Chapter 5       Examples of the application of the energy equation

 

Introduction

I have said that the validity of the energy equation lies in its usefulness. Now I need to give some examples that show how the equation is used in order to justify this assertion. Whilst doing so I can introduce the concept of using experimentally determined coefficients and rational expressions.

 

The Pitôt-static tube

Suppose that you were asked to measure the rate of flow of water through a large pipe, of say 1 metre diameter, in a way that would be acceptable in law. This is the sort of problem that crops up when liquid is sold in very large volumes and is to be delivered through a pipe. The vendor and the buyer will haggle over a price that is effectively arbitrary and then expect the quantity to be measured to some high degree of accuracy in order to monitor the contract. The time-honoured method of collecting the flow in a bucket for a given time and working out the flow from the result is hopelessly impractical in this case. In the pipe the velocity varies across the diameter being least at the wall of the pipe and it is this fact that makes it difficult to measure the throughput whilst the liquid is flowing.  However any streamlined restriction in the pipe will produce a pressure difference that could be measured and, if the flow can be measured by using a primary device, it could be calibrated.

 

We need a device that depends on primary measurements that measurements made by fundamental methods. A pressure gauge tester is one primary device among several that could be used to calibrate, say, a Bourdon pressure gauge. Its accuracy depends on the accuracy of the measurement of the piston, on the accuracy of the weights and on adherence to a set procedure. There are secondary devices that can be used to measure a flow in such a pipe but they can only be used if they are calibrated using some primary device. It follows that the problem of measuring the flow directly is not to be avoided. In the end, the only method we have relies on the basic method of dividing the area of cross-section of the pipe into many small areas and measuring the velocity through each area. As the flow through each area is the product of velocity and area the total flow is the sum of all the flows through the small areas. This leaves us with the need to measure the velocity of a flowing liquid accurately. The most commonly used device to measure the velocity is the Pitot-static tube.

 

In order to explain the design of the Pitôt-static tube we have to start with the Pitôt tube. Figure 5-1 shows the essential features of a Pitôt tube attached to a pipe of relatively small diameter. The Pitôt tube is the L shaped one that is shown set up on the axis of the pipe but could be at any radius. A second tube is fitted to the wall of the pipe in the plane of the open end of the Pitôt tube. The end of this tube is flush with the inside surface of the pipe. A suitable manometer would be connected to the external ends of the tubes at A and B. Clearly the side tube is designed to detect the pressure in the pipe at its point of connection. The Pitôt tube, which is often called the facing tube because it is facing the flow, is placed there to detect the pressure in the liquid and, in some way, a further pressure caused by the flow over the tube. The manometer will measure the pressure difference between the facing tube and the side tube.

 

We need some way of relating the pressure difference measured by the manometer and the velocity of the liquid approaching the facing tube. We could suppose the existence of a flow line, as shown in figure 5-2, in the liquid, which is in line with the axis of the Pitot tube, and along which the velocity at 1 is  and at 2 is zero.

Then, if we ignored any friction effects between 1 and 2, we could use the energy equation for steady flow and write:-

                                       

Here  equals  and we already have that  is zero. It follows that  from which we can deduce that the kinetic energy at 1 has been exchanged for an increase in pressure energy at 2.

 

The manometer is connected between 1 and 2 and will detect  on one limb and the pressure at the tapping point on the other. If this pressure equals  we have a way of measuring  because

      and, as  can be derived from the manometer reading,  and the reading can be related.

 

Now we must consider the reality of the system. Presumably the diameter of the pipe has been chosen from valid engineering considerations. If it has, then the motion of the liquid in the pipe would be one of continual mixing and injection of dye would not produce a flow line but just discolour the liquid. Nevertheless, even though the liquid is mixing as it flows, the average direction of flow through a given point is, in the absence of any general rotation, parallel to the axis of the pipe. Then if we supposed the flow to be such that we could see flow lines, they might look like those shown in Figure 5-3, where the centre one comes to rest and all the others diverge to flow round the facing tube. A feature of this flow pattern is the divergence of the flow lines before they reach the tube. This is the normal behaviour of the liquid and indicates that the pressure upstream of the tube is affected by the presence of the tube to produce forces that slow the liquid and make it diverge. Once the profiled end is passed the flow lines do not immediately flow along the surface of the tube. Instead they follow a wavy path and this leads to the pressure at A being slightly below the general pressure in this region of the pipe and the pressure at B slightly above. No attempt has been made to represent the motion in the region just inside the end of the tube because it is not known.

