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| The "pressure" of a magnetic field is $ P = B^2/2\mu_0 $, where B is the magnetic field in Teslas ( Kg /( Coul s ) ), and $ \mu_0 $ is the permiability of free space, $ \mu_0 = \pi \mult $ 4e-7 ( Kg m )/Coul^2^. When you multiply all that out, you get Kg /( m s^2^ ), or Newtons / m^2^ or Joules / m^3^ . | The "pressure" of a magnetic field is $ P = B^2/2\mu_0 $, where B is the magnetic field in Teslas ( Kg /( Coul s ) ), and $ \mu_0 $ is the permiability of free space, $ \mu_0 = \pi \times $ 4e-7 ( Kg m )/Coul^2^. When you multiply all that out, you get Kg /( m s^2^ ), or Newtons / m^2^ or Joules / m^3^ . |
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| || {{attachment:PowerLoop/MagSat.png||width=400px}} || Ordinary steel saturates at magnetic fields approaching 2 Tesla, although significant extra "H" Amp-Turns/Meter magnetization is required to raise the field that high. || There is no such thing as a "magnetic insulator" (though the Meissner effect in superconductors resembles that), so the field from the poles will cross the gap between poles, adding to the flux at the root end of the pole. Assume that we can create a field of 1 Tesla over a 5 mm gap, with alternating poles creating that field intensity over half the surface of the rotor. The resulting magnetic field pressure is approximately 200 kN/m^2^ . A clever magnet designer may be able to do better than this. |
|| [[attachment:PowerLoop/MagSat.png|{{attachment:PowerLoop/MagSat.png||width=300px}}]] || Ordinary steel saturates at magnetic fields approaching 2 Tesla, although significant extra "H" Amp-Turns/Meter magnetization is required to raise the field that high.<<BR>><<BR>> There is no such thing as a "magnetic insulator" (though the Meissner effect in superconductors resembles that), so the field from the poles will cross the gap between poles, adding to the flux at the root end of the pole. Assume that we can create a field of 1 Tesla over a 5 mm gap, with alternating poles creating that field intensity over half the surface of the rotor. The resulting magnetic field pressure is approximately 200 kN/m^2^ . A clever magnet designer may be able to do better than this.|| |
Power Loop for SusTech Notes
Flywheel Scaling
The fastest possible (non-magnetically deflected) 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.
Magnetic Field Pressure
The "pressure" of a magnetic field is P = B^2/2\mu_0 , where B is the magnetic field in Teslas ( Kg /( Coul s ) ), and \mu_0 is the permiability of free space, \mu_0 = \pi \times 4e-7 ( Kg m )/Coul2. When you multiply all that out, you get Kg /( m s2 ), or Newtons / m2 or Joules / m3 .
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Ordinary steel saturates at magnetic fields approaching 2 Tesla, although significant extra "H" Amp-Turns/Meter magnetization is required to raise the field that high. |
Some might ask, why not use superconductors? Superconductors can generate high magnetic fields, but they are fragile in many ways. They require expensive and inefficient cooling, systems that can generate as much heat as the losses in copper magnet windings. The superconducting state is easily disrupted - when that happens, the stored energy in the field turns into resistive heating in the (former) superconductor. Heat capacity is very low for materials near absolute zero, so the regions near a fault can heat up explosively. When the field changes in a superconductor, the induced voltages push non-superconducting electrons in the material, creating resistive heating, also threatening a "quench" (explosive superconductor failure). There are plenty of other challenging problems to solve first - when you hear someone suggesting superconductors, run. They are dangerous to themselves and others.
"Attractive" magnetic pressure is unstable - Earnshaw's theorem. So a control system must continuously measure the gap and adjust the magnetic field with bandwidths of hundreds of kilohertz. Superconducting field windings for 60Hz generators have (so far) proved too difficult to deploy in weight-sensitive systems such as wind turbines. It is unlikely that practical 200 kHz superconducting control magnets will be available anytime soon.
Turning and Bending the Rotor with a Magnetic Field
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