Measurement of low pressures using manometers.

I wondered about including any mention of manometers even though they were in regular use when I was lecturing. I thought that they might have been displaced by electronic devices using diaphragms to detect pressure difference. I looked on the internet and it is clear that, whilst pressure measurement by electronic instruments is widespread, manometers have not disappeared. Certainly they are in use in laboratories where, I suppose, people still have a nagging doubt about using secondary instruments that depend on calibration for accuracy. So I decided to include manometers. They do give a useful source for examination questions.

 

The physical quantities  and  can all be measured accurately. It follows that devices designed to measure pressure by applying the expression  have the potential for equal accuracy. The devices in use are collectively called manometers. There are a few precautions to be observed when designing and using manometers. Liquids have surface tension and have different behaviours when in contact with solids like glass and plastic. We have all seen the meniscus formed between mercury and glass. Any unwanted effects like this can be minimised by using tubing of sufficiently large bore but not so large a bore that the response to a change in pressure is too slow for the intended purpose. The inner surfaces must be clean. There are published standards for the construction and use of manometers and where a manometer has been constructed and operated to these standards they can be accepted as accurate in law for financial and other transactions. Presumably there are now standards for electronic pressure measuring devices.

 

The most simple is the U-tube manometer that is shown in figure 2-5 as it might be used to measure the difference between the pressure of a gas in a pipe and atmospheric pressure. It is no more than two transparent tubes joined together at the bottom, set up vertically and close together. A liquid of known density partially fills the tubes. The top of one tube is connected to the tapping ring[1] and the top of the other is open to the atmosphere. The diagram shows the manometer measuring a pressure above atmospheric producing a difference of level of  between the levels in the two tubes. It could as easily measure a pressure below atmospheric.

 

If  can be measured accurately and, if we are confident that we have no unwanted forces affecting the value of , we can calculate the pressure in the pipe. In order to do so we must make use of the observation in figure 2-2. From this we see that the pressure on the free surface in the left hand limb of the tube is equal to the pressure at the same level in the right hand limb. So the pressures at A and B are equal because the liquid between them is continuous and at rest. If the density of the gas is  and pressure at A is  and is given by :-

                      , where p is the gauge pressure of the gas in the pipe

 

In the same way the gauge pressure at B,  is equal to  where  is the density of the fluid in the manometer.

 

Therefore                     

                                                          

There are only three liquids that are suitable for use in simple manometers. They are distilled water, mercury[2] and distilled paraffin and these have densities of 1000, 13,600 and about 780 kg/m. They are all mobile liquids, more or less non-toxic and can be used safely with simple precautions. The densities of the common gases are in the range of 1 to 2 kg/m³ at atmospheric pressure with air at 1.25 kg/m³ at atmospheric pressure. Given these figures it becomes evident that, even for paraffin in the manometer, putting  involves an error of only 2/780 or 0.25%. So, providing that z is not large, as it might be if the pressure at the top of a chimney was measured at ground level, we can, for most applications, write  :-

                                                                       

The practical details of the simple U-tube manometer depend on the application, but there are some difficulties. I have said that for the accuracy of the reading there should be no unwanted forces acting on the liquid. These forces may occur at the free surfaces of the liquid. Here we have the effects of surface tension and wetting to consider. The British Standards Institution recommend the use of tubes of a minimum bore of 13 mm to make these effects negligible. However we must recognise that the B.S.I. have to make recommendations which, if followed, are acceptable in legal arbitration and these may not be essential in some applications. Then the user must assess the situation and make a choice of diameter.

 

In order for a manometer to show the reading in figure 2-6 a quantity of the gas must flow from the pipe into the manometer. The volume of this gas will depend on  and the bore of the tubes. The rate at which the gas flows into the manometer will depend on the unbalanced pressure at any instant, which as equilibrium is approached tends to zero, and on the size of any restriction in the connection between the manometer and the pipe. A manometer may take an unacceptably long time to come to equilibrium if there is a serious restriction and the bore is large. The use of a B.S. manometer in conjunction with a small pitôt-static tube would give this combination. This gives an incentive to use much smaller bores than 13 mm. Provided that the tubes are clean, so that the wetting is not affected, mercury can be used with tubes of about 3 mm bore. Water is very troublesome, because it does not easily wet tubes that are dry, but paraffin, which wets dry tubes very easily, can be used down to 2 mm.

 

These simple manometers are robust and are used extensively by, for example, the gas supply industry. A further use is to test for leaks by noting the fall in pressure following the isolation of a pipe with a leak. Its response to a leak is quick enough for the manometer to be used to check that a pipe does not leak.

 

The user of a simple manometer must make two readings and subtract one from the other. This can lead to errors and mistakes. There is an incentive to design a direct reading manometer. Figure 2-6 shows such a device. One tube is enlarged to have a cross-sectional area many times that of the other and constructed as a closed pot that is connected to the other limb by a flexible tube. The pot is mounted so that it can be moved to bring the level in the long tube to the zero of the scale. (The levels in the tube and the pot may then differ because of surface tension effects but this does not matter if the tube is clean.) The pressure to be measured is connected to the pot and this leads to a rise in level in the tube. For a tube of small bore the volume displaced is small and as the area ratio can easily be 1000 to 1 the change in level in the pot can be negligible. Then either,  can be used to calculate the pressure or, the manometer may have a direct reading scale.

