# Fluid Mechanics Applications/B44:WINDMILL AND WIND TURBINE

## INTRODUCTION

A windmill is a machine that converts the energy of wind into rotational energy by means of vanes called sails or blades.[1][2]

The reason for the name "windmill" is that the devices originally were developed for milling grain for food production; the name stuck when in the course of history, windmill machinery was adapted to supply power for many industrial and agricultural needs other than milling. The majority of modern windmills take the form of wind turbines used to generate electricity, or windpumps used to pump water, either for land drainage or to extract groundwater. A wind turbine is a device that converts kinetic energy from the wind into electrical power. A wind turbine used for charging batteries may be referred to as a wind charger.

HISTORY OF WINDMILLS: Windmills were used in Persia (present-day Iran) as early as 200 B.C.[3] The windwheel of Hero of Alexandria marks one of the first known instancesof wind powering a machine in history.[4][5] However, the first known practical windmills were built in Sistan, an Eastern province of Iran, from the 7th century. These "Panemone" were vertical axle windmills, which had long vertical drive shafts with rectangular blades.Made of six to twelve sails covered in reed matting or cloth material, these windmills were used to grind grain or draw up water, and were used in the gristmilling and sugarcane industries. Windmills first appeared in Europe during the Middle Ages. The first historical records of their use in England date to the 11th or 12th centuries and there are reports of German crusaders taking their windmill-making skills to Syria around 1190. By the 14th century, Dutch windmills were in use to drain areas of the Rhine delta. The first electricity-generating wind turbine was a battery charging machine installed in July 1887 by Scottish academic James.

## WORKING PRINCIPLE

Wind turbine operates on a simple principle. Energy from moving air, caused by temperature (and therefore pressure) differences in the atmosphere.Irradiance from the sun heats up the air, forcing the air to rise. Conversely, where temperatures fall, a low pressure zone develops. Winds (i.e. air flows) balance out the differences. Hence, wind energy is solar energy converted into kinetic energy of moving air. Wind Energy Converters (WECs) - or short: wind turbines - capture the air flow by converting it into a rotational movement, which subsequently drives a conventional generator for electricity.[6]

## TYPES OF WIND TURBINES

• Horizontal axis wind turbines: Horizontal-axis wind turbines (HAWT) have the main rotor shaft and electrical generator at the top of atower, and must be pointed into the wind. Small turbines are pointed by a simple wind vane, while large turbines generally use a wind sensor coupled with a servo motor. Most have a gearbox, which turns the slow rotation of the blades into a quicker rotation that is more suitable to drive an electrical generator.
  Advantages:
1: Higher wind speeds
2: Great efficiency
1: Angle of turbine is relevant


• Vertical axis wind turbines: Vertical-axis wind turbines (or VAWTs) have the main rotor shaft arranged vertically.[7]
 Advantages
1: Can place generator on ground
2: You don’t need a yaw mechanism for wind angle
1: Lower wind speeds at ground level
2: Less efficiency and require a push


## WIND TURBINE DESIGN AND COMPONENTS

Wind turbine design is the process of defining the form and specifications of a wind turbine to extract energy from the wind.[8] A wind turbine installation consists of the necessary systems needed to capture the wind's energy, point the turbine into the wind, convert mechanical rotation into electrical power, and other systems to start, stop, and control the turbine.

Wind turbine components

  1-Foundation
2-Connection to the electric grid
3-Tower
5-Wind orientation control (Yaw control)
6-Nacelle
7-Generator
8-Anemometer
9-Electric or Mechanical Brake
10-Gearbox
13-Rotor hub.


## FORMULA USED

A German physicist Albert Betz derived a formula to extract wind power from turbine. BETZ'S LAW:Betz's law calculates the maximum power that can be extracted from the wind, independent of the design of a wind turbine in open flow.The law is derived from the principles of conservation of mass and momentum of the air stream flowing through an idealized "actuator disk" that extracts energy from the wind stream. According to Betz's law, no turbine can capture more than 16/27 (59.3%) of the kinetic energy in wind. The factor 16/27 (0.593) is known as Betz's coefficient.[9]

PROOF

  ASSUMPTION:
1. The rotor does not possess a hub, this is an ideal rotor, with an infinite number of blades which have no drag. Any resulting drag
would only lower this idealized value.
2. The flow into and out of the rotor is axial. This is a control volume analysis, and to construct a solution the control volume must
contain all flow going in and out, failure to account for that flow would violate the conservation equations.
3. The flow is incompressible. Density remains constant, and there is no heat transfer.
4. Uniform thrust over the disc or rotor area


MATHEMATICAL MODEL:

