World's Largest Machines

Copyright ©2014 by Paul Niquette. All rights reserved.
Auxiliary Information for Solvers.

Technical Factors: Aerodynamic Interference

As noted in the puzzle formulation, solvers were encouraged to investigate aerodynamic interferences between successive blades.   With more than three blades on the rotor, one might expect the deflected wind-flow patterns to become crowded between blades.  Here is a sketch of that speculation.

 

Viewing blades in a line, we see wind-flow approaching the blades normal to the the Plane of Blade Rotation.  That can only occur at blade sections near the rotor hub where insignificant power can be captured.  Near the tip of the blade, where the most power is captured, the direction of relative air flow is dominated by tangential speed of the blade.

Rotating at 18 rpm, the tip of a blade 60 meters long will be moving tangentially at 113 m/s (368 ft/sec, 250 mph).  Driven by wind at, say, 6 m/s (20 ft/sec, 13 mph), the flow lines approach the plane of rotation at a three degree angle and would hardly be affected by closer spacing of the turbine blades.

The air pressure on the upwind side of each turbine blade is higher than the air pressure on the downstream side.  One might suppose that an aerodynamic effect of more than three blades would take the form of interfering pressure gradients from one blade to the next. This sketch attempts to visualize that speculation as it relates to blade count.


Here again, we must take cognizance of the fact that the relative wind is dominated by the tangential velocity of the blade near the tip, where the preponderance of power is captured.  Oriented nearly parallel to the plane of rotation, the pressure gradients acting on separate blades unite to form contiguous zones of differential pressure on opposite sides of the blades.
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Technical Factors: Aerodynamic Forces

As airfoils, the three blades on a wind turbine rotor are acted upon by aerodynamic forces much like an airplane wing – but with important differences.  Here is a review if the principles, starting with a sketch of the wing section of an airplane in stable horizontal flight… 

 



Four forces: [1] Aircraft Weight acting normal to the Horizon balanced by [2] Aerodynamic Lift, [3] Aerodynamic Drag acting horizontally balanced by [4] Engine Thrust.

The airfoil itself is shown has its Wing Chord maintained at an angle-of-attack α with respect to the Relative Wind, which is by convention indicated by the arrow pointing to its ‘source’.


Derivation of Drive Force

Wind turbine rotors are not propelled by Engine Thrust.  To continue the comparison, here is a sketch of the wing section for an unpowered airplane descending on a stable glide path...

 

Engine Thrust has been removed from the sketch.  Aircraft Weight continues to act normal to the Horizon Line. As indicated by the arrow, Relative Wind is coming from below the Horizon Line at a glide-angle larger than the angle-of-attack α.
Aerodynamic Lift, which always acts normal to the Relative Wind, is shown here tilted forward, projecting a replacement for Engine Thrust onto the Horizon Line, thereby enabling continued forward motion of the aircraft.
Aerodynamic Drag, which always acts directly opposite to the Relative Wind, is also projecting a component of force onto the Horizon Line tending to resist forward motion of the airplane.
As shown in the sketch, both Aerodynamic Lift and Aerodynamic Drag are projecting components of force normal to the Horizon Line, and their sum balances Aircraft Weight.
For capturing power from a moving stream of air, a wind turbine blade resembles the wing of an unpowered airplane established on a stable glide path in descending flight.  The purpose here is to show a simplified derivation of the Drive Force.



The Horizon Line is relabeled Plane of Blade Rotation, a vertical disk maintained normal to the Local Wind by the yaw-control system.  The blade motion produces the Local Tangential Velocity, which varies from zero near the rotor hub to its highest value called Tip Velocity.
 
The sketch shows the Relative Wind as the resolution of the Local Wind Velocity and the Local Tangential Velocity.  The blade is being held at an angle to the Relative Wind by the pitch-control system, and that angle is the sum of the angle-of-attack α and an angle-of-incidence β.
Aerodynamic Lift acts normal to the Relative Wind and is shown here tilted forward, projecting a force onto the Plane of Blade Rotation. 
Aerodynamic Drag acts directly opposite to the Relative Wind and is also projecting a component of force onto the Plane of Blade Rotation.
The algebraic sum of the two aerodynamic projections onto the Plane of Blade Rotation produces the Drive Force for the wind turbine blade.  Meanwhile, both aerodynamic forces exert bending moments normal to the Plane of Blade Rotation and must be balanced by the wind turbine structure.

For orientation, we see the blade is moving toward the left of the page.  By convention, wind turbines turn clockwise as viewed from the source of the wind.   Accordingly we must be looking directly upward at the Plane of Blade Rotation with the wind at our back and the blade just passing its lowest point.
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Duty Cycle
Given that the wind turbine will operate at pNP for differing amounts of time, in hours per year tNP(b), the expression 100 tNP(b) / 8,760% might be considered a 'figure of merit' for the wind turbine configuration.
Let vNP(b) = lowest wind speed at which the pPG(b) = pNP = 5,000 kW.  For each of the three configurations, the wind turbine operates at Nameplate Power only in the respective ranges of wind speeds vNP(b) < vLW <vCO.

Let tNP(b) represent the duration of time during a typical year that the wind turbine produces Nameplate Power. 
Inasmuch as, vNP(4) < vNP(3) < vNP(2), the Nameplate Duty Cycle will be found in this order tNP(2) < tNP(3) < tNP(4).  The frequency distribution of wind speeds will determine the values of tNP(b), for which this graph is provided as a guide...


For convenience, the Business Model was used to provide a cumulative distribution in a curve called an ogive, here covering wind turbine operations over a period of one year (8,760 hours).  It is derived by summing entries in the histogram of wind speed frequency, which is superimposed here for reference.  Based on Duty Cycle, then…
Two Blades: A wind turbine with two blades produces Nameplate Power for only seven hours. 
That rules out the two-blade configuration as not practical for the Design Center in the Business Model, wherein a three-blade configuration manifests a Duty Cycle 51 times greater in hours per year of operation at Nameplate Power level.
Four Blades: A wind turbine with four blades does show significant superiority over the Base Case configuration: tNP(4) = 528 hr/yr versus tNP(3) = 359 hr/yr.
The Business Model shows that with an incremental capital increase of 33%, the four-blade wind turbine produces Nameplate Power for 47% more hours per year than a conventional three blade wind turbine.  An exclamation point may seem to be appropriate.  Not so much.
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Background for World Largest Machines Puzzle

The World's Largest Machines puzzle was first drafted in 2011 with a solution appropriate for a planned release date of April 1, 2012: "Three is the closest integer approximation for pi."  It would have been the second unserious application of that non sequitur in 40 years...
The most popular airliner in the sixties was the Boeing 727 (1,832 built), which has three engines.  The Lockheed 101l Tristar and the McDonnell Douglas DC-10 also had three engines.  During the development of the DC-10, the author of this puzzle proposed an on-board computer system for real-time data analysis during flight testing, the first of its kind.  The computer and auxiliary equipment weighed a ton and took up half the cabin space, but it was justified based on reduced time-to-market for the aircraft.  During one presentation, the author of this puzzle made an admiring comment about the DC-10: “Three is the closest integer approximation for pi.”  Attendees nodded solemnly.  The proposal won the contract. 
Following residential relocation to Brittany in 2012, the author took notice of the wind turbines in his new neighborhood and became amazed at their size, captivated by their beauty, and fascinated by their technologies.  Not a joking matter, really.  Still, the Business Model used for the solution to the World's Largest Machines puzzle does accommodate non-integer values for the number of turbine blades.  One might let b = π…  Naah.

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