Michael Graham - Wind Energy Handbook

Even with no losses included in the analysis, the Betz limit is not reached because the blade design is not perfect; see Figure 3.51.

Figure 3.50 C P ‐ λ performance curve for a modern three blade turbine.

Figure 3.51 C P ‐ λ performance curve for a modern three blade turbine showing losses.

At this stage, the other principal parameter to consider is the solidity, defined as total blade area divided by the swept area. For the three blade machine, above, the solidity is 0.0345, but this can be altered readily by varying the number of blades, as shown in Figure 3.52.

The solidity could also have been changed by changing the blade chord.

The main effects to observe of changing solidity are:

1 Low solidity produces a broad, flat curve, which means that the CP will change very little over a wide tip speed ratio range, but the maximum CP is low because the drag losses are high (drag losses are roughly proportional to the cube of the tip speed ratio).

2 High solidity produces a narrow performance curve with a sharp peak, making the turbine very sensitive to tip speed ratio changes and, if the solidity is too high, has a relatively low maximum CP. The reduction in CP max is caused by stall losses.

3 An optimum solidity appears to be achieved with three blades, but two blades might be an acceptable alternative because although the maximum CP is a little lower, the spread of the peak is wider, and that might result in a larger energy capture.

It might be argued that a good solution would be to have a large number of blades of small individual solidity, but this greatly increases production costs and results in blades that are structurally weak and very flexible.

There are applications that require turbines of relatively high solidity; one is the directly driven water pump, and the other is the very small turbine used for battery charging. In both cases it is the high starting torque (high torque at very low tip speed ratios) that is of importance, and this also allows small amounts of power to be developed at very low wind speeds, ideal for trickle charging batteries.

Figure 3.52 Effect of changing solidity.

The torque coefficient is derived from the power coefficient simply by dividing by the tip speed ratio, and so it does not give any additional information about the turbine's performance. The principal use of the C Q– λ curve is for torque assessment purposes when the rotor is connected to a gearbox and generator.

Figure 3.53shows how the torque developed by a turbine rises with increasing solidity. For modern high‐speed turbines designed for electricity generation, as low a torque as possible is desirable to reduce gearbox costs. However, the multi‐bladed, high solidity turbine, developed in the nineteenth century for water pumping, rotates slowly and has a very high starting torque coefficient necessary for overcoming the torque required to start a positive displacement pump.

The peak of the torque curve is at the stall onset and occurs at a lower tip speed ratio than the peak of the power curve.

The thrust force on the rotor is directly applied to the tower on which the rotor is supported and so considerably influences the structural design of the tower.

Generally, the thrust on the rotor increases with increasing solidity, as shown in Figure 3.54. These results are computed using Eq. (3.48)without the additional contribution from the rotational wake pressure term Δ p d2[see Eq. (3.22)and the discussion following Eq. (3.48). Including this term increases the value of C Tby the order of 1% when λ = 8 (specifically 1.39% for λ = 8, a = 1/3 and blade root at r / R = 0.135).

Figure 3.53 The effect of solidity on torque.

Figure 3.54 The effect of solidity on thrust.

The majority of wind turbines currently installed generate electricity. Whether or not these turbines are grid connected, they need to produce an electricity supply that is of constant frequency else many common appliances will not function properly. Consequently, a mode favoured in the early years of wind turbine development has been operation at constant rotational speed. Connected to the grid a constant‐speed turbine is automatically controlled, whereas a stand‐alone machine needs to have speed control and a means of dumping excess power.

An alternative performance curve can be produced for a turbine controlled at constant speed. The C P –λ curve shows, non‐dimensionally, how the power would vary with rotational speed if the wind speed was held constant. The K P –1/λ curve describes, again non‐dimensionally, how the power would change with wind speed when constant rotational speed is enforced. K Pis defined as

(3.95)

The C P – λ and K P – 1/λ curves for a typical fixed‐pitch wind turbine are shown in Figure 3.55. The K P – 1/λ curve, as stated above, has the same form as the power–wind speed characteristic of the turbine. The efficiency of the turbine (given by the C P – λ curve) varies greatly with wind speed, a disadvantage of constant‐speed operation, but it should be designed such that the maximum efficiencies are achieved at wind speeds where there is the most energy available.

Figure 3.55 Non‐dimensional performance curves for constant‐speed operation.

An important feature of this K P –1/λ curve is that the power, initially, falls off once stall has occurred and then gradually increases with wind speed. This feature provides an element of passive power output regulation, ensuring that the generator is not overloaded as the wind speed increases. Ideally, the power should rise with wind speed to the maximum value and then remain constant regardless of the increase in wind speed: this is called perfect stall regulation . However, stall‐regulated turbines do not exhibit the ideal, passive stall behaviour.