Aerodynamic Design of a Windmill
Today, the design of horizontal axis windmills can be performed with good results, using inverse design methods, based on the minimum induced loss windmill, as defined by Glauert and Prandtl during the 1920s. For the analysis under off-design conditions, simple blade element methods and more complex vortex lattice methods lead to quite accurate results. The final blade geometry can be tailored exactly to the desired main operating range by using a suitable inverse design method for the airfoil sections.
|For a small horizontal axis wind turbine, the main project data were proposed by the manufacturer, based on market studies. The basic parameters of the windmill were a constant speed generator which could be switched between two velocities of rotation and that the blade angle should be constant and not adjustable. A gearbox between power generator and the rotor of the windmill permitted the decoupling of aerodynamics and the power generator. |
To avoid destruction of the windmill due to over speed caused by high wind speeds, it was decided to direct the aerodynamic layout towards a stall regulated machine. (such a windmill has no variable pitch blades to control the mill at higher wind speeds; instead, the airfoils are designed to stall sharply when the operating limits are exceeded, thus limiting the power output).
After some preliminary design studies, the parameters for the design, as listed below, were selected.
|gear 1||gear 2|
|rotor speed||80 1/min||120 1/min|
|at wind speed||6.0 m/s||9.0 m/s|
|tip Mach number||0.2||0.3|
|power loading||450 W/m²|
|max. power output||90 kW|
|useful wind speed range||5 |
|stall controlled, fixed pitch, no adjustable blade angle|
Regarding low manufacturing costs and competitive energy costs, the number of blades was limited to two. To achieve high yearly power on times, with short low power and non operating times, a specific power loading of 450 W/m² had been selected. A lower loading would increase the full power times, but was not possible due to a diameter restriction.
On the one hand, simple momentum theory correctly predicts maximum efficiency to occur at maximum diameter, but on the other hand the tip Mach number is directly proportional to the diameter for a fixed velocity of rotation. The tip Mach number was limited to 0.3, as a compromise between aerodynamics and noise constraints. Together with the available gearboxes, this lead to local Reynolds numbers of more than 500`000, which is sufficient to achieve high L/D ratios, which are essential for good performance.
The geometry of the blades is determined by the task to transform as much energy as possible from the incoming air flow into mechanical, respectively electric power. Thus the aerodynamic design of the windmill should fulfill the minimum induced loss principle. The basic aerodynamic design of the windmill was based on Glauerts optimum windmill theory, which was embedded into the framework of an existing general blade element code. This code uses two dimensional airfoil polars, which gives very good results for attached flow conditions. For cross checks, additional analysis runs were performed, using a vortex lattice code. During the preliminary design of the blades, the operating conditions for the local airfoil sections were defined in terms of Reynolds and Mach numbers as well as lift coefficient range. These conditions were used for the design of new airfoils, which were then used in the windmill design method to find the optimum blade shapes. Later additions to the code make it possible to account for the boundary layer of the ground by performing several analysis at different azimuthal blade positions.
|Besides the number of blades, the planform and the power loading, the airfoil sections are of utmost importance for the performance of a windmill. Here maximum L/D ratios are desired to maximize efficiency, taking into account, that the surface of a wind turbine will not be perfectly smooth during the whole life span of a windmill - the airfoils should have good L over D values with rough surfaces too. |
For the special case of a fixed pitch windmill, the characteristics of the airfoils are controlling the power versus wind speed performance curve. To avoid over speed conditions, the maximum power of the windmill has to be strictly limited, which can be achieved by a specially designed family of airfoils. These airfoils feature a distinct, but not necessarily hard primary stall, which leads to a limitation of the maximum power of the windmill. To avoid noise and structural problems, the airfoils have been designed to have a soft post stall plateau, followed by a soft secondary stall.
Because the preliminary windmill design and analysis defined ranges for the lift coefficient and the Reynolds number, Epplers inverse design method was ideally suited for the design of these airfoil sections.
Table 2: Blade Geometry
- blade-angle defined by plane of rotation and x-axis of airfoil-section
- Xd - position of airfoils should form a straight line to minimize torsional Loads
- airfoil-orientation is inverse (y-axis pointing with the wind)
The figure below shows the power delivered by the windmill at the two velocity of rotation settings. The lower 80 rpm value is used to start up the windmill and for wind speeds up to 8 m/s, whereas the 120 rpm setting is used for wind speeds between 8 and 14 m/s. The design power of 90 kW is reached at a wind speed of 13 m/s. For comparison, the chart also contains the power curve of a windmill using conventional airfoil sections, which result in a undesired power output at higher wind speeds.
Performance chart of the optimum stall controlled windmill.
