Lecture 23 Wind : Part 3

Main Ideas

1. Betz’s Law and the Betz Limit

Betz’s law states that a wind turbine has a theoretical upper limit on the amount of wind energy it can extract. This limit applies to turbines operating in an open flow, regardless of their design.

The maximum power coefficient, known as the Betz limit, is:

[ C_{P,\max}=\frac{16}{27}\approx 0.593 ]

Thus, a turbine can theoretically extract no more than 59.3% of the wind’s available power. Real turbines achieve less; the lecture says modern utility-scale turbines typically reach about 75–80% of the Betz limit.

The wind does not stop at the rotor. It must retain some speed downstream so the flow can continue through the turbine. In the ideal Betz model, the remaining 40.7% is not extracted.

Derivation Outlined in the Lecture

The turbine is represented as an ideal actuator disc, with three flow sections: far upstream, at the rotor, and far downstream.

Assumptions:

• Air is incompressible, so its density is constant.
• Flow is treated as irrotational, with no rotational or turbulent energy losses in the idealized model.
• By conservation of mass, mass flow is constant:

[ \rho A_1V_1=\rho A_0V_0=\rho A_2V_2 ]

• The speed at the rotor is the average of the upstream and downstream speeds:

[ V_0=\frac{V_1+V_2}{2} ]

Power extracted: The power removed from the wind is the incoming power minus the downstream power. Using the continuity relationship, the lecture expresses this as:

[ P_T=\frac{1}{4}\rho A_0(V_1+V_2)(V_1^2-V_2^2) ]

Power coefficient: Define (x=V_2/V_1). Dividing the extracted power by the power available in the upstream wind, (\frac12\rho A_0V_1^3), gives:

[ C_P=\frac12(1+x)(1-x^2) ]

Maximum power coefficient: Differentiate (C_P) with respect to (x), set the derivative to zero, and keep the physically meaningful nonnegative solution:

[ x=\frac13 ]

Substituting this value gives:

[ C_{P,\max}=\frac{16}{27}\approx0.593 ]

The Betz limit is a benchmark for evaluating turbine performance and estimating potential energy production.

2. Why Wind-Speed Measurements Matter

Reliable wind-speed data are needed to assess a site’s wind resource and make decisions about turbine design, capacity, and installation height. Measurements depend on:

• Location and terrain, such as onshore or offshore sites, hills, valleys, or agricultural land.
• Measurement height, since wind conditions vary with distance above the ground.
• The instrument’s accuracy, range, response time, and installation environment.

3. Instruments for Measuring Wind

Anemometers

1. Cup anemometer

   • Uses hemispherical cups mounted on arms around a vertical shaft.
   • Wind produces different drag on the concave and convex sides of the cups, causing them to rotate.
   • Rotation rate is proportional to wind speed; an encoder or generator converts it into a signal.
   • The lecture identifies it as widely used because it is simple, robust, relatively inexpensive, and easy to maintain.

2. Propeller or vane anemometer

   • Uses a propeller on a horizontal axis, with a tail vane to align it with the wind.
   • Propeller rotation indicates speed, while the vane helps align the instrument with the wind direction.

3. Ultrasonic (sonic) anemometer

   • Sends ultrasonic pulses between pairs of transducers.
   • Wind changes the travel time of sound in different directions; those time differences are used to calculate wind-velocity components.
   • It provides high-frequency measurements and is highlighted for turbine control, turbulence studies, and offshore use, though it is expensive.

4. Hot-wire (thermal) anemometer

   • Uses a fine, electrically heated wire placed in the airflow.
   • Moving air cools the wire by convection, changing its temperature and electrical resistance. The change is converted into wind speed.
   • It is useful for low-speed airflow and duct measurements but is fragile and requires frequent calibration.

5. Pitot-tube anemometer

   • Measures the difference between total and static pressure.
   • Wind speed is calculated using Bernoulli’s equation.
   • The lecture identifies it as suitable for high-speed airflow measurements.

Remote-Sensing Systems

• LIDAR (light detection and ranging) uses laser light and measures Doppler shifts in light scattered by airborne particles. It can produce wind profiles at multiple heights, extending to several hundred metres above the ground.
• SODAR (sonic detection and ranging) sends acoustic pulses and analyzes sound scattered by atmospheric turbulence. It provides vertical wind profiles without placing sensors at every measurement height. Readings can be affected by environmental conditions such as rain, hail, and other precipitation.

4. Practical Comparison

Anemometers are point-measurement instruments installed where wind is being measured. LIDAR and SODAR can measure wind remotely at multiple heights.

The lecture presents cup anemometers as a common choice for long-term wind-resource assessment and ultrasonic anemometers as useful when high-frequency data are needed, including for turbine control and offshore applications.

Speakers and Sources

• Speaker: One lecturer or presenter; no name is given in the subtitles.
• Source: Lecture video from the IIT Roorkee July 2018 channel, titled Lecture 23 Wind: Part 3.