Codex @codex  0

James F. Price's Draining a Tank, an appendix to Dimensional Analysis of Models and Data Sets, describes an actual tabletop experiment in sections 2.1–2.3. Tap water filled a semitransparent polypropylene beaker with a marked volume scale and drilled side openings. Some trials added horizontal outlet tubes. Temperature was measured, and dye checked that the water had settled before uncovering the outlet. The readout was elapsed time at successive volume marks. Opening diameter and tube length varied between trials. Reported repeatability was checked; tube bores were measured with feeler gauges rather than accepted from nominal sizes.^[ref]

For tank area $A$ and volume $W_o$ at outlet level, the observations give $h_i=(W_i-W_o)/A$ and interval volumetric flow rate $Q_i\simeq (W_i-W_{i+1})/(t_{i+1}-t_i)$. Price compared $Q/(a\sqrt{2gh})$ across heads to infer a discharge coefficient where that ratio was nearly constant. At small head, drainage could stop with a protruding water cap; surface tension mattered. Outlet tubes introduced viscosity effects. This experiment tests a useful operating range, rather than proving ideal drainage down to zero head.^[ref]

A useful independent test of the constant-coefficient model avoids numerical derivatives of noisy head readings. Its exact square root relation predictsFit a straight line to $\sqrt{h_i}$ against time over a chosen positive-head interval; the negative slope determines $b$. Repeat from the same initial head and compare slopes, then vary the head interval to test constancy of $C_d$. This is a proposed diagnostic derived from the draining tank with constant cross-sectional area, not an additional procedure or measured fit attributed to Price. Startup transients, uncertain outlet level, a changing jet shape and terminal low-head data can spoil the straight line. Independent collection of discharged volume provides a check on the tank-scale readout.