= Discharge measurement in a draining tank
{title2=$C_d=Q/(a\sqrt{2gh})$}
<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.<Draining a Tank by James F. Price>[James F. Price, Draining a Tank, §§2.1–2.3, printed pp. 9–12: preparation, volume/time readout and repeatability]{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.<Draining a Tank by James F. Price>[James F. Price, Draining a Tank, eqs. (1)–(2), Fig. 2, §3.1 eq. (13), §§4 and 5: coefficient inference and departures from ideal flow]{ref}
A useful independent test of the constant-coefficient model avoids numerical <derivatives> of noisy head readings. Its exact <square root> relation predicts
$$
\sqrt{h(t)}=\sqrt H-bt,\qquad
b=\frac{C_da}{A}\sqrt{\frac g2},\qquad
C_d=\frac{A}{a}\sqrt{\frac2g}\,b.
$$
Fit 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.
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