Meanwhile it may be worth considering designing the blades with a reverse pitch twist (a twist that is negative rather than the positive twist many now have) but reminding ourselves about blade startup. As can be seen, many factors come into play when blade pitch angles are adjusted and the subject is really much broader than can be covered within the scope of this treatment.

Below is one last vector diagram, the one that this whole chapter up until now has been leading up to. It shows the case of the horizontals blade airflow for the entire length of a blade pitched to a zero angle. Using this as a starting point other cases can be studied by making appropriate adjustments to it. The V vector varies from the blade root to the blade tip and so the entire blade is included by simply drawing a long line from the smallest V encountered to the largest. The rest follows with no difficulty.