Betz' law

"Please see discussion accessed from the above tab for issues related to proof."

Betz' law reflects a theory for flow machines, developed by Albert Betz. It shows the maximum possible energy — known as the Betz limit — that may be derived by means of an infinitely thin rotor from a fluid flowing at a certain speed.

In order to calculate the maximum theoretical efficiency of a thin 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:

: $dot\; m\; =\; ho\; cdot\; A\_1\; cdot\; v\_1\; =\; ho\; cdot\; S\; cdot\; v\; =\; ho\; cdot\; A\_2\; cdot\; v\_2$

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

: $F\; =\; m\; cdot\; a$

::

:: $=\; dot\; m\; cdot\; Delta\; V$

:: $=\; ho\; cdot\; S\; cdot\; v\; cdot\; (\; v\_1\; -\; v\_2\; )$

Power and Work

The work done by the force may be written incrementally as

: $dE\; =\; F\; cdot\; dx$

and the power content in the wind is

:

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

: $P\; =\; ho\; cdot\; S\; cdot\; v^2\; cdot\; (v\_1-v\_2)$

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

:

::

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

:

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

:

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 provided that they are not equal velocities (in which case no power is extracted at all). 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:

:

::

::

:: .

By differentiating (through careful application of the chain rule) $dot\; E$ with respect to for a given fluid speed "v₁" and a given area "S" one finds the "maximum" or "minimum" value for $dot\; E$. The result is that $dot\; E$ 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:

: C_p.

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

Rotor losses are the most significant energy losses in, for example, a wind mill. It is, therefore, important to reduce these as much as possible. Modern rotors achieve values for C_p in the range of 0.4 to 0.5, which is 70 to 80% of the theoretically possible.

Points of Interest

Note that the preceding analysis has no dependence on the geometry, therefore S may take any form provided that the flow travels axially from the entrance to the control volume to the exit, and the the control volume has uniform entry and exit velocities. Note that any extraneous effects can only decrease the performance of the turbine since this analysis was idealized to disregard friction. Any non-ideal effects would detract from the energy available in the incoming fluid, lowering the overall efficiencies.

There have been several arguments made about this limit and the effects of nozzles, and there is a distinct difficulty when considering power devices that use more captured area than the area of the rotor. Some manufacturers and inventors have made claims of exceeding the Betz' limit by doing just this, in reality, their initial assumptions are wrong, since they are using a substantially larger $A\_1$ than the size of their rotor, and this skews their efficiency number. In reality, the rotor is just as efficient as it would be without the nozzle or capture device, but by adding such a device you make more power available in the upstream wind from the rotor.

"Observation: If we use the middle following of the speeds"

"Vavg=2/(1/V1+1/V2)=2*V1*V2/(V1+V2)"

"To take the place of Vavg=(V1+V2)/2,""then if V2=0 then Vavg=0 for whatever value of V1 (impact without motion).""The calculation is very simple and gives a 50% output."

References

1. Betz, A. (1966) "Introduction to the Theory of Flow Machines." (D. G. Randall, Trans.) Oxford: Pergamon Press.
2. Ahmed, N.A. & Miyatake, M. "A Stand-Alone Hybrid Generation System Combining Solar Photovoltaic and Wind Turbine with Simple Maximum Power Point Tracking Control," IEEE Power Electronics and Motion Control Conference, 2006. IPEMC '06. CES/IEEE 5th InternationalVolume 1, Aug. 2006 Page(s):1 - 7.