Betz' law

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

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.

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 vavg, 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.
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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 \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:

E_{\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:

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

The "coefficient of performance" Cp (= \begin{matrix} \frac {E_{\rm max}}{E} \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] Reference

Betz, A. (1966) Introduction to the Theory of Flow Machines. (D. G. Randall, Trans.) Oxford: Pergamon Press.

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