Here is the graph for the %E obtained from this formula for various blade velocity ratios, R, and pitch angles, including 0 degrees:

Notice that at high blade speeds in relation to the wind (R values) the blade efficiency drops off for pitch angles other than zero. For the pitch angle of zero degrees it remains at nearly 100% up to an R value of 10 and beyond. Notice also that the efficiency decrement becomes sensitive to even very small pitch angle differences at these high R values. This is an additional disadvantage of running at fast blade speeds. It also demonstrates why a pitch angle of zero degrees, despite the ordinary intuitive difficulty in understanding it, can be so beneficial. Given the absence of flow separation, something often not avoidable but if so, the blade is simply more efficient at all R values.

"In relation to the wind" means what it says. Wind turbines are often compared for their ability to generate energy in low wind conditions and low wind conditions mean high R values, the same R values that pertain to fast moving blades in better wind conditions.

This, again, is a somewhat elementary view that does not take into account the airflow deflection at more remote distances from the blade surfaces. As was discussed in earlier chapters, if such additional air mass deflection is included in these calculations, negative pitch angles add to rather than subtract from the blade efficiencies up to a point. But the same conclusions, in general, can be reached for this more general case, that is, that the efficiency at high R values is sensitive to small pitch angle differences and that a zero degree pitch angle tends to maintain the efficiencies at a high value rather than allowing them to decrease. These are important conclusions.

The above graph can be presented again, for only a pitch angle of zero degrees and up to a maximum value for R of 5 rather than for a higher value. The attractive feature of this graph is that it is applicable practically universally to all sorts of wind turbines of different configurations and their blades, running at a wide range of speeds at whatever locations along their lengths and moments of time. It indicates in the clearest of terms that blade efficiencies cannot be but so great at such and such R values.

Here is what the graph looks like:

As the wind passes over the blade and is deflected it gives up its energy and this graph shows, as a maximum, just how much. Notice that, as a rule of thumb, the energy conversion can be seen to be quite good at an R value of 2 - about 95%. It is not bad at an R value of 1 either - about 83% - but it drops off very quickly at a value of .5 - about 60% - and below. If good energy conversion is required, then, it is best if all locations along the blade lengths and during as much of the rotational cycle as possible, have velocities equal to that of the wind crossing them or greater, preferably twice as great.

As mentioned earlier, this is not the same as the Betz or the overall rotor efficiency. Here we can obtain efficiencies of 100%. If these blade efficiencies average out to low values, then they risk affecting the overall rotor efficiency of the machine, lowering it below the Betz efficiency despite other measures taken to keep it high. We see the necessity now for adding blades to windmill rotors that turn slowly. It helps boost the overall machine efficiency where the efficiencies of the individual blades are low.

Considerations That Apply To The Verticals

Here are the derivations for the formulas for the energies and the %E for the verticals:

This might have been all that is needed to be said for the verticals except for several important points that must be considered when putting the above information relating to the blade velocity ratios and the blade efficiencies to use for them.

The blade velocity ratio is a useful defined quantity for use in understanding wind turbine rotor operation, all the more so because it relates directly to the blade efficiency as described above. However, for the verticals, a broader definition is applicable to cover the case of the wind direction and the blade direction being something other than at right angles to each other, such as when the blades are not on the frontal centerline.

It must be kept in mind in these instances that it must be defined as the blade velocity relative to the component of the wind velocity parallel to the blade and divided by the component of the wind velocity at right angles to the blade. This is simple enough in the case of the horizontals where, unless the rotor is not facing directly into the wind, the two velocities are always perpendicular to one other.

In all of the formulas provided above and in previous chapters, the R value used for the verticals was defined as the R value that applies at the moment the blade crosses directly in front of or behind the rotor axis. This is usually at or near the point of maximum driving force and energy production. As the blade moves away from this position the wind is no longer at right angles to the blade motion.

So the blade velocity relative to the component of the wind velocity parallel to the blade becomes (V-Wcosa), where a is the rotor angle defined previously. In addition, the component of the wind velocity at right angles to the blade velocity becomes (Wsina). This causes the effective R value to vary somewhat near the midpoint and then to increase to large values approaching infinity at the neutral points, where the sina is zero. So it can be seen that the R value used in all the previous work is near the minimum value that the rotor normally sees during its rotation.

