Wind Turbine Efficiency, Part 1: A Windy Cylinder of Power

When people think of renewable energy, probably what comes to mind is a mixture of solar panels, wave motion generators, geothermal plants, hydroelectric dams and wind turbines. Of these, the two most associated with the image of a sustainable society are probably solar panels and wind turbines. That’s why when I caught a glimpse of this wind farm while at the Jersey shore, I felt like I had been transported into some sort of futuristic society apart from our own. 

Fig. 1: Wind farm near Atlantic City (personal photo, hence the quality)

Assumedly, we all know that every energy-harvesting method has efficiency limitations, even though some people like to pretend they don’t. With solar panels, calculating efficiency can be complicated as it comes down to a matter of quantum efficiency, but the efficiency limitations of wind turbines, also called Betz’s Law, can be readily calculated with some physics. Let’s explore this.

(Note: though not particularly difficult conceptually, this derivation is math-heavy. I will do my best to explain the underlying logic as we go along. Also, this explanation will be separated into three parts so that your eyes don’t bleed from the endless stretch of math. Backbone calculations for Betz’s Law come from the Wikipedia article and this MIT presentation, but I hope you will find that my explanation, though lengthy, is more intuitive than these sources. Enjoy!)

Let’s start with calculating how much power there exists in a cylinder of wind with base area A corresponding to the arm span of the wind turbine.

Fig. 2: Cylinder with base A and wind influx at velocity v (MIT)

                          1.  KE=1/2 mv²  (kinetic energy equation)
                   2.   KE=W (work-energy equation)
                   3.   P=d/dt(W)=1/2(dm/dt)v²+mv(dv/dt)  (power equation, product rule)

For equation two, let’s assume all work is converted to electrical energy, not friction or blade deformation or other stuff. Now in equation three, what normally happens is that the dm/dt term, or the change in mass over time, is assumed to be zero, producing the familiar mv(dv/dt) or (d(mv)/dt)v or Fv term associated with power. Instead, let’s assume that the air cylinder has a constant known speed v1. Therefore, the term mv(dv/dt) goes to zero, and we are left with equation 4:

                                                                   4.  P=1/2(dm/dt)v1²

From fluid dynamics, we know that

                                                         5.  dm/dt=(m/V)(dV/dt)=r(dV/dt)

or that the mass per volume, density, multiplied by the change in volume over time is equal to the change in mass over time. Makes sense, right? This is essentially just dimensional analysis. With constant wind cylinder base A and changing length L, change in volume over time is defined as

                                                         6.  dV/dt=d/dt(AL)=A(dL/dt)=Av1

Combining equations 4, 5 and 6, we get the general power equation for a cylinder of wind to be

                                                                       7.  P=1/2rAv1³

where r is density, A is cylinder base area (also area of turbine span) and v is a constant wind velocity. 

I’ll stop the first post here since it’s a good spot to take a break. Read on to the next post to learn how to calculate the effective wind speed acting on a wind turbine, and let me know in the comments below if you have any questions.