How to prove ±0.1 K: measurement uncertainty, metrological traceability and acceptance

“Flow temperature ±0.1 K” is the most common accuracy requirement in test bench requirement specifications. It sounds unambiguous and is not. Does it refer to the control deviation or to the measurement uncertainty? Does it apply to the setpoint or to the display? And how do you tell at acceptance that it has been met, when the measuring chain itself has an uncertainty of the same order of magnitude?

This paper draws the answer from the documents that exist for the purpose — all of which are freely available. It explains the terms, shows which uncertainties are realistically achievable for each measured quantity, and describes the decision rule without which an acceptance test makes no statement.

01 · The terms, cleanly separated

The governing document is the Guide to the Expression of Uncertainty in Measurement, JCGM 100:2008, free from the International Bureau of Weights and Measures.^{[1] [2]} It defines measurement uncertainty as a “parameter, associated with the result of a measurement, that characterises the dispersion of the values that could reasonably be attributed to the measured quantity”.^[2]

The distinction between Type A and Type B is not made by the nature of the component but by the method of evaluation. In test bench construction this means, concretely: repeat measurements on the installed system are Type A; the manufacturer's specification, the class rating, the calibration certificate and the resolution are Type B.

The propagation law for uncorrelated input quantities is the sum of the squared contributions, each weighted with the sensitivity coefficient — the partial derivative of the model with respect to the input quantity concerned.^[2] For correlated quantities, covariance terms are added. For the expanded uncertainty, U = k · u_c, with k = 2 as the default value for roughly 95 % coverage.^{[2] [13]}

02 · Metrological traceability: what a calibration certificate is worth

The International Vocabulary of Metrology defines metrological traceability as the “property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty”.^[8] Two of its notes matter for test benches: in measurements with more than one input quantity, each input quantity should be traceable — not “the test bench” as a whole. And metrological traceability alone guarantees neither an adequate measurement uncertainty nor freedom from error.^[8]

Which routes to traceability are permissible is governed by the ILAC policy on metrological traceability.^[10] Two are preferred: calibration by a national metrology institute whose service is covered by the mutual recognition arrangement of the Metre Convention, or by a calibration laboratory whose service is covered by the ILAC arrangement. Other routes should apply only where these two are not possible.^[10] And the sentence most often overlooked in practice:

The German accreditation body puts it even more plainly: result reports from accredited calibration laboratories without a reference to the accreditation are not recognised as evidence of traceability.^[11] Also from the same source: measuring ranges and measurement uncertainties within the scope of accreditation cannot be made flexible — the scope is exact to the point.

On the calibration certificate itself, the ILAC policy on measurement uncertainty requires the measurement result to contain the value of the measured quantity and the associated expanded uncertainty, stated with a coverage probability of roughly 95 %. Accredited laboratories may not state measurement uncertainties smaller than their own calibration and measurement capability.^[12]

The requirements on the laboratory itself are set out in ISO/IEC 17025:2017.^[14] Clause 6.5 covers metrological traceability, clause 7.6 the evaluation of measurement uncertainty, clause 7.7 ensuring validity through intermediate checks, and clauses 7.8.6.1 and 7.8.6.2 statements of conformity together with the decision rule.^{[10] [11]}

03 · The decision rule

This is where the answer to the question in the title lies. You do not prove ±0.1 K with a measured value, but with an agreed decision rule. The terms for it are defined in JCGM 106:2012:^[15]

The guidance document ILAC-G8 tabulates the common rules with the associated risk of false acceptance:^[16]

Precisely this calculation belongs before the contract is signed, not in the acceptance test. ISO/IEC 17025 expressly requires the specification and the decision rule to be clearly defined and agreed with the customer.^[16] The corresponding standard for the inspection of measuring equipment and workpieces is ISO 14253-1:2017; the change of title since 2013 from “proving” to “verifying” is notable — the standard now speaks of verifying conformity, not of proving it.^{[17] [18]}