The effective blade velocity ratio, R, then, can be seen in the general case to be equal to:

(V - W cos a) / (W sin a)

Below is a graph that demonstrates, based on these considerations, how the effective R value varies across the face of the blade frontal swept area from one side to the other. Five cases are shown, each case having V/W values taken at the blade midpoint of 1, 2, 3, 4, and 5, in succession. As can be seen, the R values tend to be somewhat lower on the downwind side as compared to the upwind side. The lowest curve, for a V/W of 1, goes to zero on the right, as would be expected for the blade traveling at a net zero relative velocity to the wind at this point.

Blade velocity ratios less than 1.1 to 1.25 are to be avoided for efficiency reasons but, as was mentioned above, nothing much is gained for blade velocity ratios much greater than 2, either, the point of diminishing returns. If the Betz efficiency is being obtained by means of overall rotor design and operation in other respects, high blade speeds can complicate matters in such ways as being a source of noise or visual distraction or in such things as being subject to greater aerodynamic drag and faster wear and tear. For fixed speed machines, R values necessarily vary depending on the wind speed but it is clear that maintaining the blade speed to wind speed ratio at or about the relatively low value of 2 or 2.25 as shown on this graph for the verticals rotor frontal wind capture area is all that is usually necessary.

This is not the case for the horizontals. The R value most usually referenced is the R value at the blade tip, otherwise known as the TSR or tip speed ratio. This is the highest value of R the blade sees. For other locations on the blade closer to the hub, the R value is less. In maintaining an R value of 2 or above along most of its length for blade efficiency reasons, it is generally necessary for this TSR to be valued to 4 or greater in proportion. This means that, based on this reasoning alone, horizontals blades must normally run at higher velocities in their power producing regions compared with the verticals, something that, if it matters much, can be considered to be an inherent disadvantage. It is also true that, even running at these lower R values, the verticals are subject to some unavoidable aerodynamic drag during the relatively large amount of time their blades spend at locations around their circuits where energy conversion is low.

Important Formulas Based On The R Value

Now that the parameter R has been defined and its importance in relation to the efficiency of the blade in producing energy at each increment along its length has been determined, a few more formulas can be presented that are based on these blade increment R values as well. Accordingly, discussed below are formulas for the blade increment driving force, the blade increment lateral force, and the blade increment energy creation, this last being obtained from multiplying the blade driving force by the blade velocity.

Once it is clear what is happening with each blade increment along the blade lengths, then all these increments together can be viewed as "building blocks" in being assembled together in determining the total forces acting on the blades and the total energy created by them. A characteristic of most of this analysis is that it assumes no flow separation is taking place and no parasitic drag is present. These assumptions are normally not valid in practice. At low R values approaching zero, there normally is flow separation present unless the blade leading edge is made to have some camber or be round and bulbous rather than "pointy". At high R values, parasitic drag becomes noticeable, netting out some of the driving force and thus reducing the energy delivery unless, on the other hand, the blade leading edge is straight with no camber or is sharp and "pointy". Clearly, it becomes difficult to satisfy both of these conditions at the same time with the same blade.

It also is assumed that the blade increment pitch angle is zero. As we have seen in the above paragraphs, nonzero pitch angles, both positive and negative, alter the forces and energy creation in a way that is highly dependent on the value of R at its upper range of values. For the time being we want to look at the effect of R on the blade increment without the complication of pitch angle differences and using a pitch angle of a nominal value of zero does this adequately enough. The effect of pitch angle variations may be looked at separately elsewhere.

The first formula, then, is for the blade increment driving force as is shown in the below graph:

The formula is taken directly from previous chapters as reduced for a pitch angle of zero. First, it is to be noticed that, for all values of R, the driving force is generally proportional to the wind velocity squared. This is always the wind velocity the blade sees immediately and locally at right angles to the blade forward motion. For the horizontals, it means the wind velocity the blade sees following in the wake of the previous blade during the intervening time after it has removed some of its energy and velocity. For the verticals, it means substantial changes to the wind velocity at right angles to the blade as it moves around its circuit. These considerations affecting the wind velocity apply here and in the below two formulas as well.

Once the variation with wind velocity is is taken into account, then, the rest is easy. When the blade is stopped or moving very slowly (low R), the value of the multiplier factor as shown in the graph is about one. As the blade increases in speed, the multiplier factor drops to about one half and stays there no matter the blade velocity from then on (high R). It is characteristic of wind generator blades, therefore, at zero degree pitch angles that they produce about twice as much driving force or torque when they are moving slowly as when they are moving fast.