Beaufort Wind Scale
|Beaufort Number |
|Description||Effects Land / Sea|
|0||<1||<1||<1||Calm||Still, calm air, smoke will rise vertically. |
Water is mirror-like.
|Light Air||Rising smoke drifts, wind vane is inactive. |
Small ripples appear on water surface.
|Light Breeze||Leaves rustle, can feel wind on your face, wind vanes begin to move. |
Small wavelets develop, crests are glassy.
|Gentle Breeze||Leaves and small twigs move, light weight flags extend. |
Large wavelets, crests start to break, some whitecaps.
|Moderate Breeze||Small branches move, raises dust, leaves and paper. |
Small waves develop, becoming longer, whitecaps.
|Fresh Breeze||Small trees sway. |
White crested wavelets (whitecaps) form, some spray.
|Strong Breeze||Large tree branches move, telephone wires begin to "whistle", umbrellas are difficult to keep under control. |
Larger waves form, whitecaps prevalent, spray.
|Moderate or Near Gale||Large trees sway, becoming difficult to walk. |
Larger waves develop, white foam from breaking waves begins to be blown.
|Gale or Fresh Gale||Twigs and small branches are broken from trees, walking is difficult. |
Moderately large waves with blown foam.
|Strong Gale||Slight damage occurs to buildings, shingles are blown off of roofs. |
High waves (6 meters), rolling seas, dense foam, Blowing spray reduces visibility.
|Whole Gale or Storm||Trees are broken or uprooted, building damage is considerable. |
Large waves (6-9 meters), overhanging crests, sea becomes white with foam, heavy rolling, reduced visibility.
|Violent Storm||Extensive widespread damage. |
Large waves (9-14 meters), white foam, visibility further reduced.
|Hurricane||Extreme destruction, devastation. |
Large waves over 14 meters, air filled with foam, sea white with foam and driving spray, little visibility.
Saffir-Simpson Hurricane Scale
|Wind Strength - Pressure||Effects|
|65 to 83 knots |
74 to 95 mph
119 to 153 kph
> 980 mb
|Storm surge generally 4-5 ft above normal. No real damage to building structures. Damage primarily to unanchored mobile homes, shrubbery, and trees. Some damage to poorly constructed signs. Also, some coastal road flooding and minor pier damage. Hurricanes Allison of 1995 and Danny of 1997 were Category One hurricanes at peak intensity.|
|84 to 95 knots |
96 to 110 mph
154 to 177 kph
980 - 965 mb
|Storm surge generally 6-8 feet above normal. Some roofing material, door, and window damage of buildings. Considerable damage to shrubbery and trees with some trees blown down. Considerable damage to mobile homes, poorly constructed signs, and piers. Coastal and low-lying escape routes flood 2-4 hours before arrival of the hurricane center. Small craft in unprotected anchorages break moorings. Hurricane Bertha of 1996 was a Category Two hurricane when it hit the North Carolina coast, while Hurricane Marilyn of 1995 was a Category Two Hurricane when it passed through the Virgin Islands.|
|96 to 113 knots |
111 to 130 mph
178 to 209 kph
964 - 945 mb
|Storm surge generally 9-12 ft above normal. Some structural damage to small residences and utility buildings with a minor amount of curtainwall failures. Damage to shrubbery and trees with foliage blown off trees and large tress blown down. Mobile homes and poorly constructed signs are destroyed. Low-lying escape routes are cut by rising water 3-5 hours before arrival of the hurricane center. Flooding near the coast destroys smaller structures with larger structures damaged by battering of floating debris. Terrain continuously lower than 5 ft above mean sea level may be flooded inland 8 miles (13 km) or more. Evacuation of low-lying residences with several blocks of the shoreline may be required. Hurricanes Roxanne of 1995 and Fran of 1996 were Category Three hurricanes at landfall on the Yucatan Peninsula of Mexico and in North Carolina, respectively.|
|114 to 134 knots |
131 to 155 mph
210 to 249 kph
944- 920 mb
|Storm surge generally 13-18 ft above normal. More extensive curtainwall failures with some complete roof structure failures on small residences. Shrubs, trees, and all signs are blown down. Complete destruction of mobile homes. Extensive damage to doors and windows. Low-lying escape routes may be cut by rising water 3-5 hours before arrival of the hurricane center. Major damage to lower floors of structures near the shore. Terrain lower than 10 ft above sea level may be flooded requiring massive evacuation of residential areas as far inland as 6 miles (10 km). Hurricane Luis of 1995 was a Category Four hurricane while moving over the Leeward Islands. Hurricanes Felix and Opal of 1995 also reached Category Four status at peak intensity.|
|135+ knots |
|Storm surge generally greater than 18 ft above normal. Complete roof failure on many residences and industrial buildings. Some complete building failures with small utility buildings blown over or away. All shrubs, trees, and signs blown down. Complete destruction of mobile homes. Severe and extensive window and door damage. Low-lying escape routes are cut by rising water 3-5 hours before arrival of the hurricane center. Major damage to lower floors of all structures located less than 15 ft above sea level and within 500 yards of the shoreline. Massive evacuation of residential areas on low ground within 5-10 miles (8-16 km) of the shoreline may be required. There were no Category Five hurricanes in 1995, 1996, or 1997. Hurricane Gilbert of 1988 was a Category Five hurricane at peak intensity and is the strongest Atlantic tropical cyclone of record.|
The effects described in the Saffir-Simpson scale are from the
National Hurricane Center
Dvorak Current Intensity Chart
The Dvorak technique is a method using enhanced Infrared and/or visible satellite imagery to quantitatively estimate the intensity of a tropical system.
