# Environmental studies and Forestry

Wind Energy Formulas

The formulas that are used for calculating various parameters in wind energy system include;

Swept Area of Blades

Swept Area,

A= pr2 ……………………………………………………………………………………… [1]

Where;

p = 3.14159

r- Length of one blade

Wind Power

P=½?AV³ ……………………………………………………………………………………… [2]

Where;

P is power in watts

? is air density that is 1.2kg/m3 sea level and room temperature

A is swept area of the turbines in m2

V is wind speed in m/s

Betz Limit and Coefficient of Power

Cp = ratio of power extracted by turbine to total power contained in the wind resource

Cp = PT / PW ……………………………………………………………………………………… [3]

Turbines cannot operate at its maximum power because it is limited by power coefficient Cp

Therefore power total.

PT=½?AV³Cp ……………………………………………………………………………………… [4]

Turbine Equation

Typical flux alternator efficiency (ca) and wind turbine efficiencies (ct) are about 40% and 60% respectively. Therefore, wind power becomes turbine equation:

Turbine power equation:

P=½?AV³CaCt………………………………………………………………………………………………………………………. [5]

However, the power coefficient is not fixed because it varies with tip speed ratio of the wind turbine. The tip speed ratio is:

?=(Blade Tip Speed)/(Wind Speed) ………………………………………………………[6]

Where;

Blade tip speed= (rpm of the turbine *p*D)/60

Where D is the diameter of the turbine

Energy of the Wind Turbine

The energy of the turbine is product of power and time

Energy= p x t [7]

Where

P is the power of the turbine and t is the time

Capacity Factor CF

This the fraction of the year in which the turbine operate in rated peak power

CF= (Average Output)/( Peak Output) ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,[8]

Wind Pressure

It is calculated by applying the formula below;

Pressure= ½ x (air density) x (wind speed) 2 x shape factor [9]

Where;

Density of air is approximately 1.25 kg/m3

Shape factor depends on the shape of the body. It is dimensionless and has magnitude is 1

The wind speed of is measured in m/s

Therefore, pressure is expressed in kg/m/s2 or N/m2

Kinetic Energy of Wind

Wind comprise of moving air molecules that have mass. Therefore, it has kinetic energy which is expressed as follows;

Kinetic energy, KE= 0.5 x m x v2

Where;

m is mass of the wind

V is the wind velocity

Mass of the wind is represented in the equation below

M=?v=?* Avt =?*pr2 vt [10]

Where

? is the density of air

A=area swept by the blade

v is the wind velocity

r is the radius of the rotor

The mass of air hitting the blades of the turbine is represented as;

Mass/sec (kg/s) = Velocity (m/s) x Area (m2) x Density (kg/m3)

Power Hitting a Turbine

E_kin=1/2 m*v^2=p/2 ?r^2 t*v^3

The power hitting a turbine with a certain swept area is given by;

Power= [0.5 x Air Density x Swept Area x Velocity3]

Therefore;

P_Wind=E_kin/t=p/2 ?r^2*v^3 [11]

Air Density

Air density is important for wind measurement. However, air density is not the same in all places. Therefore air density is;

? = p/R.T (kg/m3) [12]

Where p is air pressure

R is Gaskonstante constant

T is temperature in Kelvin

Wind Load

Wind load is the intensity of pressure of the wind, and is also known as force.

Below is the calculation expression

F=A x p x cd [13]

Where F is force of the wind

P is the wind pressure

Cd is the drag coefficient

Shear Stress in Wind

Wind will incur a hear stress on the surface of the turbines blades. Shear stress is represented using the following equation;

t(y)=µ ?u/?y [14]

Where,

u is the velocity of the wind along the boundary;

y is the height above the boundary.

µ is the dynamic viscosity of the air;

Wind Turbine Rotation Speed

In most case, 3 blade turbines are used for wind plant. The RPM speed of these turbines is calculated using the following formula;

RPM= 60 x V x TSR/ (pD) [15]

Where TSR is tip speed ratio, which is assumed to be 6 for 3 blade turbine

V is the wind speed

D is the rotor diameter

Wind Turbine Max. Power outcome

p=1/2 c?Au3 [16]

Where- ?-airdensity ;; c-power coefficient; ; A-rotor swept;u-wind speed

standard wind speed during power generation which is equivalent to

=1/2 ?Au3.

Wind Speed Exponential deviation with Height

For any wind turbine production, the variation of wind speed in corresponding with height is often termed as the wind measured at a suggested height.

V_w (h)=V_10*(?h/h_10 )?^a, [17]

Where Vw (h)=wind’ s velocity at height h {m/s} , V10=wind^’ svelocity at h10 height, and a=Hellman’s exponent.