 

It must be evident that the real mode of flow in the vicinity of the facing tube is too complex for us to rely on the direct use of the energy equation for the highest level of accuracy but for many purposes the Pitôt tube will give an adequate measurement of velocity. The Pitôt tube could be calibrated but we want a primary instrument and for that we must turn to a Pitôt-static tube.

 

 

 

 

 

 

 

 

 

In the pitot-static tube the side and facing tubes are incorporated in a single probe to form a very versatile instrument. The construction of a Pitôt-static tube is shown in figure 5-4a and 5-4b. Figure 5-4a shows a cross-section of the head of a small Pitôt-static tube. It shows that there are two tubes one inside the other with a smoothly contoured nose to join them together. The inner tube corresponds to the facing tube of the Pitôt tube and would be connected to one limb of a manometer. The outer tube has four holes in it, all in one plane, and the annular space between this and the facing tube is connected to the other limb of the manometer.

 

The design concept is that, if the axis of the Pitôt-static tube is aligned with the direction of flow of a flowing liquid, the outer tube, which is called the static tube, will detect the pressure head of the liquid in the vicinity of the tube and the facing tube will detect the sum of this pressure and the kinetic energy head. Then the manometer will measure, in some way, the kinetic head and from this the velocity of the liquid approaching the Pitôt-static tube can be deduced as before.

 

Once more we must consider the reality of the system. If we pay no special attention to the profile of the nose-piece that joins the two tubes we must expect that the flow past the static tube of the Pitôt-static tube to behave in the same way as the flow past the Pitôt tube shown in figure 5-3. The wavy flow lines suggest that the pressure along the surface of the Pitôt tube is not likely to be uniform. This means that the pressure detected by the holes in the static tube of a Pitôt-static tube will depend on the position of the holes unless we can design a nose that eliminates the problem.

 

Conical, hemispherical, elliptical and parabolic shaped noses have been used but another difficulty influences the choice. In many applications, especially where a liquid is flowing round a solid, the direction of flow is not known. As the Pitôt-static tube has to be put somewhere to start with, it may be that the instrument is not in line with the flow. Then it is desirable that the design of the nose should make the instrument insensitive to this want of alignment. In other cases it may be more important to have a design that is sensitive to alignment.

 

It becomes clear that the detail design of the Pitot-static tube is likely to be achieved by development and this is probably best achieved by commercial organisations. This leads to a variety of designs both of the head and the stem of the Pitôt-static tube. One commercial design is shown in figure 5-4b.

 

Such designs can be used without calibration for most practical purposes and the velocity is given by  where  is the head of liquid equivalent to the pressure difference recorded by the manometer and  is put equal to unity. However, if a calibration is essential, this may be done in still water in a long tank or in a circular tank by moving the Pitôt-static tube at constant speed. This would lead to a table relating  and the reading of the manometer. It is unlikely that this table would be used subsequently because it is more convenient to put  where  is a calibration coefficient to be determined from the table of experimental data and may well be an average value. This is the first time in this text that I have used a calibration coefficient. More explanation is needed. There are those who think that a calibration coefficient or indeed any other coefficient is a “fiddle” factor. Do not be taken in. The detractors are starting from the premise that, if only we use enough mathematics we can avoid using a coefficient and anything else is less than satisfactory. They forget that, before mathematics can be used the physical properties of the system have to be described fully. No-one knows how to do that. Engineers cannot sit back and wait, they must find a way forward, and until we can do better the coefficient will have to do. It is of course a very powerful concept that is dependent on the fact that the behaviour of a Pitôt-static tube is repeatable. That is, the Pitôt-static tube behaves in the same way time after time in similar conditions. This is an essential feature of science.

 

Bodies like the British Standards Institution have publications (standards) on many devices that are used in engineering generally. Each is the result of exhaustive experimental work with the results being codified for use by anyone. There is one on the use of Pitôt-static tubes. It also gives details of the way in which the cross-section of a pipe should be divided for measurement of flow by Pitôt traverse so that the result might be acceptable in law.