|CI Number||MWS (Knots)||MSLP (Atlantic)||MSLP (Pacific)||Saffir-Simpson Category (Approximate)|
|2||30 Knots||1009 mb||1000 mb|
|2.5||35 Knots||1005 mb||997 mb|
|3||45 Knots||1000 mb||991 mb|
|3.5||55 Knots||994 mb||984 mb|
|4||65 Knots||987 mb||976 mb||1 (64-83 KTS)|
|4.5||77 Knots||979 mb||966 mb||1 (64-83 KTS); |
2 (84-96 KTS)
|5||90 Knots||970 mb||954 mb||2 (84-96 KTS); |
3 (97-113 KTS)
|5.5||102 Knots||960 mb||941 mb||3 (97-113 KTS)|
|6||115 Knots||948 mb||927 mb||4 (114-135 KTS)|
|6.5||127 Knots||935 mb||914 mb||4 (114-135 KTS)|
|7||140 Knots||921 mb||898 mb||5 (136+ KTS)|
|7.5||155 Knots||906 mb||879 mb||5 (136+ KTS)|
|8||170 Knots||890 mb||858 mb||5 (136+ KTS)|
Dvorak Chart from NOAA Satellite Services Division
Day Flags / Night Lights
|Small Craft Advisory|| |
Red over white lights
|Forecast winds of 18 to 33 knots (21 to 38 mph). |
Small Craft Advisories may also be issued for hazardous sea conditions or lower wind speeds that may affect small craft operations.
|Gale Warning|| |
White over red lights
|Forecast winds of 34 to 47 knots (39 to 54 mph)|
|Storm Warning|| |
Red over red lights
|Forecast winds of 48 knots (55 mph) or greater|
|Tropical Storm Warning|| |
Red over red lights
|Forecast winds of 48 to 63 knots (55 to 73 mph) associated with a tropical storm|
|Hurricane Warning|| |
Red over white over red
|Forecast winds of 64 knots (74 mph) or higher associated with a hurricane|
Fujita Tornado Scale
|F-Scale / Intensity Phrase||Wind Strength / Frequency||Description of Damage|
|40-72 mph |
|Minimal Damage - Some damage to chimneys, TV antennas, roof shingles and windows. Breaks branches off trees, pushes over shallow-rooted trees, damages sign boards.|
|73-112 mph |
|Moderate Damage - Automobiles overturned, carports destroyed, trees uprooted, peels surface off roofs, mobile homes pushed off foundations or overturned, moving autos pushed off the roads.|
|113-157 mph |
|Major Damage - Roofs torn off frame homes, sheds and outbuildings are demolished, mobile homes overturned or destroyed, boxcars pushed over; large trees snapped or uprooted, light object missiles generated.|
|158-206 mph |
|Severe Damage - Exterior walls and roofs blown off well-built houses, metal buildings collapsed or are severely damaged, trains overturned, forests and farmland flattened, heavy cars lifted off the ground and thrown.|
|207-260 mph |
|Devastating Damage - Few walls, if any, standing in well-built houses, structures with weak foundations blown off some distance, large steel and concrete missiles thrown far distances, cars thrown.|
|261-318 mph |
less than 1%
|Incredible Damage - Homes leveled with all debris removed, strong frame houses lifted off foundations and carried considerable distances to disintegrate. Schools, motels, and other larger structures have considerable damage with exterior walls and roofs gone, steel re-inforced concrete structures badly damaged. Automobile sized missiles fly through the air in excess of 100 meters, trees debarked.|
|319-379 mph |
less than 1%
|These winds are very unlikely. The small area of damage they might produce would probably not be recognizable along with the mess produced by F4 and F5 wind that would surround the F6 winds. Missiles, such as cars and refrigerators would do serious secondary damage that could not be directly identified as F6 damage. If this level is ever achieved, evidence for it might only be found in some manner of ground swirl pattern, for it may never be identifiable through engineering studies|
Weather Map Wind Symbols
1 knot = 1 nautical mile per hour = 6076 feet per hour = 1.15078 mph
1 mph = 1 mile per hour = 5280 feet per hour = 0.86898 knots per hour