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 \rho and maximum tensile strength S_{max} . The mass of the whole loop is 2 \pi r A \rho , and the volume is V = 2 \pi r A .

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

E = \pi r^3 A \rho \omega^2 . The maximum usable energy is a (hopefully large) fraction of that.

Consider a small element d\theta of the hoop, where \theta is the position around the loop. If the hoop is circumferential stress is S_c , then we can construct a force triangle where the deflection of the circumferential force is S_c A d\theta / 2 at each end, for a total centripedal force of S_c A d\theta . This is balanced by the centrifugal acceleration of the element, dM a = ( r A \rho d\theta ) ( \omega^2 r ) . So:

S_c A d\theta = r^2 A \omega^2 \rho d\theta

S_c = r^2 \omega^2 \rho

At maximum stress, r^2 \omega^2 = S_{max} / \rho . . . 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} } . . . half the hoop volume times the maximum tensile stress.

also, $ E = \frac{1}{2} \rho V v^2 = \frac{1}{2} V S_{max}, so

\large { v^2 = S_{max} / \rho }

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 fraction of this maximum . The maximum rim velocity, hence the maximum energy per flywheel mass, is similarly constrained.

Beacon Power flywheel maximum speeds are 670 meters per second; perhaps in time they may learn how to reliably increase this to 1000 meters per second. This will require defect-free manufacturing, and careful attention to stress management around yarn ends and voids in the binder matrix of the carbon fiber materials. I wish them the best of luck.