 

There is also a range of designs of inclined-tube manometers in which the vertical difference in level is magnified by inclining the tube. The arrangement is shown in figure 2-7 and the insert shows the way that the meniscus produces a quite sharp edge to act as a clear pointer to operate against the scale. It is enhanced by the colouring of the paraffin. A further adaptation is to have many parallel, inclined tubes connected to the pot (which must be further enlarged) and apply pressure to individual tubes rather than the pot. This gives a multi-tube manometer, for say wind tunnel work, and this can have the inclination adjustable.

 

 

 

 

 

 

 

 

There are two other manometers that are used to measure the pressure difference between two points in a pipe carrying gas or liquid. The application arises for use with certain designs of meter for measuring flow, for example, the Venturi-meter and the orifice meter.

 

The inverted U-tube manometer is used with liquids that may be under pressure. It is shown diagrammatically in figure 2-8. The U-tube is now upside down and the two pressure connections are made to the lower ends of the tubes. A non-return valve is fitted to the top of the inverted tube. In order to start the manometer the liquid in the pipe is first allowed to flow through the manometer and then air is pumped in with a bicycle pump through the non-return valve to separate the liquid in the two limbs.

 

Following the reasoning used in the simple U-tube the pressures at A and B are equal

because the air at rest above them is continuous.

 

Then     and  where   and  are the densities of the liquid and of the air.

 

Subtracting  from  gives :-

                                       , and,

                           

                                                            

This can now be reduced to  if  is small compared with .

 

The differential mercury manometer shown in figure 2-9 is used for gases or liquids in the pipe. The U-tube is now fitted with a cross connection at the top that incorporates a valve. The pressure connections are made to the tops of the limbs of the U-tube that is partly filled with mercury. In order to start the manometer the valve in the cross connection is opened and fluid allowed to flow through the connecting pipes to purge them of air. Then the valve is closed and the manometer will show a reading.

 

Once more  equals  because the mercury below them is continuous and at rest. Then:-

                       and

 where   and  are the densities of the fluid and of mercury.

Subtracting p_A from p_B gives :-

               .

As this manometer is most likely to be used in conjunction with a liquid in the pipe r_f will be significant compared with r_m and this leads to the inclusion of the descriptive word differential in the name.

 

There are many other designs of manometer including a whole range for measuring very small pressure differences. They use enlarged ends, two liquids of similar densities and other devices to improve the sensitivity. Whilst they are all primary devices, that is they do not need calibration because the accuracy depends on primary measurements, the problems of surface tension and wetting, of cleanliness, and in some cases of interaction between the surface of separation of two liquids and the walls of the containing tube, become more troublesome as the sensitivity increases. It becomes a field for the specialist supplier.

 

Unless the user of a manometer is prepared to use a stepladder and a kneeling mat the maximum value of  is limited to about 700 mm. This means that the maximum pressures that can be measured are about 0.07, 0.93 and 0.05 bar for water, mercury and paraffin as the manometric fluids. When compared with pressures of 300 bar, that are commonly met in oil hydraulic systems, these are low pressures.

 

High pressures are usually measured using pressure gauges. These have a pressure-sensing element, which is most commonly a Bourdon tube or a diaphragm, and some magnifying and display system. These systems are either mechanical or electrical and pressure gauges have evolved to be both reliable and accurate. However these gauges are secondary measuring instruments and they depend on some other device for the accuracy of their calibration.

 

The device used is the dead-weight tester and this uses the ideas of pressures in liquids at rest. In this tester the required known pressure is created by applying a known force to a known area. The area is that of an accurately machined piston working in a vertical cylinder. The known force is that exerted by the gravitational field of the Earth acting on a mass. This means that the test masses must be made to suit the strength of the gravitational field where the tester is to be used. The arrangement of the tester is shown in figure 2-10. In practice this simple device must be designed and operated so that the effects of mechanical friction between the piston and cylinder is eliminated. In the dead weight tester the gauge to be tested is connected to a closed system containing oil that can be pressurised by screwing in a plunger. The piston is a very good fit in the cylinder allowing a very small leakage of oil. The piston and the weights are carefully machined so that, however they are assembled, they interlock to form a single mass with its centre of mass on the axis of the cylinder. Then, if the cylinder is set upright using the levelling screws, there are no horizontal forces between the piston and cylinder and the piston and its masses can then be made to "float" on the oil. When the piston and masses are rotated slowly, the system is as free from friction as we know how to make it and the pressure at the face of the piston equals the applied force divided by the area. The gauge connection is regarded as the level at which the gauge measures the pressure and during calibration the level of the piston face is adjusted to that of the gauge connection.