 

Unfortunately the Pitot-tube has a non-linear characteristic which makes it unsuitable for use where low velocities have to be measured. Graph 5-1a shows how the velocity is related to the head when the head is limited to 0.75 m as would be the case for use with an inverted U-tube manometer. If, however, a differential manometer is used, and the flowing liquid is water, the range of head may be up to 10 metres and then Graph 5-1b would apply. When the U-tube manometer is used velocities below about 1 m/s do not produce a large enough reading for acceptable accuracy. Similarly, for the differential manometer velocities below about 0.4 m/s are too low to measure.

 

The hole in a tank

I choose the next example because it shows the energy equation being used to influence design.

 

There are applications of engineering where, in the design stage, it is desirable to be able to predict the rate at which liquid would flow out of a hole in the side or bottom of a tank. Such applications are, in dosing where one flowing liquid is mixed with another, in measurement of rate of flow and in estimating the time taken for a tank to empty through a hole.

 

Holes and tanks can have all sorts of shapes and sizes and the hole can be put anywhere. As we cannot make a rigorous analysis of even the most simple combination we must expect to have to experiment. However the number of possible combinations of hole and tank is much too great to justify gathering the data needed to make an accurate prediction in every case.

 

We have to limit the experimental work to something that is manageable and then devise a way to store the resulting limited data so that it can be retrieved and used to make a sensible prediction.

 

Let us start by considering the shape of the hole. Suppose that it is to be made in the plate from which the tank is constructed. The obvious hole would be made by drilling the plate and chamfering off the ragged edge produced by the drilling. A cross-section of such a hole is shown in figure 5-5. If we attempted to store data about the rate of flow through this hole we would need to be able to describe the hole in geometrical terms. For this we would need to state the thickness of the plate, the diameter of the hole, the size and angle of the chamfer and, presumably make a statement about the surface finish of the plate and the hole. This involves five dimensions, each of which may influence the rate of flow.

 

Clearly it would be the work of a life-time to gather sufficient data to be able to predict accurately the rate of flow for any given combination of these dimensions even if the shape of the hole were to be the only criterion. In practice the rate of flow is also dependent on the way in which the liquid approaches the hole and on the properties of the flowing liquid.

 

So we must reduce the geometry to something with fewer dimensions and think of a way to deal with different shaped holes. Presumably the ideal hole for experiment would have one dimension, the diameter, as shown in cross-section in figure 5-6. Such a hole would be impractical because the sharp edge would wear very quickly. In fact the sharp edge is not essential and the device that has become known as the sharp-edged orifice is made as shown in cross-section in figure 5-7. The short axial length of the orifice strengthens the sharp edge of 5-6 but the liquid, as it forms a jet, flows cleanly from the square edge, as shown in figure 5-8, just as if it is sharp. The jet that emerges from such an orifice looks like a rod of glass.

 

We must now consider how we might experiment to gather data on the rate of flow through a sharp-edged orifice. So far we have said nothing about the shape of the tank and if the orifice were to be fitted as shown in figure 5-9 or in figure 5-10 we would not expect the flow pattern of the liquid approaching the orifice to be the same as that in figure 4-8. In both cases the shape of the jet would be affected by the proximity of the walls of the tank and so presumably the rate of flow affected.

The only thing to do is to test a well-made orifice in a large tank with the orifice mounted flush with the flat, horizontal bottom of the tank so that the distant walls of the tank do not affect the flow pattern. The arrangement is shown in figure 5-11. Then the system has only two dominant dimensions, the depth  and the diameter of the orifice .

 

Others have experimented at great length and we have their data. In order to use it we must see how it was stored.

 

The rates of flow  through an orifice of diameter  were measured under a range of steady heads . Then ,  and  were related by:-

                where  is a coefficient called the coefficient of discharge and is a number. This expression defines  and, as  is the area of the orifice and  is the velocity that the liquid would have if it fell, without resistance, through a distance , the product  has the units of rate of flow. In no sense is the expression above an attempt to predict the flow, it is simply the creation of a dimensionally correct expression that involves only the measurable quantities. It is the maximum conceivable flow through the orifice and it is a comprehensible basis for the definition of a coefficient. It should be noted that if the energy equation were to be used to find the velocity of the jet it would have to be applied to the free surface and to the jet where it first becomes parallel. The position of this latter point is not easily determined and it is thought to be better to define the coefficient of discharge in terms of the measurable quantity  than to undertake to locate the end of the contraction of the jet.

