Betz’ Limit

From Wind wiki

From a macroscopic point of view, the air flow about the wind turbine is at atmospheric pressure. If pressure is constant then only kinetic energy is extracted. However up close near the rotor itself the air velocity is constant as it passes through the rotor plane. This is because of conservation of mass. The air that passes through the rotor cannot slow down because it needs to stay out of the way of the air behind it. So at the rotor the energy is extracted by a pressure drop. The air directly behind the wind turbine is at sub-atmospheric pressure; the air in front is under greater than atmospheric pressure. It is this high pressure in front of the wind turbine that deflects some of the upstream air around the turbine.

Albert Betz was together with Lancaster the first to study this phenomenon. He notably determined the maximum limit to wind turbine performance. The limit is now referred to as the Betz limit.

In order to calculate the maximum theoretical efficiency of a rotor (of, for example, a wind mill) one imagines it to be replaced by a disc that withdraws energy from the fluid passing through it. At a certain distance behind this disc, the fluid that has passed through flows with a reduced velocity.

Assumptions

1. The rotor does not possess a hub, this is an ideal rotor, with an infinite number of blades which have 0 drag. Any resulting drag would only lower this idealized value. 2. The flow into and out of the rotor is axial. This is a control volume analysis, and to construct a solution the control volume must contain all flow going in and out, failure to account for that flow would violate the conservation equations. 3. This is incompressible flow. The density remains constant, and there is no heat transfer from the rotor to the flow or vice versa.

Application of Conservation of Mass (Continuity Equation)

Applying conservation of mass to this control volume, the mass flow rate (the mass of fluid flowing per unit time) is given by:

m =ρA_1 v_1=ρSv=ρA_2 v_2

where v1 is the speed in the front of the rotor and v2 is the speed downstream of the rotor, and v is the speed at the fluid power device. ρ is the fluid density, and the area of the turbine is given by S. The force exerted on the wind by the rotor may be written as

F=ma =m dv/dt =m ̇(Δv) =ρSv(v_1-v_2)

Power and Work

The work done by the force may be written incrementally as

dE=F dx

and the power content in the wind is]

P=dE/dt=F dx/dt=Fv

Now substituting the force F computed above into the power equation will yield the power that can be extracted from the available wind:

P=ρSv^2 (v_1-v_2 )

However, power can be computed another way, by using the kinetic energy. Applying the conservation of energy equation to the control volume yields

P=ΔE/Δt =1/2 m ̇(v_1^2-v_2^2 )

Looking back at the continuity equation, a substitution for the mass flow rate yields the following

P=1/2 ρSv(v_1^2-v_1^2)

Both of these expressions for power are completely valid, one was derived by examining the incremental work done and the other by the conservation of energy. Equating these two expressions yields

P=1/2 ρSv(v_1^2-v_1^2 )=ρSv^2 (v_1-v_2)

Examining the two equated expressions yields an interesting result, mainly

or

Therefore, the wind velocity at the rotor may be taken as the average of the upstream and downstream velocities. This is often the most argued against portion of Betz' law, but as you can see from the above derivation, it is indeed correct.

Betz' Law and Coefficient of Performance

Returning to the previous expression for power based on kinetic energy:

formula here

By differentiating with respect to for a given fluid speed v1 and a given area S one finds the maximum or minimum value for . The result is that reaches maximum value when

Substituting this value results in:

The work rate obtainable from a cylinder of fluid with area S and velocity v1 is:

The "power coefficient" Cp (= ) has a maximum value of: Cp.max = = 0.593 (or 59.3%; however, coefficients of performance are usually expressed as a decimal, not a percentage).

In other words, a perfect best-possible wind turbine would be able to convert almost 59% of the power in the wind into mechanical rotating power. There are other things that limit efficiency, and efficiency may not even be the most important thing. The most important thing is generally to get the most electric power for the least money. This may require design trade-offs that limit efficiency in order to get the best overall system for the money. For example, it might not be possible to make the most efficient blade shape strong enough to hold together during a strong wind. As you can see from the formula above, being able to get power from a stronger wind is probably worth more than efficiency. On a very windy site, stronger less efficient blades might end up getting us more power.