Regenerative                                 Power     and                  Motion

# Flywheel Basics

Review of Familiar Concepts

For a very thin spinning hoop (with all its material at nearly equal tensile stress) , kinetic energy can be computed from Newton's (1642-1747) fundamental equations:

Moment of Inertia  =  (Rim Density) (Rim Volume) (Rim Radius)2
ENERGY  =  (1/2) (Moment of inertia) (Spin Speed)2
Math for Flywheel Energy Storage Design

Tensile stress at its outer radius, due to its spin speed, can be computed from:

Tensile Stress   =  (Rim Density) (Radius)2 (Spin Speed)2

A flywheel cross-section is shown at right (with inner and outer diameters ID and OD, height H) which spins about a vertical axis through its center.

Maximum stored energy, vs. weight and total volume, is usually given only for a thin hoop or solid disk.  For practical aspect ratios, it is  related to the rim's ID/OD ratio by slightly complicated relationships. The ID/OD dependence (below) was  derived by combining the above equations with the integral of the differential expression:

d(Inertia)  =  (Radius)2  (Rim Density)   d(Volume)

Max ENERGY  =  (ID/OD Factor) (Rim Volume) (Max Tensile Stress)

where:

(ID/OD Factor)  =  (1/4) {1 - (ID/OD)4} / {1 - (ID/OD)2}

From this, we get:

(Max ENERGY) / Weight  =  (ID/OD Factor) (Max Tensile Stress) / (Rim Density)

and (where Total Volume includes the space inside the rim):

(Max ENERGY) / (Total Volume)  =  (1/4) {1 - (ID/OD)4} (Max Tensile Stress)

Normalized maximum energy-to-weight, and energy-to-volume ratios, vs. flywheel ID/OD ratio, are plotted in this figure. Note that energy-to-weight ratio increases, while energy-to-volume decreases, with increasing ID/OD. A good compromise is ID/OD = 0.75 when the space inside the rim is used for a motor/generator. Besides, it is easier to maintain a vacuum in a larger volume of space, relative to outgassing materials in the enclosure. High energy-to-weight ratio is a more important factor than energy-to-volume, particularly for a thin-wall (lightweight) vacuum enclosure.

Two significant observations can be made from these equations: Flywheel weight/energy ratio  is proportional to its density (specific gravity). And energy storage is proportional to the product of  tensile strength and volume of  material at full tensile stress. So, to achieve high energy storage with light weight, rim material should be high-strength and low-density.

Precession torque (tending to tilt the flywheel spin axis), for a stationary flywheel battery application (subjected to earth rotation of one revolution per day), does not present serious problems, but is not negligible. It can be computed from:

Precession Torque  =  (Moment of  Inertia) (Spin Rate) (Precession Rate)

For a vertical spin axis, at the earth's equator, the precession rate is due to earth rotation, and is 0.00069 rpm.  Before we proceed to some practical flywheels, let's combine the above moment-of-inertia, energy, and precession torque equations. This yields the interesting result:

Precession Torque  =  2(Energy)(Precession Rate)/(Spin Rate)

It indicates that high spin-rate flywheels should have less precession torque for the energy they can store.

Gravity can effectively counter this precession torque, to maintain a vertical spin axis, of a flywheel assembly having a levitation  force (from its magnetic bearings) at its top.  The torque which maintains verticality is:

Tilt restoration torque  =  (Flywheel rotor weight)(Distance from top to CG) sin(tilt angle)

Normally, the distance from the top to the center-of-gravity (CG) is about half the rotor assembly height.  And clearly, for practical maintenance of spin-axis verticality, a long rotor assembly with relatively small diameter is best.

Practical Flywheel Batteries

With these equations (and correct unit conversions) it can be computed that...

A flywheel rim having:          Height = 1-foot        OD = 1-foot       ID/OD = .75
Max Tensile Stress = 500,000 psi (conservative design best for now)
Specific Gravity = 1.1

will store 3-kwh, weigh 23 pounds, spin up to 100,000 rpm, and incur precession torque from earth rotation of  0.12 foot-pound (causing a nominal 0.6 degree tilt). Motor/generator, bearings, a vacuum enclosure, and controller electronics will have comparable weight, with RPM's proposed siting.

Lead-acid batteries, to store 3-kwh, weigh over 250 pounds, and 5 would need to be interconnected.

A flywheel rim having:          Height = 2.5 feet      OD = 2.5 feet     ID/OD = .75
Max Tensile Stress = 500,000 psi
Specific Gravity = 1.1

will store 50-kwh, weigh 370 pounds, spin up to 44,000 rpm, and incur precession torque from earth rotation of 4.75 foot-pound (causing a nominal 0.6 degree tilt). Again, remaining parts will add a comparable weight.

Lead-acid batteries, to store 50-kwh, weigh over 4000 pounds, and over 80 would need to be installed and interconnected every few years. Their weight is at least 5x that now possible with flywheel batteries; and new flywheel materials described in the literature could enable flywheel batteries storing 10x more than lead-acid of  comparable weight.

Energy storage/weight ratio fundamentally depends on tensile strength of  lightweight composite fibers. New formulations and processing techniques have resulted in material strength that has increased steadily over the past decade.  Permanent magnets like Neodymium-Iron-Boron can also help reduce weight, by allowing smaller magnet and iron components.  Our thin-wall vacuum enclosure results in systems weighing considerably less than those designed to contain possible flywheel explosion in the vacuum enclosure.

Different Viewpoints of  Flywheel Mechanization & Potential

Over the past 20 years, flywheel system developers have taken many diverse paths:

Early (30 years ago) flywheel promoters dismissed arguments for electric interfaces. Most tried to directly engage a shaft, coupled by speedup gears to a flywheel rotor in a vacuum enclosure.  They avoided a motor/generator and electronics. They also apparently avoided research on their gear and bearing lubricants in vacuum, their O-ring seals, explosion hazard, and gyroscopic effects. As you may know, they failed on all counts.

These development short-cuts result in flywheel systems not so different from those used in punch-presses (hallmarks of our machine age) for the past century.  Most of  today's flywheel batteries can deliver power for only "tens of seconds" and standby power needed for them is 2-kw.

Others have promoted flywheels for a variety of applications. But long-term flywheel power storage for on-site UPS and building-integrated solar or wind power has not been pursued, and is essentially not now available from the dozens of existing flywheel device developers.

Lead-acid batteries are really the only practical power storage option now available for buildings.

Features of  RPM Flywheel Batteries:
Uultra-efficient motor/generator, which has virtually no idling losses while the levitated rotor is spinning even at maximum speed..
Power storage/regeneration electronics, with poly-phase sinusoidal current control, magnetic bearing servos.
Height H of the flywheel rotors is relatively high, compared to others. This minimizes rotor tilt, partly due to precession torque from earth rotation, to about 1 degree.

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