Aerodynamic Design of a Windmill

Introduction

The airfoils MH 102 to MH 110 were part of the design of an optimum windmill. The windmill itself as well as the airfoils have been designed using direct inverse design methods.

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.

Table 1: Definition of the Design Point

Technical Data
diameter	16 m
	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.

Blade Design

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.

Airfoil Design

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. The five airfoils MH 102 to MH 110 make up the new family, which shows no dramatic sensitivity with respect to surface roughness throughout the operating range. 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.

				1)	2)	3)
r/R	c/R	r	c	ß	Xd	Airfoil
[-]	[-]	[mm]	[mm]	[°]	[mm]
0.000	0.0000	0.0	0.1	84.500	.	.
0.040	0.1162	320.0	930.0	38.050	.	.
0.080	0.1094	640.0	875.1	25.844	271	MH 102
0.120	0.0988	960.0	790.4	17.883	.	.
0.160	0.0917	1280.0	733.8	12.686	.	.
0.200	0.0890	1600.0	711.9	9.077	214	MH 102
0.240	0.0842	1920.0	673.3	6.678	.	.
0.280	0.0814	2240.0	651.4	4.901	.	.
0.320	0.0783	2560.0	626.6	3.540	.	.
0.360	0.0745	2880.0	595.9	2.471	.	.
0.400	0.0717	3200.0	573.3	1.612	134	MH 104
0.440	0.0689	3520.0	551.4	1.110	.	.
0.480	0.0668	3840.0	534.8	0.725	.	.
0.520	0.0653	4160.0	522.6	0.435	.	.
0.560	0.0643	4480.0	514.1	0.219	.	.
0.600	0.0635	4800.0	508.3	0.061	115	MH 106
0.640	0.0606	5120.0	485.0	-0.207	.	.
0.680	0.0579	5440.0	463.4	-0.436	.	.
0.720	0.0553	5760.0	442.7	-0.631	.	.
0.760	0.0528	6080.0	422.1	-0.794	.	.
0.800	0.0499	6400.0	399.4	-0.938	89	MH 108
0.840	0.0468	6720.0	374.7	-1.061	.	.
0.880	0.0429	7040.0	342.9	-1.158	.	.
0.920	0.0371	7360.0	297.1	-1.247	.	.
0.960	0.0280	7680.0	223.9	-1.316	46	MH 110
1.000	0.0000	8000.0	50.0	-1.377	.	.

Table 2: Blade Geometry

1. blade-angle defined by plane of rotation and x-axis of airfoil-section
2. Xd - position of airfoils should form a straight line to minimize torsional Loads
3. airfoil-orientation is inverse (y-axis pointing with the wind)

Performance

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.

Wind Scales

Beaufort Wind Scale
Beaufort Number or Force	Wind Speed			Description	Effects Land / Sea
	mph	km/hr	knots
0	<1	<1	<1	Calm	Still, calm air, smoke will rise vertically. Water is mirror-like.
1	1-3 mph	1-5 kph	1-3 knots	Light Air	Rising smoke drifts, wind vane is inactive. Small ripples appear on water surface.
2	4-7 mph	6-11 kph	4-6 knots	Light Breeze	Leaves rustle, can feel wind on your face, wind vanes begin to move. Small wavelets develop, crests are glassy.
3	8-12 mph	12-19 kph	7-10 knots	Gentle Breeze	Leaves and small twigs move, light weight flags extend. Large wavelets, crests start to break, some whitecaps.
4	13-18 mph	20-28 kph	11-16 knots	Moderate Breeze	Small branches move, raises dust, leaves and paper. Small waves develop, becoming longer, whitecaps.
5	19-24 mph	29-38 kph	17-21 knots	Fresh Breeze	Small trees sway. White crested wavelets (whitecaps) form, some spray.
6	25-31 mph	39-49 kph	22-27 knots	Strong Breeze	Large tree branches move, telephone wires begin to "whistle", umbrellas are difficult to keep under control. Larger waves form, whitecaps prevalent, spray.
7	32-38 mph	50-61 kph	28-33 knots	Moderate or Near Gale	Large trees sway, becoming difficult to walk. Larger waves develop, white foam from breaking waves begins to be blown.
8	39-46 mph	62-74 kph	34-40 knots	Gale or Fresh Gale	Twigs and small branches are broken from trees, walking is difficult. Moderately large waves with blown foam.
9	47-54 mph	75-88 kph	41-47 knots	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.
10	55-63 mph	89-102 kph	48-55 knots	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.
11	64-72 mph	103-117 kph	56-63 knots	Violent Storm	Extensive widespread damage. Large waves (9-14 meters), white foam, visibility further reduced.
12	73+ mph	118+ kph	64+ knots	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
Category	Wind Strength - Pressure	Effects
1	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.
2	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.
3	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.
4	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.
5	135+ knots 155+ mph 249+ kph <>	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 -- Current Intensity MWS -- Mean Wind Speed MSLP -- Mean Sea Level Atmospheric Pressure in Millibars
CI Number	MWS (Knots)	MSLP (Atlantic)	MSLP (Pacific)	Saffir-Simpson Category (Approximate)
1	25 Knots
1.5	25 Knots
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

Wind Warnings
Category	Day Flags / Night Lights	Description
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
F0 Gale tornado	40-72 mph 35-62 knots 64-116 kph 29%	Minimal Damage - Some damage to chimneys, TV antennas, roof shingles and windows. Breaks branches off trees, pushes over shallow-rooted trees, damages sign boards.
F1 Moderate tornado	73-112 mph 63-97 knots 117-180 kph 40%	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.
F2 Significant tornado	113-157 mph 98-136 knots 181-253 kph 24%	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.
F3 Severe tornado	158-206 mph 137-179 knots 254-332 kph 6%	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.
F4 Devastating tornado	207-260 mph 180-226 knots 333-419 kph 2%	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.
F5 Incredible tornado	261-318 mph 227-276 knots 420-512 kph 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.
F6 Inconceivable tornado	319-379 mph 277-329 knots 513-610 kph 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