A Stirling engine is a heat engine that is operated by the cyclic compression and expansion of air or other gas (the working fluid) between different temperatures, resulting in a net conversion of heat energy to mechanical work. In this example, that mechanical work is converted to electricity.

Let's calculate the torque of a Stirling engine - a homemade model kit with a flywheel

# Torque of a Stirling engine with flywheel

Video of a Stirling engine

## Flywheel data

- mass = 0.02 kg (20 g)
- radius = 0.0275 m (2.75 cm)
- RPM = 1200
- time = 1 sec

# Flywheel Torque

We need the flywheel mass (kg), the radius (m), and the angular acceleration (rad/s^2)

\(T_{fw} = \frac{1}{2} * m * r^{2} * \alpha} [Nm]\)

\(T_{fw}\) = \frac{1}{2} * 0.02 kg * 0.00075625 m2 * 125.663706144 rad / sec) = 0.000015125 / 2 / 125.663706144 rad / sec2 = 0.000950395 [Nm]

Or dissecting the formula.

The moment of inertia of the disc is:

\(I = \frac{1}{2} * m * r^{2}\)
\(I =\)0.5 * 0.02 kg * 0.00075625 m2 = 0.000007563 kg m2

\(T_{fw} = I * \alpha [Nm]\)
\(T_{fw} = 0.000007563 kg m2 * 125.663706144 rad / sec2 = 0.000950395 [Nm]\)

## Nett inertia of the flywheel for rotation (acceleration)

\(\alpha = \frac{\Delta \omega }{\Delta t}\)
\(\alpha\) = 125.663706144 [rad] \ 1 sec = 125.663706144 rad / sec2

### Angle difference for rotation

\(\Delta \omega = \omega_{1} - \omega_{0}\)
\(\Delta \omega\) = 125.663706144 [rad] - 0 [rad] = 125.663706144 [rad/s]

### Start angle of rotation

\(\omega_{0}\) = 0 [rad]

### End angle of rotation

\(\omega_{1} = 1200 RPM = 1200 RPM * 2\pi * \frac{1}{60} sec\)
\(\omega_{1}\) = 125.663706144 [rad]

# Power of a Stirling engine with flywheel

Usually you can convert that torque and RPM to watts, which could give you a rough estimate power output of your engine.
You can multiply the amount of torque in Newton-meters (Nm) by the rotational speed (rad/sec) in order to find the power (watts).
Although it does not account for losses and the such.

A basic formula would be:

\(\P_{W} = \(T_{fw} * speed [rad/s] = 0.000950395 [Nm] * 125.663706144 [rad/s] = 0.1194301580007269 [W]\)

Now, if we want to generate electricity using a brushed motor for electric generator (dynamo), our engine outputs 0.119 W and we can choose a motor of 0.119 W or a bit less and start from there.