Power-Law Head-Discharge Rating

Parshall flumes and many other measuring devices are rated by fitting a curve to calibration pairs of head and measured discharge. When the points fall on a straight line on log-log paper, the curve is a power law, $Q = C h^{n}$, or in logarithms $\log Q = n \log h + \log C$. The slope of the line is the exponent $n$ and its intercept gives the coefficient $C$. Once you have them, the head read on the device gives you the discharge.

Each term has a plain meaning. Set $h = 1$ m and $Q = C$ whatever $n$ is, so on this page $C$ is the discharge in cubic metres per second at a head of one metre. The head is measured above the zero-flow level of the device, the crest or the throat floor. The exponent belongs to the device and is found from the fit: near 1.5 for a flume and 2.5 for a V-notch weir. The page's example sets $C = 2.5$ and $n = 1.8$ at a head of 1 m and returns 2.5 m³/s, which is $C$ itself.

The US Bureau of Reclamation's Water Measurement Manual publishes such ratings for Parshall flumes in its Table 8-6, as $Q = C h^{n}$ in cubic feet per second and feet (equation 8-3). This page takes $C$ in cubic metres per second at a head of one metre, so a manual coefficient is converted before it is entered. The 3-inch flume, with the manual's $C = 0.992$, $n = 1.55$ and a head of 0.20 ft, gives 0.082 ft³/s, as the manual quotes. The 20-foot flume at 3.25 ft gives about 503 ft³/s, and read backwards, 503 ft³/s returns a head of 3.25 ft. For the 50-foot flume at 5.7 ft the manual gives 3,000 ft³/s to two figures, and the equation gives 3,027.

The point that matters most in using a fitted rating is how to judge it. The manual's Chapter 3 gives the percent deviation of the discharge, the equation against the measured flows (equation 3-9), and that, not the correlation coefficient of the logarithms, is the test: a log correlation of 0.999 can sit on top of deviations of 10 percent in discharge. A line can therefore look excellent on log-log paper and still misstate the flow.