 

How attractive this expression would be if it transpired that  is independent of  and  for all known liquids! Of course this is not the case and values of  are stored in the form of graphs like graph 5-2 for each liquid. For water there is a region above  = about 450 mm for which  for each orifice is independent of . As liquids that are inflammable or toxic would not be allowed to flow with a free surface, the practical range of liquids is relatively small and graph 5-2 could be used where the liquid has a similar mobility to that of water.

 

The testing of an orifice under steady flow conditions involves an inflow and this will prevent the water being free from eddies as it flows to the orifice. No information is given by the experimenters on the state of the water in the test tank but in fact the disturbance of the water as a result of filling has no detectable effect on the flow. Indeed even a fully-developed, air-entraining vortex as shown in picture 5-1 makes little difference to the rate of flow. (In picture 5-1 you can see the vortex at the top of the picture that was taken when the level in the tank was low. The vortex could be established easily at any depth.) It seems that in convergent flow disturbances are quickly damped out.

 

The flow through an orifice is not significantly different to that through the nozzle that has already been considered in this chapter, so the typical value of  of 0.63 is lower than might have been expected. However figure 5-8 shows that the jet is smaller in diameter than the orifice and it is this contraction, which reduces the area of the jet to less than that of the orifice, which accounts for the low value of . A further useful piece of information for the engineer is that the loss in the flow as it converges to form the jet varies much more with diameter than with head. The loss ranges from about 9% for  =20 mm to 1% for  = 60 mm.

 

We can now consider our original problem of the tank with a hole. We now know that the energy loss in the converging flow can be related to hole size and that it is the effective area of the jet which dominates . We also know that the effective area depends on the flow pattern upstream of the hole. We might then expect that the flows through the orifices in figures 5-9 and 5-10 would be greater than for an orifice that is not affected by the walls of the tank. Figure 5-12 shows the flow pattern through a short pipe with a square entry. The flow separates at the sharp edge and then contracts just as it does in the sharp-edged orifice but, in the short pipe, it expands again to fill the pipe. Unfortunately divergent flow usually breaks down into small eddies and this happens in this short pipe with considerable additional loss of energy.  varies with the ratio of l/d and, for l/d = 3,  is approximately 0.83.

 

The contraction and divergence of the short pipe can be eliminated by changing the square edge to a radius as shown in figure 5-13. In this case  varies with three dimensions and typically, for r/d = 0.5 and l/d = 3,  is about 0.95.

 

The data given above would be adequate to make a useful estimate of  in most cases. Only the designer would know whether the application justified any further commitment of time and money to get an accurate value.

 

 

 

The Venturi meter

The best-known application of the energy equation must be in the flow measurement device called the Venturi meter. It was invented by Herschel but named after Venturi. It exploits the drop in pressure that occurs when a liquid or a gas that flows along a pipe passes through a smooth contraction to make it into a flow meter. This is simple in principle but has to be applied in practice. All the work needed to make a practical flow meter was done at least 100 years ago and nothing has changed. You can find it all in Gibson’s book of 1912 called “Hydraulics and its applications”.

 

The success of the simple contraction shown in figure 5-14 depends on the existence of an expression on which a coefficient can be defined so that the volume flow can be related to the dimensions of the contraction and the pressure drop between sections 1 and 2. Then the coefficient can be found by experiment.

For the contraction we can write using the energy equation that :-

        .

Then:-

             .

 

 We know that :-

 and substitution gives:-

       . However, if we let  this reduces to :-

                                              or:-

                                                              

This expression is very simple. It says that the difference in head between inlet and throat is equal to the kinetic energy at inlet multiplied by a number that depends on the ratio of the diameters at inlet and throat. This makes sense.

 

We can explore this expression using say Mathcad because the range of  is not great in practical pipes and will not normally exceed say 3 m/s.

 

I have run this out in graph 5-1 and it is evident that it will be necessary to choose a diameter ratio to match the instrument used to make the measurement to the manometric head developed between the inlet and throat. Further the Venturi meter is not suitable for measuring flows at low velocity e.g. 0.3 m/s.

