22.3. VARIABLE-AREA FLOWMETERS

regime exhibits laminar behavior. The large number of small-diameter tubes packed in the wide area of the element provide adequate wall surface area for the fluid’s viscosity to act upon, creating an overall pressure drop from inlet to outlet which is measured by the di erential pressure transmitter. This pressure drop is permanent (no recovery of pressure downstream) because the mechanism of pressure drop is friction: total dissipation (loss) of energy in the form of heat.

Another common form of laminar flow element is simply a coiled capillary tube: a long tube with a very small inside diameter. The small inside diameter of such a tube makes wall-boundary e ects dominant, such that the flow regime will remain laminar over a wide range of flow rates. The extremely restrictive nature of a capillary tube, of course, limits the use of such flow elements to very low flow rates such as those encountered in the sampling networks of certain analytical instruments.

A unique advantage of the laminar flowmeter is its linear relationship between flow rate and developed pressure drop. It is the only pressure-based flow measurement device for filled pipes that exhibits a linear pressure/flow relationship. This means no “square-root” characterization is necessary to obtain linear flow measurements with a laminar flowmeter. The big disadvantage of this meter type is its dependence on fluid viscosity, which in turn is strongly influenced by fluid temperature. Thus, all laminar flowmeters require temperature compensation in order to derive accurate measurements, and some even use temperature control systems to force the fluid’s temperature to be constant as it moves through the element31.

Laminar flow elements find their widest application inside pneumatic instruments, where a linear pressure/flow relationship is highly advantageous (behaving like a “resistor” for instrument air flow) and the viscosity of the fluid (instrument air) is relatively constant. Pneumatic controllers, for instance, use laminar restrictors as part of the derivative and integral calculation modules, the combination of “resistance” from the restrictor and “capacitance” from volume chambers forming a sort of pneumatic time-constant (τ ) network.

22.3Variable-area flowmeters

An variable-area flowmeter is one where the fluid must pass through a restriction whose area increases with flow rate. This stands in contrast to flowmeters such as orifice plates and venturi tubes where the cross-sectional area of the flow element remains fixed.

31This includes elaborate oil-bath systems where the laminar flow element is submerged in a temperature-controlled oil bath, the purpose of which is to hold temperature inside the laminar element constant despite sudden changes in the measured fluid’s temperature.

22.3.2Weirs and flumes

A very di erent style of variable-area flowmeter is used extensively to measure flow rate through open channels, such as irrigation ditches. If an obstruction is placed within a channel, any liquid flowing through the channel must rise on the upstream side of the obstruction. By measuring this liquid level rise, it is possible to infer the rate of liquid flow past the obstruction.

The first form of open-channel flowmeter is the weir, which is nothing more than a dam obstructing passage of liquid through the channel. Three styles of weir are shown in the following illustration; the rectangular, Cippoletti, and V-notch:

A rectangular weir has a notch of simple rectangular shape, as the name implies. A Cippoletti weir is much like a rectangular weir, except that the vertical sides of the notch have a 4:1 slope (rise of 4, run of 1; approximately a 14 degree angle from vertical). A V-notch weir has a triangular notch, customarily measuring either 60 or 90 degrees.

The following photograph shows water flowing through a Cippoletti weir made of 1/4 inch steel plate:

At a condition of zero flow through the channel, the liquid level will be at or below the crest (lowest point on the opening) of the weir. As liquid begins to flow through the channel, it must spill over the crest of the weir in order to get past the weir and continue downstream in the channel. In order for this to happen, the level of the liquid upstream of the weir must rise above the weir’s crest height. This height of liquid upstream of the weir represents a hydrostatic pressure, much the same as liquid heights in piezometer tubes represent pressures in a liquid flowstream through an enclosed pipe (see section 2.11.15 for examples of this). The height of liquid above the crest of a weir is

analogous to the pressure di erential generated by an orifice plate. As liquid flow is increased even more, a greater pressure (head) will be generated upstream of the weir, forcing the liquid level to rise. This e ectively increases the cross-sectional area of the weir’s “throat” as a taller stream of liquid exits the notch of the weir33.

Greater level upstream of the weir

33Orifice plates are variable-pressure, constant-area flowmeters. Rotameters are constant-pressure, variable-area

flowmeters. Weirs are variable-pressure, variable-area flowmeters. As one might expect, the mathematical functions describing each of these flowmeter types is unique!

