A compound beam pumping engine uses high- and low-pressure steam cylinders coupled to a rocking beam. The beam transmits motion to pumping equipment and, in a rotative arrangement, a crank and flywheel. This illustration shows inclined steam cylinders and a separate single-acting water plunger. The calculator estimates water displacement and average delivery from the pump dimensions.
Compound Beam Pumping Engine Interactive Calculator
Explore a compound beam engine driving one single-acting water plunger. Enter the water-pump diameter, full stroke, complete cycles per minute and estimated volumetric efficiency. The beam and rods move through their pinned joints while the pump proportions follow your dimensions.
Equation Used
- One single-acting water plunger, one delivery stroke per full cycle.
- Constant entered volumetric efficiency estimates average delivery; no instantaneous leakage or valve-flow model.
- US liquid gallons, not Imperial gallons.
- Schematic inclined compound steam cylinders and flywheel. Steam valve timing and thermodynamic performance are not modeled.
- Beam and connecting rods retain fixed lengths. Pump dimensions set the bore-to-stroke ratio; the rest of the illustration is not dimensioned.
Single-acting water-side displacement estimate. N is complete pump cycles per minute, with one delivery stroke per cycle. The compound steam engine is schematic; no steam-side performance is calculated.
Following the engine and pump motion
The overhead beam rocks on a fixed central bearing. Connecting rods link it to the two inclined steam-cylinder sliders, the rotating crank and the vertical pump crosshead. The flywheel and crank complete one revolution during each full pump cycle.
The water-side cutaway shows a plunger entering and withdrawing from a chamber. On entry it displaces water toward the delivery valve; on withdrawal the inlet opens to refill the chamber. This is a single-acting representation with one delivery stroke per complete cycle.
Compounding concerns the steam side: steam leaving the high-pressure stage is used by a lower-pressure stage. The display identifies the two cylinders but does not simulate steam admission, expansion or exhaust timing. Hiscox’s Figure 504 provides the reference for the inclined-cylinder arrangement; the illustrated proportions and slider links are schematic rather than a reconstruction of that engine.
Compound beam engines in pumping service
Historical compound beam engines were used for waterworks pumping. Their steam-cylinder arrangement, pumping equipment and valve systems varied. The inclined arrangement shown in Hiscox leaves room for the pump and crank connections.
The present calculation describes one single-acting water plunger. A double-acting pump, multiple plungers or a differential pump needs a different effective displacement calculation. Historical gallon ratings also need their gallon convention checked before comparison.
Water delivery from pump dimensions
For plunger diameter d in inches and full stroke S in feet, the displaced volume per delivery stroke is V = π(d/12)²S/4 cubic feet. Multiply by 7.48051948 to express V in US liquid gallons.
With N complete cycles per minute and one delivery stroke per cycle, Q = VNηv gives average US gallons per minute. The entered volumetric efficiency ηv is a fraction. Daily delivery at uninterrupted operation is 1440Q, and mean absolute rod speed is 2SN feet per minute because the rod travels one stroke in each direction.
These inputs do not establish pressure head, required steam power, fuel use, suction performance, valve capacity or a safe operating speed. The efficiency input is an assumption, not a result calculated from the mechanism.
Example using the calculator defaults
A 36-inch water plunger has a 3-foot diameter and an area of approximately 7.069 square feet. With an 8-foot stroke, its swept volume is about 56.549 cubic feet, or 423.0 US gallons per delivery stroke.
At 10 complete cycles per minute and an assumed 90% volumetric efficiency, delivery is approximately 3,807 US gallons per minute, or 5.48 million US gallons per uninterrupted day. The mean absolute rod speed is 2 × 8 × 10 = 160 feet per minute. These are displacement estimates for the entered pump, not historical performance claims.
Interpreting the estimate
Diameter changes displacement with its square. Stroke and cycle rate change average displacement linearly. Reducing the assumed volumetric efficiency reduces calculated delivery without changing the rigid linkage motion.
Whether an actual installation can deliver the calculated flow depends on the available power, pumping head, water supply, valve behavior and mechanical condition. Operating limits and maintenance intervals depend on the particular installation.
Questions about the beam-pump calculation
Does one cycle mean one upstroke or an up-and-down pair?
It means a complete up-and-down pair and one full crank revolution. The single-acting pump delivers once during that cycle.
Does the calculator apply to a double-acting pump?
No. Both working sides and any rod-area deduction must be included separately for a double-acting design.
Why does efficiency change the flow but not the animation stroke?
Efficiency adjusts the estimate of delivered water volume. It does not change the entered geometry or rigid linkage motion.
Are the gallons US or Imperial?
They are US liquid gallons. Historical specifications may use a different convention.
Is the displayed rod speed its peak speed?
No. It is the total distance traveled per minute, averaged over the complete cycle. Instantaneous rod speed varies with crank position.
References
- Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, printed page 139, Figure 504: inclined high- and low-pressure cylinders in a compound beam pumping engine.
- Smithsonian: Model of McNaught Type Compound Steam Engine, explanation of compounding and an alternative beam-cylinder arrangement.
The calculator’s water-volume equations follow from circular area and swept distance. Its illustration is not a dimensional model of the museum engine.
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