… can they be calibrated by dimensioning alone?

Purpose

The purpose of this guidance note is to explain why it is not possible to claim formal traceability for the pressure measurements made by a pressure balance, at anywhere approaching state-of-the-art levels of measurement uncertainty, when such traceability is based on locally obtained dimensional measurements of individual pistons and cylinders. In the process of doing this it also tries to clarify a misunderstanding that pressure balances are inherently primary standards.

Piston-cylinder section

Introduction to pressure balances and the SI base quantities from which pressure is derived

Pressure balances – sometimes known as piston gages or dead-weight testers - are widely used to generate reference pressures in a range that extends from about 3 kPa to 1 GPa and most are used to calibrate other pressure-measuring instruments. They consist of finely machined pistons, with diameters ranging from about 1 mm to 50 mm, mounted vertically in closely fitting cylinders as shown in the diagram opposite. When the pressure in the fluid (shown black) within the cylinder is just sufficient to support the weight of the piston plus the additional weights its value is calculated from the relationship:

where the effective area of the piston-cylinder combination is the area bounded by the neutral surface in the fluid between the piston and the cylinder – roughly the average of the piston area and the slightly larger cylinder area.

Thus to be able to calculate the pressure being generated by such a device it is necessary to know three parameters: the mass of the floating components (ie the piston and weights), the local value of gravitational acceleration and the effective area of the piston-cylinder combination. These parameters invoke the dimensions of length, mass and time – the three SI base quantities from which pressure is derived – as shown in the table below. (Also see how does 'pressure' fit into the International System of units?)

 

Quantity	SI unit	Symbol	Dimensions
mass	kilogram	kg	mass
area	metres squared	m2	length2
gravitational acceleration	metres per second per second	m/s2	length/time2
pressure	pascal (or newtons/metre2)	Pa or N/m2	(mass × length)/time2

Dimensions of mass, area, ‘g’ and pressure 

Fundamental (methods) versus primary (standards)

So pressure balances neatly combine the three base quantities from which pressure is derived (most other pressure measuring instruments cannot do this) and in so doing provide an inherently fundamental method of measuring pressure. Indeed, the ISO definition of a fundamental method of measurement could have been written with pressure balances in mind, namely: … a method of measurement in which the value of a measurand is obtained by measurement of the appropriate base quantities.

But the phrase is sometimes falsely equated with primary standard and manufacturers, for example, have been known to use this term erroneously to describe the pressure balances they sell. No pressure balances are inherently primary by their nature and quite literally no one is able to make one that immediately starts life with this title – even though some of the primary standards developed at national measurement institutes, like NPL, are indeed based on pressure balances. The reason is to be found in the definition of the term and in appreciating of the degree of effort needed to achieve primary standard status.

A primary standard is … one designated or widely acknowledged as having the highest metrological qualities and whose value is accepted without reference to other standards of the same quantity. This last phrase … without reference to other standards of the same quantity … does essentially mean that primary standards must also be fundamental in the way they work. It is not valid to argue the converse, however, and instruments employing fundamental measurement methods are not inherently primary standards. For example, the measurement method employed in the crude water U-tubes sometimes used by plumbers to measure the nominal pressure in a gas boiler is fundamental but obviously such devices do not have ... the highest metrological qualities ... and are certainly not primary standards. But also, as said above, even potentially far more accurate devices like pressure balances are not inherently primary standards either.

Turning a pressure balance into a primary standard

This is a tough task and success is by no means guaranteed. Determining the mass values of the components (ie the piston and weights) and the local value of gravitational acceleration are the easy bits; the stumbling block is always determining the effective area of the piston-cylinder combination and how it varies with pressure etc. First it is necessary to consider all the imaginable effects that can alter a pressure balance’s effective area – anything left out will detract from it having the highest metrological qualities and, at least as importantly, will lead to the measurement uncertainty ascribed to the value of effective area being too low (too optimistic). The effects include:

• the dimensions of the piston (unpressurised) including:
  • its roundness
  • how its roundness changes with position along the piston
  • straightness
  • any other deviations from non-simple shape
• the dimensions of the cylinder (unpressurised) including:
  • its roundness
  • how its roundness changes with position along the cylinder
  • straightness
  • any other deviations from non-simple shape
• the distortion of the piston with increasing pressure
  • this depends critically on the exact design details, dimensions and the elastic properties of the piston material (see spot the difference below)
• the distortion of the cylinder with increasing pressure
  • this depends critically on the exact design details, dimensions and the elastic properties of the cylinder and also its clamping arrangements (see spot the difference below)
• the temperature coefficient of the piston
• the temperature coefficient of the cylinder
• the properties of the fluid including:
  • surface tension
  • density
  • pressure-viscosity characteristics
• how all the above combine.

Developing a mathematical model

Armed with the best information available on all the above points, a mathematical model then has to be developed to compute the effective area of the piston-cylinder from the combined effects, and also the measurement uncertainties associated with it (see note 1 below). This is usually a most difficult aspect of evaluating a pressure balance and, perhaps surprisingly, there is no definitive way to accomplish it; at this level of metrology the inherently unique characteristics of each instrument demand individual attention.

Spot the difference
Piston and cylinder unpressurised.	Piston and cylinder pressurised.
Pressure balance pistons and cylinders are manufactured to state-of-the-art tolerances but even so they are not perfectly round or straight. Determining their effective area is not straight forward and depends very much on the microscopic deviations in geometry, from an ideal shape, of a particular piston and cylinder combination.

