DP flow (√)
Flow from the differential pressure across an orifice — the square-root relationship that means a quarter of the DP is half the flow, and why low-flow readings on a DP meter get so coarse.
Flow from differential pressure · orifice / DP element
Q = Qfs × √(DP / DPfs) · flow ∝ √DP
What this gives you
A DP flow element — an orifice plate, a flow nozzle, a venturi or an averaging pitot — does not measure flow directly. It measures the differential pressure the restriction creates, and that DP rises with the square of the flow. So the flow you actually want is recovered by taking a square root: Q = Qfs × √(DP ÷ DPfs). Enter the DP your transmitter reads, the DP that corresponds to full-scale flow, and the flow at full scale, and this returns the flow rate plus what percent of span you are running. Because the physics is flow ∝ √DP, a quarter of the DP is half the flow.
Why the square root, and why it matters at low flow
The restriction converts velocity into a pressure drop, and kinetic energy goes with velocity squared, so DP tracks flow squared. Reading it backward, flow tracks the square root of DP. That single curve is why 25 percent DP is 50 percent flow and 56.25 percent DP is 75 percent flow. It also explains why a DP meter gets coarse at the bottom of its range: at 10 percent flow the DP has collapsed to just 1 percent of span, so a tiny DP error or a hair of zero drift turns into a large flow error. That steep low end is why plain orifice measurement is usually held to about a 3:1 or 4:1 turndown and why a low-flow cutoff is applied to keep noise near zero from producing jumpy readings.
Field note — do not square-root twice
The most common mistake is double extraction. A smart DP transmitter can be configured to output either the raw differential pressure (linear in DP) or the square root already taken (linear in flow). If the transmitter is doing the √ and the DCS block is also set to square-root the signal, your flow reads far too low across the whole range — roughly the square root of the true fraction instead of the fraction itself. Confirm which device owns the extraction, do it in exactly one place, and make sure the DP-at-full-scale you enter here matches the calibrated span of the cell, not the pipe's mechanical rating.
Worked example
A cell is ranged 0–100 inH2O for a meter run rated 0–100 gpm, and it reports 56.25 inH2O. The DP fraction is 56.25 ÷ 100 = 0.5625, its square root is 0.75, so the flow is 100 × 0.75 = 75.0 gpm, or 75 percent of full scale. Notice the DP is only 56 percent of span while the flow is 75 percent — the square root pulls mid-range DP up. Drop the reading to 25 inH2O and the flow is not 25 percent but √0.25 = 50 percent; that same behavior, read near zero, is exactly why the low end is hard to trust.
Working backward, and units
Switch the mode to DP from flow to run the relationship in reverse: enter the flow you want and it returns the DP the cell must read, from DP = DPfs × (Q ÷ Qfs)². That is the sizing question — “what DP corresponds to the flow I need to see?” — and it is why 75 percent flow needs 56.25 percent of span in DP while 50 percent flow needs only 25 percent. The DP-units toggle (inH2O, kPa, mbar) applies to both the reading and the full-scale span, and the flow-units toggle (gpm or m³·h⁻¹) applies to both the full-scale flow and the answer; because the math only ever uses the ratios DP÷DPfs and Q÷Qfs, changing units simply relabels — it never changes the number, as long as the reading and its full-scale span carry the same unit.