The Rope Pump Depth Bound: Slip, Washer Spacing, and the Depth at Which Output Falls Away

A rope pump lifts water on an endless rope carrying washers through a rising main, and it is sometimes described as though depth were a secondary consideration. This study models a rope pump under stated assumptions, with a rising main of inner diameter 0.040 m, washers of diameter 0.038 m, a rope speed of 0.70 m/s, and washer spacing of 0.20 m, giving an ideal displacement of 3.167 m^3/h. Slip is modelled as a lumped flow under the full lift head through an effective leak area derived from the annular gap, and operator power is limited to a sustainable 80 W at an overall efficiency of 0.65. Delivered flow is 2.924 m^3/h at a lift of 1 m, defined here as the surface output, and 2.625 m^3/h at a lift of 5 m, after which the power limit governs: delivered flow falls to 1.908 m^3/h at a lift of 10 m, 1.272 m^3/h at a lift of 15 m, and 0.954 m^3/h at a lift of 20 m. Delivered flow reaches 0.75 of surface output at a depth of 8.7 m, one half of surface output at a depth of 13.05 m, and one quarter of surface output at a depth of 26.1 m. Holding the ideal flow at a lift of 20 m would require 265.5 W, far beyond the assumed sustainable power, which disproves the proposition that the device as specified works equally at any depth. A sizing rule is originated that joins a spacing condition, a diameter condition, and a depth condition, and worked values are given: a target of 20.0 L/min requires a diameter of at least 24.6 mm by the diameter condition and has a governing depth bound of 15.9 m, while a target of 30.0 L/min requires at least 30.2 mm and has a governing bound of 10.6 m.