Betz' Law

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Betz' Law

Schematic of fluid flow through a disk-shaped actuator.

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 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.

Let v₁ be the speed of the fluid in front of the rotor and v₂ that of the fluid downstream from it. The mean flow velocity through the disc representing the rotor is v_avg, where

$v_{\rm avg} = \begin{matrix} \frac12 \end{matrix} \cdot (v_1 + v_2)$

With the area of the disc equal to S, and with ρ = fluid density, the mass flow rate (the mass of fluid flowing per unit time) is given by:

$\dot m = \rho \cdot S \cdot v_{\rm avg} = \frac{\rho \cdot S \cdot (v_1 + v_2)}{2}$

The power delivered is the difference between the kinetic energies of the flows approaching and leaving the rotor in unit time:

$\dot E = \begin{matrix} \frac12 \end{matrix} \cdot \dot m \cdot (v_1^2 - v_2^2)$

$= \begin{matrix} \frac14 \end{matrix} \cdot \rho \cdot S \cdot (v_1 + v_2) \cdot (v_1^2 - v_2^2)$

$= \begin{matrix} \frac14 \end{matrix} \cdot \rho \cdot S \cdot v_1^3 \cdot (1 - (\frac{v_2}{v_1})^2 + (\frac{v_2}{v_1}) - (\frac{v_2}{v_1})^3)$ .

The horizontal axis reflects the ratio , the vertical axis is the "coefficient of performance" Cp.

The horizontal axis reflects the ratio $\begin{matrix} \frac{v_2}{v_1} \end{matrix}$ , the vertical axis is the "coefficient of performance" C_p.

By differentiating $\dot E$ 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 area S and velocity v₁ is:

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

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

Observation: If we use the middle following of the speeds

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

To the place of Vavg=(V1+V2)/2 because 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 International Volume 1, Aug. 2006 Page(s):1 - 7.

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia Encyclopedia article "Betz' Law"

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