World's Largest Machines -- Puzzle

World's Largest Machines
Copyright ©2014 by Paul Niquette. All rights reserved.

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Wind turbines are arguable the world’s largest machines. “Arguably” – because indeed there may be an argument. The largest wind turbine in service is the Enercon E-126, which produces 7.6 megawatts of electrical power. Its linear dimensions (height x length x width) in meters are 225 x 12 x 127 (731 ft x 39 ft x 413 ft)...

According to World’s Largest Machines, the TAKRAF RB293 Bucket-Wheel Excavator, used in the strip-mining of coal, has these linear dimensions in meters : 94 x 220 x 50 meters (306 ft x 715 ft x 163 ft)...

So, then, which is larger? Taking simple sums of linear dimensions to represent their respective sizes, the E-126 and TAKRAF RB293 are arguably the same size, 364 meters! Nevertheless, wind turbines hold primacy in this puzzle by virtue of the plural in its title World's Largest Machines. The TAKRAF RB293 is merely a one-off.

Solvers might watch one or more of several time-lapse films showing the construction of a typical wind turbine to obtain a perspective about the size of these magnificent machines.

Wind turbines are now installed by the thousands all over the world on hillsides and in farm fields, at off-shore sites and in forests. Whatever their locations and nationalities, whatever their sizes and power ratings – indeed, whatever their respective brands and technical designs, all wind turbines (with extremely few exceptions) share one common attribute -- three blades. Why? That is arguably the world's most obvious question...

Technical references often conceal design criteria that may be regarded as proprietary by designers. For this most obvious feature of wind turbines, an elementary explanation remains undisclosed on the web -- even here.

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Solvers will be well advised to watch an extraordinary 10-minute animation entitled What’s inside a Wind Turbine, which is a vital resource for anybody with a technical interested in the World’s Largest Machines. The film concludes with the sobriquet “Windmill on a Stick.”

Let the World's Largest Machines puzzle address the question of optimum blade count as an exercise in science, keeping in mind...

Science has a modern definition -- "reverse engineering nature."

-- Paul Niquette Sophisticated: The Magazine 1996

...in this case, as in others (Tin-Can Mystery, Express vs Local), we shall be reverse engineering engineering.

In The Rational Process, solvers will find a recommendation "to refract every problem through three lenses...

Technical Factors
Economic Issues
Human Considerations

...and there's a neat word for this analytic discipline, prescinding."

Technical Factors

With their three blades, wind turbines are unexcelled in their ability to concentrate power. References on the web deal superficially, though, with the blade-count question using technical aspects that include rotating speed and auditory noise, blade thickness and solidity (the ratio of rotor blade surface area to the swept area that the rotor passes through), cyclical loading and peak torques in the drive train and gear box, and overall reliability. One might also study potential aerodynamic interferences between successive blades.

The construction of a wind turbine takes energy, which raises a question: How long does it take after the machine is put in service to provide that amount of energy in the form of electricity. Such an interval in time might be called energy breakeven.

Economic Issues

Obviously the number of blades on a wind turbine must be a first-order determinant of capital cost. Though lowest in cost, a one-blade design might not be practical because of balance and wobble issues. A few wind turbines with two blades have been put into service, but with little solidity, they are relatively ineffective in capturing wind energy. Three blades are more costly to build but also more effective in delivering electrical power.

Apparently that extra cost for three blades can be justified, so why not four?

Nota bene, the puzzle solution need not analyze five or more blades if the greater capital cost of four blades cannot be justified by a commensurate delivery of electrical power.

Human Considerations

In addition to auditory noise generated by turbine blades mentioned above, we have one consideration that is probably the most human: "Aesthetics can be considered... some people find that the three-bladed rotor is more pleasing to look at than a two-bladed rotor." Same for a four-bladed configuration, one might suppose.

What is your explanation for three blades?

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Power Concentration

Renewable energy from the wind arrives at the wind turbine through an invisible disk swept by the turbine blades. For the Enercon E-126, that disk has a diameter of 127 meters, thereby defining a cross-sectional area of 12,667 square meters.

In round numbers, for a rated power of 7.6 megawatts, something like 600 watts per square meter of wind must be passing through the turbine blades. The electrical power is delivered through three 1-inch diameter wires with a total cross-sectional area of 0.0015 square meters. That corresponds to 5 gigawatts per square meter, a 'power concentration ratio' of 8.3 million-to-one.

Compare that ratio with power concentration of other technologies that capture energy from sustainable sources. Photovoltaics, for example, can capture 200 watts per square meter from the sun -- at high noon on a clear day -- and deliver electrical power through those same-sized electrical conductors, a power concentration ratio of 6.6 million-to-one.

Energy Breakeven

The concept of ‘breakeven’ is so familiar in economics that one would expect the calculation to be appropriate in the realm of energy.

Let P_WindTurbine megawatts be the electrical power produced by a wind turbine. One might estimate that its construction requires C_WindTurbine megawatt-hours of energy. The ‘breakeven’ time T_WindTurbine, which is the operating interval required for the wind turbine to produce the equivalent of its own construction energy, would be given simply by…

T_WindTurbine = C_WindTurbine / P_WindTurbine megawatt-hours / megawatts = hours.

For comparison, the breakeven calculation for electrical power produced by a coal-fired plant would seem to comply with the same formula…

T_CoalFired = C_CoalFired / P_CoalFiredmegawatt-hours / megawatts = hours.

However, for an energy system that consumes fossils, one must take into account energy yield, which is the ratio of electrical power produced P_CoalFired megawatts to the corresponding fossil fuel consumption rate F_CoalFired megawatts…

Y_CoalFired = P_CoalFired / F_CoalFired, (dimensionless), where Y_CoalFired < 1.

Throughout the interval T_CoalFired, the plant is consuming fossil fuel…

F_CoalFired= P_CoalFired / Y_CoalFired megawatts.

That amounts to an energy ‘expenditure’ of F_CoalFiredx T_CoalFired megawatt-hours. In fairness for the comparison, that fossil energy should be treated as if it were added hour by hour to the plant’s construction energy, as seen here…

C_CoalFired + F_CoalFiredx T_CoalFired megawatt-hours

…thus increasing the breakeven time interval. Hmm. It appears we have a little recursion sequence going on here…

T_CoalFired(0) = C_CoalFired / P_CoalFired
T_CoalFired(1) = C_CoalFired / P_CoalFired + P_CoalFired /Y_CoalFired x T_CoalFired(0)
T_CoalFired(2) = C_CoalFired / P_CoalFired + P_CoalFired /Y_CoalFiredx T_CoalFired(1)

…and so forth.

Inasmuch as T_{CoalFired(n

+ 1)} > T_CoalFired(n), this series can never converge. In the realm of energy, a breakeven time for the coal-fired power plant can never be reached! Same for other fuels, since thermodynamics limits Y < 1. Which is not to say that such plants are economically unprofitable. But one sees clearly here that the breakeven concept may not be appropriate in the realm of energy.

Some baffling results for economists arise in energy calculations (see for example an anecdote from 1973 entitled Yellowcake).