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Discharge over a sharp crested notch or weir

 

A notch is an opening in the side of a measuring tank or reservoir extending above the free surface. It is, in effect, a large orifice which has no upper edge, so that it has a variable area depending upon the level of the free
surface. A weir is a notch on a large scale, used, for example, to measure the flow of a river, and may be sharp-edged or have a substantial breadth in the direction of flow.

The method of determining the theoretical flow through a notch is the same as that adopted for the large orifice.

Sharp crested weirs (or notches) are generally used to measure the discharge in small open channels where accuracy is required. Because such weirs can be accurate to ±2% or even ±l %, they are often used as measuring devices in hydraulics laboratories, but they also have practical applications. For instance, the seepage through a dam may be measured by channelling it over a sharp crested weir. Because the weirs have thin, sharp crests they are not suitable for measuring the discharge in large rivers where they would be prone to damage by the impact of floating debris. Concrete structures like the broad crested weir are used for this, and they operate on a totally different principle from those described here. The two types of weir should not be confused.

A sharp crested weir is usually formed from a sheet of metal which will not rust. A notch is then cut out of the plate, the shape of which defines the geometry of the weir  (See diagrams below).

 
Common shapes for sharp crested weirs (see diagram 1) are rectangular, triangular and another shape is trapezoidal (Cipolletti Weir). The weir must have an accurately finished square upstream edge, a crest width of less than 2mm with a bevel on the downstream side, (see diagram 2). With prolonged use the crest may become worn and rounded, and this can adversely affect the accuracy.

The weir plate must be installed with the upstream face vertical. Normally the length of the weir crest is less than the width of the channel (b < B) so, in plan, the flow has to contract to pass through it. Similarly, the crest is usually set above the bottom of the channel, so the streamlines have to rise upwards to pass over the weir (See diagram below). The crest, or sill, of the weir has to be high enough for the water to fall freely into the downstream channel, so that the flow over the weir is not affected by the downstream water level. The water
flowing over the crest, that is the nappe, should spring clear of the downstream face of the weir plate. This is the 'free' condition in which both the upper and lower surfaces of the nappe are exposed to the atmosphere.  This enables a convenient assumption to be made, that is the pressure distribution throughout the nappe is close to atmospheric. 

If the kind of weir plate in diagram below is used, then the free condition should exist naturally at all but the smallest of discharges. However, if the weir crest spans the full width of the channel so that b = B, this is called the suppressed condition. This maximises the discharge over the weir for a particular channel width, which may be desirable in some circumstances, but means that the air under the nappe is now trapped. Gradually the air becomes entrained  in the nappe and is carried away. This leaves air at low pressure under the nappe, which enables the backwater to rise. When most or all of the air has been removed, the nappe collapses and adheres to the face of the weir plate, forming a clinging nappe (as in diagram 3 below).
This is undesirable because the weir will not function as an accurate measuring device like this. To prevent the formation of a clinging nappe with the suppressed condition, it is necessary to provide an air vent or pipe to admit air to the underside of the nappe. 

If the discharge in a relatively wide channel is to be measured, then more than one weir plate may be used. In this case the individual weir plates would be attached to vertical posts.
It may then be desirable to set the weir crests at different heights, forming a compound weir. This is often done to improve accuracy: when the discharge is small all the water passes over only the lowest crest, but as the flow increases the other weirs come into use. 

The relationship that is always sought with a weir is between the head, H, over the weir crest and the discharge, Q. Note that H is always the head above the weir crest (not the total depth of water in the channel). Note also that because the weir has a smaller cross-sectional area of flow than the approach channel, the continuity equation Q = AV dictates that the velocity over the weir crest must be higher than the velocity in the approach channel. This increase in velocity means that the velocity head increases so, assuming that the total energy line is horizontal, the water surface falls towards the weir as the flow accelerates.
Consequently the head over the weir is usually measured some distance upstream where the velocity head is still relatively small. A distance of at least 4 x Hmax is desirable, where Hmax is the maximum head that will occur over the weir. The head is usually measured in a stilling well or tube which is located to one side of the channel and connected to it by a pipe. 

 

Derivation of the discharge equation for a rectangular weir 

The equation is derived in the same way as that for a large rectangular orifice, this simply being the situation where the water surface has fallen below the top of the opening. The basis of the method is to apply the energy equation (Bernoulli) to two points on a streamline, point 1 being on the water surface some distance upstream of the weir, and point 2 being in the nappe as it passes over the weir crest at a depth, h, below the water surface. There are a number of assumptions that should be listed in connection with the derivation, since they are of significance later on. They are:

(i)             That the water discharges over the weir from the surface of a large reservoir, so it can be assumed that the velocity of approach is negligible and the pressure is atmospheric. In other words, V₁ = 0 and P₁ = 0.

 

(ii)            That the nappe is at atmospheric pressure. Thus if atmospheric pressure is used as a datum, P₂ = 0.

 

(iii)          There are no energy losses.

 

(iv)          The velocity in the nappe varies with depth, h, that is V = (2gh)^1/2 but there is no variation in velocity across the

                 length, b, of the weir crest.

 

(v)            The nappe is as wide as the weir crest, that is it also has a length, b.

 

(vi)           The streamlines are horizontal as they pass over the weir crest.

 

If a thin horizontal strip of length, b, and thickness, δh, is taken across the nappe at a depth, h, see diagram below, then:

Area of the strip, δA = bδh

Velocity of flow through the strip = (2gh)^1/2
Discharge through the strip, δQ = area x velocity

                                                            = b(2gh)^1/2δh

 

 

To determine the total theoretical discharge, QT, the above expression must be integrated to obtain the sum of all the horizontal strips covering the entire depth of the nappe as defined by the limits h = 0 and h = H. Note that b and g are both constants.

QT = b(2g)^1/2

 
QT =b(2g)^1/2H^3/2

 The 2/3 arises from the integration. To obtain the actual discharge Q_A, a coefficient of discharge, C_D, is introduced so that: