Flumes: Theory & Application

Flumes are open-channel flow sections that force flow to accelerate. Acceleration is produced by converging the sidewalls, raising the bottom, or a combination of both. When only the bottom is raised with no side contractions, the flume is commonly called a broad-crested weir. When the downstream depth is shallow and enough convergence exists between the upstream and downstream channels, the flow passes through critical depth. Therefore, flumes are sometimes called critical-depth flumes. When flow passes through critical depth, a unique water surface profile occurs within the flume or broad-crested weir for each discharge. This condition is known as free flow. For this case, upstream heads at one location relative to the control bottom elevation near the region of critical depth can be used to determine a usable head versus discharge relationship for flow measurement.

Flume head loss is less than about one-fourth of that needed to operate a sharp-crested weir having the same control width, and in some long-throated flumes, may be as low as one-tenth. Another advantage compared to most standard weirs is that for a properly designed and installed flume, the velocity of approach is a part of the calibration equations. Unauthorized altering of the dimensions of constructed flumes to obtain an unfair share of water is difficult and, therefore, not likely. Velocity of flow can usually be designed to minimize sediment deposition within the structure. Gradual convergence sections at the entrance tend to improve velocity distribution of approach flow and the passage of floating debris. Some flumes can be more expensive than sharp-crested weirs or submerged orifices in unlined channels.

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Selecting a Primary Measuring Device: Weir vs. Flume

The selection of a primary measuring device for a particular flow measurement installation usually involves a series of three decisions:

§ Which primary measuring device to use: a weir or a flume?

§ Which specific type of primary device to use?

§ What is the exact size of the primary device to be installed at the location in question?

Weirs and flumes each have decided advantages and disadvantages. A weir is the simplest device that can be used to measure flow in open channels. It is low in cost, relatively easy to install, and quite accurate when properly used. However, it normally operates with a rather significant loss in head, and its accuracy can be affected by variations in the approach velocity of the liquid in the flow channel. Moreover, it must also be periodically cleaned to prevent deposits of sediment or solids in the upstream side of the wear, which will adversely affect its accuracy.

On the other hand, a flume tends to be self-cleaning since the velocity of flow through it is high and there is no obstruction across the channel. It can also operate with a much smaller head loss than a weir, which can be important for many for many applications where the available head is limited. In addition, a flume is relatively less sensitive to varying approach velocity. Nonetheless, a fume is much more costly than a weir, and its installation is more difficult and time consuming. Flumes are also generally less accurate that weirs.

It should be noted that the initial installation cost is not the only expense that should be considered when choosing a primary device. As mentioned earlier, flumes tend to be self-cleaning, whereas weirs must be periodically cleaned. Lower maintenance costs associated with a flume may eventually outweigh the higher initial cost.

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Critical Flow Theory

In reference to J. A. Robertson & C. T. Crowe, “Engineering Fluid Mechanics – 6^th Edition”, 1997

q Energy Equation Applied to Open-Channel Flow

The one-dimensional energy equation for open channels (see Fig. 1-a) is:

(1)

It’s evident from Fig. 1-a that the following equalities hold:

Where S₀is the slope of the channel bottom, and y is the depth of flow. Assuming α₁= α₂= 1, we can write Eq. (1) as:

(2)

Considering the special case where the channel bottom is horizontal (S₀= 0) and head loss is negligible (h_L= 0), Eq. (2) becomes:

(3)

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q Specific Energy

The sum of the depth of flow and the velocity head is defined as specific energy:

(4)

Thus Eq. (3) states that the specific energy at section 1 is equal to the specific energy at section 2. The continuity equation between section 1 and 2 is:

(5)

Accordingly, Eq. (3) can be expressed as:

(6)

Since A₁& A₂are both functions of the depth y, the magnitude of the specific energy at the section 1 or 2 is solely a function of the depth at each section. If, for a given channel and given discharge, depth versus specific energy is plotted, a relationship such as that shown in Fig. 1-b is obtained.

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For a given value of specific energy, we can see that the depth may be either large or small. Physically, this means that for the small depth, the bulk of the flow energy is in the form of kinetic energy (Q²/2gA²); whereas for a larger depth, most of the energy is in the form of potential energy.

Flow under a sluice gate (Fig. 1-c) is an example of flow in which two depths occur for a given value of specific energy. The large depth and low kinetic energy occurs upstream of the gate, while the low depth and large kinetic energy occurs downstream. The depths are called alternate depths.

If we maintain the same rate of flow but set the gate with a larger opening, the upstream depth will drop, and the downstream depth will rise. Thus we have different alternate depths and a smaller value of specific energy than before. This is consistent with the diagram in Fig. 1-b.

Finally, it can be seen in Fig. 1-b that a point will be reached where the specific energy is minimum and only a single depth occurs. At this point, the flow is termed critical. Thus one definition of critical flow is the flow that occurs when the specific energy is minimal for a given discharge. The flow for which the depth is less than critical is (velocity is greater than critical) is termed supercritical flow, and the flow for which the flow is greater than critical is termed subcritical flow. In Fig. 1-c, subcritical flow occurs upstream and supercritical flow occurs downstream of the sluice gate. (It should be noted that some engineers refer to subcritical and supercritical flow as tranquil and rapid flow.)

