Wind Turbine: Terms, Types, and Theories

Turbine Terms, Types, and Theories

• Wind power generation arises from the wind's kinetic energy
• Kinetic energy formula: 1/2 * m * v^2 (m = air mass, v = velocity)
• Power: rate of change of energy (d/dt of E)
• Power formula with kinetic energy: d/dt (1/2 * m * v^2) = 1/2 * rho * A * v^3 (rho = air density, A = area, v = velocity)

Wind Turbine Operating Characteristics

• Cut-in speed: Minimum wind speed for blades to generate useful power (typically 10-16 km/h)
• Rated speed: Minimum wind speed at which the turbine generates its designated rated power
• Power output increases with wind speed between cut-in and rated speeds
• Power curves: Graphs provided by manufacturers showing wind turbine performance vs. wind speed
• Cut-out speed: High wind speed (typically 72-128 km/h) at which the turbine shuts down to prevent damage
• Automatic brakes or blade twisting mechanisms are used to shut down the turbine at cut-out speed

Betz Limit

• Theoretical maximum amount of energy extractable by a wind turbine rotor: ~59%
• Turbine efficiency is typically 35-45% due to losses in rotor, transmission, generator, storage, etc
• Complete wind energy systems deliver 10-30% of the original energy available in the wind

Design Function Requirements for Wind Turbines

• High stiffness and strength are desirable to resist vibrations and periodic loads
• Weight saving is achieved using composite materials
• Realistic safety margins are required to maintain safety in the blades
• Blades must resist impacts from foreign bodies and mishandling during servicing
• Leading edges of blades require abrasive-resistant materials
• Blades require corrosive-resistant materials to reduce maintenance costs
• Design optimization is needed to satisfy cost requirements, including initial, operating, and maintenance costs
• High reliability and less maintenance needed
• Electrically conductive paths are needed to discharge high voltage during lightning strikes

Betz Limit Derivation

• Energy available in wind: kinetic energy (1/2 * m_w * V_infinity^2) (m_w = air mass, V_infinity = free stream velocity)
• Power available in wind: 1/2 * rho * A * V_infinity^3 (rho = air density, A = rotor area, V_infinity = free stream velocity)
• The rotor is considered as an actuated disc
• Upstream velocity: Vi, and downstream velocity: Vo
• Mass flow rate through the disc: rho * A_D * V_average (A_D = area of disc, V_average = (Vi + Vo) / 2)
• Power output: P_out = 1/2 * m_w * (Vi^2 - Vo^2) = 1/4 * rho * A * (Vi + Vo)^2 * (Vi - Vo)
• Derivation leads to the Betz limit of ~59%, where only 59% of the wind power is extractable for conversion

One-Dimensional Momentum Theory and Betz Limit

• The one-dimensional model predicts the performance of ship propellers
• Assumptions:
  • Homogenous, incompressible, steady-state fluid flow
  • Control volume analysis
  • Turbine is represented as a uniform actuated disc, where a discontinuity in pressure occurs

Assumptions of the Actuator Disc Model

• The model assumes homogeneity, meaning no different phases are involved
• It assumes incompressibility, implying density changes are moderate with respect to temperature and pressure
• The analysis is based on steady-state conditions, with no changes occurring over time
• Friction drag is disregarded in the model
• The model assumes an infinite number of blades, which negates tip losses in the derivation of the Betz limit
• Thrust is considered uniform over the rotor area
• The wake is non-rotating in the model
• The static pressure far upstream and downstream of the rotor is equivalent to the undisturbed ambient static pressure

Betz Limit Derivation

• The derivation involves a 1-dimensional linear momentum theory to calculate maximum available power
• Linear momentum theory equates the net force on the control volume to the thrust force of the wind on the turbine
• The thrust force is expressed as the mass flow rate (ρAU) multiplied by the change in velocity (U1 - U4)
• For steady-state flow, the mass flow rate at the upstream end equals the mass flow rate at the downstream end
• The thrust (T) is given by the mass flow rate multiplied by the difference between upstream (U1) and downstream (U4) velocities (T = m(U1 - U4))

Bernoulli's Equation and Thrust

• Bernoulli's equation relates pressure and velocity for the upstream side: P1 + 0.5ρU1^2 = P2 + 0.5ρU2^2
• For the downstream side, it's written as: P3 + 0.5ρU3^2 = P4 + 0.5ρU4^2
• Thrust can also be expressed as the area (A2) multiplied by the pressure difference between the upstream and downstream sides of the disc (P2 - P3)
• To find the pressure difference, the assumption P1 = P4 is used, meaning upstream and downstream pressures are equal
• It's also assumed that U2 = U3, meaning velocity across the disc remains the same

Deriving the Thrust Equation

• By applying the assumption P1 = P4 and U2 = U3 to Bernoulli's equations, P2 - P3 is derived as 0.5ρ(U1^2 - U4^2)
• Substituting this pressure difference into the thrust equation yields T = 0.5ρA2(U1^2 - U4^2)
• Another thrust equation is substituted, and further manipulation leads to U2 = (U1 + U4) / 2, indicating the average velocity is the mean of upstream and downstream velocities

Axial Induction Factor

• An axial induction factor (a) is introduced, defined as a = (U1 - U2) / U1
• U2, the velocity at the rotor plane, can be written in terms of 'a' as U2 = U1(1 - a)
• The downstream velocity (U4) is defined as U1(1 - 2a)
• If a = 0, U4 = U1, indicating no energy harvested
• If a = 0.5, U4 = 0, which is not practically achievable due to thermodynamic losses
• In practical systems, 'a' should lie between 0 and 0.5
• The term U1 * a is the induced velocity at the rotor

Power Output

• Power output equation P = Thrust times Velocity
• Power out (P out) is expressed as 0.5 * rho * A2 * U2 (U1^2 - U4^2), where A2 is the rotor area
• By substituting the previously defined relationships for U2 and U4, power output can be expressed in terms of 'a' and U1
• The derived power output formula is P_out = 0.5 * rho * A * U^3 * 4a * (1 - a)^2, using A for rotor area and U for upstream velocity

Power and thrust coefficients

• Wind turbine rotor performance is characterized by power coefficient (C p)
• C p defined as rotor power divided by power in the wind
• The power coefficient (Cp) is given by 4a(1 - a)^2
• To maximize Cp, it's differentiated with respect to 'a' and set to 0
• Solving the equation results in a = 1/3 for maximum power coefficient
• C p max equals 16/27 or 0.593
• Substituting in your axial induction factor, a=1/3
• Rotor velocity should be two-thirds of the free stream velocity for maximum power production
• Substituting a = 1/3 into the thrust equation results in a thrust coefficient of 8/9 for maximum power protection

Lift and drag coefficients

• Lift coefficient (C L) to F L upon half rho A U square where F L is lift force
• Tip speed ratio is defined as the speed of the blade tip up free stream wind speed
• Lift force is perpendicular to incoming wind flow due to unequal pressure differences on the surface blades
• Omega R rotational velocity upon is nothing FreeStream velocity applies only for horizontal axis machine
• Drag coefficient is parallel to direction of airflow created by friction force at surfaces of the blade and due to pressure differences
• Today's discussion includes the maximum theoretical wind power extraction (59%)
• Discussed 1 dimensional momentum analysis in depth, which considers assumptions
• How to calculate lift coefficient and drag coefficient, derived the coefficient
• Tip speed ratio