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| $ \large { E = \frac{1}{2} V S_{max} } $ . . . half the hoop volume times the maximum tensile stress. | $ \large { E = {1 \over 2} V S_{max} } $ . . . half the hoop volume times the maximum tensile stress. |
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| 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 . | also, $ E = {1 \over 2} \rho V v^2 = {1 \over 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. |
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 
rArA
If the flywheel rotates at angular frequency 
r
r3A2
Consider a small element 
2 d)(2r)
=r2A2d
2
At maximum stress, 2=Smax
rA(r22)=rA(Smax)=rASmax
also, $ E = {1 \over 2} \rho V v^2 = {1 \over 2} V S_{max}, so
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.