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q Characteristic of Critical Flow

It has been proven that critical flow occurs when the specific energy is minimum for a given discharge. The depth for this condition may be determined if we solve for dE/dy from E = y + Q²/2gA²and set dE/dy equal to zero:

(7)

However, dA = Tdy, where T is the width of the channel at the water surface. Then Eq. (7), with dE/dy = 0, will reduce to:

(8,9)

Introducing the hydraulic depth D, which is defined as A/T, then Eq. (9) becomes:

(10,11)

Upon dividing Eq. (11) by D_c and taking the square root of it, we get

(12)

In open channel hydraulics, the Froude number is a very important nondimensional parameter. The Froude number, Fr, is the ratio of inertia force to gravity force, which simplifies to:

(The Froude number is equal to unity when critical flow prevails. If the Froude number is less than unity, the flow is subcritical and has a low velocity, which is often described as tranquil and streaming. On the other hand, if the Froude number is greater than unity, the flow is supercritical and has a high velocity, which is usually described as rapid, shooting, and torrential.)

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If a channel is of rectangular cross section (as in the case of a Parshall Flume), then A/T is the actual depth, and Q²A² = q²/y², so the formula for critical depth, Eq. (9), becomes:

(13)

Were q is the discharge per unit width of the channel.

Critical flow may also be examined in terms of how the discharge in a channel varies with depth for a given specific energy. For example, consider in a rectangular channel where

If we consider a unit width of the channel and let q = Q/B, then the above equation becomes

If we determine how q varies with y for a constant value of specific energy, we see that the critical flow occurs when the discharge is maximum (see Fig. 1-d)

Originally, the term critical flow probably related to the unstable character of the flow for this condition. Referring to Fig. 1-b, we notice that only a slight change in specific energy will cause the depth to increase or decrease a significant amount; this is a very unstable condition. In fact, observations of critical flow in open channels show that the water surface consists of a series of standing waves. Because of the unstable nature of the depth in critical flow, designing canals so that normal depth is either well above or well below critical depth is usually best. The flow in canals and rivers is usually subcritical; however, the flow in steep chutes or over spillways is supercritical.

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Theory Applied to Flumes

Flumes are designed to force a transition from sub-critical to super-critical flow. The transition is caused by designing flumes to have a narrowing at the throat, raising of the channel bottom, or both. Such a transition causes flow to pass through critical depth at the flume throat. At the critical depth, energy is minimized and there is a direct relationship between water depth and flow rate. However, it is physically very difficult to measure critical depth in a flume because its exact location is difficult to determine and may vary with flow rate. Through mass conversion, the upstream depth is related to the critical depth. Therefore, flow rate can be determined by measuring the upstream depth, which is a highly reliable measurement.

q General Discharge Vs. Head Formula

This equation and the specific energy equation are the basic critical flow relationships for any channel shape.

In order to derive the basic form of the discharge equation for various flumes, assume that we have a channel of rectangular cross section. As mentioned earlier, all flumes are designed with the intent of producing critical depth in the flume throat and thereby creating a unique depth-discharge relationship. If this critical depth is measured, the rate of flow can easily be computed from:

where B is the width of the channel normal to the flow direction.

Because y_c/2 = v²_c/(2g), it is easily shown thaty_c= 2/3E, where again, E is the total head, y + v²_approach/(2g); hence the preceding equation can be written as:

Had the velocity of approach been neglected, the equation can be rewritten as:

If the height of the flume is relatively small, then the velocity of approach may be significant and the discharge produced will be greater than that given by the preceding equation. To account for these effects, a discharge coefficient C is defined as:

Then

(14)

This sums up the basic derivation of the discharge vs. head formula for most flumes. However, extensive experimental research has been done and most flumes have a published set of empirical formulas that are function of channel widths or other key dimensions. In the case of the Parshall flume, for example, different flume widths (with the corresponding different dimensions) had their discharge vs. head results plotted and fitted to power functions with K and n as the variable parameters. Replacing the constant parts of Eq. (14) by K (which also encompasses the throat width), and n as the head power (a value that circulates around 1.5) the general formula becomes:

The various contributing variables to this formula are the velocity of approach, channel width, non-horizontal channel bottom, and the desired units. This generalized equation is to be elaborately discussed in the to the Parshall flume section of this paper.

q Submergence

All flumes have a minimum needed head loss to assure that free flow exists and that only an upstream head measurement is needed to determine discharge rate. This required head loss is usually expressed as a submergence limit defined by the ratio of the downstream head to the upstream head, both referenced to the flume throat bottom. The term "modular limit" is defined as this limiting submergence ratio for a particular flow module, which causes no more than a 1% deviation in the upstream head reading for a given discharge. When these limits are exceeded, an additional downstream head measurement is sometimes used to extend the measurement range of a flume, particularly for Parshall and cutthroat flumes, but at considerable loss of accuracy. Submergence also increases upstream channel depth, decreasing the upstream velocity, which may aggravate sedimentation problems.

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Flume Classes

As mentioned earlier, the two basic classes or forms of flumes are long-throated flumes and short-throated flumes; they are discussed in depth below:

q Long-Throated Flumes:

Long-throated flumes (Fig. 2-a) control discharge rate in a throat that is long enough to cause nearly parallel flow lines in the region of flow control. Parallel flow allows these flumes to be accurately rated by analysis using fluid flow concepts (energy principle, critical depth relationships, and boundary layer theory). Long-throated flumes can have nearly any desired cross-sectional shape and can be custom fitted into most canal-site geometries. The modified broad-crested