An aeolipile, also called Hero’s engine, rotates through the reaction of escaping steam jets. Two bent outlets on opposite sides of a hollow rotor direct their jets in the same rotational sense, so their reaction torques add. Explore how force, radius and nozzle direction affect this torque; the animation does not predict the speed of a physical steam engine.
Aeolipile Interactive Calculator
Compare reaction torque from two equal jets, the effect of nozzle direction and an entered resisting torque. Radial-component sums are not a net bearing-load calculation.
Equation Used
The aeolipile torque calculation uses the article relationship tau = 2 x F x r, because two opposed tangential steam jets both add torque about the pivot. Nozzle angle error changes tangential force by cos(alpha). The radial-component card adds component magnitudes under the illustrated symmetric error convention; it does not calculate the vector sum of forces on the bearings.
- Two opposed nozzles produce additive torque.
- Nozzle angle alpha is the error away from perfectly tangential flow.
- Jet force is per nozzle.
- Pivot drag is subtracted from generated torque.
Why the sphere turns
Steam enters the rotor and leaves through bent outlets. The outward-moving fluid carries angular momentum, and the resulting reaction acts on the rotor. Opposed outlets can exert forces in opposite spatial directions while producing torque in the same rotational direction.
For a given force and distance from the axis, the tangential component creates torque about that axis. A radial component has no moment about the axis. With perfectly opposed equal jets, radial components can cancel as external forces even though each outlet and its support carry local loads.
Principal parts
- Rotor or globe: contains the steam passages and turns about its axis.
- Bent outlets: direct discharge to produce reaction torque.
- Supports and pivots: locate the rotor and permit rotation.
- Steam supply: provides the working fluid through the appropriate connection.
The drawing shows a fixed steam supply feeding through a hollow pivot. Its dimensions, vessel construction and heating arrangement are schematic. The calculator does not specify a boiler, vessel wall, nozzle bore or operating pressure.
A visible example of jet reaction
Hero’s aeolipile is useful for explaining reaction torque and the relationship between force direction and rotation. Museum reconstructions, including the Kotsanas Museum’s presentation, make the historical principle easier to see.
A rotating aeolipile can in principle deliver shaft work to a suitable load. That does not establish a practical power rating or a universal efficiency. Steam generation, exhaust losses, rotor speed, friction and the applied load must be included to assess performance.
The interactive model is a force-and-torque illustration. It does not provide construction or pressure-setting instructions for a working steam apparatus.
Two-jet torque calculation
Let F be reaction force per nozzle, r the distance from the rotation axis and α the angle away from a tangential direction. With two equal nozzles whose torques add, ideal tangential torque is 2Fr and the direction-adjusted value is 2Fr cos α. Use r in meters and F in newtons for N·m; multiply by 1,000 for mN·m.
Net Torque is the direction-adjusted torque minus the entered resisting torque. A zero or negative result means the entered jets do not overcome that resistance in this simplified comparison. It does not imply that friction drives the rotor backward. The animation holds the rotor still in this condition.
Radial Sum is 2F sin α under the illustrated common error convention. It describes added radial component values, not the vector resultant on the rotor or bearing. Opposed symmetric components may cancel.
The input F is supplied directly. Estimating it from steam conditions requires a flow model, including momentum flux and any exit-pressure contribution. The simplified F = mass-flow rate × exit velocity relation is insufficient when its neglected terms matter; rotor motion also affects a complete angular-momentum analysis.
Torque alone does not determine rpm. Rotor inertia governs acceleration, and steady speed requires the speed-dependent driving and resisting torques to balance. Power at a specified speed is torque times angular speed; neither speed nor thermal efficiency is solved here.
Worked example: direction and resistance
Take two equal reaction forces of 0.40 N each at radius 40 mm, or 0.040 m. With tangential jets, ideal torque is 2 × 0.40 × 0.040 = 0.032 N·m, or 32 mN·m.
At a 10° nozzle error, direction-adjusted torque is 32 cos(10°) ≈ 31.5138 mN·m. With entered drag 5 mN·m, Net Torque is approximately 26.5138 mN·m. The directional reduction is about 1.519% of ideal torque, not a fixed percentage per degree.
The radial-component sum is 2 × 0.40 × sin(10°) ≈ 0.1389 N. It is not a prediction of net bearing load. If entered resistance rises to 35 mN·m, the torque comparison becomes approximately −3.4862 mN·m, and the animation stops the rotor.
The assumed nozzle forces do not establish a boiler pressure, exhaust velocity, safe vessel design, run time or steady rpm.
Different ways to turn steam energy into motion
| Mechanism | Motion-producing principle | What a performance model needs |
|---|---|---|
| Aeolipile | Reaction from outlets on the rotating body. | Flow, rotor motion, losses and load. |
| Impulse turbine | A jet transfers momentum to rotor blades. | Nozzle expansion, velocity triangles, blade losses and operating conditions. |
| Reciprocating steam engine | Pressure acts on a piston linked to a crank. | Pressure cycle, valve events, expansion, friction and load. |
Efficiency and useful power cannot be assigned from these mechanism names alone. The simple aeolipile calculation compares torque components rather than ranking complete power systems.
Aeolipile questions
Their forces act at opposite positions. When the discharge directions have the same rotational sense, the reaction moments about the axis have the same sign even if the net translational force cancels.
No. In this fixed-force model, torque changes by cos α. A 10-degree error reduces it by about 1.519%; a 5-degree error reduces it by about 0.381%.
No. The output adds radial component values. A bearing reaction requires vector equilibrium and all applied forces, including their directions and any asymmetry.
No. It accepts nozzle force and resisting torque as inputs. Steam thermodynamics, flow, inertia and speed-dependent losses are not solved.
The entered resistance exceeds the calculated jet torque. Here that means no sustained forward rotation from the specified balance, rather than friction causing reverse rotation.
No. That depends on the supply and resulting pressure and flow conditions. This calculator assumes exactly two nozzles and does not solve how the supply responds to a changed outlet area.
Not generally. Nozzle geometry, pressure ratio, heat transfer, phase behavior and losses determine the result. The force input should come from an appropriate analysis or measurement, not a universal efficiency factor.
References and further reading
Kotsanas Museum: The Aeolipile of Heron presents the historical mechanism. NASA Glenn: General Thrust Equation explains the momentum and pressure terms relevant when estimating jet reaction. A rotating aeolipile additionally requires angular-momentum analysis for performance prediction.
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