This double-acting differential pump uses a lower bucket piston with a check valve and a smaller solid upper plunger connected to it. Both move through the same stroke. Their different swept areas make the delivery chamber receive liquid on both the upward and downward movements.
Double-acting Differential Pump Interactive Calculator
Compare the delivery of the two strokes in Hiscox’s differential pump: a valved lower bucket and a smaller upper solid plunger move together. Enter both diameters and their common stroke. Equal delivery requires the upper plunger area to be half the lower bore area.
Equation Used
- Upper plunger diameter is strictly smaller than lower bucket bore.
- Both pistons are rigidly connected and have the same stroke.
- Incompressible liquid, ideal check valves, no leakage or trapped gas in working spaces.
- The connecting rod has equal area in the adjoining chambers, so it cancels from the ideal volume balance.
- Reported volumes are geometric displacement into the delivery chamber, not a time-resolved outlet-flow prediction.
This is the two-piston differential arrangement in Hiscox 512, not an ordinary four-check-valve double-acting cylinder. The smaller dimension is the upper plunger diameter, not the thin connecting rod diameter. Invalid diameter combinations remain visible but are not assigned usable delivery volumes.
Two pistons, one connecting rod
Hiscox’s mechanism 512 combines a lifting-pump lower section with a solid piston in a sleeve descending from the cover. The surrounding upper casing acts as an air chamber. The cutaway exposes the lower bucket, upper plunger, open sleeve and surrounding air chamber.
On the upstroke, the lower inlet check opens and the bucket valve closes. The bucket raises liquid into the delivery chamber. At the same time, the upper plunger withdraws in its sleeve and creates space below itself. The net volume available for delivery is the difference between the two swept volumes.
On the downstroke, the inlet check closes and the bucket valve opens. Liquid below the bucket transfers through it. The descending upper plunger displaces liquid from its sleeve into the common delivery chamber.
The animation shows the lower inlet and bucket valves separately, with flow arrows that follow their operating states. The centre rod moves both pistons through the same distance. The upper sleeve remains open to the atmosphere above its piston.
Compare the two stroke deliveries
The useful calculation is an area and volume balance. It shows whether the two strokes deliver similar volumes or whether one dominates. Equal volume per stroke does not mean constant instantaneous flow: both pistons stop at their reversal points.
An air cushion can moderate delivery variations, but predicting its response needs pressure, gas volume and downstream flow conditions. Those inputs and dynamics are not part of this calculator.
Differential swept volume
Let A be the lower bore area and a the upper solid plunger area. The upstroke delivers (A−a)S and the downstroke delivers aS. Their sum is AS, not the (2A−a)S formula for an ordinary four-valve double-acting pump.
The thin connecting rod occupies equal area in the adjoining spaces. Its effects cancel in this ideal volume balance; it is not the upper-plunger diameter input.
Equal stroke volumes require A−a=a, or a=A/2. For circular sections this gives d=D/√2. The reported imbalance is the absolute difference between the stroke volumes divided by their sum: 100|A−2a|/A.
The requirement 0<d<D ensures both modeled stroke deliveries are positive. An upper plunger equal to or larger than the lower bore is shown as an invalid area pair.
An 80 mm bucket bore
For D=80 mm, the lower area is approximately 5,026.55 mm². A 100 mm stroke gives 502.65 mL total displacement per complete cycle.
An upper plunger of 80/√2≈56.569 mm gives half that volume on each stroke, approximately 251.33 mL. The slider’s 56.6 mm default is a rounded approximation, so a small nonzero imbalance is expected.
Doubling the stroke doubles both delivered volumes without changing their percentage imbalance. Changing the upper diameter redistributes the two stroke volumes while leaving their sum unchanged, provided the area pair remains valid.
What this drawing does and does not predict
The cutaway distinguishes the upper atmospheric sleeve, surrounding air cushion, common delivery chamber, lower barrel and two one-way valves. Bore widths follow the entered diameters; axial stroke is scaled separately to keep the mechanism readable. The picture is not a manufacturing drawing.
The display does not assign a pressure rating, suction-lift limit, valve clearance, leakage allowance or air-vessel size. It also does not make the outlet flow constant merely because the pump displaces liquid on both strokes.
Differential pump questions
Why are there only two check valves?
The lower bucket itself contains one check; the other is at the inlet. The upper plunger connects directly to the delivery chamber.
Is d the connecting rod diameter?
No. It is the diameter of the upper solid pumping plunger.
Why is full-cycle delivery AS?
The two contributions are (A−a)S and aS. Adding them cancels a.
What diameter balances delivery?
d=D/√2 gives equal areas a and A−a.
Does balanced delivery eliminate pulsation?
No. Instantaneous displacement rate still changes through each stroke and reaches zero at reversal.
Reference
Hiscox, Mechanical Movements, Powers, Devices and Appliances, mechanism 512, printed page 141 (PDF page 151): lower bucket, connected upper solid piston, open sleeve and surrounding air chamber.
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