Power Loop for SusTech Notes

Flywheel Scaling

The fastest possible non-magnetic flywheel is a thin hoop of material at maximum stress. Assume a thin flywheel hoop rotating around a vertical axis, with radius r, cross section A, made of a material with a mass density of and maximum tensile strength Smax. The mass of the whole loop is 2rA, and the volume is V=2rA.

If the flywheel rotates at angular frequency , the rim velocity v=r, and the stored energy is

E=r3A2. The maximum usable energy is a (hopefully large) fraction of that.

Consider a small element d of the hoop, where is the position around the loop. If the hoop is circumferential stress is Sc, then we can construct a force triangle where the deflection of the circumferential force is ScAd2  at each end, for a total centripedal force of ScAd . This is balanced by the centrifugal acceleration of the element, dMa=(rAd)(2r). So:

SaAd=r2A2d

Sa=r22

At maximum stress, r22=S  Plugging this into the stored energy equation, we get:

E = \pi r A \rho ( r^2 \omega^2 ) = \pi r A \rho ( S_{max} / \rho ) = \pi r A S_{max}

\large { E = \frac{1}{2} V S_{max} }

Real flywheels will have rotating structure that does not contribute to this strength, and flywheels thicker than a thin shell will have inner material at less than maximum stress and storing less than maximum energy. So the simple energy equation above is overoptimistic; if you include the container, the empty volume, the axial motor, the magnetic bearings, a decent safety factor, etc, then the maximum stored energy may be a small fra