A lever turns a central pinion between two opposed racks connected to air-pump pistons. One piston rises while the other falls, so their intake and delivery strokes alternate. The calculator reports swept volume and theoretical displacement rate per cylinder, with a separate illustrative loss allowance.
Air-pump Motion Interactive Calculator
Calculate swept volume and theoretical displacement rate per cylinder from bore, stroke and cycle rate. A separate assumed seal-gap curve shows how a chosen loss allowance changes the displayed flow; it is not a measured leakage model.
Equation Used
Swept volume is piston area times effective stroke. Multiplying it by complete cycles per minute gives theoretical displacement rate per cylinder. The displayed net flow applies an illustrative loss curve based only on the seal-gap input. That curve does not predict leakage for a particular seal or operating condition.
- Stroke is the effective piston stroke after lever or rack geometry.
- One discharge event occurs per reciprocating stroke cycle.
- The loss allowance is an illustrative function of entered gap; pressure, valve behavior and seal geometry are not inputs.
- At a gap input of 0.05 mm, the assumed loss allowance is 15%; this is a comparison assumption, not a universal seal-leakage value.
One pinion drives two opposed racks
The lever rocks the central pinion. Teeth on opposite sides engage two vertical racks, each attached to a piston rod. Rotation of the pinion raises one rack and lowers the other by an equal distance.
Each piston changes the volume inside its cylinder. Inlet and delivery valves establish the flow direction: the inlet admits air as chamber volume increases, and the delivery valve passes air outward during compression once the required pressure is reached.
The animation shows the alternating strokes and keeps the racks connected to the pinion. Bore and full stroke follow the inputs; guides, teeth and other proportions are illustrative. The calculation is per cylinder even though the drawing shows two cylinders.
Parts to follow
- Lever and pinion: provide the rocking input.
- Opposed racks: produce equal and opposite linear motion.
- Pistons and cylinders: define swept volume.
- Inlet and delivery valves: direct the air flow.
Comparing displacement and alternating delivery
The opposed-rack arrangement is useful for studying how one input coordinates two reciprocating pistons. Their opposite phases distribute the delivery strokes through the cycle.
Use bore, effective stroke and cycle rate to compare geometric displacement. A working pump additionally needs valve, seal, pressure, drive-load and thermal data. The model does not establish the performance of a toy, medical device or animatronic installation.
Geometric flow and the assumed loss curve
For bore D and full stroke S in millimetres, swept volume per cylinder is V = πD²S/4000 millilitres. With N complete cycles per minute and one delivery stroke per cycle, theoretical displacement rate is Qgross = VN/1000 litres per minute.
The displayed net value is Qgross multiplied by the assumed efficiency fraction. The illustrative gap curve uses a 3% loss for gap c up to 0.02 mm, rises linearly to 15% at 0.05 mm, and then rises by 20 percentage points per additional 0.07 mm, capped at 45% loss. Efficiency is 100% minus this allowance.
This curve is not a general leakage law. It ignores pressure difference, seal type, passage length, gas state, clearance volume and valve dynamics. Actual delivered air flow requires specified measurement conditions and pump data. Changing cycle rate in this calculator does not change its assumed efficiency.
Worked per-cylinder comparison
For a 25 mm bore and 30 mm full stroke, V = π × 25² × 30 / 4000 = 14.726 mL per delivery stroke. At 60 complete cycles per minute, the theoretical displacement rate is 0.884 L/min per cylinder.
At a 0.05 mm gap input, the illustrative curve applies a 15% loss allowance, or 85% efficiency. The resulting comparison value is 0.884 × 0.85 = 0.751 L/min per cylinder.
Doubling cycle rate doubles both displayed flow values with this unchanged allowance. Two identical cylinders would have twice the combined theoretical displacement rate, but actual combined delivery and pulsation depend on valve action, pressure and pipework. These numbers do not establish a relief setting, outlet size or bellows movement.
What the mechanism and calculation establish
The rack geometry coordinates piston displacement. It does not determine pressure rating, drive torque, seal life, noise or delivery efficiency. A comparison with a diaphragm or rotary pump needs the specifications of the actual equipment and required duty.
The geometric swept volume remains useful even when the loss allowance is unsuitable for a real pump. Use measured or manufacturer-specified flow data for operating decisions; do not treat the gap slider as a validated way to set running clearances.
Lever-and-rack pump questions
Are the flow results for one cylinder or both?
They are per cylinder, with one discharge event per complete reciprocating cycle. The animation shows two opposed cylinders.
Does seal gap alone determine actual leakage?
No. Pressure, seal geometry, gas conditions, clearance volume and valve behavior also matter. The gap curve is an illustrative assumption rather than a measured pump model.
Why does the assumed efficiency not change with speed?
The implemented allowance depends only on the entered gap. Speed multiplies displacement rate; speed-dependent valve and leakage effects are outside this calculation.
Does gross flow equal delivered air at discharge pressure?
Not necessarily. Gross flow is swept volume times cycle rate. A delivered flow rating needs defined pressure and temperature conditions and pump performance data.
What does stroke mean here?
It is the effective full piston travel after the lever and rack geometry. The formula uses that distance directly to calculate displaced volume.
Does opposite phasing guarantee smooth flow?
No. Opposite piston strokes alternate delivery, but valve opening, compressibility and downstream volume determine the resulting pulsation. Those effects are not calculated.
References & Further Reading
- Wikipedia contributors. Reciprocating pump. Wikipedia
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