The Hellmann exponent is a function of site topography, the coastal location, and the air stability

Wind Shear Formula

The speed of wind can also be found using wind shear formula. In this regard, the speed of wind at a certain level above the ground is give by

U=Uref (z/zref) a [18]

Where U and Uref represent the average speed of wind at the levels Z and Zref respectively

This formula is used in estimating the speed of wind U at a greater elevation (z) by use of a surface (10m) or measurements towers for the speed of winds represented by Uref at reference height of z

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Wind Energy Formulas

The formulas that are used for calculating various parameters in wind energy system include;

Swept Area of Blades

Swept Area,

A= pr2 ……………………………………………………………………………………… [1]

Where;

p = 3.14159

r- Length of one blade

Wind Power

P=½?AV³ ……………………………………………………………………………………… [2]

Where;

P is power in watts

? is air density that is 1.2kg/m3 sea level and room temperature

A is swept area of the turbines in m2

V is wind speed in m/s

Betz Limit and Coefficient of Power

Cp = ratio of power extracted by turbine to total power contained in the wind resource

Cp = PT / PW ……………………………………………………………………………………… [3]

Turbines cannot operate at its maximum power because it is limited by power coefficient Cp

Therefore power total.

PT=½?AV³Cp ……………………………………………………………………………………… [4]

Turbine Equation

Typical flux alternator efficiency (ca) and wind turbine efficiencies (ct) are about 40% and 60% respectively. Therefore, wind power becomes turbine equation:

Turbine power equation:

P=½?AV³CaCt………………………………………………………………………………………………………………………. [5]

However, the power coefficient is not fixed because it varies with tip speed ratio of the wind turbine. The tip speed ratio is:

?=(Blade Tip Speed)/(Wind Speed) ………………………………………………………[6]

Where;

Blade tip speed= (rpm of the turbine *p*D)/60

Where D is the diameter of the turbine

Energy of the Wind Turbine

The energy of the turbine is product of power and time

Energy= p x t [7]

Where

P is the power of the turbine and t is the time

Capacity Factor CF

This the fraction of the year in which the turbine operate in rated peak power

CF= (Average Output)/( Peak Output) ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,[8]

Wind Pressure

It is calculated by applying the formula below;

Pressure= ½ x (air density) x (wind speed) 2 x shape factor [9]

Where;

Density of air is approximately 1.25 kg/m3

Shape factor depends on the shape of the body. It is dimensionless and has magnitude is 1

The wind speed of is measured in m/s

Therefore, pressure is expressed in kg/m/s2 or N/m2

Kinetic Energy of Wind

Wind comprise of moving air molecules that have mass. Therefore, it has kinetic energy which is expressed as follows;

Kinetic energy, KE= 0.5 x m x v2

Where;

m is mass of the wind

V is the wind velocity

Mass of the wind is represented in the equation below

M=?v=?* Avt =?*pr2 vt [10]

Where

? is the density of air

A=area swept by the blade

v is the wind velocity

r is the radius of the rotor

The mass of air hitting the blades of the turbine is represented as;

Mass/sec (kg/s) = Velocity (m/s) x Area (m2) x Density (kg/m3)

Power Hitting a Turbine

E_kin=1/2 m*v^2=p/2 ?r^2 t*v^3

The power hitting a turbine with a certain swept area is given by;

Power= [0.5 x Air Density x Swept Area x Velocity3]

Therefore;

P_Wind=E_kin/t=p/2 ?r^2*v^3 [11]

Air Density

Air density is important for wind measurement. However, air density is not the same in all places. Therefore air density is;

? = p/R.T (kg/m3) [12]

Where p is air pressure

R is Gaskonstante constant

T is temperature in Kelvin

Wind Load

Wind load is the intensity of pressure of the wind, and is also known as force.

Below is the calculation expression

F=A x p x cd [13]

Where F is force of the wind

P is the wind pressure

Cd is the drag coefficient

Shear Stress in Wind

Wind will incur a hear stress on the surface of the turbines blades. Shear stress is represented using the following equation;

t(y)=µ ?u/?y [14]

Where,

u is the velocity of the wind along the boundary;

y is the height above the boundary.

µ is the dynamic viscosity of the air;

Wind Turbine Rotation Speed

In most case, 3 blade turbines are used for wind plant. The RPM speed of these turbines is calculated using the following formula;

RPM= 60 x V x TSR/ (pD) [15]

Where TSR is tip speed ratio, which is assumed to be 6 for 3 blade turbine

V is the wind speed

D is the rotor diameter

Wind Turbine Max. Power outcome

p=1/2 c?Au3 [16]

Where- ?-airdensity ;; c-power coefficient; ; A-rotor swept;u-wind speed

standard wind speed during power generation which is equivalent to

=1/2 ?Au3.

Wind Speed Exponential deviation with Height

For any wind turbine production, the variation of wind speed in corresponding with height is often termed as the wind measured at a suggested height.

V_w (h)=V_10*(?h/h_10 )?^a, [17]

Where Vw (h)=wind’ s velocity at height h {m/s} , V10=wind^’ svelocity at h10 height, and a=Hellman’s exponent.

The Hellmann exponent is a function of site topography, the coastal location, and the air stability

Wind Shear Formula

The speed of wind can also be found using wind shear formula. In this regard, the speed of wind at a certain level above the ground is give by

U=Uref (z/zref) a [18]

Where U and Uref represent the average speed of wind at the levels Z and Zref respectively

This formula is used in estimating the speed of wind U at a greater elevation (z) by use of a surface (10m) or measurements towers for the speed of winds represented by Uref at reference height of z

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