AJP/0600

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H6Flow over a Notch

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CONTENTSSection Page

1 INTRODUCTION 1

2 DESCRIPTION OF THE APPARATUS 3Description 3Installation and Preparation 3Routine Care and Maintenance 3

3 THEORY OF FLOW OVER SHARP-EDGED WEIRS 5

4 ADDITIONAL NOTCHES AND ASSOCIATED THEORY 7

5 EXPERIMENTS AND RESULTS 9Results and Calculations 9Rectangular Notch - Typical Calculation 10V-Notch – Typical Calculations (θθθθ = 15°°°°) 10Discussion of Results 10Questions for Further Discussion 11

TQ Flow Over a Notch

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SECTION 1 INTRODUCTION

In hydraulic engineering, weirs are commonly used toregulate flow in rivers and other open channels. In somecases the relationship between the water level upstreamof the weir and the discharge over it is known, so thatthe discharge at any time may be found by observing theupstream water level. Figure 1 shows two differentshapes of weir, one formed by cutting a rectangularnotch, the other by cutting a V-shaped notch, in verticalplates. Such notches usually have sharp edges so thatthe water springs clear of the plate as it passes throughthe notch.

Figure 1 Rectangular and V-notches

It is the purpose of these experiments to deriverelationships between the head on the weir anddischarge for both rectangular and V-shaped notches.Two V-shaped notches are supplied with angle θ of 15°and 45° respectively.

TQ Flow Over a Notch

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SECTION 2 DESCRIPTION OF THE APPARATUS

DescriptionFigure 2 shows the arrangement in which water fromthe bench supply valve feeds through a flexible hose toa pipe which serves to distribute the water fairly evenlyin the enlarged end of the tank. A contraction sectionleads the water to a short channel, into which either therectangular or V-notched weir plates may be fitted.

Water which flows over the notch is collected in theexit tank, the outlet of which leads, via a drain port, tothe weigh tanks of the Hydraulic Bench.

Figure 2 Arrangements of apparatus for measuringover weirs

The water level in the short approach channel may bemeasured with the height gauge placed in the recesseson the top edge of the tank. Zero the gauge by looseningthe nut on the top of the gauge rack.

Installation and Preparation1. Check for transit damage and thoroughly clean all

traces of packing material.2. Assembly the plastic inlet pipe assembly and the

pad of rubber sponge into the tank as shown. (Thesponge pad acts as a ‘water settler’. See Figure 2).

3. Place the depth gauge in the recesses on the topflanges of the tank.

4. Place the apparatus onto the Hydraulic Bench topwith the widest (inlet) end to the rear of the benchand the discharge over the bench weigh tank.

5. Connect the Hydraulic Bench supply hose to theapparatus.

6. Apply silicon compound (supplied) to the sides,lower corners and bottom edges of the plate to befitted into the channel. Fit the plate into the groove,ensure it is properly seated and smear the siliconcompound around the join, adding more ifnecessary, to give a full seal. Remove any excess.

7. Connect the Hydraulic Bench to a suitable electricalsupply. (Refer to the electrical nameplate mountedon the contractor box).

Routine Care and MaintenanceAfter use, the apparatus should be drained of all waterand dried with a lint-free cloth. The glass fibre tank hasa white pigmented interior and can easily be cleanedwith any good quality car body cleaner and polished.

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SECTION 3 THEORY OF FLOW OVER SHARP-EDGED WEIRS

Figure 3

Figure 3 indicates the essential features of flow overrectangular or V-notches. Consider the motion of aparticle of fluid from a position M some distance up-stream of the weir to its subsequent position N in theplane of the vertical weir plate. If there is no energyloss, according to Bernoulli’s equation:

N

2N

2N

M

2M

2M

22z

wp

guz

wp

gu ++=++

(1)

Now, provided that the approach channel has a muchlarger cross-sectional area than the notch, the fluid inthe vertical plane containing M will be comparatively atrest. It is in almost a hydrostatic condition for which thetotal head of all points has the same value of H relativeto the datum shown. Making the further (and lessjustifiable) assumption that pN = 0, i.e. the staticpressure is atmospheric at N, Equation (1) simplifies to:

Hzg

u=+ N

2

N

2(2)

Now

hzH =− N

(3)

As may be seen from the diagram, so

hgu

=2

2N

(4)

This velocity is the same as that which would beattained by a particle falling freely from the level of theupstream surface to the position of N.

