Wind Turbine Efficiency, Part 2: Too Many V Terms

Welcome back! If you haven’t already, please read Wind Turbine Efficiency, Part 1: A Windy Cylinder of Power.

All calculations thus far have dealt with air in front of the wind turbine. We should take a look at what happens behind it as well. From fluid dynamics we know the following equations to be true through logical progression:

                               8.     V1=V2 (V1 is air volume before turbine, V2 is air volume after)

                               9.     dV1/dt=dV2/dt

                               10.     dV/dt=Av (equation 5)

                               11.     A1v1=A2v2

Equations 9-11 simply state that the volume of wind entering our cylinder from Part 1 must change at the same rate as the volume of air exiting the turbine from behind. This is true if we assume no changes in temperature or atmospheric pressure before to after the turbine (beyond those induced by the slowing wind, which we are calculating for now). From energy conservation, we know that if work is performed on the wind turbine, then the kinetic energy of the wind after the turbine, and accordingly the wind velocity, must be less than that before. In other words, v1>v2. The result is that A2>A1, so the wind after the turbine must effect a cone of area expanding away form the turbine blades.

Fig. 3: Air flow through a wind turbine (Danish Wind Industry Association)

Let’s try to piece apart the maximum power obtainable from the wind cylinder calculated above. Wind force is defined as follows:

                                     12.     F=ma=d(mv)/dt=(dm/dt)v+m(dv/dt) (force equation)

For simplicity’s sake, we are assuming that the wind before the turbine has velocity v1 and slows to velocity v2 after performing work on the turbine, both of which are know constant values. When wind blows through our wind cylinder, the mass of air, and as a result the effective length of the cylinder since base A is constant, changes with time. Therefore, we are not concerned with how force changes with velocity, so the m(dv/dt) term can go to zero and we can evaluate the change in wind force over the turbine as follows:

                                  13.     ΔF=(dm/dt)Δv=(dm/dt)(v1-v2)

                                  14.     ΔF=r(dV/dt)(v1-v2)=rAve(v1-v2) (from equations 4 and 5)

A new variable has been introduced in equation 14, namely the effective velocity ve, which corresponds to the wind velocity that the turbine experiences as part of work production (in other words, if the wind were to lose all kinetic energy after contacting the turbine, a wind velocity of ve would generate a quantity of power the same as from wind with velocity v1 slowing to v2). Concerning the power obtained from the change in force, we know that since the change in wind force is equal to the force exerted on the turbine, power can be calculated as follows concerning the wind contacting the turbine:

                         15.     W=FL (instantaneous work equation, L is analogous displacement here)

                         16.     dW/dL=F, therefore dW=FdL

                         17.     P=dW/dt=d/dt(FdL)=F(dL/dt)=Fve (power equation)

                         18.     P=rAve²(v1-v2) (from equation 14)

From expanding equation 4 to accept that the wind velocity after the turbine is not zero, we obtain the equation:

               19.    P=1/2(dm/dt)(v1²-v2²) (this is just the difference in kinetic energy with changing mass)

               20.    P=1/2rAve(v1²-v2²) (from equations 5 and 6, parallel to equation 7)

Now we have two different equations for power, one from the work equation and the other from the kinetic energy equation! Setting these equal to each other, we obtain the expression:

                   21.     P=1/2rAve(v1²-v2²)=rAve²(v1-v2)

                   22.   v=1/2(v1+v2) (simplification of equation 21 using (v1²-v2²)=(v1+v2)(v1-v2) identity)

This equation tells us what the relationship between the effective velocity ve and the initial and final wind velocities v1 and v2 are. Specifically, the relationship is that the effective velocity is the average of the initial and final wind velocities. Using this relationship, let’s redefine the kinetic energy-derived power equation in terms of v1 and v2 in preparation for calculating maximum power.

                            23.    P=1/2rAv(v1²-v2²)=1/2rA(1/2(v1+v2))(v1²-v2²) (from equation 20)

                            24.    P=1/4rAv1³(1-(v2/v1)²+(v2/v1)-(v2/v1)³) (simplification of equation 23)

Things are starting to get exciting! Read on to Part 3 to finally start calculating the external efficiency limitation of wind turbines, and let me know any questions in the comments below.