Measurement uncertainty worksheet for compressed-air tests

Why uncertainty matters in compressed-air decisions

No measured value is exact. A flow meter reading, a pressure gauge, a power logger and a thermometer all carry some doubt. More importantly, a calculated result – specific power, pressure drop, leak rate, energy saving – inherits uncertainty from every input that went into it.

This matters when money and equipment decisions rest on the numbers. If an air audit says a compressor runs at 6.8 kW per m³/min and a replacement would achieve 6.3, the decision depends on whether that difference is larger than the uncertainty in both figures. If the uncertainty is ±0.5, the “saving” may not exist.

The worksheet answers three practical questions:

• How much confidence can reasonably be attached to this result?
• Which input contributes most to the doubt?
• Would more measurement work – a better instrument, a longer logging period – materially improve the decision?

It is not a substitute for a calibration certificate, instrument manual or test standard. It is the place where those documented inputs are brought together on one basis. The method follows the internationally used approach set out in the Guide to the Expression of Uncertainty in Measurement (GUM).

Start with the measurement model

Every uncertainty budget begins with the equation linking the reported result y to its inputs: y = f(x₁, x₂, … x_n). Without it, uncertainties cannot be propagated correctly.

Write the model at the top of the worksheet. Anyone checking the result can then see exactly which inputs matter and how.

Put every source on the same basis

Uncertainty information arrives in different forms. Each must be converted to a standard uncertainty – equivalent to one standard deviation – before combining.

Uncertainties evaluated from repeated readings are called Type A; those from certificates, specifications and judgement are Type B. Both are treated equally once converted. Check whether a manufacturer’s specification is stated as a percentage of reading or of full scale – a “±1% of full scale” transmitter is much less precise at low readings than the headline figure suggests.

Sensitivity coefficients and combining

Each input affects the result by a different amount. The sensitivity coefficient c_i is the partial derivative of the model with respect to that input – how much y changes per unit change in x_i. For independent inputs:

1. Contribution of each input: u_i(y) = |c_i| × u(x_i).
2. Combined standard uncertainty: u_c(y) = √(Σ u_i(y)²).
3. Expanded uncertainty: U = k × u_c, with k = 2 giving roughly 95% coverage when the result is approximately normal.

For products and quotients, relative uncertainties are simpler. For specific power y = P ÷ Q, the relative standard uncertainty is √((u_P/P)² + (u_Q/Q)²). For a difference such as pressure drop, the absolute uncertainties combine, which is why small pressure drops measured with two separate gauges are often very uncertain – a common trap in distribution audits.

Where inputs are correlated – for example two pressure transmitters calibrated against the same reference – correlation terms are added. A practical option is to measure both points with one instrument in turn, or with a differential transmitter.

Worked example: specific power

An audit measures package power and delivered flow on a screw compressor over a stable loaded period. The figures are illustrative.

Specific power = 90.0 ÷ 14.0 = 6.43 kW per m³/min. Combined relative standard uncertainty = √(0.87² + 0.17² + 1.73² + 1.15²) ≈ 2.26%. Expanded at k = 2 gives about ±4.5%, or ±0.29 kW per m³/min.

So the result is 6.43 ± 0.29 kW per m³/min at roughly 95% confidence. The budget shows flow dominates; a longer straight pipe run or a better-installed meter would improve the result far more than a better power logger.

Sampling and representativeness

Instrument uncertainty is often not the biggest problem. An average taken over a period that does not represent normal operation can be precise and still wrong. The worksheet adds a representativeness contribution when:

• Logging covered only part of the week, missing weekend or night patterns.
• Ambient temperature during the test was unusual – a cool morning reading will not represent a 45 °C summer afternoon.
• Demand was cycling, so a short test captured an unusual mix of load and unload.
• The logger sampling interval was too long to capture short pressure dips.

For time-series averages, split the data into blocks, calculate the spread of block averages and use it as a Type A contribution. Where a pattern is plainly unrepresentative, collect more data rather than inflating the uncertainty.

Using the result to decide

When comparing before and after, for example in a leak-reduction project, uncertainty applies to both measurements. Using the same instruments in the same locations removes much of the systematic error from the difference.

Limits of the worksheet

• It assumes the model is correct; a wrong boundary, such as leaving out the dryer, is not an uncertainty but an error.
• k = 2 for 95% assumes a near-normal result with enough effective degrees of freedom; very small samples need a larger factor.
• It relies on calibration information being current and applicable at the measured range.
• Large non-linear models may need numerical (Monte Carlo) methods rather than first-order sensitivity coefficients.

How Reddy Compressors helps

Reddy Compressors, at Tarnaka in Secunderabad, has worked with industrial compressed air since 2002. We sell, rent and service compressors of ELGi, Atlas Copco, Ingersoll Rand, KAESER, Chicago Pneumatic and many other makes as an independent multi-brand specialist, not as an authorised dealer, and our service team covers all of Telangana and Andhra Pradesh.

• Plan audit measurements – instrument choice, location, straight runs, logging period – so results are good enough to decide on.
• Carry out specific-power and pressure-drop assessments using the ISO 11011 method, with uncertainty stated alongside the results. We do not issue conformity certificates.
• Review supplier performance claims and acceptance test results on a consistent basis.
• Recommend and supply new or used/refurbished compressors, dryers and controls where the evidence supports a change.

What to send us

• What you are trying to decide and the results you already have.
• Instrument details: make, model, range and calibration certificates.
• Where each instrument was installed, with photos of the pipework.
• Logging period, sampling interval and shift pattern during the test.
• Compressor make, model, serial number and nameplate photo.

Frequently asked questions

What is measurement uncertainty in simple terms?

It is the range within which the true value of a measurement is expected to lie, with a stated confidence. A result of 6.43 ± 0.29 kW per m³/min at 95% means the true specific power is very probably between 6.14 and 6.72. It comes from instrument accuracy, resolution, repeatability, installation and how representative the test period was.

How do I calculate expanded uncertainty?

Convert each source to a standard uncertainty, multiply each by its sensitivity coefficient, square the contributions, add them, and take the square root to get the combined standard uncertainty. Multiply that by a coverage factor, usually k = 2, for an expanded uncertainty giving about 95% confidence when the result is close to normally distributed.

Why divide a manufacturer’s accuracy figure by the square root of three?

A plus-or-minus specification usually only tells you the limits, not how likely values are within them. Treating every value inside the limits as equally likely is a rectangular distribution, and its standard deviation is the half-width divided by the square root of three. This puts the specification on the same basis as other contributions.

How accurate is a compressed-air flow meter?

It depends on the meter type, the range, the installation and calibration. A specification is often stated as a percentage of reading or of full scale, and installation effects such as short straight runs, bends and moisture can add significant extra uncertainty. In many compressed-air audits the flow measurement is the dominant contributor to the uncertainty of specific power.

Is uncertainty the same as error?

No. Error is the difference between a measured value and the true value, which is usually unknown. Uncertainty describes the doubt about the result after known corrections have been applied. A mistake such as measuring at the wrong boundary or using the wrong units is not covered by uncertainty; it must be found and corrected.

When is my audit result good enough to make a decision?

When the difference between the options you are comparing is clearly larger than the combined uncertainty of both results. If the difference and the uncertainty are of similar size, improve the dominant contributor first, often flow measurement or the logging period, then measure again before committing to equipment or project spending.

Related pages

• Measurement, verification and energy performance
• Audit reporting and verification method
• ISO 11011 assessment framework
• Compressor performance acceptance worksheet
• Motor input and electrical measurement worksheet
• Compressed air knowledge and calculation centre