The discharge over each weir may now be found byintegration. For the rectangular weir of width b, the areaof an element having height δh is bδh, so that the flowrate δQ through it is:

hbghhbuQ δ=δ=δ 2N

(5)

The total flow rate, Q, obtained by integrating betweenzero and H gives a result which neglects the lowering ofthe surface level in the plane of the weir, and is:

�=H

0

2 dhbghQ

or

2/332 2 HbgQ =

(6)

for the rectangular weir.

For the V-notch of angle 2θ, the width of an element is2(H − h) tanθ, so that the area of the element havingheight δh is 2(H − h) tanθδh. The flow rate through it is:

( ) ( ) hhHghhhHuQ θδ−=θδ−=δ tan22tan2N

(7)

so that, integrating as above,

( )� θ−=H

0

tan22 dhhHghQ

or

25158 tan2 HgQ θ=

(8)

for the V-notch.

There is, in fact, a considerable contraction of thestream as it passes through the notch. This takes placeboth in the vertical and the horizontal planes. In thevertical plane the upper surface slopes downwards overthe notch and the lower surface springs from the crest of

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the notch in an upward direction. In the horizontalplane, the water leaves the edges of the weir in a curvewhich reduces the width of the stream. This contractionis similar to that previously observed at a sharp-edgedorifice and has the same effect of reducing thedischarge. It is therefore customary to rewrite theequations in the form:

2332

d 2 bHgCQ =(9)

for the rectangular notch, and:

25158

d tan2 HgCQ θ=(10)

for the V-notch, in which Cd is a coefficient ofdischarge of the notch (which is not necessarilyindependent of H and may be determined byexperiment).

A convenient way of finding Cd, and the exponent ofH, in either of these expressions is as follows. Either ofEquation (9) or Equation (10) may be written in theform:

Q = kHn

(11)

or

logQ = logk + nlogH(12)

By plotting the experimental results on a graph havingaxes of log H and log Q, then, providing that k and n areconstant over the range of the results, they will lie on astraight line of slope n and intercept log k on the axis oflog Q. This is as indicated in Figure 4.

Figure 4

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SECTION 4 ADDITIONAL NOTCHES AND ASSOCIATED THEORY

The Cipoletti NotchThe most familiar kinds of sharp-edged notched usedfor flow measurement are rectangular and V-shaped.However, notches of other shapes have been suggested,and among these is one developed by Cipoletti, asillustrated in Figure 5. This slight modification to arectangular profile is intended to produce a constantvalue of discharge coefficient over a wide range ofheads.

Experiments on rectangular notches show that thevalue of the discharge coefficient falls slightly withincreasing head, and thus with increasing value of ratioH/b. By using sloping sides, the mean width of thenotch increases with head, thereby providingcompensation for the reduction in discharge coefficient.

To check the validity of this concept, the Cipolettinotch should be tested in exactly the same manner as therectangular one, and the two head-dischargecharacteristics compared on a logarithmic basis.

Figure 5 Profile of the Cipoletti notch

The Proportional WeirThe proportional or Sutro weir is an ingenious devicefor producing a flow rate Q which is linearlyproportional to head H. Therefore, in the expressionQ = KHn the value of n is to be n = 1. Now we knowalready that for a notch of constant width (a rectangularnotch), n = 1.5 nearly, and for a V-notch, where thewidth increases with head, n = 2.5 nearly. It wouldtherefore be expected that to obtain a result of n = 1, thenotch width would need to decrease in some way withincreasing head. This indeed proves to be the case. Therequired shape is illustrated in Figure 6.

Figure 6 The proportional or Sutro notch profile

Discharge RateWhen the flow over the crest is very small, the notchbehaves as if it were a simple rectangular one. Forhigher flow rates:

( )0HHKQ −=(13)

in which the theoretical value of H0 is given by:

30 HH =(14)

and K is the constant defined as:

gABCK 2d=(15)

Cd is a discharge coefficient to be determined byexperiment. A value of H0 may also be determinedexperimentally, and checked against the theoreticalvalue given by Equation (14).

The head-flow relationship is shown in Figure 6,where head, H, is plotted vertically and flow rate, Q,horizontally. Experimental values shown in this wayshould lie on a straight line. From this line we canobtain values of H0 and Cd using Equations (13), (14)and (15).

