The energy a flywheel stores per kilogram is fixed by its material and the safety factor, not by its size: for a thin ring it is half of the allowable stress divided by the density, which for iron and steel at the safety factors of 4 to 8 used for machine parts comes to about 0.25 to 2.13 Wh per kilogram. Banking 1 kWh of usable energy therefore takes roughly 626 kg of high-strength steel at the best case and over 5 tonnes of weak cast iron at the worst, so speed can only be bought with strength, and mass can only be bought with weight.
Part of Flywheel Energy Storage
How to size a flywheel when the speed limit comes from the rim stress: the energy per kilogram, the mass for a target energy, and the rpm that follows.
Size Does Not Buy Energy Density
For a thin ring the rim stress is σ = ρ × v² (derived in Burst Energy Calculation) and the energy per kilogram is ½ × v². Eliminating v:
E per kg = ½ × σ_allowable ÷ ρ
Neither the radius nor the rpm appears. A 3 m ring and a 0.3 m ring at the same tip speed hold the same energy per kilogram. What a larger wheel changes is the rpm: n = v ÷ (2π × r) revolutions per second. The only ways to store more are more mass, a stronger or lighter material, or a smaller safety factor.
Worked Example: Energy Per Kilogram For The Corpus Materials
Inputs: tensile strengths from Tensile Strength (cast iron 15,000 to 30,000 psi = 103 to 207 MPa; mild steel 50,000 to 70,000 psi = 345 to 483 MPa); densities 7,200 kg/m³ for cast iron (corpus) and 7,850 for steel (handbook value, not from the corpus); safety factors 4 and 8 on stress, the range Safety Factors gives for machine components under repeated loading.
| Material, strength | Safety factor | σ allowable | Tip speed limit | Energy per kg |
|---|---|---|---|---|
| Cast iron, 103 MPa | 8 | 12.9 MPa | 42 m/s | 898 J = 0.25 Wh |
| Cast iron, 103 MPa | 4 | 25.9 MPa | 60 m/s | 1,795 J = 0.50 Wh |
| Cast iron, 207 MPa | 4 | 51.7 MPa | 85 m/s | 3,590 J = 1.00 Wh |
| Mild steel, 345 MPa | 8 | 43.1 MPa | 74 m/s | 2,744 J = 0.76 Wh |
| Mild steel, 345 MPa | 4 | 86.2 MPa | 105 m/s | 5,489 J = 1.52 Wh |
| Mild steel, 483 MPa | 4 | 120.7 MPa | 124 m/s | 7,685 J = 2.13 Wh |
Row check, the 345 MPa steel at safety factor 4: 345 ÷ 4 = 86.2 MPa; v = √(86.2e6 ÷ 7,850) = 105 m/s; E per kg = ½ × 105² = 5,489 J.
What That Costs For A Target
To bank 1 kWh and be able to draw it all you must store more than 1 kWh, because the drive stops being useful as the wheel slows. Running from full speed down to half speed delivers 75% of the energy, so the wheel must hold 1,000 ÷ 0.75 = 1,333 Wh at full speed.
- Best case, 483 MPa steel at safety factor 4: 1,333 ÷ 2.13 = 626 kg.
- Mild steel at 345 MPa and safety factor 4: 1,333 ÷ 1.52 = 877 kg.
- Weakest, cast iron at 103 MPa and safety factor 8: 1,333 ÷ 0.25 = 5,300 kg.
One kilowatt-hour runs a 100 W load for ten hours. A village flywheel that carries a household overnight is a multi-tonne object, and that is the arithmetic behind the carrier’s advice to use batteries for anything longer than minutes to hours.
The Safety Factor Is Squared Into The Energy
A safety factor of 4 on stress is a factor of 2 on speed (stress goes as speed squared) and a factor of 4 on stored energy. Running at half the burst speed leaves only a quarter of the burst-limit energy in the wheel. That quarter is the price of a margin against flaws in a cast or welded rim. Cutting the margin to save weight is the single most dangerous economy in this design space.
Choosing Between More Speed And More Mass
- Set the energy target and add the 25% that cannot be recovered: E_full = E_usable ÷ 0.75.
- Pick the material and a safety factor of at least 4, and read the tip speed limit from the table.
- Mass = 2 × E_full ÷ v². Check: 1,333 Wh = 4.8 MJ and v = 74 m/s (mild steel, safety factor 8) gives 2 × 4.8e6 ÷ 5,476 = 1,753 kg.
- Choose the radius for convenience; rpm = v × 60 ÷ (2π × r). At 0.6 m radius and 74 m/s that is 1,178 rpm, and at 1.5 m radius it is 471 rpm.
- Bigger radius means lower rpm and easier bearings, but a larger rim to cast or roll and more air to push (see Reducing Windage Losses).
Common Mistakes
- Raising rpm until the bearings complain, then stopping. The bearings are not the limit; the rim is. Compute the tip speed first.
- Taking a strength figure from one casting and applying it to another. Cast iron is brittle and variable; use the lowest figure the corpus gives.
- Comparing wheels by rpm. Compare by tip speed. Two wheels at the same rpm have rim stresses in the ratio of their radii squared.
Safety
The energy table is a ceiling, not a target
These figures come from the lowest tensile strength the corpus gives for each material and assume a sound, unjointed ring. A rivetted, bolted or welded joint carries less, and the corpus gives no joint factor, so do not run a jointed rim at a stress this table allows for sound metal. Read Safety & Containment before you choose a speed.
What’s Next
- Learn next: Cast Iron & Steel Disc — turning the numbers into a casting or a forging
- Combine with: Burst Energy Calculation — the derivation behind these limits