Betz' law

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Schematic of fluid flow through a disk-shaped actuator.

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

Betz' law is a theory about the maximum possible energy to be derived from a "hydraulic wind engine", or a wind turbine such as the Éolienne Bollée (patented in 1868), the Eclipse Windmill (developed in 1867), and the Aermotor (first appeared in 1888 to pump water for cattle, and is still in production). Decades before the advent of the modern 3-blade wind turbine that generates electricity, Betz's law was developed in 1919 by the German physicist Albert Betz[1]. According to Betz's law, no turbine can capture more than 59.3 percent of the kinetic energy in wind.

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[edit] Three Independent Discoveries of the Turbine Efficiency Limit

The British scientist Lanchester derived the same maximum already in 1915. The leader of the Russian aerodynamic school, Zhukowsky, also published the same result for an ideal wind turbine in 1920, the same year as Betz did.[2] It is thus an example of Stigler's Law.

[edit] Economic Relevance

Because some modern wind turbines approach this potential maximum efficiency, once practical engineering obstacles are considered, Betz' Law shows a limiting factor for this form of renewable energy. Engineering constraints, energy storage and transmission losses and other factors mean that even the best modern turbines operate at efficiencies substantially below the Betz Limit.

[edit] Proof

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.

[edit] 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.

[edit] 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 = \rho \cdot A_1 \cdot v_1 = \rho \cdot S \cdot v = \rho \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

\begin{align} F & = m \cdot a \\ & = m \cdot \tfrac {dv} {dt} \\ & = \dot m \cdot \Delta v \\ & = \rho \cdot S \cdot v \cdot \left ( v_1 - v_2 \right ) \\ \end{align}

[edit] 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

P = \begin{matrix} \frac{dE}{dt} \end{matrix} = F \cdot \begin{matrix} \frac{dx}{dt} \end{matrix} = F \cdot v

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

P = \rho \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

P = \begin{matrix} \frac{\Delta E}{\Delta t} \end{matrix}
= \begin{matrix} \frac12 \end{matrix} \cdot \dot m \cdot (v_1^2 - v_2^2)

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

P = \begin{matrix} \frac12 \end{matrix} \cdot \rho \cdot S \cdot v \cdot (v_1^2 - v_2^2)

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 = \begin{matrix} \frac12 \end{matrix} \cdot \rho \cdot S \cdot v \cdot (v_1^2 - v_2^2) = \rho \cdot S \cdot v^2 \cdot (v_1-v_2)

Examining the two equated expressions yields an interesting result, mainly

\begin{matrix} \frac12 \end{matrix} \cdot (v_1^2-v_2^2) = \begin{matrix} \frac12 \end{matrix} \cdot (v_1 - v_2) \cdot (v_1 + v_2) = v \cdot (v_1-v_2)

or

v = \begin{matrix} \frac12 \end{matrix} \cdot (v_1 + v_2)

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.

[edit] Betz' Law and Coefficient of Performance

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

\dot E = \begin{matrix} \frac12 \end{matrix} \cdot \dot m \cdot (v_1^2 - v_2^2)
= \begin{matrix} \frac12 \end{matrix} \cdot \rho \cdot S \cdot v \cdot (v_1^2 - v_2^2)
= \begin{matrix} \frac14 \end{matrix} \cdot \rho \cdot S \cdot (v_1 + v_2) \cdot (v_1^2 - v_2^2)
= \begin{matrix} \frac14 \end{matrix} \cdot \rho \cdot S \cdot v_1^3 \cdot (1 - (\frac{v_2}{v_1})^2 + (\frac{v_2}{v_1}) - (\frac{v_2}{v_1})^3) .
The horizontal axis reflects the ratio \begin{matrix} \frac{v_2}{v_1} \end{matrix} , the vertical axis is the "coefficient of performance" Cp.

By differentiating (through careful application of the chain rule) \dot E with respect to \begin{matrix} \frac{v_2}{v_1} \end{matrix} 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 \begin{matrix} \frac {v_2}{v_1} = \frac13 \end{matrix} .

Substituting this value results in:

P_{\rm max} = \begin{matrix} \frac{16}{27} \cdot \frac{1}{2} \end{matrix} \cdot \rho \cdot S \cdot v_1^3 .

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

P = \begin{matrix} \frac12 \end{matrix} \cdot \rho \cdot S \cdot v_1^3 \cdot Cp.

The "coefficient of performance" Cp (= \begin{matrix} \frac {P}{P_{\rm max}} \end{matrix} ) has a maximum value of: Cp.max = \begin{matrix} \frac{16}{27} \end{matrix} = 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 Cp in the range of 0.4 to 0.5, which is 70 to 80% of the theoretically possible.

[edit] 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 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 A1 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

V_{avg} = \frac{2}{\frac{1}{V_1} + \frac{1}{V_2}} = \frac{2 V_1 V_2}{V_1 + V_2}

To take the place of V_{avg} = \frac{V_1+V_2}{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.

[edit] Modern Development

In 1935 H. Glauert derived the expression for turbine efficiency, when the angular component of velocity is taken into account, by applying an energy balance across the rotor plane. [3] Due to the Glauert model, efficiency is below the Betz limit, and asymptotically approaches this limit when the tip speed ratio goes to infinity.

In 2001, Gorban, Gorlov and Silantyev introduced an exactly solvable model (GGS), that considers non-uniform pressure distribution and curvilinear flow across the turbine plane (issues not included in the Betz approach).[4] The GGS model predicts that peak efficiency is achieved when the flow through the turbine is approximately 61% which is very similar to the Betz result of 2/3rd,but the GGS predicted peak efficiency is much smaller: 30.1%.

Recently, a Direct numerical simulation (DNS) approach based on Computational Fluid Dynamics (CFD) is applied to wind turbine modelling and demonstrated satisfactory agreement with experiment. [5] Computed optimal efficiency is, typically, between the Betz limit and the GGS solution.

[edit] References

  1. ^ Betz, A. (1966) Introduction to the Theory of Flow Machines. (D. G. Randall, Trans.) Oxford: Pergamon Press.
  2. ^ Gijs A.M. van Kuik, The Lanchester–Betz–Joukowsky Limit, Wind Energ. 2007; 10:289–291
  3. ^ White, F.M., Fluid Mechanics, 2nd Edition, 1988, McGraw-Hill, Singapore
  4. ^ Gorban' A.N., Gorlov A.M., Silantyev V.M., Limits of the Turbine Efficiency for Free Fluid Flow, Journal of Energy Resources Technology - December 2001 - Volume 123, Issue 4, pp. 311-317.
  5. ^ Hartwanger, D., Horvat, A., 3D Modelling of A Wind Turbine Using CFD, NAFEMS UK Conference 2008 "Engineering Simulation: Effective Use and Best Practice", Cheltenham, UK, June 10-11, 2008, Proceedings.
  • 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 International, Volume 1, Aug. 2006 Page(s):1 - 7.


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