Theory of Flow over Broad-Crested Weirs

Figure 7 The broad-crested weir

The weir shown in Figure 7(a) differs from a notch, inthat the crest is not sharp, but is long in relation to thehead H. Such a weir is therefore referred to as broad-crested. In the acceleration along the length of the crest,the flow tends towards a uniform condition. Figure 7(b)shows an idealisation in which the flow has becomecompletely uniform and horizontal at the outlet sectionC from the crest. At this section, the depth of flow is zcand the uniform flow velocity is uc. The flow rate isgiven by:

ccbzuQ =(16)

where b is the width of the weir (in the direction normalto the figure).

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In this, uc may be found by applying Bernoulli’sequation to the streamline in the water surface:

( )ccc 22 zHgghu −==(17)

Substituting for uc in Equation (17) from Equation (16):

( ) cc2 bzzHgQ −=(18)

This result appears to show that the flow rate Q dependson both variable H and variable zc. However, it isobvious that zc is not truly independent, as it mustdepend in some way on H. With any chosen fixed valueof H on a given weir, zc will settle to some valuesomewhere in the range between zero and H. This valueis in fact the one that maximises Q for the chosen valueof H. Now, for fixed values of g, b and H, Q has amaximum when:

0ccc

=− zzHdzd

Performing the differentiation:

( ) ( ) 021cc

21c2

1 =−+−− − zHzzH(19)

which leads to:

Hz 32

c =

Substituting in Equation (18) leads to the result:

23233

2 bHgQ =

(20)

This is the theoretical flow rate. To take account of thesimplifications used in deriving this, the dischargecoefficient Cd is introduced as before:

23d 2

332 bHgCQ =

(21)

The value of Cd is found by experiment, having a valuewhich is usually somewhat less than unity. It may varyslightly with H, as the ratio of head to crest breadthchanges. A logarithmic plot would not be a straight lineof slope 1.5. It is therefore recommended that Equation(21) be used to compute the value of Cd for each pair ofobservations of H and Q, and these values of Cd beplotted against the ratio of head to crest breadth.

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SECTION 4 EXPERIMENTS AND RESULTS

The apparatus is first levelled and the depth gaugezeroed.

Figure 8 Notches

To do this, water is admitted from the bench supply tothe apparatus until the level is approximately correct,and then carefully baled out or in, using a small beaker,until the crest of the weir lies just in the surface. For therectangular notch this can be checked as illustrated inFigure 8(a) by placing a steel rule on the crest. For theV-notch, the reflection of the V in the surface serves toindicate whether the level is correct or not as illustratedin Figure 8(b). When the correct level has been obtainedthe gauge should be set to coincide with the free watersurface and the dial set to read zero. The followingprocedure relates to use of the height gauge:

1. Set scale to whole number on gauge.2. Lock gauge.3. Loosen nut and slide down depth pointer to touch

water - lock nut.4. Set dial on depth gauge to zero.5. Datum is now set at whole number on the depth

gauge.

A series of measurements of discharge and head on theweir are then taken, the flow being regulated at thebench supply valve. It is recommended that the firstreading is taken at maximum discharge, and subsequentreadings with approximately equal decrements in head.Readings should be discontinued when the level hasfallen to a point at which the stream ceases to spring

clear of the notch plate; this is likely to occur when thehead has been reduced to about 10 mm for a rectangularnotch and about 20 mm for a V-notch. About 8 differentdischarges for each notch should be sufficient.

The width of the rectangular notch and the angle ofthe V-notches (best found by measuring the depth andwidth of the V) should be recorded.

Results and CalculationsResults given in this section are typical of thoseobtainable from the equipment supplied. There will,however, be slight differences between individual units.

Gaugereading

mm

Gmm

104 x Qm3/s

Log Q Log H

3.93 0 – – –62.61 58.68 7.62 –3.1180 –1.231557.12 53.19 6.70 –3.1739 –1.274249.92 45.99 5.43 –3.2652 –1.337343.07 39.14 4.28 –3.3686 –1.407439.34 35.41 3.68 –3.4342 –1.450931.98 28.05 2.58 –3.5884 –1.552028.01 25.08 2.06 –3.6861 –1.600722.38 18.45 1.44 –3.8402 –1.735017.06 13.13 0.78 –4.1051 –1.8817

Table 1 Results with Rectangular Notch

Table 1 shows a suitable results table and includescolumns for recording measurements of head, H, anddischarge, Q, together with log H and log Q. From theseresults, a graph of discharge rate Q against head H andlog Q against log H should be plotted.

