The Emptying Bucket: How Falling Head Drifts a Gravity Drip Kit's Flow, the Uniformity Bound It Implies, and the Constant-Head Fix Sized to Beat It

Bucket drip kits — a raised bucket feeding low-pressure drip laterals by gravity — are sold as complete irrigation for a small plot, and their one moving part is the water level itself. As the bucket empties, the head driving every emitter falls, and emitter flow falls with it. This paper computes that drift for a stated kit: a 20-litre bucket (0.30 m diameter) on a 1.00 m stand, feeding two 12 m laterals with 60 emitters in total over 12.0 m², with emitter law q = 0.6325 h^0.5 litres per hour (2.0 litres per hour at 10 m head). Head falls from 1.283 m when full to 1.014 m at a 1-litre residual — a swing of 0.269 m — and emitter flow falls from 0.716 to 0.637 litres per hour, a decline of 11.1 percent across a single bucketful, which takes 28.1 minutes to drain. Friction along the laterals, by contrast, is negligible at these flows: head loss is 0.00114 m and the first-to-last emitter flow ratio is 0.9996, so position along the line costs almost nothing while time in the run costs 11.1 percent. Combining space and time, the application-uniformity bound — worst emitter discharge over best — is 0.8887, which fails the stated criterion of 0.90: stated plainly, the plain bucket kit does not meet it. A farmer who times the run by the clock at the full-bucket rate under-delivers by a further 5.2 percent. The paper then sizes the fix: for flow to stay within ±5 percent with an emitter exponent of 0.5, head must be held within a band 0.230 m wide (1.038 to 1.268 m about a 1.15 m working head), which the bucket's own 0.269 m swing exceeds by 0.039 m; a Mariotte bottle or float-valve header holding 1.15 m delivers 0.678 litres per hour per emitter, 40.7 litres per hour in total, restores the uniformity bound to 0.9996, and must be reservoir-sized by the rule V ≥ 1.25 × Q × t — 150.0 litres of reservoir for a 10 mm irrigation of the stated plot, applied in 2.95 hours, or seven refills of a 19-litre Mariotte vessel. The study is simulated, not measured.