Instrumentation

Square Root Extraction Explained

Differential pressure across a flow element is proportional to flow squared, so flow is the square root of differential pressure. Learn why 25% ΔP is 50% flow, where extraction happens, and the low-flow cutoff.

TypeEngineering guide — concept explainer

Definition

Square-root extraction is the step that turns a differential-pressure (DP) signal into a flow signal. Across a flow element — an orifice plate, nozzle, or venturi — the differential pressure produced is proportional to the square of the flow rate. Flow is therefore proportional to the square root of the differential pressure. Extraction applies that square root so the output reads in flow units rather than DP units: flow% = √(ΔP% / 100) × 100, where ΔP% is the differential pressure as a percent of its calibrated span.

Why it matters

DP flow is one of the most common flow-measurement methods in process plants, and the square-root relationship is the single most misunderstood part of it. Because the relationship is non-linear, a given percentage of differential pressure does not correspond to the same percentage of flow. At 25% of DP span the flow is already at 50% — not 25%. If extraction is applied twice, or not applied at all, every flow reading and every flow-based total is wrong, often by a large margin in the lower half of the range. Knowing where the square root is applied — and confirming it is applied exactly once — is essential to a trustworthy flow measurement.

Formula

DP across element

ΔP ∝ flow²

Flow from ΔP

flow ∝ √ΔP

Flow percent from ΔP percent

flow% = √(ΔP% / 100) × 100

ΔP percent from flow percent

ΔP% = (flow% / 100)² × 100

Units involved

•ΔP% — differential pressure as a percent of the calibrated DP span (0–100%)
•flow% — flow as a percent of the calibrated flow range (0–100%)
•flow fraction — flow% expressed as a fraction (0–1)
•low-flow cutoff — the flow percent below which the extracted output is forced to zero

Concept diagram

Worked example

A DP transmitter on an orifice plate reads 25% of its calibrated differential-pressure span. What flow does that represent?

01flow fraction = √(ΔP% / 100) = √(25 / 100) = √0.25 = 0.5
02flow% = 0.5 × 100 = 50%
03Check the reverse: ΔP% = (flow% / 100)² × 100 = (0.5)² × 100 = 25%

Result

25% of differential pressure corresponds to 50% of flow (and 50% flow back to 25% ΔP).

Common mistakes

•Treating the DP percent as the flow percent. They are only equal at 0% and 100%; everywhere in between the flow is higher than the DP percent.
•Applying square-root extraction twice. If the transmitter is already in square-root mode and the DCS extracts again, the displayed flow is square-rooted twice and reads far too low across most of the range.
•Applying no extraction at all. If the transmitter is in linear (DP) mode and nothing downstream extracts, the "flow" reading is actually proportional to flow squared.
•Ignoring the low-flow cutoff. Near zero ΔP the square root amplifies noise, so a small, jittery DP can produce a visibly unstable flow reading unless a cutoff forces it to zero.

When to use the calculator

Use the Square Root Extraction calculator to convert between differential-pressure percent and flow percent in either direction, and to check whether a flow sits below a low-flow cutoff. Use the DP Flow Signal calculator to take the relationship all the way to the 4–20 mA signal for an extracted transmitter. For the underlying linear signal scaling, the 4–20 mA scaling guide and the mA to Process Value and Process Value to mA calculators cover the signal-to-value step.

Square Root Extraction Calculator →DP Flow Signal Calculator →mA to Process Value Calculator →Process Value to mA Calculator →

FAQ

Why does 25% differential pressure give 50% flow?

Because differential pressure rises with the square of flow. Flow is therefore the square root of DP: √0.25 = 0.5. The square root maps the lower DP values up — half the DP span is reached at only a quarter, but half the flow span is reached at a quarter of the DP. Concretely, 25% DP → √0.25 = 50% flow.

Where is the square root actually applied?

It can be done in the transmitter (square-root output mode), in the DCS or PLC (linear transmitter, extraction in the control system), or in a local indicator. The key rule is that it must be applied exactly once. Always confirm which device owns the extraction so it is neither doubled nor missed.

What is a low-flow cutoff?

Near zero differential pressure, the square root has a very steep slope, so small noise on the DP signal produces large swings in the calculated flow. A low-flow cutoff forces the flow output to zero below a set percentage (often a few percent) to keep the reading stable and to avoid integrating noise into a flow total.

Does this size the orifice plate or flow element?

No. This is only the square-root signal relationship. Sizing the orifice, nozzle, or venturi — including beta ratio, density, and installation effects — is a separate flow-element design task that requires datasheets and standards.