 

Generally a Venturi meter is used to measure flow and the expression can be rearranged to give volume flow for a given difference in head. It becomes:-

                         

 

This is an expression for flow in terms of the measurable quantities and the real flow will be given by multiplying by a coefficient  to be found by experiment.

                                                                 

 

The application of the energy equation and continuity has yielded an expression that tells us that this contraction can be used as a flow meter. In order to get it we did not analyse the behaviour of liquid as it flows though the contraction, we simply applied the equations in a mechanistic way to get an expression that is suitable for gathering and storing data. Now we have to look at the engineering design that will make it into a flow-meter.

 

Others, and Herschel was the first, have found out what we need to do. Normally the meter is made with a conical contraction and a conical divergence and the two are joined with a profile that is an easy curve to produce a throat. Clearly an essential feature of the design is the method of detecting the pressures at the inlet to the contraction and at the throat.

 

I have drawn just the contraction in figure 5-15. The proportions come from Gibson who suggests a length equal to 2.5 times the pipe diameter and a throat diameter equal to 1/3 of the pipe diameter. The manometric connections are made to two piezometer rings that are annular rings like an egg band on an earthworm. The rings are cast as part of the instrument and at least four holes are drilled through the wall of the contraction so that liquid fills the rings and any pipes connected to the rings. The holes must not be so small that they clog with dirt or so big that they disrupt the flow over them. They must be free from any rag or burr. Then, if the contraction follows a straight run of pipe at least 15 pipe diameters in length and the diameter of the pipe is greater than 50 mm the value of the coefficient lies between 0.97 and 1. In effect, for most purposes the Venturi meter can be used without calibration.

 

This result means that the flow in the contraction has not caused any significant loss of energy to internal friction. However the water is moving at high speed in the throat and if we just have an abrupt change in diameter back to that of the pipe all that kinetic energy will be lost. This may not matter but, if the water is being pumped along the pipe there will be financial cost to be met and it may be significant. Certainly its importance is set to grow. So a transition must designed to reduce this potential loss if not to minimise it.

 

If the transition is conical, and this is certainly the easiest shape to make, it turns out that the included angle of the cone for minimum loss is 5.1°. I have drawn the profile of such a Venturi meter in figure 5-16 and this is a very slow taper and I find it to be counter intuitive. I say this because one would think that, as the loss is caused by the high velocity, it would be best to reduce the velocity as soon after the throat as possible. Massey gives a graph of loss versus included angle of the cone and it is clear that the loss is small over a range of angle from 3° to about 11° with the lowest at this 5.1°. Somewhere about 20° the flow breaks down and the losses increase rapidly with increase in included angle. The consequence is that, if lowest running cost is an important factor, the Venturi meter will be about 10 diameters long.

 

There is nothing in the application of the energy equation that would be affected if the convergence were to be much shorter and the high value of the coefficient means that the convergent flow is almost free from losses so the convergence could be shortened without affecting the calibration very much. The divergence could be shortened as well with a consequent increase in running cost.[1] Balancing all these factors is what engineers are paid to do.

 

Taken all round the Venturi meter is a versatile device capable of being used for pipes of any size from 50 mm and upwards. It can often be used without calibration and be physically adaptable to circumstances.

 

It depends on a simple expression derived from the energy equation. I like to call such expressions rational expressions because they are not the outcome of some mathematical analysis but of a search for a simple expression that contains only measurable quantities, has the correct dimensions, and can be used with a coefficient to create a storage and retrieval system for data.

 

Energy exchange within a flowing fluid

I want to fit another use of the energy equation in at this point. In my experience people find it difficult to explain how some fundamental devices actually work. I recall someone asking a collection of graduates in engineering how the force is generated on a rocket in space. Everyone knew about momentum and could calculate the force but no one knew how the force was actually applied to the casing of the rocket. It is not an unreasonable question because the casings of booster rockets are required to have a minimum weight yet to be strong enough to withstand whatever forces are applied to them when they are in use. We can make a start at an explanation with the following example.

 

Figure 5-14 is familiar to every student of applied mathematics. It shows a tank fitted with a nozzle on one side and mounted on wheels so that the whole tank can move across a horizontal plane. The tank is filled with water and a jet of water flows from the nozzle. Given the rate of flow  from the nozzle and the area of cross-section of the jet , the student is asked to calculate the force exerted on the tank and perhaps, given the weight and size of the tank and the depth of water, the acceleration of the tank. It is primarily an exercise in the use of force equals rate of change of momentum.