The following table shows the definition of various variables used in this model:

E = Kinetic Energy(J)

ρ = Density(kg/m3)

m = Mass (kg)

A = Swept Area(m2)

v = Wind Speed(m/s)

Cp = Power Coefficient

P = Power (W)

dm/dt =Mass flow rate(kg/s)

x = distance (m)

dE/dt =Energy Flow Rate (J/s)

t = time (s)

Under constant acceleration, the kinetic energy of an object having mass m and velocity v is equal to the work done W in displacing that object from rest to a distance s under a force F , i.e.:

${\displaystyle E=W=Fs}$

According to Newton’s Law, we have: ${\displaystyle E=ma}$

Hence, ${\displaystyle E=mas}$  … (1)

Using the third equation of motion:

${\displaystyle v^{2}=u^{2}+2as}$

we get:

${\displaystyle a=(v^{2}-u^{2})/2s}$

Since the initial velocity of the object is zero, i.e.

u = 0 , we get:

   ${\displaystyle a=v^{2}/2s}$


Substituting it in equation (1), we get that the kinetic energy of a mass in motions is:

${\displaystyle E=(1/2)mv^{2}}$

The power in the wind is given by the rate of change of energy:

${\displaystyle P={\frac {dE}{dt}}=(1/2)v^{2}{\frac {dm}{dt}}}$  ...(3)

As mass flow rate is given by:

${\displaystyle {\frac {dm}{dt}}=\rho \cdot A{\frac {dx}{dt}}}$

and the rate of change of distance is given by:

${\displaystyle {\frac {dx}{dt}}=v}$

we get:

${\displaystyle {\frac {dm}{dt}}=\rho \cdot A\cdot v}$

Hence, from equation (3), the power can be defined as:

 ${\displaystyle P={\begin{matrix}{\frac {1}{2}}\end{matrix}}\cdot \rho \cdot A\cdot v^{3}.}$    ...(4)


A German physicist Albert Betz concluded in 1919 that no wind turbine can convert more than 16/27 (59.3%) of the kinetic energy of the wind into mechanical energy turning a rotor. To this day, this is known as the Betz Limit or Betz' Law. The theoretical maximum power efficiency of any design of wind turbine is 0.59 (i.e. no more than 59% of the energy carried by the wind can be extracted by a wind turbine). This is called the “power coefficient” and is defined as:

 ${\displaystyle Cpmax=0.59}$


Also, wind turbines cannot operate at this maximum limit. The value is unique to each turbine type and is a function of wind speed that the turbine is operating in. Once we incorporate various engineering requirements of a wind turbine - strength and durability in particular - the real world limit is well below the Betz Limit with values of 0.35-0.45 common even in the best designed wind turbines. By the time we take into account the other factors in a complete wind turbine system - e.g. the gearbox, bearings, generator and so on - only 10-30% of the power of the wind is ever actually converted into usable electricity. Hence, the power coefficient needs to be factored in equation (4) and the extractable power from the wind is given by

   ${\displaystyle P_{\rm {avail}}=C_{\mathrm {p} }\cdot {\begin{matrix}{\frac {1}{2}}\end{matrix}}\cdot \rho \cdot A\cdot v^{3}.}$    ...(5)


The swept area of the turbine can be calculated from the length of the turbine blades using the equation for the area of a circle:

${\displaystyle A=\pi r^{2}.\,}$  ...(6)

## Characteristic parameters

The coefficient of power is the most important variable in wind turbine aerodynamics. Buckingham π theorem can be applied to show that non-dimensional variable for power is given by the equation below. This equation is similar to efficiency, so values between 0 and less than one are typical. However, this is not exactly the same as efficiency so in practice some turbines can exhibit greater than unity power coefficients. In these circumstances one cannot conclude the first law of thermodynamics is violated because this is not an efficiency term by the strict definition of efficiency.

${\displaystyle C_{P}={\frac {P}{{\frac {1}{2}}\rho AV^{3}}}}$

(CP)

where: ${\displaystyle C_{P}}$  is the coefficient of power, ${\displaystyle \rho }$  is the air density, A is the area of the wind turbine, finally V is the wind speed.

