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

Approaching maximum stress, v^2 = 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 < {1 \over 2} V S_{max} } . . . half the hoop volume times the maximum tensile stress.

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. Also, the hoops will need enough radial stiffness (and damping!) so that small radial vibrations do not create increased stresses; that material adds little to the energy storage. I wish them the best of luck.