 

It is easy to calculate the force, it equals  where  is the instantaneous mass flow, which is equal to , and  is the velocity of the jet where the student is expected to ignore friction and put equal to . Then, as  equals  it also equals  and the force equals  or

.

However the student is seldom asked to explain how the force is actually exerted on the tank.

 

One might start by noting that if there were to be no hole in the side of the tank, the tank would be in equilibrium because, for each small force exerted on a given small area of the inside wall of the tank, there would be an equal and opposite force on an equal area on the other side. The presence of the hole destroys this balance. Now as the jet does not expand sideways or contract as it emerges from the nozzle it follows that the jet emerges at the same pressure as its surroundings, that is at atmospheric pressure. It might then be supposed that the unbalanced force is simply caused by the difference between the atmospheric pressure acting on the area  of the jet and the pressure  acting on an equal area on the opposite side of the tank. This would equal . But this force is only a half of the force calculated from momentum above. This must mean that the force has another component that we have yet to consider. Somewhere another area is subject to pressure that is lower than that on the corresponding area on the other side of the tank. This can only be the internal surface of the nozzle and the region surrounding the nozzle where the water is accelerating to form the jet. The conclusions we can draw are that the pressure of the water in the immediate vicinity of the nozzle falls progressively from its static value to that of the atmosphere and that there is an associated exchange of pressure energy for kinetic energy. Figure 4-14a shows more or less hemispherical surfaces of uniform pressure around the nozzle each surface being at a lower pressure as the water approaches the nozzle. In 4-14b the junction of the surfaces and the wall of the tank are shown and successive circles are at lower pressures.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

We can now see that in the tank there has been an exchange of potential energy for pressure energy as the liquid descends through the tank and then, in the relatively small region near the nozzle, an exchange of potential energy and pressure energy for kinetic energy.

 

 and the flow through the nozzle =  where  is the diameter of the jet.

In figure 5-15a I have shown imaginary hemispheres centred on some point just inside the nozzle and, as I did in figure 3-1, supposed the flow to be radial. Using the fact that the surface area of a hemisphere is  the velocity through a hemisphere of radius  is given by :-

                                                                     

The sum of the potential energy, the pressure energy and the kinetic energy is the same for every point in the flow. As the kinetic energy increases as the water approaches the nozzle the sum of the potential energy and the pressure energy must fall. As the potential energy of the water is the same at any given level the increase in kinetic energy near the nozzle will reduce the pressure compared with an identical point on the other side of the tank. It follows that if we find the drop in pressure at radius  and then the force on an elemental ring of radial width  as shown in figure 5-15b we can integrate to find the reduction in force on the nozzle side of the tank.

 

We have . It then follows that the force on the elemental area is given by :-

                                             which reduces to:-

                                                                  

We now have to decide on some limit for the integral and the lower limit must be  the other limit is not so easy to decide so I will look at the outcome of the integral. It is :-

                                                                   

The lower limit will be  and if the upper limit is large the term becomes vanishingly small.

 

The result is that the reduction in force on the side of the tank round the nozzle is  and we have the other half of the force.

 

The booster rocket works this way but there we have a burning gas flowing through a nozzle that first converges and then diverges and is much more complicated. There are two forces, one on the top end of the casing and one on the exit “cone” of the nozzle

 

You may think that this is just a fuss about nothing but, if you do, explain the children’s toy rocket. In this toy a soft drinks bottle is half filled with water and a nozzle fitted to replace the cap. Then air is compressed into the top half using a cycle pump and the toy launched upwards. The air drives out the water and the bottle rises to a respectable height. The child gets wet and all is well. If the air in the bottle is at 3 bar try deciding on the best ratio for the volume of water to the volume of air.

 

Notches

It must be becoming evident to you that the use of the energy equation is in some way tied up with finding rational expressions on which to base coefficients to be determined experimentally in order to store data relating to the performance of the various devices that we want to use. I think that the method is very well illustrated by the use of notches for measuring flows of water that is flowing with a free surface. A notch is a specially-shaped gap in a plate. The basic shapes in use are the triangle, the rectangle and the trapezoid which can be thought of as a combination of a rectangle and a triangle. All of them have an edge having the profile shown in figure 5-17. They are not often seen in places that are used by the public but there are many in use in places like sewage treatment plants and water treatment work. I did come across one high in some hills that had been fitted for the remote measurement of the flow in a small stream. I wondered what possible purpose it could serve except feeding bureaucratic paperwork. Anyway they are useful to us in this context.

