Flywheel Energy Storage Calculator

Compute kinetic energy, moment of inertia, and mass for rotating flywheels. Essential for mechanical design and energy storage systems.

Key formula: E = ½ I ω², where I = moment of inertia, ω = angular velocity (rad/s). For solid cylinder: I = ½ m r² , m = ρ π r² h. For thin hoop: I = m r², m = ρ · π ( (r + t/2)² - (r - t/2)² ) h (t = radial thickness).

Steel Ø1m×1m, 5000 RPM
Carbon r=0.3m, h=0.5m, 10000 RPM
Aluminum r=0.6m, h=0.8m, 3000 RPM
Thin hoop steel r=0.5m, t=0.02m, h=0.2m, 2000 RPM
Computing...

Understanding Flywheel Energy Storage

A flywheel stores kinetic energy in a rotating mass. The energy is proportional to the moment of inertia and the square of rotational speed. Flywheels are used for grid stabilization, UPS systems, and regenerative braking.

Fundamental equations:

Kinetic energy: E = ½ I ω² (Joules)

Moment of inertia (solid cylinder): I = ½ m r² = ½ ρ π r⁴ h

Moment of inertia (thin hoop): I = m r² = ρ π ( (r + t/2)² - (r - t/2)² ) h · r²

Angular velocity: ω = 2π · RPM / 60 (rad/s)

Why the shape matters

The distribution of mass significantly affects the moment of inertia. For the same mass and outer radius, a thin hoop stores twice the energy of a solid cylinder. However, the hoop experiences higher hoop stress for a given rotational speed, making material strength a critical design factor.

Material properties & stress limits

The maximum energy a flywheel can store is ultimately limited by the tensile strength of the material. The characteristic parameter is the maximum tangential speed v_max = ω·r. The specific energy (energy per unit mass) can be expressed as:

E/m = K · (σ/ρ)

where σ is the allowable stress, ρ the density, and K a shape factor (0.5 for solid cylinder, 1.0 for thin hoop). This shows that materials with high strength-to-density ratio (e.g., carbon composites) are ideal for high-performance flywheels.

Material Density (kg/m³) Tensile strength (MPa) Max tip speed (m/s)* Shape factor K Max specific energy (Wh/kg)**
Steel (AISI 4340) 7800 ~1200 ~400 0.5 (solid) ~5
Aluminum 7075 2700 ~500 ~430 0.5 (solid) ~10
Carbon composite (T700) 1550 ~2000 ~1100 0.5 (solid) / 1.0 (hoop) ~35 (solid) / ~70 (hoop)

* Approximate max tangential speed before burst (rotor geometry dependent). ** Theoretical maximum specific energy at speed limit (excluding safety factors).

Design considerations for real-world systems

  • Vacuum enclosure: To reduce aerodynamic drag, flywheels are often operated in a vacuum.
  • Bearings: Magnetic bearings eliminate mechanical friction and enable high speeds.
  • Safety: Burst containment is critical; composite rotors are designed to fail gradually (e.g., through delamination) rather than shatter.
  • Motor/generator: A integrated electrical machine accelerates and decelerates the flywheel.

Applications in depth

1
Grid frequency regulation – Flywheels respond in milliseconds to smooth out fluctuations from renewable sources, providing inertia to the grid.
2
Uninterruptible power supplies (UPS) – High-power, short-duration backup for data centers and industrial processes, bridging the gap until generators start.
3
Automotive KERS – Kinetic energy recovery systems in Formula 1 and hybrid vehicles capture braking energy and release it during acceleration.
4
Spacecraft attitude control – Reaction wheels (a form of flywheel) use momentum exchange to orient satellites without thrusters.

Calculator assumptions: This tool assumes uniform density, perfect geometry, and neglects stress limits. Use it for preliminary sizing only. For detailed design, consult stress analysis and material data.

Frequently Asked Questions

Divide by 3.6 million (1 kWh = 3.6 × 10⁶ J). The calculator does this automatically.

For the same mass and radius, a thin hoop (mass at rim) has twice the moment of inertia (I = mr²) compared to a solid cylinder (I = ½mr²), thus stores more energy but experiences higher hoop stress.

Because ω is proportional to RPM, and energy depends on ω². Doubling RPM quadruples stored energy.

The ultimate limit is the material's tensile strength. As speed increases, centrifugal stress grows, and eventually the rotor would burst. Practical designs include safety margins and burst containment.