Windmills
Windmills are structures that harness wind energy for various practical purposes, including energy production and food processing. Historically, they have evolved from simple, broad designs with an even number of sails to modern, sleek wind turbines featuring three blades and advanced technology for optimizing energy capture. The earliest known windmills originated in ancient Persia, serving essential functions like grinding grain and pumping water, and their use spread across the Middle East, Asia, and eventually into Europe by the twelfth century.
Windmills can be classified into two primary designs: vertical-axis and horizontal-axis. Vertical-axis windmills rotate perpendicularly to the ground and can capture wind from any direction, while horizontal-axis windmills, more commonly recognized, are aligned with the prevailing wind for maximum efficiency. The number of blades on a windmill influences its power generation, with additional blades providing marginal increases in output.
Mathematics plays a crucial role in both the design and efficiency modeling of windmills, with principles such as the "Betz limit" defining the theoretical maximum energy capture from wind. As renewable energy sources gain importance, the role of windmills and turbines in generating electricity continues to expand, often seen in wind farms where multiple turbines work together to supply energy to electrical grids.
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Windmills
Summary: The amount of power that a windmill can harness can be determined mathematically according to its size and design.
For centuries, windmills have captured peoples’ imaginations through their form, function, and romantic appeal. Immortalized by Miguel de Cervantes in his book Don Quixote, windmills have transformed over the years from broad, short structures with an even number of sails to tall, sleek, three-sailed structures equipped with turbines for capturing energy from the wind. Windmills utilize natural power sources to perform a variety of functions, including energy production and food processing. Wind-driven prayer wheels have been used since the fourth century in Tibet and China. Historians believe that people in ancient Persia built the first practical windmills for both grinding grain and pumping water. From there, they spread through the Middle East and parts of Asia, as well as to India. They can be documented in Europe by the twelfth century. Wind turbines developed primarily in the twentieth century. Mathematics has been important for both the design of windmills and in calculating and modeling their output. Interestingly, English mathematician and physicist George Green was also a miller, and he is believed to have done much of his mathematics work in his windmill.
Designs
Windmills have had a wide variety of designs and appearances. Some of the earliest windmills rotated along a vertical axis, with the main rotor placed vertically in relation to the ground and giving a look similar to a helicopter. Some modern wind turbines have retained this engineering design in areas where wind direction is variable. This design is advantageous because vertical-axis windmills have an axis of rotation perpendicular to the ground, so the sails react similarly to all wind directions. On the other hand, horizontal-axis windmills have an axis of rotation that is parallel to the ground, resembling the more common image of a windmill such as that found in Don Quixote. The structure of horizontal-axis windmills gives the advantage of allowing their potential work to be maximized with respect to a specific wind direction. It is important to place a horizontal-axis windmill in line with the prevailing wind.
Windmills have traditionally been designed symmetrically, including an even number of sails. Historically, workers would place food and other substances in special locations inside the windmill to be ground by stones or other clashing materials. The grinding materials were sometimes connected to a system of gears and pulleys to increase the power beyond the mere rotation of the sails. Most modern wind turbines continue to have a sleek, symmetric design but have three sails. The insides of these turbines are devoted mostly to the attainment of electric power.
Number of Blades
The number of blades on a windmill is in direct correlation to the power generated, although the coefficient is quite small. The amount of power generated increases nearly linearly with each additional blade but the increase in power beyond just two or three blades is quite small for modern wind turbines. Physicists have determined that the power generated by a wind turbine is proportional to the cube of the wind speed and can be found algebraically by
where E is the power efficiency of the rotor, A is the swept area, d is air density, and v is wind speed. The swept area relates to the circle created by a rotation of a sail, calculated by
where l is the length of the sail. The theoretical maximum of E, known as the “Betz limit,” is 0.59. The Betz limit is named for Albert Betz, a German physicist who was also interested in wind power. However, this theoretical value is reduced significantly when common physical constraints, including friction and drag on the rotors, are considered. One can calculate the maximum power produced by a windmill algebraically as
It is difficult to put tight parameters on the variables that determine the amount of power produced by a wind turbine. However, a good estimate of the production of power for a 10-foot diameter sail in 12 miles per hour average winds is 2300 kilowatts of power. In a wind farm, several turbines are interconnected by a power collection system and communications network to pool their output and connect to a power grid. Probabilistic mathematical models are used to estimate and describe the output of networks of wind turbines.
Bibliography
Betz, A. Introduction to the Theory of Flow Machines. Translated by D. G. Randall. Oxford, England: Pergamon Press, 1966.
Brooks, L. Windmills. New York: Metro Books, 1989.
Gipe, Paul. Wind Power, Revised Edition: Renewable Energy for Home, Farm, and Business. New York: Chelsea Green Publishing, 2004.
Gorban, A. N., A. M. Gorlov, and V. M. Silantyev. “Limits of the Turbine Efficiency for Free Fluid Flow.” Journal of Energy Resources Technology, 123, no. 4 (2001).