The estimate has two input paths: the installed pressure sensor supplies P1, and the flowmeter supplies Q to the characterized valve-drop function. The calculation then subtracts ΔP(Q) from P1. Follow both paths separately before combining their uncertainties.
Where does the pressure estimate travel?
The calculation is Pfilter = P1 - g(Q), with g(Q) = A·Q² + B·Q + C. The pressure sensor is upstream of the electrovalve; the required pressure is downstream of that valve at the filter inlet. The external pressure sensor provides the downstream reference during characterization rather than during normal operation.
| Path element | Reading or operation | Error contribution | Check |
|---|---|---|---|
| Installed pressure sensor | P1 |
Specified accuracy of ±50 mmHg
|
Confirm range, zero, pressure tap, and whether the specification is a limit or statistical uncertainty |
| Flowmeter | Q |
Changes the calculated valve drop through the polynomial sensitivity | Express its accuracy in the same flow units used to fit A, B, and C
|
| Characterized model | g(Q) |
Coefficient uncertainty, calibration-instrument error, and unexplained residual variation | Retain the calibration data, residuals, and coefficient covariance |
| Subtraction | P1-g(Q) |
Combines the two paths; subtraction does not cancel uncertainty automatically | Select worst-case limits or statistical propagation before calculating a result |
Is the hydraulic path still the path that was characterized?
Layer one first. A numerical uncertainty calculation is valid only while the physical valve-drop relationship remains applicable. Match the electrovalve opening, tubing arrangement, pressure-tap locations, fluid, and relevant fluid condition to the characterization setup. Inspect for trapped gas, compliant tubing movement, restrictions, leaks, valve hysteresis, and pressure pulsation. These effects can change the pressure relationship without producing a sensor fault.
Use the curve only inside its characterized flow range. A polynomial can interpolate measured behavior but does not prove valid extrapolation. If measurements cross different flow regimes, one coefficient set may not describe the complete range. Do not delete B or C merely because an idealized valve relationship is quadratic: the fitted terms can represent non-ideal behavior, offsets, or the limited calibration range.
A loading filter changes the circuit operating point. Its increasing restriction can change flow and pressures over time, but the target calculation subtracts the electrovalve pressure drop, not the filter pressure drop. Recheck the valve curve if filter loading drives operation outside the characterized range or changes the surrounding hydraulic conditions enough to alter valve behavior.
Are the inputs steady and synchronized?
A steady-state curve cannot correct a dynamic mismatch between P1 and Q. Compare timestamps or acquisition order, then plot both signals through a flow change. If one channel filters more heavily, updates more slowly, or arrives later, the calculation combines values from different hydraulic states and creates a transient pressure error.
| Observation | Meaning | Next check |
|---|---|---|
| Calculated error appears only during flow changes | Channel delay, filtering, or hydraulic transients dominate | Align samples or calculate only after both readings settle |
| Error changes with flow but is repeatable | Curve shape or flow calibration is wrong | Compare residual error against Q
|
| Error is nearly constant across flow | Pressure zero or polynomial intercept is suspect | Check sensor zeros and C
|
| Error changes between repeated runs at the same flow | Physical repeatability or measurement noise is limiting | Repeat points in both increasing and decreasing flow directions |
How does flowmeter accuracy affect calculated pressure?
The flowmeter accuracy cannot be added directly in flow units to a pressure accuracy. Convert it through the slope of the valve curve. The local sensitivity is dg/dQ = 2A·Q + B. For a small flow uncertainty uQ, the corresponding standard pressure uncertainty is approximately uQ→ΔP = |2A·Q+B|·uQ.
For a stated maximum flow error UQ, evaluate the nonlinear endpoints instead of relying on the derivative:
UQ→ΔP = max(|g(Q+UQ)-g(Q)|, |g(Q-UQ)-g(Q)|)
Perform this calculation at each operating flow. The same flowmeter accuracy can produce different pressure uncertainty because the curve slope changes with Q. First convert percentage-of-reading, percentage-of-range, resolution, and any zero term from the flowmeter specification into one bound or one standard uncertainty. Do not mix those two interpretations.
Where does characterization uncertainty enter?
At each calibration point, the observed valve drop is the upstream pressure minus the external filter-inlet pressure. Its uncertainty therefore contains both pressure measurements. The external sensor is not an unrelated term to append after fitting; it enters through the calibration values used to estimate A, B, and C.
For coefficient vector β=[A,B,C] and x=[Q²,Q,1], parameter uncertainty at a selected flow is u²parameter(Q)=x·Cov(β)·xᵀ. Add the residual prediction variance when the objective is uncertainty for a new pressure estimate rather than uncertainty of the fitted mean. If calibration flow also has material uncertainty, use a fitting method that accounts for error in both axes or propagate the calibration inputs with repeated sampling; ordinary least squares that treats measured flow as exact can understate the result.
Separate repeatable bias from random variation. If the same installed pressure sensor contributed to both the curve characterization and the online value, the two terms may be correlated. Statistical propagation then includes -2·Cov(P1,g). Use calibration records to quantify that covariance. If it is unavailable, report a conservative worst-case bound instead of claiming independent random errors.
How should the final accuracy band be calculated and verified?
Choose the calculation that matches the meaning of the input specifications. A manufacturer accuracy limit such as ±50 mmHg is not automatically a one-standard-deviation value.
| Method | Use when | Output |
|---|---|---|
| Worst-case limits | Inputs are maximum tolerances or their probability distributions are unknown | Ufinal = 50 mmHg + UQ→ΔP + Ucurve(Q) |
| Root-sum-square | Inputs have been converted to independent standard uncertainties | ufinal = √(uP1² + ((2A·Q+B)uQ)² + ucurve²) |
| Covariance propagation | Calibration and online measurements share sensors or fitted coefficients are correlated | Include all covariance terms rather than adding variances as independent |
- Define the operating flow range and use the same pressure and flow units as the fitted curve.
- Record the installed sensor limit of
±50 mmHgand obtain the flowmeter and external-sensor accuracy definitions from their specifications. - Recalculate calibration-point pressure drops, inspect residuals versus flow, and retain the coefficient covariance and prediction residual.
- At each required
Q, calculateg(Q), the flow-induced pressure term, and the curve prediction uncertainty. - Combine terms by worst case or by statistical propagation, label the method, and report a flow-dependent band rather than one unsupported constant.
- Reconnect the external sensor at the filter inlet, test several points across the characterized range in both flow directions, and confirm that
P1-g(Q)differs from the reference by no more than the stated band.
Frequently Asked Questions
Can I add the pressure and flowmeter accuracy specifications directly?
No. Convert the flow error into pressure through |2A·Q+B|, or evaluate g(Q±UQ) for a finite accuracy limit, before combining it with ±50 mmHg.
Does the external pressure sensor accuracy count twice?
No. It enters the uncertainty of the characterized curve through the calibration pressure differences, coefficient covariance, and residual prediction error. Add it separately only if it also supplies an independent online measurement used by the final calculation.
Can I use one accuracy value for every flow rate?
Only if the calculated worst case over the entire characterized range is reported as that value. Verify it by comparing P1-g(Q) with the external filter-inlet measurement at multiple flows across that range.