Having done this, it is most important to test the tailored theory against rigorous practical pressure measurements as these often highlight inconsistencies, and therefore probable weaknesses and errors, in the model. It is not uncommon, for example, to find that the newly computed theoretical characteristics of a pressure balance do not agree – within calculated measurement uncertainties - with results from practical comparisons with other standards of nominally the same performance. Problems of this type are usually most obvious when a new standard is compared, at the upper and lower ends of its operating range, with the primary standards already in use at higher and lower pressures. An apparent discontinuity in the pressure scale is obviously absurd but it indicates how hard it is to model a pressure balance properly (see note 2 below).

The new model has to be examined very closely before it can be concluded that the new standard’s results were ‘more correct’ than the other standards - and clear evidence of the failings of the former are needed too. There has invariably to be an iterative cycle of practical work and modifications to the theory until there is persuasive convergence.

Some potential primary standards do not survive this – they are found to be unsuitable. But even those instruments that are eventually deemed to be primary standards are still not formally and internationally considered to have proven their equivalence with the SI pascal – and hence the measurements made in other parts of the world. They are thus not automatically recognised as providing traceability to the SI.

Obtaining formal recognition of measurement ‘equivalence’

Since its inception in 1999, nearly fifty countries have signed a Mutual Recognition Arrangement set up under the auspices of the International Committee on Weights and Measures (CIPM). It commits signatories to taking part in extensive measurement comparisons - known as key comparisons - designed to show, for the first time, a comprehensive picture of how measurement standards in the national measurement institute (NMI) in each country formally compare with others – in other words their metrological equivalence. Across pressure and vacuum there are nine key comparison pressure ranges. The information will soon be publicly available at the website of the International Bureau of Weights and Measures (BIPM). Some information is already available there, including the related Calibration Measurement Capabilities (CMCs) of the participating NMIs but this is just the first stage in the process of enabling the metrological equivalence of NMIs to be measured.

Participation in key comparisons (covering the range of measurement services offered by a given NMI) is not optional. Failure to participate in one of the periodic comparisons is deemed as having not shown the equivalence of its measurements to the appropriate SI units - the pascal in the case of pressure. In the near future, when the comparison database is complete, it will be possible over the internet not only to see how closely one NMI’s measurements agreed with the reference values used in a particular comparison but whether or not the combined measurement uncertainties of any two participants cover the differences between their respective measurement values.

Thus no one can now formally claim to be making measurements that are equivalent to others unless their measurements have been verified via key comparisons. With rare exception, only NMIs take part in key comparisons – leaving dissemination of measurement scales to their calibration services.

Can primary standards be less than state-of-the-art?

The terminology in this heading is not strictly self-consistent - by international definition a primary standard embodies the highest metrological qualities… - but, ignoring this point, it is a reasonable question. And indeed there is no reason why more modest metrological instrumentation should not be evaluated from first principles, but there are some issues - including an economic one - that make such an approach questionable

Less than state-of-the-art accuracies can be obtained by reduced analytical effort but this in turn makes it more likely that pressure balance idiosyncrasies will be missed and the estimate of measurement uncertainties will be too low (too optimistic). Paralleling the example above, it is not uncommon for two similarly designed pressure balances to undergo the same analytical process only to find the two theoretically obtained values of effective area are not consistent with the physical comparison of their pressure-generating abilities. Pressure balance distortion is particularly difficult to evaluate and, although modern finite element techniques are helping considerably, they are still mostly not capable of resolving such problems (see note 3 below).

So if less effort is used to evaluate a pressure balance, the corresponding uncertainty components have to increase quite substantially to be sure that the physical effects otherwise ignored are properly accounted for in the uncertainty budget. It is not possible to say precisely by how much it will increase but it will be at least a factor of several times larger (ie less accurate) than a state-of-the-art primary pressure balance evaluated in an NMI - or indeed having its effective area determined via a conventional cross-float calibration.

But if an instrument’s effective area is evaluated in this fashion it will still have to be audited, by means of a pressure calibration against a traceable standard, before its values can be formally accepted (assuming that such acceptance is required). Thus using the dimensional route will not bypass the pressure calibration step – as is sometimes assumed - and relying solely on the pressure calibration and ignoring the dimensional data would provide lower uncertainties!

Conclusion

• Although NMIs use dimensional metrology as part of the process for determining the effective area of pressure balances being considered for primary use, such measurements alone cannot provide formally recognised traceability to the SI pascal.
• Primary standard pressure balances, operating at state-of-the-art levels of measurement uncertainty, cannot be produced in isolation. They need extreme levels of data, extensive and unique mathematical modelling, and have to be extensively compared with other instruments of proven performance to ensure the pressure values they generate are compatible and consistent with the SI pascal. Also, to keep a check on how various parameters change with time, this part of the process is never ending. The apparatus required to do this is only likely to be available in a national measurement institute such as NPL.
• The cost of attempting to set up a less-than-state-of-the-art primary standard pressure balance from dimensional measurements is still much higher than accepting secondary standard status afforded by a conventional hydrostatic cross-float calibration and at the same time leads to measurement uncertainties considerably higher (worse).

Notes

1. The uncertainty associated with distortion can be large and, at high pressures it can dominate an uncertainty budget. It is sometimes said that such contributions to uncertainty can be ignored in instruments that are designed to have a low or zero distortion coefficient. This argument is entirely false; just because a distortion coefficient is designed to be zero does not mean that it is zero. But even if it was determined to be so, zero is not a special value and it is not exempt from uncertainty in its estimation.
2. Primary standard pressure balances are not immune from the problems of reproducibility – changes in performance with time – that affect all instruments. They therefore have to be periodically re-evaluated and one of the best ways to do this is to regularly compare an instrument with higher- and lower-pressure standards where the ranges just overlap. This again illustrates why a primary standard cannot be evaluated and operated in isolation.
3. The problem of determining the elastic distortion of a pressure balance is perhaps put into perspective by appreciating that, at primary standard level, the dimensional uncertainties needed in predicting distortion can be equivalent to just a few atomic diameters!