Report

Wind Turbine Project Recap Wind Power & Blade Aerodynamics Wind Turbine Project Turbines tested indoors under controlled conditions A single metric for success - amount of electricity generated Design will be executed using theoretical calculationsbuild and test ONCE at end! (with one trial fitting) Harnessing available power in wind Max available power How can we predict blade performance? Blade aerodynamics Rotor performance Power coefficient How well is our turbine performing? Rotor power Cp = Power in the wind At best only 45% can be captured by real turbines (theoretical limit). requires blade and rotor physics Project estimates – class exercise (5 min) Available power Estimating maximum Pgenerated Project estimates – class exercise (5 min) Available power P = 60 W Estimating maximum Pgenerated Atlantic City estimates – class exercise (5 min) Now assuming the offshore wind velocity is12 m/s The diameter of a turbine is 73 m, there are 5 turbines Estimate of maximum Pgenerated Blade aerodynamics Turbine blades are airfoils We need to understand blade aerodynamics to determine effectiveness and performance Airfoil terminology R Free stream velocity U∞ W α Relative wind velocity C Airfoil types NACA airfoils National Advisory Committee for Aeronautics NACA 2412 maximum camber of 2% located 40% from the leading edge with a maximum thickness of 12% of the chord NACA 0012 symmetrical airfoil, 00 indicating no camber.12 indicates that the airfoil has a 12% thickness to chord Airfoil function – generation of lift lift thrust drag weight ‘suction’ side ‘pressure’ side Airfoil forces Lift force perpendicular to airflow Drag force parallel to the airflow Calculating lift and drag Power = Force x Velocity Force in the wind geometric factor Force generated by airfoil Coefficients of lift and drag Lift Lift coefficient CL = how effectively the wing turns available dynamic pressure (kinetic energy) into lift Drag force Drag coefficient CD = how much of the pressure (kinetic energy) is converted to drag Coefficients of lift and drag Coefficients of lift and drag 1.75 Depend on: airfoil shape angle of attack Empirically determined 1.50 Lift/Drag Coefficient Geometric factors CD and CL lift coefficient 1.25 1.00 0.75 0.50 drag coefficient 0.25 0 5 10 15 20 25 Angle of Attack (degrees) 30 Airfoil behavior Performance parameters Drag Thrust Lift Torque Rotational Speed Direction of translation b a Relative wind velocity Free stream Wind velocity Wind turbine performance based on • lift and drag coefficients • Pitch angle, b - angle btwn chord line and plane of rotation • Angle of attack, a - angle btwn blade and relative wind, which changes depending on speed of blade and wind speed Lift and drag on translating air foil What force actually provides useful work to rotate the turbine? A) B) C) D) Lift Drag F1 F2 K.L. Johnson (2006) Lift and drag on translating air foil K.L. Johnson (2006) F1 is force to rotate the turbine Tower must be strong enough to withstand thrust force F2 Connection to wind turbines lift and drag cause the rotor to spin angle of attack changes over the span of the blade lift and drag forces also change over the span of the blade Next How to calculate torque generated from lift and drag on each blade? Complications Free stream characteristics change approaching and across blades Rotation of blades causes counter rotation in wind Things vary with r Must use conservation of mass Conservation of momentum Conservation of energy Things vary with r : Blade Element Theory (BET) Blade divided into sections, on which momentum is applied Result is nonlinear equations that can be solved iteratively *Does not consider shed tip vortex. Some flow assumptions made breakdown for extreme conditions when flow becomes stalled or a significant proportion of the propeller blade is in windmilling configuration while other parts are still thrust producing. http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/propeller/prop1.html Free stream characteristics change… Variables r – density (constant) A – cross-section area U – wind speed p – pressure T – thrust of wind on turbine Circular tube of air flowing through ideal wind turbine (K.L. Johnson 2006) If a tube of air is moving with diameter d1, speed u1, and pressure p1 as it approaches turbine, the air speed decreases, causing