An aeolipile, or Hero’s steam engine, turns through the reaction of steam leaving bent outlets on a hollow rotor. The two jets act at opposite sides of the rotor but produce torque in the same rotational direction. This interactive diagram illustrates that principle and estimates ideal momentum-flux torque from entered flow and velocity. Its angle penalty is an illustrative rule, not a physical prediction of nozzle losses.
Aeolipile or Hero's Steam Engine Interactive Calculator
Compare two-jet momentum flux and torque from entered flow, velocity and radius. The angle control applies an illustrative penalty, not a measured nozzle-loss law.
Equation Used
The first force output is twice the entered per-nozzle mass flow times jet velocity. The angle control applies an illustrative penalty of 1.5 percentage points per degree, capped at 95%. This penalty is not the geometric projection of a nozzle force: that projection varies with cos(angle). The adjusted force and torque therefore describe this assumed penalty only, not measured steam-engine performance.
- Mass flow is entered per nozzle and the aeolipile has two matched nozzles.
- Nozzles are intended to be tangent to the rotation circle.
- Angle penalty is an illustrative rule: 1.5 percentage points per degree, capped at 95%. It does not represent a physical nozzle-loss law.
- Bearing friction, windage, pressure drop, and acceleration inertia are not included.
How the jets turn the rotor
Steam reaches the hollow rotor and leaves through bent outlets. The jets carry angular momentum away, producing a reaction torque on the rotor. The drawing shows an external supply through a hollow pivot, rather than water being heated inside the rotating globe.
Equal, opposed jet forces can cancel as translational forces while their moments add. If both outlets discharge in the same rotational sense, equal flow strengthens the driving torque; it does not make the rotor stop. Reversing one outlet can oppose the other outlet’s torque.
Main components
- Rotor: the hollow rotating body containing the steam passages.
- Outlets: bent passages directing the jets around the axis.
- Pivots and frame: support and locate the rotor.
- Steam connection: carries the working fluid from the fixed supply.
The illustration does not specify a pressure vessel, nozzle bore, wall thickness, material, bearing tolerance or heating arrangement. Those require a separate engineering design; no safe pressure or construction dimension follows from this torque calculation.
Reaction torque made visible
Aeolipiles are useful historical and teaching examples of jet reaction. The Kotsanas Museum presents Hero’s mechanism as an early steam-engine concept. Seeing the outlet directions and resulting rotation helps distinguish force from moment about an axis.
An aeolipile can in principle turn an attached load if its driving torque exceeds resistance. The mechanism name alone does not determine efficiency, useful power or speed. This diagram is an explanation of the principle, not a design for a boiler or a rated power source.
What the calculator computes
Mass flow is entered in grams per second per nozzle and converted to kilograms per second. With two equal nozzles, the ideal momentum-flux force sum is F = 2ṁv. Taking the nozzle radius as half the entered sphere diameter gives ideal torque T = 2ṁvr. This assumes tangential discharge and neglects exit-pressure thrust and the effect of rotor motion on the flow.
The angle control applies p = min(95, max(0, 1.5α)) in percent, then calculates adjusted force F(1 − p/100) and adjusted torque Fr(1 − p/100). It is an illustrative penalty rule. The 95% cap leaves 5% of ideal force even at large entered angles; it is not a physical residual-thrust prediction.
For a fixed nozzle force inclined by α from the tangent, the actual geometric torque component is Fr cos α. At 10°, this component is about 98.481% of its tangential value, a reduction of approximately 1.519%. That differs from the calculator’s assumed 15% penalty at the same angle.
A more complete jet-force model includes momentum and exit-pressure terms; a rotating engine additionally needs angular-momentum analysis. Torque alone cannot predict rpm. Inertia determines acceleration, while steady speed depends on how driving torque and resistance vary with speed. The animation uses illustrative rotation rather than solving that balance.
Worked example: flow and an assumed penalty
Use hypothetical inputs of 0.15 g/s per nozzle, jet velocity 180 m/s and sphere diameter 70 mm. These are calculation inputs, not a construction or operating specification. Convert the flow to 0.00015 kg/s and the assumed radius to 0.035 m.
The two-jet momentum-flux sum is 2 × 0.00015 × 180 = 0.054 N. With zero angle penalty, torque is 0.054 × 0.035 = 0.00189 N·m, or 1.89 mN·m.
At a 10° angle input, this calculator assigns a 15% penalty. Adjusted force is 0.054 × 0.85 = 0.0459 N and adjusted torque is 1.6065 mN·m. These are the results of the stated illustrative rule.
For comparison, projecting the same fixed force geometrically would give 1.89 cos(10°) ≈ 1.8613 mN·m. Neither calculation predicts steam pressure, nozzle dimensions, heating power, run time or rotor rpm.
Comparing steam mechanisms
| Mechanism | How motion is produced | What determines performance |
|---|---|---|
| Aeolipile | Jet reaction on the rotating body. | Flow, rotor speed, geometry, losses and load. |
| Reciprocating steam engine | Pressure on a piston connected to a crank. | Pressure cycle, valve events, friction and load. |
| Steam turbine | Momentum and pressure changes through bladed stages. | Steam conditions, stage design, losses and operating point. |
There is no universal rpm, efficiency, service life or cost for any row. Those depend on the particular engine and its operating conditions.
Aeolipile questions
The forces act at opposite positions. With matching discharge directions around the axis, both reaction moments have the same rotational sign. Equal jets may cancel net translational force while adding torque.
Not from direction alone. The component of a fixed force varies with cosine of the angle; 10 degrees gives about 1.519% reduction. The angle penalty in this calculator is an illustrative rule, not that physical projection.
No. Rotor inertia, speed-dependent flow, bearing resistance, windage and applied load are needed to determine acceleration and steady speed.
In principle yes, if the driving torque can overcome the applied resistance. This calculator does not establish a power rating or thermal efficiency.
No. It accepts flow, velocity and a radius approximation as inputs. It does not solve the steam supply, pressure vessel or nozzle design.
The calculator assumes two equal nozzles and multiplies the entered single-nozzle momentum flux by two. It does not calculate how a boiler’s output divides when nozzle geometry changes.
It shows opposed discharge directions and reaction-driven rotation. The illustrated rotation is not a prediction of the speed, pressure or temperature of a working apparatus.
References and further reading
Kotsanas Museum: The Aeolipile of Heron illustrates the historical mechanism. NASA Glenn: General Thrust Equation describes momentum-flux and pressure contributions to jet thrust. Neither reference establishes the illustrative angle-penalty rule used in this calculator.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
