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| Fig. 1: A modern flywheel system, the NASA G2 at the Glenn Research Center. The flywheel spins at 6 × 104 rotations per minute. [6] (Source: Wikimedia Commons. Courtesy of NASA) |
Flywheels have been used for centuries to store useful energy for a variety of applications. In modern times, flywheels attached an electric motor (as opposed to an engine) can be used to both store and generate energy. This is done by using the motor to spin up the flywheel, converting the electricity into kinetic energy that can later be used. To take advantage of this stored electricity, one simply lets the flywheel drive the motor which will produce an electric current that can be used again. In this way, the flywheel system can act as a battery. An example of a modern flywheel system can be seen in Fig. 1.
The energy content is determined by a variety of factors, and has some fundamental limitations. The energy E of the flywheel system is given by
where I is the moment of inertia, and ω is the angular velocity. In the case of a flywheel with all of its mass concentrated along the outer radius, we would have a hollow cylinder with moment of inertia given by I = M r2 where M is the mass of the flywheel and r is its radius. The flywheel is ultimately limited by the material properties of the flywheel itself as well as the motor inducing the torque. If we use the tensile strength of the material, we can calculate the maximum angular velocity using the equation σ = ρ ω2 r2, where σ is the material tensile strength and ρ is the material density. [1] This means the maximum energy of the flywheel is independent of the chosen radius, and purely a function of the material properties and our choice of the value of M.
If we assume a reasonable value for the tensile strength of stainless steel around 800 MPa and density of approximately 8000 kg/m3, we have that [2-3]
We have a reasonable estimate of the price per kilogram of stainless steel of roughly $4 kg-1. Using this, we can estimate a materials cost per stored joule of $2.0 × 10-6 J-1. [4] Converting the energy unit to 1 kWh = 3.6 × 106 J traditionally used in industry, we find $72 kWh-1. A reasonable estimate for the cost of lithium ion batteries in 2018 is about $300 kWh-1, so we see that purely from a cost perspective the flywheel solution is roughly a quarter the price if we assume a flywheel system with no energy loss due to friction, Earth's rotation, and various other sources of energy loss. [5]
Let's consider an example. If we want to store an amount of energy necessary for the LA metropolitan area for 8 hours, we could need about 14 GW × 8 hours = 1.1 × 108 kWh. This would translate to about 4 million tonnes of steel. If we assume a radius of each flywheel of roughly a quarter of a meter, we would have a maximum RPM of 12,000 RPM (Note: this is not achievable in real life scenarios as it cannot be design to operate at its failure limit). If all 4 million tonnes of this steel was spinning at this maximum speed, we get an energy content of 2 × 1014 J. For context, Little Boy, the atomic bomb dropped over Hiroshima in 1945, was around 15 kilotonnes of TNT or 6.3 × 1013 J. This would clearly be very dangerous if these flywheels failed catastrophically as the energy content is even greater than this devastating nuclear weapon. This is good indicator that flywheels likely do not have a place for energy storage on the order of hours on such large scales, but would probably be more appropriate for smaller time scales depending on the energy requirements over that period of time.
© Joshua Barnett. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] B. Wheeler, "Flywheel Energy Storage," Physics 240, Stanford University, Fall 2010.
[2] "Stainless Steel Grade Chart," Atlas Steels Metal Distribution, November 2000.
[3] K. J. R. Rasmussen, "Full-range Stress-Strain Curves for Stainless Steel Alloys," J. Constr. Steel Res. 59, 47 (2003).
[4] "Specialty Stainless Sheet and Strip Stainless Price Book," AK Steel, November 2019.
[5] K. Mongird et al., "Energy Storage Technology and Cost Characterization Report," U.S. Department of Energy, PNNL-28866, July 2019.
[6] R. H. Janson and T. P. Dever, "G2 Flywheel Module Design," U.S. National Aeronautics and Space Administration, NASA/CR-2006-213862, August 2006.
