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.