In the automotive supply chain the same system is regulated through measurement process capability. There the capability ratio is formed from the expanded uncertainty of the measuring system and the tolerance, with a limit of 15 %, and as a guard band g = 1,65 · u is applied — which corresponds to a risk of false acceptance of 5 % per side and thus matches ISO 14253-1.^[19] The corresponding international standard is ISO 22514-7:2021.^[20]

04 · What is achievable for each measured quantity

Temperature

For platinum resistance thermometers, IEC 60751 applies in its third edition of 2022.^[21] The tolerance classes are temperature-dependent formulae, not fixed values:

The most useful German-language source for the uncertainty budget in practice is the calibration guideline DKD-R 5-1 in its 2023 edition — freely available and with concrete figures for the contributions you would otherwise have to estimate:^[23]

For thermocouples, IEC 60584-1 applies in its 2013 edition.^[24] The tolerance limits are class values: for type T in class 1 the greater of ±0.5 °C and 0.004 · |t|, in class 2 the greater of ±1.0 °C and 0.0075 · |t|; for type K in class 1 ±1.5 °C or 0.004 · |t| respectively.^[25] The same source documents effects that bite in everyday test bench work: green rot in type K in an oxygen-poor environment between about 800 and 1050 °C, a thermoelectric voltage error of up to 0.8 mV — roughly 5 °C — on slow cooling in the range 400 to 600 °C, and the rule that the errors of the thermocouple and of the extension cable add up.^[25]

Pressure

For Bourdon tube pressure gauges, EN 837-1 applies.^[26] The decisive property is not in the class number but in what it refers to: the class refers to the measuring span, not to the measured value.^[27] An instrument of class 1.6 with a range of 0 to 100 bar has ±1.6 bar across the whole range — at an operating point of 10 bar that is 16 % of the measured value.

For calibration, the guideline DKD-R 6-1 is authoritative and freely available.^[28] It defines three calibration sequences according to the uncertainty aimed for: sequence A for below 0.1 % of the span with nine measuring points, three preloads, two minutes of dwell time and two measurement series; sequence B for 0.1 to 0.6 %; sequence C for above 0.6 % with five points, one preload and one minute of dwell time. As contributions it names resolution, the uncertainty of the measurement standard, zero offset, repeatability, hysteresis span and reproducibility.^[28] The physical upper limit is set by the pressure balance: the effective piston area can be determined with relative uncertainties below 5 · 10⁻⁶, the applied force with 1 · 10⁻⁶ or better.^[29]

Flow

For differential pressure devices, the ISO 5167 series applies, parts 1 to 4 in the 2022 edition.^{[30] [31]} On the achievable uncertainty, a figure from the previous edition of part 2 shows the order of magnitude: the relative uncertainty of the discharge coefficient is already 0.5 % for diameter ratios between 0.20 and 0.60, and additional terms for small pipe diameters and low Reynolds numbers are added arithmetically.^[32]

For Coriolis meters there is a selection and installation standard in ISO 10790^[33] and a freely available guide from a national test laboratory with sound figures from practice.^[34] Manufacturers' figures start at 0.05 % for mass flow, but the guide names the conditions for that: zero stability has a relatively strong effect at low flow rates — oversized measuring points in part-load operation are the classic mistake. Temperature makes the measuring tubes softer and leads to over-reading, pressure increases the stiffness and leads to under-reading, both linearly. And a calibration at 50 mm²/s viscosity, applied at 300 mm²/s, produces deviations of over 0.5 %.^[34] The core recommendation of the guide: calibrate as close to the operating conditions as possible.

As a reference for the national upper limit: the hydrodynamic test field of the PTB states a target uncertainty of ±0.02 %.^[35]

Electrical power

There are no accuracy classes here as there are for pressure gauges; the manufacturer's specification is what counts. As an example of the upper end: a precision power analyser states ±(0.01 % of reading + 0.02 % of range) for active power, voltage and current between 45 and 66 Hz, and ±(0.02 % + 0.05 %) for direct current. Reference conditions are 23 °C ± 5 °C, power factor one and about 30 minutes of warm-up time; the temperature coefficient is ±0.01 % of reading per degree.^[36]

05 · The budget for a heat output

The most common derived value in test bench construction is the heat output from mass flow, specific heat capacity and temperature difference. For a pure product the relative uncertainties add in quadrature — the sensitivity coefficients cancel out.^[2]

The message of the curve is unambiguous: the uncertainty of the temperature difference is an absolute quantity, and the spread is in the denominator. At 5 K spread and 0.1 K uncertainty that is 2 % on the output — more than mass flow and material property together. At 30 K it is 0.33 %.

