Regenerative
Power and
Motion |

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}

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:

This leads to the result: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.

More RPM web pages:

Flywheel Battery
Comparison with Others

Electric Vehicles with
In-transit Power from Highways

EV with Onboard
Batteries, Charger, PV, Motor-wheels, Pedal Power

Technology:
Public and Business Policy

Flywheel Facts and
Fallacies

Future Options for
Clean and Sustainable Power

If you have questions, comments or suggestions, email *fradella@earthlink.net*

Aug 2015