Wind Turbine Efficiency, Part 3: The 40.3% That Got Away

You made it to Part 3! If you haven’t already please go back to read Part 1 and Part 2.

Let's continue. From calculus, we know that to find a maximum of a given single-variable function, we take the derivative, or the slope of the function, with respect to the variable and find where the derivative equals to zero. This is because the slope of a function is zero where the function changes from increasing to decreasing or decreasing to increasing output values (aka peaks and valleys). Equation 24 from Part 2 is rather ugly, however, so let’s set the derivative with respect to k=v2/v1 (the ratio of final and initial velocity) as our plan of attack (reminder: ve, v1 and v2 are all known constants in this expression). Let’s do this now.

            25.     P=1/4rAv1³(1-k²+k-k³) (substitution of k=v2/v1 into equation 24)

            26.     dP/dk=1/4rAv1³(0-2k+1-3k²)=1/4rAv1³(-2k+1-3k²)=0

            27.     -2k+1-3k²=0 ((1/4rAv1³) is a giant multiplicative non-zero constant term, so it can be
             dumped into the zero never to emerge again (except by integration))

            28.     k=1/3, -1 (solutions to equation 27, obtained by factoring the second-degree polynomial and
             letting each set of terms equal to 0 since if ab=0, either a=0 or b=0 makes the statement true)

Obviously, a negative value of k=v2/v1 does not make sense since this requires that the wind hit the turbine and flow backwards, so we will only keep the k=1/3 value. Plugging the value for k into equation 25, we get:

                                           29.     Pmax=1/4rAv1³(1-1/9+1/3-1/27)=16/27(1/2rAv1³)

To interpret this result, all that must be remembered is that the total power in the original cylinder of wind was found to be Pwind=1/2rAv1³ in equation 7. Substituting in this expression reveals the following relationship:

                                                                      30.     Pmax=16/27Pwind

In other words, only 16/27, or 59.3%, of the total power in a column of wind can ever be extracted by wind turbines (this is assuming 100% turbine internal efficiency relative to the 59.3% limitation). This means that there is an inherent limitation in the production efficiency of wind power relative to available energy to be harvested.

Fig. 4: Turbine efficiency cartoon (Wikimedia)

And there you have it, Betz's Law in only 30 steps.

So, knowing this, should we abandon wind power? The answer is a definite no. Wind turbines have an installation cost in both money and materials and also severe limitations as to practical operation, such as a decreasing efficiency in high wind velocities to prevent structural damage, but they are an important member of a small set of what I would consider to be true renewable energy strategies. I won’t talk about this now since I feel guilty about the density of the topic I just covered, but I will make this the topic of my next post.

If you have any questions, leave them in the comments section below and I’ll try to answer them as best I can.