The correlation advantage of matched sensor pairs

One point regularly missing from budgets: if the flow and return sensors are calibrated together against the same measurement standard and through the same measuring card, their deviations are correlated. The covariance term of the propagation law then reduces the uncertainty of the difference considerably compared with the naive root-two calculation.^[2] That is precisely why legal metrology certifies sensor pairs and not individual sensors.

The system behind this is shown by the technical guideline on the verification of heat meters.^[37] It breaks the meter down into exactly the three terms of the model, with maximum permissible errors for the flow sensor, the calculator and the temperature sensor pair, which are then added arithmetically — maximum permissible errors are not uncertainties. Both temperature-dependent terms contain the ratio of the smallest to the actual spread, and thus reproduce exactly the dominance shown above.^[37]

The same guideline provides the citable source for a rule that in plant engineering is usually passed on only by word of mouth: the measurement uncertainty of the test equipment must not exceed a certain fraction of the maximum permissible error at the test points — a fifth or a third, depending on the type.^[37]

If you are looking for a fully worked budget as a model: the document EA-4/02 of the European accreditation organisation is freely available and contains, among other things, worked examples for the calibration of a thermocouple at 1000 °C, a temperature block calibrator at 180 °C and a water meter.^[13] For calorimetric measurements, the most extensive freely accessible work is a NIST report on a 20 MW calorimeter, which works through two models completely with tabulated percentage contributions.^[38]

06 · Calibration intervals and intermediate checks

The authoritative guidance is ILAC-G24, or OIML D 10, in the 2022 edition — freely available.^[39] Since that edition it is called “recalibration intervals of measuring equipment”; the older designation from 2007 is superseded.

Five methods are available: automatic adjustment by calendar time, the control chart, sizing by operating hours instead of calendar months, frequent checking of critical parameters with full calibration only when a limit is exceeded, and further statistical methods. The guidance states expressly that each laboratory decides for itself which of them it applies — on the basis of its needs and its risk assessment.^[39]

In concrete terms: a defined reference point in the circuit, a stored setpoint, and an evaluation that tracks the trend over time. Anyone who builds that in extends the calibration intervals with evidence — and notices a drift before it spoils a test result.

07 · What belongs in an acceptance agreement

A checklist follows from the above. It is our conclusion.

1. Separate what is meant. Control deviation, measurement uncertainty and reproducibility are three different numbers. A requirement of “±0.1 K” without stating which of them is meant cannot be verified.
2. Agree the decision rule. Shared risk or guard band — and if a guard band, with which factor.^{[15] [16]}
3. Name the measuring chain. Which sensors, which card, which calibration laboratory, which calibration scope. Metrological traceability applies to each input quantity, not across the board.^[8]
4. Enclose the uncertainty budget. Following the model of EA-4/02: quantity, estimate, standard uncertainty, distribution, sensitivity coefficient, contribution.^[13]
5. Define the test points and the conditions. At which temperature, which flow, which ambient condition the demonstration is made.
6. Define the intermediate checks. What is checked how often against which reference, and what happens if it is exceeded.^[39]
7. Check the calibration certificates — for measurement uncertainty and for the reference to accreditation.^{[10] [11] [12]}

For acceptance measurements on plants with redundant measuring points there is a further tool that is used too rarely in test bench construction: data reconciliation over balance equations, described in VDI 2048 Part 1.^[40] Where energy and mass balances can be closed, it reduces the uncertainty considerably in some cases and exposes faulty measuring points. Methodologically that is the most demanding acceptance evidence, but also the most convincing.

08 · Sources

The fundamental metrological documents — GUM, VIM, EA-4/02, the DKD guidelines, ILAC-G8 and OIML D 10 — are all available free of charge. Anyone preparing an acceptance test will find more substance there than in most textbooks.