The Leaking Pump Delivers Nothing: A Leak-Area Bound for Loss of Delivery in a Piston Hand Pump

A piston hand pump with a worn cup seal and a leaking foot valve loses part of every stroke back down the rising main, and beyond a definite leak area it loses the whole stroke. This study models net delivery as swept volume minus leak-back, under stated assumptions for a pump of piston diameter 0.075 m and stroke length 0.35 m, giving a swept volume of 1.546 L per stroke. At the baseline condition of a lift of 10 m and a stroking rate of 40 strokes per minute, net delivery reaches zero when the total leak area reaches 113.2 mm^2, which is 2.56 percent of the piston area of 4417.9 mm^2 and corresponds to an equivalent annular gap of 0.481 mm around the piston. The bound tightens with depth and with slower stroking: at a lift of 25 m and the same stroking rate the zero-delivery leak area is 71.6 mm^2, or 1.62 percent, while at a lift of 10 m and a slower rate of 20 strokes per minute it is 56.6 mm^2, or 1.28 percent, with an equivalent gap of 0.24 mm. The slower-stroking effect is exhibited directly: with a fixed leak of 1.0 percent at a lift of 10 m, slowing from 40 to 20 strokes per minute reduces net delivery per stroke from 0.943 L to 0.34 L and delivered flow from 37.72 L/min to 6.79 L/min, because each stroke leaks for longer. A maintenance rule is originated in measurable terms: replace the cup seal when the strokes needed to fill a 20 L vessel reach 20 strokes, corresponding to a delivery fraction of 0.647 and, at the baseline condition, an implied leak area of 40.0 mm^2, or 0.91 percent, with an equivalent gap of 0.17 mm.