This also applies to horizontals blade driving forces along their lengths from root to tip (low R to high R). The driving force drops somewhat but remains quite high at a constant value all the way out to the tip and this, of course, encourages the lengthening of the blades in the by now classic horizontals wind generator upgrade design process. The point of diminishing returns occurs when the parasitic drag effects at high R values become significant. The lateral forces on the blade, described next, also play a role in limiting blade lengths. But until then little stands in the way of rotors of larger and larger diameters that profit from this additional blade length.

The second formula is for the blade increment lateral force as shown in the below graph:

This formula is also taken directly from previous chapters as reduced for a pitch angle of zero. Again, it is to be noticed that, for all values of R, the lateral force is generally proportional to the wind velocity squared. It is important to minimize the lateral force and help is available in either designing the blade for stall regulation or in depitching it (increasing the pitch angle in a positive sense) to avoid excessive lateral forces when necessary. When the blade is stopped or moving very slowly (low R), the value of the multiplier factor as shown in the graph is about one. As the blade increases in speed, the multiplier factor increases and thereafter assumes a somewhat linear increase with blade velocity (high R). This, of course, is despite the pitch angle being zero. Again, it must be emphasized that, for a fixed wind speed, the blade "bend-back" (lateral) force increases linearly with blade velocity, something not often realized.

This also applies to horizontals blade lateral forces along their lengths from root to tip (low R to high R). While the driving forces remain level out to the blade tips, as noted above, the lateral forces, which are typically greater to begin with, go on to increase in a linear fashion and so can reach values that are much higher by large factors. These forces must be absorbed and accommodated since, after all, this is how wind generator blades do their work in stopping or slowing the wind.

This point bears repetition. While the blade driving forces remain constant with blade speeds and along blade lengths out to their tips, the lateral forces increase in a linear fashion and can reach much greater values.

The third formula is for the blade increment energy creation as shown in the below graph:

This formula is also taken directly from previous chapters as reduced for a pitch angle of zero. It is seen, of course, to be equal to the formula for the driving force provided earlier multiplied by the blade forward velocity. It is to be noticed that, for all values of R, the energy creation is generally proportional to the wind velocity cubed. When the blade is stopped or moving very slowly (low R), the value of the multiplier factor as shown in the graph is zero or close to zero. This is also the place where the driving force is at its maximum as discussed earlier. It is one of those odd things about wind generator blades that those times when the blade torque is at its highest are also the times when the energy delivery is at its lowest. This should help in facilitating startups when necessary if measures are also taken to eliminate flow separation.

The energy delivery by the blade increases, then, as seen above, in a linear fashion with increasing R (high R). This same effect is observed by sailboats when sailing close hauled in that their sails see greater apparent winds with greater forward speeds and so, in succinct terms and to a certain extent, the faster they go the faster they go. Unlike sailboats, however, as mentioned above under the formula for the driving force, wind generator blades can only rotate so fast before the wind velocity seen by each blade following in the wake of the previous blade, which has removed some of the wind energy, begins to be reduced from that of the ambient wind velocity, bringing to an end any further gains.

But this still applies to horizontals blade energy creation along the blade lengths from root to tip (low R to high R). It is to be expected that the tips would provide the most significant energy input since they cover the greatest distance while the blades go around their circuits. Again, the considerations mentioned above apply and other factors become involved with increasing blade lengths.

Finally

A thought that should be kept in mind is that the Betz analysis states that for wind turbine rotors the wind speed seen at the plane of the swept area is equal to the average of the upwind (ambient) and downwind air speeds. If the machine is running at the maximum efficiency, the downwind wind speed is one third of the upwind wind speed and the average between them is two thirds. So when calculating the R value for the blades, this lower value should be used and not the full ambient windspeed. This increases R values, normally a good thing. And it applies to both the horizontals and verticals configurations.

No one ever said that these analyses would be easy reads and often this material is entered upon with a certain amount of hesitation. But in going through it, there is no doubt that some valuable insights are gained. One of the most ambitious is that wind energy is still in its infancy as of this writing. Let's hope that some of the obstacles to development that have stood in the way are removed as things unfold and the potentials that are there are realized in greater measure.