Remember the Nilometer? (Figure 1) It's the first known level meter, simply a huge staff gauge (ruler-like device mounted at the measurement point), that was used to measure the flood flow of the Nile River for thousands of years. What does it do? It measures the level of the Nile River, so planting season could be planned. With calculation, measured level can be used to derive gravity flow in open channels, even ones like the Nile.
In 1889, Irish engineer Robert Manning produced an equation for calculating flow in open conduits from the level in the channel.
V = Rh2/3 • S1/2
Q = VA, where
Q = Volumetric flow
A is the cross-sectional area of the wetted perimeter,
V is the cross-sectional average velocity,
k is the units conversion factor,
n is the coefficient of friction,
Rh is the hydraulic radius,
S is the slope of the channel.
Manning's equation works best in man-made conduits because the hydraulic radius, slope and coefficient of friction can be better known or estimated. It will work in any open channel, but the error expands as these terms are less well known. Manning's equation, and its successors, are often used for calculating the flow in sewers and in man-made irrigation channels.
Although tables of Manning's "n" are common, it is nearly impossible to accurately calculate "n," so it is almost always an educated guess, based on the engineer's experience and expertise. A small change in "n" can result in a large change in "V" and thus a large change in "Q."
For real accuracy, the use of a primary device is required. These are restrictive devices placed in the flow stream that raise the height of the water behind the restriction to a predictable level based on volumetric flow. Typically, these are flumes and weirs.
Flumes and Weirs
Early in the last century, hydraulic engineers discovered that there was a relationship between the head height behind a constriction in the channel and the flow rate through the constriction. This relationship takes the form of:
Q = KHn
where Q is volumetric flow, K is a units constant, H is the head height, and n is a coefficient that is related to the size and type of constriction.
These constrictions can be divided into two basic groups, flumes and weirs. There are various types of each.
Weirs have been known for centuries, and were used as primitive spillways from small earthen dams. Measurement weirs come in several configurations: v-notch, rectangular and trapezoidal, among others (Figure2). The U.S. Dept. of the Interior's Bureau of Reclamation publishes an online version of its comprehensive "Water Measurement Manual," which details all the various types of weirs and flumes, and how to install them and maintain them.
In a v-notch weir, all the water flows through the weir, and the level is measured at a known point behind (upstream) of the weir. This level forms the "H" in our formula.
Sometimes, if very high accuracy is required at the lowest flow rates, a compound weir can be used. This is often a rectangular weir with a sharp v-notch forming the bottom of the weir. Thus, the rectangular weir equation can be used for high flows, and once the flow reduces to below the zero point of the rectangular weir, the v-notch equation is used. Many open channel flowmeters have the ability to switch equations like this.