Super Rotors to Power Starshots

Previously I have tweeted on super-strong carbon nanotube fibres for use in energy storing flywheels, based on this News story.

First, a review of the underlying physics.

From basic rotational physics, we can describe the flywheel rotor as a solid cylinder of even composition and constant density \(\rho\). For any rotating object the important figures of merit are the Moment of Inertia \(I\) and the angular velocity \(\omega\). For a cylinder of mass \(m\), length \(l\) and circular radius \(r\) the Moment of Inertia is:

$$I = \frac12 mr^2 = \frac12 \rho\pi r^2lr^2 = \frac12 \rho\pi r^4l$$

Rotation Kinetic Energy is:

$$E = \frac12 I\omega^2 = \frac14 mr^2\omega^2$$

Material strength limits the flywheel rotor’s performance. Stress in the flywheel’s material is from the centrifugal reaction force that is acting to explode the rotor. The rotor material’s molecular structure must counter that with its tensile strength, the force that the material exerts on itself to keep it together. In the case of the purported nanotube fibre it’s internal strength is measured as upwards of 80 billion pascals or 80 gigapascals (GPa). Steels typically have 0.25 GPa tensile strength, so the nanotube material is 320 times stronger.

In a spinning rotor the stress to be countered by material strength at its maximum radius is:

$$\sigma_t = \rho r^2 \omega^2$$

The maximum the rotor can safely spin is when that stress equals its tensile strength. Past that point and the material will eventually ‘fail’, pulling itself apart violently due to all its kinetic energy, likely vapourising it in the process. To operate safely the rotor should be run at a maximum of some fraction of that limit. A factor of 50% is considered reasonable, allowing wiggle room for fluctuations. Thus the maximum operating stress should be about 2/3 of its maximum – in this example 2/3 x 80 GPa = 54 GPa.

Notice that the stress and the rotational kinetic energy look very similar. In fact their relationship is simply:

$$E = \frac{m\sigma_t}{4\rho}$$

This allows the energy Figure of Merit, the Specific Energy Density or stored energy per unit mass – to be derived as:

$$\frac Em = \frac{\sigma_t}{4\rho}$$

For the carbon nanotube material, with a density of about 1,300 kg/m3, and an operating maximum stress of 54 GPa, that means a specific energy density of 10 MJ/kg.

Consider the power storage needs of the Starshot interstellar sail, which masses 2 grams and cruises to Alpha Centauri at 0.25 c. About 60 terajoules per Starshot is needed, expended over about 20 minutes. Assuming near perfect conversion from rotational energy to laser power, the mass of spinning rotors needed per shot is about 6,000 tonnes. This can be an arrangement of multiple flywheels, hooked up to a massive solar array farm or a high efficiency nuclear reactor, that can be powered up over several hours.

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