Equation (1) shows two important dependents. The first is the speed (U) that the machine is going at. The speed at the tip of the blade is usually used for this purpose, and is written as the product of the blade radius and the rotational speed of the wind (U=omega*r, where omega = rotational velocity in radians/second).[please clarify] This variable is nondimensionalized by the wind speed, to get the speed ratio:

${\displaystyle \lambda ={\frac {U}{V}}}$

The force vector is not straightforward, as stated earlier there are two types of aerodynamic forces, lift and drag. Accordingly there are two non-dimensional parameters. However both variables are non-dimensionalized in a similar way. The formula for lift is given below, the formula for drag is given after:

${\displaystyle C_{L}={\frac {L}{{\frac {1}{2}}\rho AW^{2}}}}$

(CL)

${\displaystyle C_{D}={\frac {D}{{\frac {1}{2}}\rho AW^{2}}}}$

(CD)

where: ${\displaystyle C_{L}}$  is the lift coefficient, ${\displaystyle C_{D}}$  is the drag coefficient, ${\displaystyle W}$  is the relative wind as experienced by the wind turbine blade, A is the area but may not be the same area used in the power non-dimensionalization of power.

The aerodynamic forces have a dependency on W, this speed is the relative speed and it is given by the equation below. Note that this is vector subtraction.

${\displaystyle {\vec {W}}={\vec {V}}-{\vec {U}}}$

## Maximum power of a drag based wind turbine

Equation (1) will be the starting point in this derivation. Equation (CD) is used to define the force, and equation (RelativeSpeed) is used for the relative speed. These substitutions give the following formula for power.

${\displaystyle P={\frac {1}{2}}\rho AC_{D}\left(UV^{2}-2VU^{2}+U^{3}\right)}$

The formulas (CP) and (SpeedRatio) are applied to express (DragPower) in nondimensional form:

${\displaystyle C_{P}=C_{D}\left(\lambda -2\lambda ^{2}+\lambda ^{3}\right)}$

(DragCP)

It can be shown through calculus that equation (DragCP) achieves a maximum at ${\displaystyle \lambda =1/3}$ . By inspection one can see that equation (DragPower) will achieve larger values for ${\displaystyle \lambda >1}$ . In these circumstances, the scalar product in equation (1) makes the result negative. Thus, one can conclude that the maximum power is given by:

${\displaystyle C_{P}={\frac {4}{27}}C_{D}}$

Experimentally it has been determined that a large ${\displaystyle C_{D}}$  is 1.2, thus the maximum ${\displaystyle C_{P}}$  is approximately 0.1778.

## Maximum power of a lift based wind turbine

The derivation for the maximum power of a lift based machine is similar, with some modifications. First we must recognize that drag is always present, thus cannot be ignored. It will be shown that neglecting drag leads to a final solution of infinite power. This result is clearly invalid, hence we will proceed with drag. As before, equations (1), (CD) and (RelativeSpeed) will be used along with (CL) to define the power below expression.

${\displaystyle P={\frac {1}{2}}\rho A{\sqrt {U^{2}+V^{2}}}\left(C_{L}UV-C_{D}U^{2}\right)}$

Similarly, this is non-dimensionalized with equations (CP) and (SpeedRatio). However in this derivation the parameter ${\displaystyle \gamma =C_{D}/C_{L}}$  is also used:

${\displaystyle C_{P}=C_{L}{\sqrt {1+\lambda ^{2}}}\left(\lambda -\gamma \lambda ^{2}\right)}$

(LiftCP)

Solving the optimal speed ratio is complicated by the dependency on ${\displaystyle \gamma }$  and the fact that the optimal speed ratio is a solution to a cubic polynomial. Numerical methods can then be applied to determine this solution and the corresponding ${\displaystyle C_{P}}$  solution for a range of ${\displaystyle \gamma }$  results. Experiments have shown that it is not unreasonable to achieve a drag ratio (${\displaystyle \gamma }$ ) of approximately 0.01 at a lift coefficient of 0.6. This would give a ${\displaystyle C_{P}}$  of about 889. This is substantially better than the best drag based machine, hence why lift based machines are superior.

## WIND MILL APPLICATIONS

 -Farm Windmill
-Golf Course Aeration
-Cattle Farm Windmill
-Pond Aeration
-Residential Water Aeration
-Fish Ponds and Hatcheries
-West Nile Virus Prevention
[10]

## References

1. "Mill definition". Thefreedictionary.com. Retrieved 2013-08-15.
2. "Windmill definition stating that a windmill is a mill or machine operated by the wind". Merriam-webster.com. 2012-08-31. Retrieved 2013-08-15.
3. "Part 1 — Early History Through 1875". Retrieved 2008-07-31.
4. A.G. Drachmann, "Heron's Windmill", Centaurus, 7 (1961), pp. 145–151
5. Dietrich Lohrmann, "Von der östlichen zur westlichen Windmühle", Archiv für Kulturgeschichte, Vol. 77, Issue 1 (1995), pp. 1–30 (10f.)
6. http://energy.gov/eere/wind/how-does-wind-turbine-work