Figure 9 Variations of discharge with head forrectangular and V-notches

Figures 9 and 10 show the form of the graphs expected.The slope n and intercept k on the axis of the log Q

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scale should be determined and used to derive therelationship between Q, the discharge rate, and H, thehead.

Figure 10 Variations of log Q with log H forrectangular and V-notches

Rectangular Notch - Typical CalculationWidth of notch b: 30 mm.From graph of log H against log Q slope n = 1.50.Intercept on log Q axis (i.e. when log H = 0) = 2.72 (byextrapolation).

The relationship between log Q and log H is thus:

log Q = 2.72 + 1.50 log H

so that the relationship between Q and H is:

Q = 0.0525H1.50

(22)

Comparing this with the expression derived previously,

2332

d 2 bHgCQ =

we note that the exponent of H is the same in both, andthat Cd is given by:

59.0

030.081.920525.0

20525.0

32

32d

=

××==

bgC

V-Notch - Typical Calculations (θθθθ = 15°°°°)Width across top of V = 55 mmDepth of V = 102 mm

Tan θ = 1022

55×

= 0.270 = 15.1°

Gaugereading

mm

Hmm

104 x Qm3/s

log Q log H

1.94 0 – – –85.88 83.94 7.46 –3.1273 –1.076075.28 73.34 5.77 –3.2388 –1.134765.35 63.41 3.96 –3.4023 –1.197856.37 54.43 2.78 –3.5560 –1.264845.98 44.04 1.69 –3.7721 –1.356142.79 40.35 1.34 –3.8729 –1.388835.60 33.60 0.83 –4.0799 –1.4729

Table 2 Results with V-notch = 15°

From graph of log H against log Q:

Slope n = 2.50

Intercept on log Q axis = 1.60 (by extrapolation).

The relationship between log Q and log H is:

log Q = 1.60 + 2.50 log H

so that the relationship between Q and H is:

Q = 0.398H2.50

(23)

Comparing this with the expression derived previously:

25158

d tan2 HgCQ θ=(10)

we note that the exponent of H is the same in both, andthat Cd is given by:

62.0

270.081.92398.0

tan2398.0

158

158d

=

××=

θgC

Discussion of ResultsThe results show that discharge over the rectangularweir may be represented by the equation:

2332

d 2 bHgCQ =(9)

in which Cd = 0.59 over the range of the experiment.

The expression 2332 2 bHg is the calculated discharge,

neglecting losses and the contraction of the jet as itpasses through the notch, and Cd is the empirical factorto take account of these effects.

The discharge over the V-notch is represented by theequation:

23158

d tan2 HgCQ =(10)

in which Cd = 0.62 over the range of the experiment.

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In this case 25158 ant2 Hg θ represents the calculated

discharge, and Cd again takes account of losses and ofcontraction of the jet.

It should be noted that, over the range of theexperiments, slightly different exponents of H, withcorresponding different values of Cd, could be fitted tothe results. For example, the equation:

47232

d 2 bHgCQ =(24)

in which Cd = 0.54 fits the results for the rectangularweir almost as well as Equation (9) with Cd = 0.59. Awider range of H would be required to differentiatebetween the various alternatives, but without anyevidence to suggest that the exponents 1.50 and 2.50 forthe rectangular V-notches do not apply exactly inpractice, it is reasonable and convenient to take thesevalues. Moreover, the values of Cd associated with theseexponents are close to the value obtained for flowthrough an orifice.

Questions for Further Discussion1. What suggestions have you for improving the

apparatus?2. How would you interpret results, which, when

plotted logarithmically, as in Figure 10, fall on aline which is not straight, but slightly curved?

3. To what extent does the experiment confirm thetheoretical treatment? Has the dependence on b (forthe rectangular notch) or θ (for the V-notch) beenestablished?

4. A suggested project is to plan a series of tests toexplore the dependence on b, using a set of notchesor by partially covering the width of the onesupplied with a sharp-edged metal strip. What rangeof b should be chosen? What is the best way topresent the results? Is there a recognisedmodification to the form of Equation (9) that allowsfor the effect of the contractions at the side? A V-notch with an angle, θ, of 45° is supplied. Whateffect does this change in angle from the 15°standard have on the equations? Why?

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