 

The sharp edged notch

One device that is frequently used is the triangular notch. I choose this because it shows us how the energy equation has been used to define a calibration coefficient when the flow is very complex and quite beyond our powers of analysis. The plate might form part of a vertical side of a tank or simply be set up vertically across a stream. Liquid then backs up behind the plate and flows over the notch. The shapes that are taken up by the liquid vary but they are all attractive to look at and interesting to study. They are called nappes, a word that describes the shape of the liquid in the same way as jet. I shall attempt to describe the nappe for a triangular notch so that we can see why it is beyond analysis.

 

Pictures 5-2 and 5-3 show water flowing over a triangular notch fitted into a channel with its axis of symmetry vertical. In truth the notch is in a piece of commercial educational equipment that is nothing like as good as the college-built rig that it replaced. In the college-built equipment the notch was in the side of a big tank that was fitted with screens to reduce the eddying. The commercial equipment placed the notch at the end of a very short channel and fed it from a deeper tank through smooth curves. You can see that the approach flow is not smooth and the result is evident in the ruffled shape of the nappe. The approach channel is also too narrow and the width affects the nappe more and more as the width of the nappe increases with depth. The accountants who forced the change did not know this.

 

The shape of this size of nappe is partly dependent on the effect of surface tension. Engineers do not pay much attention to surface tension but it acts on freely falling streams of water. Look at picture 5-1. (I repeat it for your convenience here.) You can see the air-entraining vortex through the Perspex tank. The water is rotating and, when it comes through the orifice it has a hole in it. For the want of a radially inward force the hollow jet tries to spread but then surface tension starts to provide a force to stop the spreading and is indeed sufficient to reverse it. The inward flow superimposed on the jet leads to a second expansion and sometimes a third. It was great fun to play with this tank and students enjoyed doing so. The important thing is that the surface tension of water in not to be totally ignored. It is quite capable of imposing shape on moving water if the size of the system is not too big. Full sized tugs produce a white foaming moustache of a bow wave but a scale model of say 40² in length running at the “correct” speed never produces foam because surface tension holds the eddies together.

 

In figure 5-16 I have drawn the essential feature of the flow and we need some explanation of the shape.

 

As the triangular notch has no other function than as a measuring instrument and we know that the edge of the notch is carefully made we end up with a nappe that has its shape as a result of the strength of the gravitational field, the physical properties of water and two dimensions the angle of the notch and the vertical distance between the general level of the free surface and the apex of the notch.

 

The flow pattern in the nappe is symmetrical about a plane that is normal to the plane of the notch and through its axis of symmetry. Figure 5-17 shows the flow pattern in this plane. The pressure all over the free surface of the nappe is the same, ie atmospheric. The liquid in the free surface falls as it approaches the plane of the notch and exchanges some potential energy directly for kinetic energy. The liquid that emerges at the apex will be exchanging pressure energy for kinetic energy and potential energy as it flows towards the apex. Once it has cleared the apex the liquid will be moving more quickly than that in the free surface because it has given up more potential energy to kinetic energy. At the highest point it reaches, it has still exchanged much more potential energy for kinetic energy than that at the free surface. It moves more quickly than the liquid above it.[2]

Figure 5-19a shows the paths of two objects that have been projected from point O and allowed to move freely. Path OA is for a slow moving object projected horizontally and OB for a faster one projected outwards and upwards. (You could run these out using Mathcad and see the shapes.) The liquid on the free surface of the nappe is trying to follow a path like OA whilst the liquid emerging at the apex is trying to go upwards and outwards to follow a path like OB. They are shown in their relative positions in figure 5-18b. Not surprisingly the liquid at the top of the nappe falls sideways, the centre comes up and the nappe spreads. This spreading would continue if it were not for surface tension which holds the nappe together and which, for water and other liquids with a high enough surface tension, brings the sides back in. The cross-section of the nappe is also shown in figure 5-16.

