Flywheel Energy Storage Calculator

Flywheel Energy Storage Calculator finds the energy stored in a spinning rotor from mass, radius, shape and RPM, with usable energy, rim stress, burst speed and run time on a load.

What the Flywheel Energy Storage Calculator Shows

The Flywheel Energy Storage Calculator finds the energy stored in a spinning rotor. Its default is a 50 kg solid steel disk with a 0.3 m radius at 6,000 RPM, which holds 123.4 Wh, or 444.1 kJ. Of that, 92.53 Wh is usable if the speed is only allowed to fall to half.

The Calculate list has four choices. Stored Energy is the main one. The other three start from a target energy and return the speed, the mass or the outer radius that would hold it. A Flywheel Shape list sets how the mass is spread, from a solid disk to a thin rim.

Four cards in the Flywheel Energy Storage Calculator then answer the practical questions. They cover how much of the energy can be drawn off, the inertia and shaft torque, the stress in the rotor with its safety factor, and how long the flywheel can feed a load. A stored figure alone says little without those.

How to Calculate Energy Stored in a Flywheel

A flywheel stores rotational kinetic energy. The flywheel energy equation is half the moment of inertia times the angular velocity squared, and the inertia depends on the mass, the radius and the shape.

$$E = \frac{1}{2} I \omega^2$$

$$I = k m r^2$$

Here E is in joules, I is in kg·m², ω is in radians per second, m is in kilograms and r is in meters. The factor k is 0.5 for a solid disk and 1 for a thin rim. These are the same formulas that Sandia National Laboratories uses in its Energy Storage Handbook chapter on flywheels.

For the default, 6,000 RPM is 628.3 rad/s. The inertia is 0.5 × 50 × 0.3², or 2.25 kg·m². The energy is ½ × 2.25 × 628.3², which is 444,132 J. Dividing by 3,600 turns joules into watt-hours and gives 123.4 Wh.

Speed matters far more than mass. Doubling the mass doubles the energy, but doubling the speed gives four times as much. The Energy at Other Speeds table in the Flywheel Energy Storage Calculator shows this for your rotor.

The same sum works in a spreadsheet. In Excel, with k, mass, radius and RPM in cells A1 to D1, the formula for watt-hours is =0.5*A1*B1*C1^2*(D1*2*PI()/60)^2/3600. Entering 0.5, 50, 0.3 and 6000 returns 123.37.

Flywheel Shape and the Factor k

The shape decides how much of the mass sits far from the axis, where it moves fastest. A thin rim puts all of it at the edge, so it stores twice the energy of a solid disk of the same mass, radius and speed.

For a hollow cylinder the factor is ½ × (1 + (r₁ ÷ r)²), where r₁ is the inner radius. The extra energy has a price. The rim and the bored cylinder run at more than twice the stress of the solid disk, so the solid disk can spin faster before it fails.

A second k appears in flywheel texts, and the two are often mixed up. The stress shape factor is the share of a material’s strength-to-density ratio that a shape can turn into stored energy.

Sandia lists it as 0.606 for a disk and 0.5 for a thin-walled cylinder. It is not an inertia factor, and using 0.606 in I = k m r² overstates a solid disk’s energy by 21%.

The Flywheel Energy Storage Calculator keeps the two apart. The shape list sets the inertia factor, and the Max Specific Energy row applies the stress factor. A Custom Shape Factor choice accepts any inertia factor up to 1 for a spoked or tapered wheel.

Usable Energy and Runtime

A flywheel cannot be run down to a standstill in service. The motor-generator needs speed to deliver power, so the rotor works between full speed and a lowest usable speed. The energy between those two speeds is what counts.

$$\Delta E = \frac{1}{2} I \left( \omega_1^2 – \omega_2^2 \right)$$

This is also the formula for the maximum fluctuation of energy in a flywheel. With the lowest speed at 50% of full, three quarters of the energy is usable. The default gives 92.53 Wh out of 123.4 Wh. In the Flywheel Energy Storage Calculator, set Lowest Usable Speed under Load, Material, Units to match your drive.

Runtime is the usable energy divided by the load power. The default feeds a 5 kW load for 66.62 s. The Discharge Efficiency field starts at 100%, and lowering it to 90% for motor-generator and drive losses cuts the runtime to 59.96 s. That short figure is typical. Flywheels deliver high power for seconds to minutes, which suits bridging a power cut until a generator starts.

The same card shows why. At 2.467 Wh/kg, the default rotor holds far less energy per kilogram than a battery. Its strength is that it can give that energy up, and take it back, very fast and a great many times.

Flywheel Torque and Acceleration

The second card of the Flywheel Energy Storage Calculator gives the moment of inertia, the spin-up time and the peak shaft torque. Torque at a given power is the power divided by the angular velocity.

$$\tau = \frac{P}{\omega}$$

For 5 kW at 6,000 RPM that is 7.958 N·m. Torque rises as the rotor slows under a constant load, so at the 50% lowest speed the same 5 kW needs 15.92 N·m. The card shows that peak figure, because it is the one that sizes the shaft and the motor-generator.