the tube of air to increase to d2. Air pressure rises in front of turbine and drops behind the turbine. Part of the kinetic energy (KE) of air is converted to potential energy (PE) to create the pressure increase and more KE is converted to PE after the turbine to return the pressure to atmospheric. Wind speed decreases until pressure is in equilibrium and u4 = u1. BET Limitation – Axial Induction factor Axial Induction factor a u1 u2 u1 u2 u1 (1 a) u4 u1 (1 2a) accounts for wind speed reduction as wind approaches turbine Consider the limits: a0 a1 No reduction in wind speed 2 Wind stops downstream, model invalid Power and Power coefficient 2 Theoretical P Tu2 1 rA2 u12 u4 2 u2 1 rA2u13 4a1 a 2 2 Power P rotor_ power 2 4a1 a Coefficient of Power CP 1 3 r u A power_ in _ wind 2 dCP 1 0 a ,1 da 3 Theoretical max Cp, set Sub 1/3 into Cp to get max of 16/27 = 0.5927 (Betz Limit) only 59% of max theoretically possible. Value of 1 invalidates model (not btwn 0 and ½) Counter rotation of wind: Blade Momentum Theory Angular Induction factor a w 2W accounts for reduction due to rotational wake Consider the limits: Rotor induces rotation in opposite direction of blade rotation W – Rotor rotational velocity w – Induced wind rotational velocity a 0 a 1 No induced rotation 2 Induced rotation, w equal and opposite to rotor rotation Angular velocity of rotor affects local wind at blade Drag Lift T Q wr1 a b W U 1 a w r 1 a 2 2 2 2 a 2 W U 1 a b a arcsin W 1 dQ rW 2 * B * c * C L sin C Dcos * r * dr 2 1 dT rW 2 * B * c * C L cos C D sin * r * dr 2 U 1 a Power Generated by Turbine Power = Torque * rotational velocity P Qw R 1 Q dQ rW 2 * B * c * C L sin C D cos * r * dr 2 r0 Solidity ratio Bc Closed versus open area 2pr B*c = net chord length of ALL blades 2pr = total circumference at radius, r Constraints and Materials Max diameter of wind turbine = 1 meter Max number of blades is 12 Hub is given and has a radius of 0.05 meter made of plastic Must be a horizontal axis wind turbine With blades that are thin flat plates (remember that our model is also developed for aerodynamics of blades/airfoils that are thin flat plates), so we’ll use foam board Attach blades to hub with wooden dowel rods Parameters and/or Variables Primary Pitch of blades, which in turn affects angle of attack Cord/shape of blades Constant cord – to make simple rectangular blades Variable cord – to make another shape (triangle, parallelogram, etc.) Secondary Number of blades <=12 Radius <= 0.5 meter Plot theoretical results of Coefficient of Power (Cp) versus angular velocity of the hub and determine the conditions for which a max occurs (note, power is related to performance, how well does your turbine perform) On test day, we will measure electrical output (voltage and current, recall P(elect) = V*I) and angular velocity. You’ll see how well results match predictions. Just as in the bottle rocket project, that’s what matters to find a max for your conditions, predict it and achieve it. Cp, Coefficient of Power Performance metrics and evalutation w, Rotational Speed Definitions W – relative wind speed Uinf - free stream wind speed a – angle of attack b – blade pitch a – axial induction factor a’ – angular induction factor – relative angle of wind B – number of blades CL – coeficient of lift CD – coefficient of drag Q, dQ- total blade torque, torque on differential element Cp - coefficient of power Matlab Pseudo Code: Find these steps! Inputs: number of blades N, chord length c, blade span R, blade angle δ For a range of rotational speeds ψ For a range of blade elements dr up to the blade span R While a and a’ converge Calculate relative wind velocity W using Calculate a using Eq. Calculate angle of attack χ using Use the empirical data to evaluate CL and CD for the χ Calculate new a and a’ using End Calculate the differential blade torque dQ for the blade element Sum the elemental contributions dQ to the total torque Q End Calculate power by the product of total torque Q and rotational speed ψ Calculate coefficient of performance Cp for the rotational speed ψ End Plot coefficient of performance as a function of rotational speeds ψ Generator Performance Curves Recall that losses occur converting mechanical power from the turbine to electric power by the generator Test or find specifications for generator performance