# Flywheel Energy Storage

## Benjamin Wheeler October 24, 2010

### Submitted as coursework for Physics 240, Stanford University, Fall 2010

There are many renewable energies currently utilized and in development around the world. Some of these methods include harnessing solar, wind, hydro, and thermal energies. The only problem is there are no efficient methods of storage. To be able to convert and use renewable energy as electricity there needs to be a process for storing it. The focus of this report is on the feasibility of using flywheels to store rotational energy and convert it to electric energy when necessary. I have chosen to approach this from a small vehicle perspective, rather than determine if flywheels can store the energy needed to supply a city or country. If flywheels are capable of the energy density to power a vehicle effectively for the average citizen's needs then a huge portion of the demand for oil and the pollution of the environment can be lifted.

To simplify calculations, the capabilities and outputs of the Tesla Roadster will be used to judge what a flywheel should be capable of. A majority of drivers here in the US would be more than satisfied with the 200 mile range of the Roadster's 450 kg, 53 kWh Li-ion battery pack. [1] Thus we will determine if a flywheel of similar mass can store energy equivalent to this battery. The following equations can be found in most physics textbooks and flywheel books.

First, determine the expression for the energy of a rotational system. Our flywheel will be a hollow cylinder, which gives us Mr2 for moment of inertia. E-energy. I-inertia. M-mass. r-radius. w-angular velocity.

Second, determine the limits to angular velocity due to material used: ρ = density, r = radius, ω = angular velocity, σ = tensile stress (maximum before breaking).

Third, substitute the maximum angular velocity into energy equation.

Material M (kg) σ (pascals) ρ (kg/m3 Emax (joules) Emax (kWh) Emax/M (J/kg)
Titanium 450 8.8 x 108 4506 4.4 x 107 12 9.8 x 104
Carbon Fiber 450 4.0 x 109 1799 5.0 x 108 139 1.1 x 106
Steel 450 6.9 x 108 8050 1.9 x 107 5 4.3 x 104
Aluminum 450 5.0 x 108 2700 4.2 x 107 12 9.2 x 104
Table 1: Maximum flywheel energy storage of various materials. (Material properties produced from commercial material suppliers. [3-5])

These calculations do not account for frictional losses or efficiency in transforming electric to kinetic energy and back. Even if a carbon fiber flywheel is only 50% efficient it has the ability to store and provide more energy than Tesla's Li-ion battery with comparable mass. There would also be additional mass needed to house the flywheel and mechanisms, but these should be small compared to the maximum limit of energy storage. While metal flywheels do not perform to standards, a carbon fiber flywheel is a viable option for storing electricity for vehicles and many other applications such as back up grid power.

© Benjamin Wheeler. The author grants permission to copy, distribute and display this work in unaltered form, with the attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

## References

[1] G. Berdichevsky et al, "The Tesla Roadster Battery System," Tesla Motors, August 2006.

[2] Books LLC, Tesla Motors Vehicles: Tesla Roadster (Books LLC, 2010), pp. 1-40.

[3] James Zerbe, Practical Mechanics for Boys (M.A. Donohue & Company, 1914), Ch. 17.

[4] J. M. Corum et al., "Basic Properties of Reference Crossply Carbon Fiber Composite," Oak Ridge National Laboratory, ORNL/TM-2000/29, February 2000.

[4] C. Chung, Carbon Fiber Composites (Butterworth-Heinemann, 1994), pp. 65-66, 102, 164.