A variation on the theme of a weir is another open-channel device called a flume. If weirs may be thought of as open-channel orifice plates, then flumes may be thought of as open-channel venturi tubes:

Like weirs, flumes generate upstream liquid level height changes indicative of flow rate. One of the most common flume design is the Parshall flume, named after its inventor R.L. Parshall when it was developed in the year 1920.

The following formulae relate head (upstream liquid height) to flow rate for free-flowing Parshall flumes34:

Where,

Q = Volumetric flow rate (cubic feet per second – CFS)

L = Width of flume throat (feet)

H = Head (feet)

Flumes are generally less accurate than weirs, but they do enjoy the advantage of being inherently self-cleaning. If the liquid stream being measured is drainageor waste-water, a substantial amount of solid debris may be present in the flow that could cause repeated clogging problems for weirs. In such applications, flumes are often the more practical flow element for the task (and more accurate

34It is also possible to operate a Parshall flume in fully submerged mode, where liquid level must be measured at both the upstream and throat sections of the flume. Correction factors must be applied to these equations if the flume is submerged.

over the long term as well, since even the finest weir will not register accurately once fouled by debris).

Once a weir or flume has been installed in an open channel to measure the flow of liquid, some method must be employed to sense upstream liquid level and translate this level measurement into a flow measurement. Perhaps the most common technology for weir/flume level sensing is ultrasonic (see section 20.5.1 beginning on page 1472 for more information on how this technology works). Ultrasonic level sensors are completely non-contact, which means they cannot become fouled by the process liquid (or debris in the process liquid). However, they may be “fooled” by foam or debris floating on top of the liquid, as well as waves on the liquid surface.

The following photograph shows a Parshall flume measuring e uent flow from a municipal sewage treatment plant, with an ultrasonic transducer mounted above the middle of the flume to detect water level flowing through:

Once the liquid level is successfully measured, a computing device is used to translate that level measurement into a suitable flow measurement (and in some cases even integrate that flow measurement with respect to time to arrive at a value for total liquid volume passed through the

R

element, in accordance with the calculus relationship V = Q dt + C).

A technique for providing a clean and “quiet” (still) liquid surface to measure the level of is called a stilling well. This is an open-top chamber connected to the weir/flume channel by a pipe, so the liquid level in the stilling well matches the liquid level in the channel. The following illustration shows a stilling well connected to a weir/flume channel, with the direction of liquid flow in the

channel being perpendicular to the page (i.e. either coming toward your eyes or going away from your eyes):

FT

ultrasonic

Water in weir/flume channel

To discourage plugging of the passageway connecting the stilling well to the channel, a small flow rate of clean water may be introduced into the well. This forms a constant purge flow into the channel, flushing out debris that might otherwise find its way into the connecting passageway to plug it up. Note how the purge water enters the stilling well through a submerged tube, so it does not cause splashing on the water’s surface inside the well which could cause measurement problems for the ultrasonic sensor:

FT

ultrasonic

Valve

Water

supply

Water in weir/flume channel

Sediment flushed out of passageway

A significant advantage that weirs and flumes have over other forms of flow measurement is exceptionally high rangeability: the ability to measure very wide ranges of flow with a modest pressure (height) span. Another way to state this is to say that the accuracy of a weir or flume is quite high even at low flow rates.

Earlier in this section you saw a three-image representation of liquid flow through a rectangular weir. As fluid flow rate increased, so did the height (head) of the liquid upstream of the weir:

Effective notch area

Effective notch area

The height of liquid upstream of the weir depends on the flow rate (volumetric Q or mass W ) as well as the e ective area of the notch through which the fluid must pass. Unlike an orifice plate, this area changes with flow rate in both weirs and flumes. One way to envision this by comparison is to imagine a weir as acting like an elastic orifice plate, whose bore area increases with flow rate. This flow-dependent notch area exhibited by both weirs and flumes means that these devices become more sensitive to changes in flow as the flow rate becomes smaller.

A comparison of transfer function graphs for closed-pipe head elements such as orifice plates and venturi tubes versus weirs and flumes shows this striking di erence in characteristics:

Looking at the orifice plate / venturi tube graph near the lower-left corner, you can see how small changes in flow result in extremely small changes in head (di erential pressure), because the