Wind power generation is rapidly growing worldwide, and with that growth, demand for wind turbine design engineers is also growing.  However, an engineer who has experience designing turbines in most applications, will often have trouble translating their hard-won skills for general turbine design, into the wind turbine design. Why? 

A Field of Wind Turbines

 

The primary difficulty is recognizing that there is practically no pressure ratio across a wind turbine; all the energy comes from the kinetic energy of the wind. The velocity of the wind drops across the wind turbine, but the static pressure is the same far upstream and downstream. It is just the opposite for other types of turbine designs where the velocity is typically unchanged between the upstream and downstream locations (assuming no exit diffuser and zero exit swirl), but there is a large change in pressure and temperature.

 

It is instructive to look at a plot of the distribution of static pressure and axial velocity through an ideal wind turbine in a 20 mph (9 m/s) wind.  An ideal wind turbine will operate at the well-known Betz limit, which means that the axial velocity of the wind coming out of the stream tube that passes through the turbine is 1/3 of the upstream velocity. Thus, the axial induction factor (a) is 1/3. Using the Bernoulli equation independently on both sides of the rotor, the pressure distribution difference from atmospheric from far upstream to far downstream can be computed. The results show a pressure rise to the rotor. Across the rotor, there is a sudden pressure drop to below atmospheric as power is extracted from the wind. Then the pressure rises again until the pressure equals atmospheric far downstream. Notice that the pressure changes are very small, and the total-to-static pressure ratio is only 1.0005.

 

The distribution of static pressure implies that far upstream, the inlet to the stream tube of flow that is captured by the wind turbine is smaller than the swept area of the rotor. The stream tube area increases to the swept area right at the rotor and then continues to expand far downstream.  This is essentially a virtual diffuser, which enables a static pressure drop across the rotor, so that power can be extracted. The virtual diffuser is the essence of the Betz limit and why even an ideal wind turbine cannot produce more than ~59% of the available power in the swept area of the rotor.

 

The practical implication of the Betz limit is that the wind turbine rotor must be designed for an inlet velocity determined by the induction factor, which is, in turn, dependent on the design of the rotor. Other turbine designs start with a fixed inlet state that does not depend on the actual design of the turbine.  Because of this difference, turbomachinery engineers must modify how they think about wind turbine design.

 

In upcoming blogs, I will go into more detail about the specifics of rotor design for wind turbines, look at ways to optimize the design, and explore ducted wind turbines to see if a ducted turbine can be used to exceed the Betz limit, as some people think.

 

Wind Turbine Betz Limit