Measurement Uncertainty Worksheet

Route: /engineering-tools/measurement-uncertainty-worksheet/ (proposed until route reconciliation). It supports compressed-air audit and performance calculations without presenting estimated precision as certainty.

1. Purpose and Engineering Decision

A measured flow, pressure, power or temperature value is never perfectly exact. More importantly, a calculated result such as specific power or pressure drop inherits uncertainty from every input used to create it.

The worksheet is not a substitute for a calibration certificate, instrument manual or test method. It is the controlled place where those documented inputs are converted to comparable standard uncertainties and combined with sampling and model contributions.

2. Measurement Model Comes First

The worksheet begins with a measurement model: the equation that links the reported result y to the input quantities x1, x2 ... xn.

y = f(x1, x2, ... , xn)

For a direct pressure reading, y may be the corrected pressure itself. For specific power, y = P/Q. For pressure drop, y = P_up - P_down.

3. Standard Uncertainty - Put Different Sources on One Basis

Uncertainty contributions may be documented in different forms: an expanded calibration uncertainty, a plus/minus instrument specification, a resolution increment, a standard deviation from repeated readings or an engineering estimate. They must be converted to standard uncertainty before combination.

4. Uncertainty Budget Template - Required Asset

5. Sensitivity Coefficients

c_i = partial derivative of y with respect to x_i

u_i(y) = |c_i| x u(x_i)

Sensitivity coefficients convert an input uncertainty into the units of the final result. For a direct measurement y = x, c = 1. For y = P/Q, the sensitivity to power is 1/Q and the sensitivity to flow is -P/Q^2.

6. Combine Independent Contributions by Root-Sum-Square

u_c(y) = sqrt[ Sum( c_i^2 x u_i^2 ) ]

Independent contributions are not normally added arithmetically. Root-sum-square combination reflects the fact that independent errors do not all reach their maximum in the same direction at the same time. A separate worst-case mode may sum limits arithmetically, but it must be labelled as worst-case rather than statistical combined uncertainty.

7. Correlated Inputs

When two inputs share a common calibration, common environmental effect or common reference, their uncertainties may be correlated. Ignoring correlation can either understate or overstate uncertainty.

u_c^2 = Sum(c_i^2 u_i^2) + 2 Sum(c_i c_j rho_ij u_i u_j)

rho is the correlation coefficient between -1 and +1. If correlation is unknown and potentially material, the result should be flagged for review.

8. Expanded Uncertainty and Coverage Factor

U = k x u_c

The worksheet should report both combined standard uncertainty u_c and expanded uncertainty U. The coverage factor k is an explicit user input or method-specific value.

9. Relative Uncertainty

Relative Expanded Uncertainty (%) = U / |y| x 100

Relative uncertainty is useful for comparing measurement quality across different magnitudes.

10. Sampling and Data-Representativeness Uncertainty

A highly accurate meter can still produce a poor plant estimate if the logging window is not representative.

11. Avoid Double Counting

The same physical effect should appear only once in the budget. For example, if a calibration certificate already includes instrument resolution under the exact configuration used, adding a separate resolution term can double count it.

12. Reference Test Case A - Pressure Measurement

A pressure transmitter reports 7.20 bar(g). Calibration expanded uncertainty is +/-0.030 bar with k = 2. A separate documented installation contribution is +/-0.020 bar with rectangular distribution, and display resolution is 0.01 bar.

Reported screening result: 7.20 bar(g) +/- 0.0383 bar at the stated coverage factor and assumptions.

13. Reference Test Case B - Specific Power

Measured package input is 100.0 kW with standard uncertainty 1.5 kW. Delivered flow is 18.0 m3/min with standard uncertainty 0.45 m3/min. Assume independent inputs.

SP = P / Q

The flow term dominates this example. Improving power-meter precision would therefore have limited value unless flow uncertainty is also improved.

14. Reference Test Case C - Pressure Drop from Two Sensors

Upstream pressure is 7.10 bar and downstream pressure is 6.85 bar, giving DeltaP = 0.25 bar. Each sensor has standard uncertainty 0.020 bar.

Because the measurement model is upstream minus downstream, positive common-mode correlation can reduce uncertainty in the difference.

15. Reference Test Case D - Time-Series Mean with Sampling Contribution

A 24-hour flow study reports a mean of 12.0 m3/min. Instrument standard uncertainty is 0.24 m3/min. Comparison of repeated representative daily windows gives a sampling/representativeness standard uncertainty of 0.60 m3/min.

This example shows why a long decimal display from a precise instrument does not prove that the annual or plant-wide demand estimate is equally precise.

16. Contributor Ranking

Contribution Share_i (%) = [u_i(y)^2 / u_c(y)^2] x 100

This turns the budget into an action tool: calibrate, improve installation, extend logging, synchronize datasets or leave the measurement system unchanged when the dominant uncertainty lies elsewhere.

17. Decision Margin and Guard Band

When a measured result is compared with a limit, guarantee or acceptance criterion, uncertainty should not be ignored.

The tool must not invent a universal guard band.

18. Input Model

19. Confidence Model

20. Validation and Guardrails

21. Output Specification