Spin-Up is the time to charge the rotor from rest at constant power, which is the stored energy divided by the power. The default takes 88.83 s at 5 kW. It is longer than the 66.62 s runtime because charging has to supply all 123.4 Wh, while discharge only draws the usable 92.53 Wh.

To reach speed in a set time instead, work from torque. The torque needed is the inertia times the angular acceleration. Taking the default rotor from rest to 6,000 RPM in 60 s needs 10.47 rad/s² and 2.25 × 10.47, or 23.56 N·m. The Angular Acceleration Calculator works from torque and inertia directly.

The step-by-step panel also lists the angular momentum, 1,414 kg·m²/s for the default. That is why a spinning flywheel resists being tilted. The Angular Momentum Calculator covers that quantity, and the Angular Velocity Calculator converts RPM to rad/s.

Rim Speed, Stress and Burst Speed

Spinning stretches the rotor. The stress depends on the material density and the square of the rim speed, not on the mass or the size alone. For a solid disk the peak is at the center, and the third card reports it with the rim speed.

$$\sigma = \frac{3 + \nu}{8} \rho v^2$$

Here ρ is the density, v is the rim speed and ν is Poisson’s ratio, taken as 0.3. The default rim moves at 188.5 m/s, and the stress in steel of 7,850 kg/m³ is 0.4125 × 7,850 × 188.5², or 115.1 MPa. Against a 500 MPa strength, the safety factor is 4.346.

For a thin rim the stress is simply ρ v², which is 278.9 MPa at the same rim speed. Burst speed is where the stress reaches the strength. The default disk would reach it at 12,508 RPM, a little over twice its running speed. A rotor is run well below that.

Material sets the ceiling. Sandia gives maximum surface speeds of 200 to 400 m/s for high-performance steel and 500 to 1,000 m/s for carbon composite. Composite rotors store more per kilogram because they are both stronger and lighter, and because energy rises with the square of that speed.

$$\frac{E}{m} = K \frac{\sigma}{\rho}$$

This is the limit the Max Specific Energy row of the Flywheel Energy Storage Calculator reports, with K of 0.606 for a disk. For 500 MPa steel it is 10.72 Wh/kg. The default runs at 2.467 Wh/kg, less than a quarter of that. The 500 MPa and 7,850 kg/m³ entries are sample values for steel, so replace them with the figures for your own material.

Speed, Mass or Radius for a Target Energy

The three reverse choices in the Flywheel Energy Storage Calculator size a flywheel for a job. Each takes a target energy, 250 Wh by default, in watt-hours, kilowatt-hours, joules or foot-pounds. The flywheel mass calculation is the one most often needed.

Check the stress card after each one. Reaching 250 Wh by speed alone takes the default disk to 8,541 RPM, which cuts the safety factor to about 2.1. Adding mass keeps the rim speed and the stress where they were, at the cost of a rotor twice as heavy.

How a Flywheel Stores Energy in Practice

The working principle is simple. A motor spins the rotor up to store energy, and the same machine runs as a generator to take it back, slowing the rotor. The energy is mechanical, held as rotational kinetic energy, with no chemical change.

Flywheel energy storage is real and in service. Sandia describes a 20 MW grid plant built from 200 units. Each unit stores 25 kWh and delivers up to 100 kW from a 2,000 lb composite rotor with a surface speed up to 600 m/s. Together they hold 5 MWh, which is 15 minutes at full power.

Grid-scale flywheel energy storage is used for frequency regulation, where power must swing in and out within milliseconds. Other uses are backup power for data centers, recovering braking energy at metro stations and launching roller coasters. High-speed rotors run in a vacuum on low-friction bearings to cut drag.

Rotors come in two broad types. Steel rotors are cheaper per kilogram and run at lower surface speeds. Carbon composite rotors run faster and store more per kilogram. The same Sandia chapter puts rotor material cost near $1,200 per kWh for carbon composite and $4,000 per kWh for 600 MPa steel.

So flywheel energy storage is expensive per unit of energy. Sandia notes that battery production costs are below $100 per kWh. Flywheels compete on power and life instead. The chapter gives a cycle life above ten million cycles, where few developers forecast less than one million.

The disadvantages follow from the numbers above. Energy per kilogram is low, the cost per kWh is high, and no long-duration use is commercially viable yet. A rotor failure also releases its energy at once, so safety rests on a wide stress margin. That is the reason to watch the safety factor in the results.

A bicycle wheel shows the scale at the other end. A 1.3 kg wheel of 0.356 m radius, treated as a thin rim at 60 RPM, stores 3.25 J. Energy in a straight line follows the same half-times-mass-times-speed-squared pattern, which the Kinetic Energy Calculator covers. More rotation tools are on the Rotational Motion Calculators hub.