 

Add to this the side contractions that are evident in picture 5-3 and we have a very complex pattern of flow that might yield to analysis by finite element methods if only we could include the surface tension but that looks to be especially difficult and certainly not worth the cost and effort involved. We must expect to experiment as we did for the orifice.

 

For the triangular notch the only measurable quantities are , the head  above the apex and the angle  of the notch and once more we must test with the sides and bottom of the tank far enough away to have no effect on the flow pattern.

 

The obvious liquid to use in testing is water and data is available for the commonly used angles of 90° and 60°. This data can be stored in terms of a coefficient of discharge but for the triangular notch there is no obvious expression, like that for an orifice, on which to base the coefficient. We have to see how others have contrived a suitable expression.

 

In the case of the orifice we found an expression for the maximum conceivable flow on which to base a coefficient. The maximum flow that we can imagine for the Vee notch would be when the water emerged horizontally through a vertical triangle with the each particle of water moving with velocity  where  is the depth of the particle below the free surface. In unit time a volume having the shape shown in figure 5-20 would flow through the notch. We need to find this volume. If the vertical section at the notch plate is divided into elements as shown in figure 5-19 we can find an expression for the flow through the element and then it is only a matter of integration to find an expression having the dimensions of rate of flow and in terms of  and  both of which are measurable.

 

 

The flow through an element =  and

                                               = 

                                               = 

Therefore the notional volume passing the plane of the notch per second :-

                                               = 

                                               = 

                                               = 

                                               = 

Then, if  is the measured flow, a coefficient of discharge, , can be defined by :-

                                            = .

 

The most commonly-used triangular notch is the 90° notch (  = 45°) and it is normally used for values of  from about 25 mm to 200 mm. Below 25 mm the nappe clings to the plate and above about 200 mm the nappe becomes too wide at the surface and the flow varies too quickly with head to be useful. It is then better to change to a suitable trapezoidal notch or a weir.

 

If a triangular notch is used in an approach channel that is more than  wide and more than  deep below the apex, the shape of the nappe is not affected by the walls of the channel. Then, if  is measured at a point in the free surface that is close to the notch but far enough away for the surface not to have fallen measurably, the value of  is 0.62 and is effectively independent of  for the normal range. Then the calibration expression becomes  cubic metres/ second where  is in metres. We could not have wished for a better outcome to our search for a useful expression.

 

Let me repeat my original statement. Here we have defined a coefficient of discharge for the triangular notch using the same idea as was used for the sharp-edged orifice and if these two devices had no practical use, they would still be valuable to show, in a simple way, how a science is constructed. It is important to realise that the acceptability of the process by which these coefficients have been set up does not lie in the validity of the process but in the usefulness of the result. We shall meet this time and time again and it is much easier to understand if you can recognise what is being done.

 

The rectangular notch

I have said that there are other shaped notches and the rectangular notch has interest for us. It is just a rectangular shape in a plate in place of the triangle. It is shown in figure 5-21. I have shown the nappe and it is clear that the water flows upwards and outwards from the lower edge, (the sill) and inwards and outwards from the sides. This is another flow that we cannot analyse. However we can use the same method as before to find an expression using the two measurable quantities  the head over the sill and  the width of the sill. The flow through the element is given by  and integrating between  and  gives . Then, if the real flow is  we can define a coefficient of discharge  using:-

                                                           

The figure above shows that the nappe has two “side contractions”. These side contractions vary in size with . Figure 5-23 shows the difference in the shape of the nappe for a large flow and a small flow. We cannot expect  to be independent of . It has a value that varies around 0.6 depending on the net effect of the side contractions.

 

Inevitably there have been experiments to find out how  varies with  and  these are readily available in several empirical expressions.[3]

 

One quite practical device is the Cippoletti notch which has sides inclined at 14°. This widening with increasing head above the sill offsets the increasing contraction and gives the notch a fairly constant value of .

 

The weir.

If a plate is fitted across a channel of rectangular section to form a level sill the side contractions do not occur and the device is called a weir. I have already drawn a weir in figure 3-2. It is not a weir that might be used to measure flow because the wooden boards can be removed to control the river. However weirs of similar construction are used to measure flow and the important feature is that the nappe is ventilated to let it break free pf the weir wall. Needless to say these devices are sometimes overwhelmed by the excessive flow and river gauging is a study in its own right. Nevertheless it is still dependent on the steady flow energy equation.

 

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